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The
Principle of Relativity
with applications to
Physical Science
The
Principle of Relativity
with applications to
Physical Science
BY
A^ N. WHITEHEAD, Sc.D., F.R.S.
Hon. D.Sc. (MANCHESTER), Hon. LL.D. (Sx ANDREWS)
Fellow of Trinity College, Cambridge, and Professor of
Applied Mathematics in the Imperial College
of Science and Technology
M
CAMBRIDGE
AT THE UNIVERSITY PRESS
1922
TO MY WIFE
WHOSE ENCOURAGEMENT AND COUNSEL
HAVE MADE MY LIFE S WORK POSSIBLE
PRINTED IN GREAT BRITAIN
PREFACE
THE present work is an exposition of an alternative
rendering of the theory of relativity. It takes its
rise from that awakening from dogmatic slumber to
use Kant s phrase which we owe to Einstein and
Minkowski. But it is not an attempt to expound either
Einstein s earlier or his later theory. The metrical for
mulae finally arrived at are those of the earlier theory,
but the meanings ascribed to the algebraic symbols are
entirely different. As the result of a consideration of
the character of our knowledge in general, and of our
knowledge of nature in particular, undertaken in
Part I of this book and in my two previous works* on
this subject, I deduce that our experience requires and
exhibits a basis of uniformity, and that in the case of
nature this basis exhibits itself as the uniformity of
spatio-temporal relations. This conclusion entirely cuts
away the casual heterogeneity of these relations which
is the essential of Einstein s later theory. It is this
uniformity which is essential to my outlook, and not
the Euclidean geometry which I adopt as lending itself
to the simplest exposition of the facts of nature. I should
be very willing to believe that each permanent space
is either uniformly elliptic or uniformly hyperbolic, if
any observations are more simply explained by such a
hypothesis.
It is inherent in my theory to maintain the old
division between physics and geometry. Physics is the
* The Principles of Natural Knowledge^ and The Concept of Nature,
both Cambridge Univ. Press.
a3
vi PREFACE
science of the contingent relations of nature and geo
metry expresses its uniform relatedness.
The book is divided into three parts. Part I is con
cerned with general principles and may roughly be
described as mainly philosophical in character. Part II
is devoted to the physical applications and deals with
the particular results deducible from the formulae assumed
for the gravitational and electromagnetic fields. In
relation to the spectral lines these formulae would require
a limb effect and a duplication or a triplication of indi
vidual lines, analogous to phenomena already observed.
Part III is an exposition of the elementary theory of
tensors. This Part has been added for one reason because
it may be useful to many mathematicians who may be
puzzled by some of the formulae and procedures of Part
II. But this Part is also required by another reason.
The theory of tensors is usually expounded under the
guise of geometrical metaphors which entirely mask the
type of application which I give to it in this work. For
example, the whole idea of any fundamental tensor is
foreign to my purpose and impedes the comprehension
of my applications.
The order in which the parts should be studied will
depend upon the psychology of the reader. I have placed
them in the order natural to my own mind, namely,
general principles, particular applications, and finally
the general exposition of the mathematical theory of
which special examples have occurred in the discussion
of the applications. But a physicist may prefer to start
with Part II, referring back to a few formulae which
have been mentioned at the end of Part I, and a
mathematician may start with Part III. The whole
evidence requires a consideration of the three Parts.
PREFACE vii
Practically the whole of the book has been delivered
in the form of lectures either in America at the College
of Bryn Mawr, or before the Royal Society of Edinburgh,
or to my pupils in the Imperial College. I have care
fully preserved the lecture form and also some redupli
cation of statement, particularly in Part I.
The exposition of a novel idea which has many reactions
upon diverse current modes of thought is a difficult
business. The most successful example in the history
of science is, I think, Galileo s Dialogues on the Two
Systems of the World. An examination of that masterly
work will show that the dialogue form is an essential
element to its excellence. It allows the main expositor
of the dialogues continually to restate his ideas in
reference to diverse trains of thought suggested by the
other interlocutors. Now the process of understanding
new conceptions is essentially the process of laying the
new ideas alongside of our pre-existing trains of thought.
Accordingly for an author of adequate literary ability
the dialogue is the natural literary form for the pro
longed explanation of a tangled subject. The custom
of modern presentations of science, and my own diffi
dence of success in the art of managing a dialogue,
have led me to adopt the modified form of lectures in
which the audiences real audiences, either in America,
Edinburgh or South Kensington are to be regarded
as silent interlocutors demanding explanations of the
various aspects of the theory.
Chapter II was originally delivered* in Edinburgh
as a lecture to the Royal Society of Edinburgh when
it did me the honour of making me the first recipient of
the James-Scott Prize for the encouragement of the
* June 5, 1922.
viii PREFACE
philosophy of science. Chapter IV was originally
delivered* at the College of Bryn Mawr, near Phila
delphia, on the occasion of a festival promoted by
the former pupils and colleagues of Prof. Charlotte
Angus Scott in honour of her work as Professor of
Mathematics at the college since its foundation.
My thanks are due to my colleague, Assistant-Pro
fessor Sillick, for the figure on p. 31. I am also further
indebted to him for a series of beautiful slides containing
the mathematical formulae of Chapter IV; even the
admirable printing of the Cambridge University Press
will not compensate readers of this book for the loss of
the slides as used in the original lecture.
In acknowledging my obligations to the efficiency
and courtesy of the staff of the University Press, I take
the opportunity of paying a respectful tribute to the
work of the late Mr A. H. Waller as secretary of the
Press Syndicate. The initial negotiations respecting
this book were conducted through him and he died just
as the printing commenced. The loss of his wisdom, his
knowledge, and his charm will leave a gap in the hearts
of all those who have to deal with the great Institution
which he served so well.
* April 18, 1922.
A. N. W.
15 September, 1922.
TABLE OF CONTENTS
PART I. GENERAL PRINCIPLES
PAGES
CHAPTER I. PREFATORY EXPLANATIONS 3-12
Scope of doctrine of relativity : the two gauges : philosophy :
Poyn ting s aphorism : time and space : physical objects : the planet
Mercury: spectral lines, shift, limb effect and duplication: two novel
magnetic forces in steady fields : temperature effect.
II. THE RELATEDNESS OF NATURE . . . 13-39
1 James-Scott Prize : Hume and relatedness : a ground of uni
formity : fact and factors : awareness, cogitation and entities : fact
not an entity : concreteness, embeddedness, factuality : limitation,
fmitude, canalisation : examples : the significance of factors : active
and passive cognisance, cognisance by adjective and cognisance by
relatedness : full awareness : spatio-temporal relationships : sense-
awareness and nature : nature significant of ideality : nature, time
and space, events : nature as a closed system : objection to doctrine
of relatedness, truth incompatible with any i^imrancc: essential and
contingent relationships : significance concerned with essential rela
tionships : every factor uniformly significant : patience of fact for
each factor : examples : uniform significance of events, spatio-temporal
structure: dissent from Einstein: stratification, patience of fact for
finite consciousness: objects and recognition: a field: two-termed
relation between universals and concrete particulars : observer s mind :
solipsism : adjectives of events : contraction of sphere of contingency :
structure of the continuum of events : past, present, future, causal past,
causal future, co-present region : spatial routes, historical routes : per
vasive adjectives and adjectival particles: individuality and process:
pseudo-adjectives and sense-objects : atomic field of an adjectival
particle : scientific objects qualify future events : permanence and field :
critical velocity c : velocity of light : tubes of force : obstructed fields :
ether and the apparent world identical: ether of events: events sup
plant stuff: Descartes on space and time : necessity of definite meaning
for symbols.
III. EQUALITY 40-60
Importance of equality: equality, congruence, quantity, measure
ment, identity, diversity: Euclid s axiom: ros: characteristics of the
equality relation: matching: congruence of stretches: axiom of
Archimedes: Sophus Lie: multiplicity of inconsistent congruence
relations : gap in classical theory : necessity of structure : straight
lines, planes, order, parallelism, rectangularity : space, bodies, events :
origin of parallelism: origin of perpendicularity: impossibility of
deriving time from permanent bodies : Einstein and alternative time-
systems : order in space derivative from order in time.
x TABLE OF CONTENTS
<*
PAGES
IV. SOME PRINCIPLES OF PHYSICAL SCIENCE 41-88
The apparent world : bifurcation of nature: two-fold cognisance,
cognisance by adjective, cognisance by relatedness : systematic
coherency of nature : events and two-fold cognisance : perception : the
contingency of appearance : the uniform significance of events : the
yard-measure: doctrine of time: time a stratification of nature: no
unique system of temporal stratification : Einstoin : simultaneity
fundamental : appearance is nature : time and space, their assimila
tion and distinction : spatial and historical routes : individuality of
adjectival particles derivative from their historical routes: time-
systems and parallelism: Euclidean assumption, slight evidence:
permanent space of a time-system derivative from rest and motion :
permanent points : spatial and temporal relations of permanent bodies
arise mediately : the physical field : limitation of the contingency of
appearance: intermediate distribution of character attachable to
future events : physical field not cause of perception : physical field
atomic: physical atomic character exemplifies Aristotelian attribute:
knowledge impossible without atomicity: systematic relatedness,
intelligibility due to uniformity and to atomicity of the contingent:
sense in which physical field is perceived: adjectival particles: their
definition: pervasion: they involve stretches of route: kinematic
elements: mass-particles: kinematic past, kinematic future, region
co-present, causal past, causal future: Faraday s tubes of force:
metrical formulae : impetus : potential mass impetus, potential electro
magnetic impetus: realised impetus: stationary property of realised
impetus: gravitational field: an integral law of gravitation: com
parison with Einstein s law: conditions fulfilled: four alternative
laws satisfying all conditions at present known : rotation : the genius
of Einstein and Minkowski.
PART II. PHYSICAL APPLICATIONS
V. THE EQUATIONS OF MOTION .... 91-92
Derivative forms of the equations: the pure centrifugal gravitational
terms : the composite centrifugal gravitational terms : the pure
gravitational term.
VI. ON THE FORMULA FOR dJ* .... 93-100
Fundamental formulae : the general potential : the tensor potential :
the first associate potential: the second associate potential: the
contemporary positions : the associated space.
VII. PERMANENT GRAVITATIONAL FIELDS 101
VIII. APPARENT MASS AND THE SPECTRAL
SHIFT . . ... 102-103
IX. PLANETARY MOTION . 104-105
TABLE OF CONTENTS xi
PAGES
X. ELECTROMAGNETIC EQUATIONS . . 106-107
XI. GRAVITATION AND LIGHT WAVES . 108-111
XII. TEMPERATURE EFFECTS ON GRAVITA
TIONAL FORCES 112-113
XIII. THE ELECTROSTATIC POTENTIAL AND
SPECTRAL SHIFT 114-116
XIV. THE LIMB EFFECT 117-120
XV. PERMANENT DIRECTIONS OF VIBRA
TION AND THE DOUBLING EFFECT . 121-126
XVI. STEADY ELECTROMAGNETIC FIELDS . 127-131
XVII. THE MOON S MOTION ... . . 132-136
PART III. ELEMENTARY THEORY OF TENSORS
XVIII. FUNDAMENTAL NOTIONS . . . 139-147
SECTION 1. Coordinates: 2. Scalar Characters and Invariant Ex
pressions: 3. Physical Characters of the First Order: 4. Tensors
of the First Order: 5. Covariant and Contra variant First-Order
Tensors : 6. Characters and Tensors of Higher Orders : 7. Tensor-
Invariance of Formulae.
XIX. ELEMENTARY PROPERTIES . . . 148-152
SECTIONS. Test for Tensor Property : 9. Sum of Tensors : 10. Pro
duct of Tensors : 11. Representation of a Tensor as a Sum of
Products.
XX. THE PROCESS OF RESTRICTION . . . 153-159
SECTION 12. Definition of Restriction : 13. Multiple Restriction :
14. Invariant Products: 15. The Tensor ||/||: 16. Restriction of
a Single Mixed Tensor: 17. Argument from Products [Restricted
or Unrestricted] to the Tensor Property: 18. Differential Forms.
XXL TENSORS OF THE SECOND ORDER . . 160-165
SECTION 19. Symmetric Tensors : 20. Skew Tensors: 21. The
Determinants: 22. Associate Tensors: 23. Derivative Tensors.
XXII. THE GALILEAN TENSOR .... 166-172
SECTION 24. Galilean Tensors : 25. Galilean Differential Forms :
26. The Linear Equations of Transformation : 27. Cartesian Group :
28. Associate Galilean Tensors and Galilean Derivative Tensors :
29. Galilean Derivative Tensors of the First Order.
xii TABLE OF CONTENTS
PAGES
XXIII. THE DIFFERENTIATION OF TENSOR
COMPONENTS 173-183
SECTION 30. The Christoffel Three-Index Symbols: 31. Differentia
tion of Determinants of Tensors : 32. The Standard Formulae :
33. Covariant Tensors of the First Order : 34. Contravariant Tensors
of the First Order: 35. An Example: 36. Tensors of the Second
Order: 37. Tensors of the Third Order.
XXIV. SOME IMPORTANT TENSORS . . . 184-190
SECTION 38. The Riemann-Christoffel Tensor: 39. The Linear
Gravitational Tensor : 40. Cyclic Reduction : 41. Some Cartesian
Group-Tensors.
PART I
GENERAL PRINCIPLES
W. R.
CHAPTER I
PREFATORY EXPLANATIONS
THE doctrine of relativity affects every branch of natural
science, not excluding the biological sciences. In general,
however, this impact of the new doctrine on the older
sciences lies in the future and will disclose itself in ways
not yet apparent. Relativity, in the form of novel
formulae relating time and space, first developed in con
nection with electromagnetism, including light pheno
mena. Einstein then proceeded to show its bearing on
the formulae for gravitation. It so happens therefore
that owing to the circumstances of its origin a very
general doctrine is linked with two special applications.
In this procedure science is evolving according to its
usual mode. In that atmosphere of thought doctrines
are valued for their utility as instruments of research.
Only one question is asked : Has the doctrine a precise
application to a variety of particular circumstances so
as to determine the exact phenomena which should be
then observed? In the comparative absence of these
applications beauty, generality, or even truth, will not
save a doctrine from neglect in scientific thought. With
them, it will be absorbed.
Accordingly a new scientific outlook clings to those
fields where its first applications are to be found. They
are its title deeds for consideration. But in testing its
truth, if the theory have the width and depth which
marks a fundamental reorganisation, we cannot wisely
confine ourselves solely to the consideration of a few
happy applications. The history of science is strewn
with the happy applications of discarded theories. There
12
4 PREFATORY EXPLANATIONS [OH
are two gauges through which every theory must pass.
There is the broad gauge which tests its consonance with
the general character of our direct experience, and there
is the narrow gauge which is that mentioned above as
being the habitual working gauge of science. These
reflections have been suggested by the advice received
from two distinguished persons to whom at different
times I had explained the scheme of this book. The
philosopher advised me to omit the mathematics, and
the mathematician urged the cutting out of the philo
sophy. At the moment I was persuaded: it certainly
is a nuisance for philosophers to be worried with applied
mathematics, and for mathematicians to be saddled with
philosophy. But further reflection has made me retain
my original plan. The difficulty is inherent in the
subject matter.
To expect to reorganise our ideas of Time, Space, and
Measurement without some discussion which must be
ranked as philosophical is to neglect the teaching of
history and the inherent probabilities of the subject.
On the other hand no reorganisation of these ideas can
command confidence unless it supplies science with added
power in the analysis of phenomena. The evidence is
tw r o-fold, and is fatally weakened if the two parts are
disjoined.
At the same time it is well to understand the limita-^ 1
tions to the meaning of philosophy in this connection.
It has nothing to do with ethics or theology or the
theory of aesthetics. It is solely engaged in determining
the most general conceptions which apply to things
observed by the senses. Accordingly it is not even
metaphysics : it should be called pan-physics. Its task
is to formulate those principles of science which are
i] PREFATORY EXPLANATIONS 5
employed equally in every branch of natural science. |
Sir J. J. Thomson, reviewing in Nature* Poynting s
Collected Papers, has quoted a statement taken from
rone of Poynting s addresses :
I have no doubt whatever that our ultimate aim
must be to describe the sensible in terms of the sensible.
Adherence to this aphorism, sanctioned by the
authority of two great English physicists, is the keynote
of everything in the following chapters. The philosophy J
of science is the endeavour to formulate the most general I
characters of things observed. These sought-for charac
ters are to be no fancy characters of a fairy tale enacted
behind the scenes. They must be observed characters!
of things observed. Nature is what is observed, and the
ether is an observed character of things observed. Thus
the philosophy of science only differs from any of the
special natural sciences by the fact that it is natural
science at the stage before it is convenient to split it up
into its various branches. This philosophy exists because
there is something to be said before we commence the
process of differentiation. It is true that in human
thought the particular precedes the general. Accord
ingly the philosophy will not advance until the branches
of science have made independent progress. Philosophy
then appears as a criticism and a corrective, and what
is now to the purpose as an additional source of evi
dence in times of fundamental reorganisation.
This assignment of the role of philosophy is borne
out by history. It is not true that science has advanced
in disregard of any general discussion of the character
of the universe. The scientists of the Renaissance and
their immediate successors of the seventeenth century,
* Dec. 30, 1920.
PREFATORY EXPLANATIONS
[CH
to whom we owe our traditional concepts, inherited
from Plato, Aristotle and the medieval scholastics. It
is true that the New Learning reacted violently against
the schoolmen who were their immediate predecessors;
but, like the Israelites when they fled from Egypt, they
borrowed their valuables and in this case the valuables
were certain root-presuppositions respecting space, time,
matter, predicate and subject, and logic in general. It
is legitimate (as a practical counsel in the management
of a short life) to abstain from the criticism of scientific
foundations so long as the superstructure works. But
to neglect philosophy when engaged in the re-formation
of ideas is to assume the absolute correctness of the
chance philosophic prejudices imbibed from a nurse or
a schoolmaster or current modes of expression. It is to
enact the part of those who thank Providence that they
have been saved from the perplexities of religious en
quiry by the happiness of birth in the true faith. The
truth is that your available concepts depend upon your
philosophy. An examination of the writings of John
Stuart Mill and his immediate successors on the pro
cedure of science writings of the highest excellence
within their limitations will show that they are ex
clusively considering the procedure of science in the
framing of laws with the employment of given concepts.
If this limitation be admitted, the conclusion at once
follows that philosophy is useless in the progress of
science. But when once you tamper with your basic
concepts, philosophy is merely the marshalling of one
main source of evidence, and cannot be neglected.
But when all has been said respecting the importance
, of philosophy for the discovery of scientific truth, the
I narrow-gauged pragmatic test will remain the final
i] PREFATORY EXPLANATIONS 7
arbiter. Accordingly I now proceed to a summary
account of the general doctrine either implicit or explicit
in the following pages or in my two previous books *
on this subject, and to detail the facts of experience
which receive their explanation from it or should be
observed if it be true.
A relativistic view of time is adopted so that an in
stantaneous moment of time is nothing else than an
instantaneous and simultaneous spread of the events of
the universe. But in the concept of instantaneousness
the concept of the passage of time has been lost. Events
essentially involve this passage. Accordingly the self-
contradictory idea of an instantaneous event has to be
replaced by that of an instantaneous configuration of
the universe. But what is directly observed is an event.
Thus a duration, which is a slab of time with temporal
thickness, is the final fact of observation from which
moments and configurations are deduced as a limit which
is a logical ideal of the exact precision inherent in
nature. This process of deducing limits is considered
in detail in my two previous books under the title
Extensive Abstraction. But it is an essential assump
tion that a concrete fact of nature always includes
temporal passage.
A moment expresses the spread of nature as a con
figuration in an instantaneous three dimensional space.
The flow of time means the succession of moments, and
this succession includes the whole of nature. Rest and
motion are direct facts of observation concerning the
relation of objects to the durations whose limits are the
moments of this flow of time. By means of rest a
* The Principles of Natural Knowledge, and The Concept of
Nature, both Camb. Univ. Press.
8 PREFATORY EXPLANATIONS [en
permanent point is defined which is merely a track of
event-particles with one event-particle in every moment.
Refined observation (in the form of the Michelson-
Morley experiment and allied experiments) shows that
there are alternative flows of time or time-systems,
as they will be called, and that the time-system
actually observed is that one for which (roughly
speaking) our body is at rest. Accordingly in different
circumstances of motion, space and time mean different
things, the moments of one time-system are different
from the moments of another time-system, the per
manent points of one time-system are different from
those of another time-system, so that the permanent
space of one time-system is distinct from the permanent
space of another time-system.
The properties of time and space express the basis of
uniformity in nature which is essential for our know
ledge of nature as a coherent system. The physical field
expresses the unessential uniformities regulating the
contingency of appearance. In a fuller consideration of
experience they may exhibit themselves as essential;
but if we limit ourselves to nature there is no essential
reason for the particular nexus of appearance.
Thus times and spaces are uniform.
Position in space is merely the expression of diversity
of relations to alternative time -systems. Order in space
is merely the reflection into the space of one time-system
of the time-orders of alternative time-systems.
A plane in space expresses the quality of the locus of
intersection of a moment of the time-system in question
(call it time-system A ) with a moment of another time-
system (time-system B).
The parallelism of planes in the space of time-system
A means that these planes result from the intersections
i] PREFATORY EXPLANATIONS 9
of moments of A with moments of one other time-
system B.
A straight line in the space of time-system A perpen
dicular to the planes due to time-system B is the track
in the space of time-system A of a body at rest in the
space of time-system B.
Thus the uniform Euclidean geometry of spaces,
planeness, parallelism, and perpendicularity are merely
expressive of the relations to each other of alternative
time-systems.
The tracks which are the permanent points of the
same time -system are also reckoned as parallels.
Congruence and thence, spatial measurement is
defined in terms of the properties of parallelograms and
the symmetry of perpendicularity.
Accordingly, position, planes, straight lines, paral
lelism, perpendicularity, and congruence are expressive
of the mutual relations of alternative time-systems.
The symmetrical properties of relative velocity are
shown (in The Principles of Natural Knowledge] to
issue in a critical velocity c, which thus is defined with
out reference to the velocity of light. However experi
ment shows that for our purposes it must be a near
approach to that velocity. The final result is the geo
metry and kinematic which are explained in Chapter iv
of the present volume.
A physical object, such as a mass-particle or an elec
tron, expresses the character of the future so far as it
is determined by the happenings of the present. The
exact meaning of an object as an entity implicated in
events is explained. The track of an object amid events
is determined by the stationary property of the impetus
realised by the pervasion of the track by the object.
This impetus depends partly on the intrinsic character
10 PBEFATORY EXPLANATIONS [OH
of the object e.g. its mass or its electric charge and
partly on the intrinsic potential impetus of the track
itself. This potential impetus arises from the physical
character of the events of the region due to the presence
of other objects in the past. This physical character is
partly gravitational and partly electrical.
This dependence of physical character on antecedent
objects is directly expressed by the formula here adopted
for the gravitational law. This law also gives the most
direct expression to the principle that the flux of time
is essential to the concrete reality of nature, so that a
loss of time-flux means a transference to a higher ab
straction. It gives this expression by conceiving the
attracting body as pervading an element of its track
and not as at an event-particle. This law gives the
Einstein expression for the revolution of the perihelion
of Mercury.
The electromagnetic equations adopted are Maxwell s
equations modified by the gravitational tensor compo
nents in the well-known way. Light is given no privi
leged position, and all deductions concerning light follow
directly from treating it as consisting of short waves of
electromagnetic disturbance. In this way Einstein s
assumption that a ray of light follows the path
[i.e. in Einstein s notation
ds 2 = 0]
can be proved as an approximation due to the shortness
of the waves.
The bending of the light rays in a gravitational field
then follows.
With regard to the shift of spectral lines, there are
i] PREFATORY EXPLANATIONS 11
three effects to be considered: (i) Einstein s predicted
shift due to the gravitational potential, (ii) the limb
effect which has been observed in the case of light from
the sun, (iii) the doubling or trebling of spectral lines
observed in the spectra due to some nebulae. Neither
of the effects (ii) or (iii) has hitherto been explained.
As to (i) this is traced to the combination of two
causes, one being the change in the apparent mass due
to the gravitational potential and the other being the
change in the electric cohesive forces of the molecule
due to the gravitational field. The total result is that the
period of vibration is changed from T to T+ST 7 , where
T 6c 2
i// 4 being the gravitational potential. Einstein s result
is t/>4/ c % so that the two formulae are practically identical
for observational purposes.
With regard to effects (ii) and (iii) reasons are given
for believing that the molecules will separate into three
groups sending a distant observer light of changing
relative intensities as we pass from the centre of the
disc of the emitting body (sun or nebulae) to the edge.
One group has the above-mentioned shift, another has
the shift
~
(where 77 is probably about 1/10, but may be nearly 1/5),
and the third group has the shift
8T_2+2
T * "WVi-
Under circumstances such that all or two of the groups
send separately observable light, the trebling or doubling
effects are explained to the extent of demonstrating the
12 PREFATORY EXPLANATIONS [OH i
existence of causes for the multiplication of lines, other
than those due to the motions of the matter of the nebulae.
Under other circumstances (e.g. light from the sun s disc)
in which the influence of the grouping is effective but
not separately observable the shift approximates to
^ T 1
j + l -
where yS x varies from zero at the centre of the disc of
the sun to ?r/2 at its edge. But there will be various
intermediate circumstances between these extreme as
sumptions as to the observability of the grouping effect.
Finally in a steady electromagnetic field the electro
magnetic equations predict two novel magnetic forces
due to the gravitational field. These forces are exces
sively small: (i) A steady electric force at a point on
the earth s surface (F in electrostatic units) should be
accompanied by the horizontal magnetic force
r2xlO~ 9 x.Fsma (gausses)
perpendicular to its direction and to the vertical, where
a is the angle between these directions.
(ii) A steady current (/ in electromagnetic measure)
in a straight wire making an angle /3 with the vertical
should produce at a point distant R from the wire the
parallel magnetic force (i.e. in a direction parallel to the
wire),
1 27
- x 10~ 9 x cos < sin 2/3 x -^ (gausses),
where < is the angle between the vertical plane through
the wire and the plane through the wire and the point.
The temperature of an attracting body should augment
its gravitational field by an amount which is probably
outside the limits of our observational powers.
CHAPTER II
THE RELATEDNESS OF NATURE
"Threads and floating wisps
Of being, ..."
CLEMENCE DANE S Will Shakespeare, Act i.
You have conferred upon me the honour of becoming
the first recipient of the James-Scott Prize, and have
at the same time assigned to me the duty of delivering
a lecture upon the subject which this prize is designed
to foster. In choosing the topic of a lecture which is to
be the first of a series upon the philosophy of science,
it seems suitable to explore the broadest possible aspect
of the subject. Accordingly I propose to address you
upon Relatedness and, in particular, upon the Related-
ness of Nature. I feel some natural diffidence in speaking
upon this theme in the capital of British metaphysics,
haunted by the shade of Hume. This great thinker
made short work of the theory of the relatedness of
nature as it existed in the current philosophy of his
time. It is hardly too much to say that the course of
subsequent philosophy, including even Hume s own
later writings and the British Empirical School, but
still more in the stream which descends through Kant,
Hegel and Caird, has been an endeavour to restore
some theory of relatedness to replace the one demolished
by Hume s youthful scepticism. If you once conceive
fundamental fact as a multiplicity of subjects qualified
by predicates, you must fail to give a coherent account
of experience. The disjunction of subjects is the pre
supposition from which you start, and you can only
14 THE RELATEDNESS OF NATURE [OH
account for conjunctive relations by some fallacious
sleight of hand, such as Leibniz s metaphor of his monads
engaged in mirroring. The alternative philosophic posi
tion must commence with denouncing the whole idea
of subject qualified by predicate as a trap set for
philosophers by the syntax of language. The conclusion
which I shall wish to enforce is that we can discern in
nature a ground of uniformity, of which the more far-
reaching example is the uniformity of space-time and
the more limited example is what is usually known
under the title, The Uniformity of Nature. My argu
ments must be based upon considerations of the utmost
generality untouched by the peculiar features of any
particular natural science. It is therefore inevitable that
at the beginning my exposition will suffer from the
vagueness which clings to generality.
Fact is a relationship of factors. Every factor of fact
essentially refers to its relationships within fact. Apart
from this reference it is not itself. Thus every factor of
fact has fact for its background, and refers to fact in a
way peculiar to itself.
I shall use the term awareness for consciousness of
factors within fact. A converse mode of statement is
that awareness is consciousness of fact as involving
factors. Awareness is itself a factor within fact.
I shall use the term cogitation for consciousness of
factors prescinded from their background of fact. It is
the consciousness of the individuality of factors, in that
each factor is itself and not another. A factor cogitated
upon as individual will be called an entity. The
essence of cogitation is consciousness of diversity. The
prescinding from the background of fact consists in
limiting consciousness to awareness of the contrast of
ii] THE RELATEDNESS OF NATURE 15
factors. Cogitation thus presupposes awareness and is
limited by the limitations of awareness. It is the re
finement of awareness, and the unity of consciousness
lies in this dependence of cogitation upon awareness.
Thus awareness is crude consciousness and cogitation
is refined consciousness. For awareness all relations
between factors are internal and for cogitation all
relations between entities are external.
Fact in its totality is not an entity for cogitation,
since it has no individuality by its reference to any
thing other than itself. It is not a relatum in the
relationship of contrast. I might have used the term
totality instead of fact ; but c fact is shorter and
gives rise to the convenient term * factor. Fact enters
consciousness in away peculiar to itself. It is not the sum
of factors ; it is rather the concreteness (or, embedded-
ness) of factors, and the concreteness of an inexhaustible
relatedness among inexhaustible relata. If for one
moment I may use the inadmissible word Factuality, it
is in some ways better either than fact or totality for
the expression of my meaning. For fact suggests one
fact among others. This is not what I mean, and is a sub
ordinate meaning which I express by factor. Also
totality suggests a definite aggregate which is all that
there is, and which can be constructed as the sum of all
subordinate aggregates. I deny this view of factuality.
For example, in the very conception of the addition of
subordinate aggregates, the concept of the addition
is omitted although this concept is itself a factor
of factuality. Thus inexhaustibleness is the prime
character of factuality as disclosed in awareness ; that
is to say, factuality (even as in individual awareness)
cannot be exhausted by any definite class of factors.
16 THE KELATEDNESS OF NATURE [CH
After this explanation I will now relapse into the use
of fact in the sense of factuality.
The finiteness of consciousness, the factorisation of
fact, the individualisation of entities in cogitation, and
the opposition of abstract to concrete are all exhibitions
of the same truth of the existence of limitation within
fact. The abstract is a limitation within the concrete,
the entity is a limitation within totality, the factor is a
limitation within fact, and consciousness by its reference
to its own standpoint within fact limits fact to fact as
apprehended in consciousness. The treatment of the
whole theory of limitation has suffered by the introduc
tion of metaphors derived from a highly particular form
of it, namely, derived from the analogy between extended
things, such as that of whole to part and that of things
mutually external to each other.
I use the term * limitation for the most general
conception of fmitude. In a somewhat more restricted
sense Bergson uses the very convenient term canalisa
tion. This Bergsonian term is a useful one to keep in
mind as a corrective to the misleading associations of
the terms external and internal/ or of the terms
whole and part. It adds also a content to the
negative term limitation. Thus a factor is a limitation
of fact in the sense that a factor refers to fact canalised
into a system of relata to itself, i.e. to the factor in
question. The mere negative limitation, or finitude, in
volved in a factor is exhibited in cogitation, wherein the
factor degenerates into an entity and the canalisation
degenerates into a bundle of external relations.
Thus also finite consciousness is a limitation of fact,
in the sense that it is a factor canalising fact in ways
peculiar to itself. We must get rid of the notion of
ii] THE RELATEDNESS OF NATURE 17
consciousness as a little box with some things inside it.
A better metaphor is that of the contact of conscious
ness with other factors, which is practically Hume s
metaphor impression. But this metaphor erroneously
presupposes that fact as disclosed in awareness can be
constructed as an entity formed by the sum of the
impressions of isolated factors.
Again cogitation is a further limitation of fact in that
it is a canalisation of consciousness so as to divest it of
the crudeness of awareness. This illustrates that in
limitation there is a gain in clarity, or definition, or
intensity, but a loss of content.
For example, the factor red refers to fact as canalised
by relationships of other factors to red, and the entity
red is the factor red in its capacity as a relatum in the
relationship of contrast, whereby it is contrasted with
green or with sound or with the moon or with the
multiplication table. Thus the factor red, essentially
for its being, occasions the exhibition of a special aspect
of fact, and the entity red is a further limitation of this
aspect. Similarly the number three is nothing else than
the aspect of fact as factors grouped in triplets. And
the Tower of London is a particular aspect of the
Universe in its relation to the banks of the Thames.
Thus an entity is an abstraction from the concrete,
which in its fullest sense means totality.
The point of this doctrine on which I want to insist
is that any factor, by virtue of its status as a limitation
within totality, necessarily refers to factors of totality
other than itself. It is therefore impossible to find any
thing finite, that is to say, any entity for cogitation,
which does not in its apprehension by consciousness
disclose relationships to other entities, and thereby dis-
w. R. 2
18 THE RELATEDNESS OF NATURE [CH
close some systematic structure of factors within fact.
I call this quality of finitude, the significance of factors.
This doctrine of significance necessitates that we admit
that awareness requires a dual cognisance of entities.
There can be awareness of a factor as signifying, and
awareness of a factor as signified. In a sense this may
be represented as an active or a passive cognisance of
the entity. The entity is either cognised for its own
sake, that is to say, actively, or it is cognised for the
sake of other entities, that is to say, passively. If an
entity is cognised actively, it is cognised for the sake
of what it is in itself, for the sake of what it can make
of the universe. I will call this sort of awareness of a
factor, cognisance by adjective; since it is the character
of the factor in itself which is then dominant in conscious
ness. Although in cognisance by adjective an entity is
apprehended as a definite character in its relations to
other entities, yet in a sense this type of cognisance
marks a breakdown in relatedness. For the general
relatedness of the character to other factors merely
marks the fullness of its content, so that in effect the
character is cognised for what it is in itself. Relation
ships to other factors occur in such cognisance only
because the character is not itself apart from that
ordering of fact.
When an entity is cognised passively, we are aware
of it for the sake of some other factor. We are conscious
passively of factor A, because factor B of which we are
actively aware would not be what it is apart from its
relatedness to A. Thus the individual character of A
is in the background, and A becomes a vague something
which is an element in a complex of systematic related-
ness. The very nature of the relatedness may impose
n] THE RELATEDNESS OF NATURE 19
on A some character. But the character is gained
through the relatedness and not the relatedness through
the character. Accordingly A gains in consciousness
the very minimum foothold for the relationship of
contrast, and is thus the most shadowy of entities. I
will call this sort of awareness of a factor, cognisance
by relatednoss. For example the knowledge of events
inside another room is to be gained by their spatial and
other relationships to events of which we have cognisance
by adjective.
Thus cognisance of one factor by relatedness pre
supposes cognisance of other factors by adjective; and
conversely, cognisance of one factor by adjective pre
supposes cognisance of other factors by relatedness.
It is possible to be aware of a factor both in cognisance
by adjective and cognisance by relatedness. This will
be termed full awareness of the factor and is the usual
form of awareness of factors within the area of clear
apprehension when intrinsic characters and mutual
relations are jointly apparent. Perception will be the
name given to the consciousness of a factor when to
full awareness cogitation of it as an entity is also
superadded.
But cogitation does not necessarily presuppose full
awareness. For the contrast involved in cogitation may
simply fall on the quality of the individualities of the
factors, as when green as such is contrasted with red as
such. In such a case merely awareness by adjective is
presupposed. But the contrast may also fall on the
specific relationships of each of the two factors to other
factors, as when we contrast an event in the interior of
the moon with another event in the interior of the
earth. The spatio-temporal relationships of the two
22
20 THE KELATEDNESS OF NATUKE [CH
events are then contrasted ; and it is from contrasts of
this type that the two events gain their definite
individuality as entities.
At this point in the discussion I will confine the scope
of the remainder of my lecture strictly to the considera
tion of the relatedness of nature. This requires us to
recognise another limitation within awareness which
cuts across those already mentioned. I mean the limita
tion of awareness to sense-awareness. Nature is the
system of factors apprehended in sense-awareness. But
sense-awareness can only be defined negatively by
enumerating what it is not.
Divest consciousness of its ideality, such as its logical,
emotional, aesthetic and moral apprehensions, and what
is left is sense-awareness. Thus sense-awareness is
consciousness minus its apprehensions of ideality. It is
not asserted that there is consciousness in fact divested
of ideality, but that awareness of ideality and sense-
awareness are two factors discernible in consciousness.
The question as to whether either the one or the other,
or both jointly may not be a factor necessary for
consciousness is beyond the scope of the present dis
cussion. The finiteness of individual consciousness means
ignorance of what is there for knowledge. There is
limitation of factors cognised by adjective, and equally
there is limitation of factors cognised by relatedness.
So it is perfectly possible to hold, as I do hold, that
nature is significant of ideality, without being at all
certain that there may not be some awareness of nature
without awareness of ideality as signified by nature.
It would have, I think, to be a feeble awareness. Per
haps it is more likely that ideality and nature are dim
together in dim consciousness. ~It is unnecessary for us
to endeavour to solve these doubts. My essential premise
nj THE RELATEDNESS OF NATURE 21
is that we are conscious of a certain definite assemblage
of factors within fact and that this assemblage is
what I call nature. Also I entirely agree that the
factors of nature are also significant of factors which
are not included in nature. But I propose to ignore
this admitted preternatural significance of nature, and
to analyse the general character of the relatedness of
natural entities between themselves.
Nature usually presents itself to our imagination as
being composed of all those entities which are to be
found somewhere at some time. Sabre-toothed tigers
are part of nature because we believe that somewhere
and at some time sabre-toothed tigers were prowling.
Thus an essential significance of a factor of nature is its
reference to something that happened in time and space.
I give the name event to a spatio-temporal happening.
An event does not in any way imply rapid change ; the
endurance of a block of marble is an event. Nature
presents itself to us as essentially a becoming, and any
limited portion of nature which preserves most com
pletely such concreteness as attaches to nature itself is
also a becoming and is what I call an event. By this I
do not mean a bare portion of space- time. Such a
concept is a further abstraction. I mean a part of the
becomingness of nature, coloured with all the hues of
its content.
Thus nature is a becomingness of events which are
mutually significant so as to form a systematic structure.
We express the character of the systematic structure
of events in terms of space and time. Thus space and
time are abstractions from this structure.
Let us now examine more particularly the significance
of events in so far as it falls within nature. In this way
we are treating nature as a closed system, and this I
22 THE EELATEDNESS OF NATURE [OH
believe is the standpoint of natural science in the strict
sense of the term.
But before embarking on the details of this investiga
tion I should like to draw your attention to an objection,
and a very serious objection, which is urged by opponents
of the whole philosophic standpoint which I have been
developing. You admit, it is said, that a factor is not
itself apart from its relations to other factors. Ac
cordingly to express any truth about one entity you
must take into account its relations to all entities. But
this is beyond you. Hence, since unfortunately a pro
position must be either right or wrong or else unmeaning
and a mere verbal jangle, the attainment of truth in
any finite form is also beyond you.
Now I do not think that it is any answer to this
argument to say that our propositions are only a little
wrong, any more than it is a consolation to his friends
to say that a man is only a little dead. The gist of the
argument is that on our theory any ignorance is blank
ignorance, because knowledge of any factor requires no
ignorance. A philosophy of relatedness which cannot
answer this argument must collapse, since we have got
to admit ignorance.
Obviously if this argument is to be answered, I must
guard and qualify some of the statements which have
been made in the earlier portion of this lecture. I have
put off the job until now, partly for the sake of simplicity,
not to say too much at once, and also partly because
the line of argument is most clearly illustrated in the
case of nature, and indeed the application to nature is
the only one in which for the purposes of this lecture
we are interested. So I have waited until my discourse
had led me to the introduction of nature.
ii] THE KELATEDNESS OF NATURE 23
The answer can only take one road, we must dis
tinguish between the essential and the contingent
relationships of a factor. The essential relationships of
a factor are those relationships which are inherent in
the peculiar individuality of the factor, so that apart
from them the factor is not the special exhibition of
finitude within fact which it is. They are the relation
ships which place the factor as an entity amid a definite
system of entities. The significance of a factor is solely
concerned with its essential relationships. The con
tingent relationships of a factor are those relationships
between that factor and other factors which might be
otherwise without change of the particular individuality
of the factor. In other words, the factor would be what
it is even if its contingent relationships were otherwise.
Thus awareness of a factor must include awareness
of its essential relationships, and is compatible with
ignorance of its contingent relationships.
It is evident that essential and contingent relationships
correspond closely to internal and external relations. I
hesitate to say how closely, since a different philosophic
outlook radically affects all meanings.
We still have to explain how awareness of a factor
can exclude ignorance of the relationships involved in
its significance. For, on the face of it, this doctrine
means that to perceive factor A we require also to per
ceive factors B, C, D, etc., which A signifies. In view
of the possibilities of ignorance, such a doctrine appears
to be extremely doubtful. This objection ignores the
analysis of awareness into cognisance by adjective and
cognisance by relatedness. In order to perceive A we do
not require to be conscious of B, (7, D, with cognisance
by adjective. We only require cognisance by relatedness.
24 THE RELATEDNESS OF NATURE [CH
In other words we must be conscious of B, C, D, ... as
entities requisite for that relatedness to A, which is in
volved in A s significance. But even this explanation
asks for too much. It suggests that we must be conscious
of B, C, D, ... as a definite numerical aggregate of
entities signified by A. Now it is evident that no factor
A makes us conscious of the individual entities of such
an aggregate. Some necessary qualification of the doc
trine of significance has been omitted. The missing
principle is that any factor A has to be uniformly
significant. [Every entity involves that fact shall be
patient of it.JJThe patience of fact for A is the converse
side of the significance of A within fact.j This involves
a canalisation within fact ; and this means a systematic
aggregate of factors each with the uniform impress of
the patience of fact for A. A can be, because they are.
Each such factor individually expresses the patience of
fact for A.
Thus the knowledge required by the significance of A
is simply this. In order to know A wejnust ^now how
other factors express the patience of fact for A^ We
need not be aware of these other factors individually,
but the awareness of A does require an awareness of
their defining character. There is no such entity as mere
A in isolation. A requires something other than itself,
namely, factors expressing the patience of fact in respect
to factor A.
Let us now apply to nature this doctrine of uniform
significance. We commence by taking the case of the
colour green. When we perceive green, it is not green
in isolation, it is green somewhere at some time. The
green may or may not have the relationship to some
other object, such as a blade of grass. Such a relation
n] THE RELATEDNESS OF NATURE 25
would be contingent. But it is essential that we see it
somewhere in space related to our eyes at a certain
epoch of our bodily life. The detailed relationships of
green to our bodily life and to the situations in which
it is apparent to our vision are complex and variable and
partake of the contingence which enables us to remain
ignorant of them. But there can be no knowledge of
green without apprehension of times and places. Green
presupposes here and there, and now and then. In other
words, green presupposes the passage of nature in the
form of a structure of events. It may be merely green
associated vaguely with the head, green all about me;
but green is not green apart from its signification of
events with structural coherence, which are factors ex
pressing the patience of fact for green.
A blade of grass is an object of another type which
signifies nature as a passage of events. In this respect
it only differs from green in so far as its contingent
relations to some definite events are perhaps sharper
and capable of more precise determination.
The significance of events is more complex. In the
first place they are mutually significant of each other.
The uniform significance of events thus becomes the
uniform spatio-temporal structure of events. In this
respect we have to dissent from Einstein who assumes
for this structure casual heterogeneity arising from con
tingent relations. Our consciousness also discloses to
us this structure as uniformly stratified into durations
which are complete nature during our specious presents.
These stratifications exhibit the patience of fact for
finite consciousness, but then they are in truth charac
ters of nature and not illusions of consciousness.
Returning to the significance of events, we see that
26 THE RELATEDNESS OF NATURE [OH
there is no such thing as an isolated event. Each event
essentially signifies the whole structure. But further
more, there is no such entity as a bare event. Each
event also signifies objects, other than events which are
in essential "relation to it. In other words the passage
of an event exhibits objects which do not pass. I have
termed the natural factors which are not events but are
implicated in events objects, and awareness of an object
is what I have termed recognition. Thus green is an
object and so is a blade of grass, and awareness of green
or of a blade of grass is recognition. Thus an event
signifies objects in mutual relations. The particular
objects and their particular relations belong to the sphere
of contingence ; but the event is essentially a field, in
the sense that without related objects there can be no
event. On the other hand related objects signify events,
and without such events there are no such objects.
The celebrated two-termed relationship of universals
to the concrete particulars which they qualify is merely
a particular example of the general doctrine of signifi
cance and patience. The universals are significant of
their particulars, and the particulars are factors ex
hibiting the patience of fact for those universals.
But in the apparent world, that is to say, in the world
of nature disclosed by sense-awareness, no example of
the simple two-termed relationship of a universal signi
fying its particular is to be found. Green appears to an
observer in a situation distinct from that of the observer,
but simultaneous with it. Thus there is essential refer
ence to three simultaneous events, the event which is
the bodily life of the observer, called the percipient
event, and the event which is the so-called situation of
the green at the time of observation, and to the time of
n] THE RELATEDNESS OF NATURE 27
observation which is nothing else than the whole of
nature at that time. Under the obsession of the logical
theory of universals and concrete particulars the per
cipient event was suppressed, and the relation of green
to its situation represented as universal qualifying par
ticular. It was then noted that this relation only holds
for the particular observer, and that furthermore account
must be taken of contingent circumstances such as the
transmission of something, which is not the colour green,
from an antecedent situation to the percipient event.
This process, of first presupposing a two-termed re
lation and then finding that it is not true, has led to the
bifurcation which places green in the observer s mind,
qualifying a particular also in the observer s mind ; while
the whole mental process has some undetermined rela
tion to another system of entities variously described
either as an independent physical universe in some
causal relation to mind or as a conceptual model.
I have argued elsewhere in detail that this result is
untenable. Here I will only remark that if we incline
to adopt the physical universe, we can find no shred of
evidence for it, since everything apparent for conscious
ness has been accounted for as being in the observer s
mind; while, if we turn to the conceptual model, it is
also the model for the same consciousness. Accordingly
whichever choice we make there will be no shred of
evidence for anything other than the play of that con
sciousness at one moment of self-realisation. For recol
lection and anticipation are merely the play of immediate
consciousness. Thus on either alternative, solipsism
only is left and very little of that.
Meanwhile the whole difficulty has arisen from the
initial error of forcing the complex relations between
28 THE RELATEDNESS OF NATURE [OH
green and the structure of events into the inadequate
form of a two-termed relation.
Yet after all the search for universals to qualify events
in the simple two-termed manner does represent a
justifiable demand. We want to know what any par
ticular event A is in itself apart from its reference to
other events. By this I mean, we want to determine
how A can enter into a two- termed relation of contrast
with any other factor X without having necessarily to
enlarge the relationship by including other events B,
C, D, by way of determining A. For example, the
colour green is in itself different from red, and we do
not have to specify green or red by their diverse relation
ships amid events in order to appreciate their contrast.
Now we want to do much the same thing for events, so
as to feel that an event has a character of its own. We
have seen that the immediate objects of the apparent
world such as colours do not satisfy the requisite
conditions since their reference to events involves the re
lations of the percipient event to the so-called situation.
I call such objects of immediate appearance, sense-
objects. Colours, sounds, smells, touches, pushes, bodily
feelings, are sense-objects. But after all, the way we
do connect these sense -objects, as I call them, with
their situations shows that awareness of an event carries
with it apprehension of that event as patient of a
character qualifying it individually. In fact every event
signifies a character for itself alone, but what exactly
that character may be lies within the sphere of con
tingency and is not disclosed in our immediate conscious
ness of the apparent world. I will call such a character
an adjective of its event. An adjective marks a break
down in relativity by the very simplicity of the two-
ii] THE KELATEDNESS OF NATURE 29
termed relation it involves. The discovery of these
missing adjectives is the task of natural science. The
primary aim of science is to contract the sphere of con
tingency by discovering adjectives of events such that
the history of the apparent world in the future shall be
the outcome of the apparent world in the past. There
obviously is some such dependence, and it is the pur
pose of science to express this dependence in terms of
adjectives qualifying events. In order to understand
this procedure of science, there are three concepts which
we must understand. They are
(i) The structure of the four dimensional continuum,
(ii) Pervasive adjectives and adjectival particles, *
(iii) The atomic field of an adjectival particle.
I will conclude this lecture by considering them in order.
(i) The structure of the continuum of events
This structure is four-dimensional, so that any event
is a four-dimensional hyper-volume in which time is the
fourth dimension. But we should not conceive an event
as space and time, but as a unit from which space and
time are abstracts.
An event with all its dimensions ideally restricted is
called an event-particle/ and an event with only one
dimension of finite extension is called a route or path/
I will not in this lecture discuss the meaning of this ideal
restriction. I have investigated it elsewhere under the
name of extensive abstraction.
The structure is uniform because of the necessity for
knowledge that there be a system of uniform relatedness,
in terms of which the contingent relations of natural
factors can be expressed. Otherwise we can know
30 THE RELATEDNESS OF NATURE [OH
nothing until we know everything. If P be any event-
particle, a moment through P is a system of event-
particles representing all nature instantaneously con
temporaneous with P. According to the classical view
of time there can be only one such moment. According
to the modern view there can be an indefinite number
of alternative moments through P, each corresponding
to a different meaning for time and space. A moment
is an instantaneous three-dimensional section of nature
and is the entity indicated when we speak of a moment
of time.
The aggregate of event-particles lying on moments
through P will be called the region co-present with P.
The remainder of the four-dimensional continuum is
divided by the co-present region into two regions, one
being P s past and the other being P s future. The three-
dimensional boundary between P s past and P s co-
present region is P s causal past, and the corresponding
boundary between P s future and P s co-present region
is P s causalfuture. The remaining portion of P s future
is P s kinematic future.
A route lying entirely in one moment is called a
spatial route, and a route which lies entirely in the past
and future of each one of its event-particles is called a
historical* route.
(ii) Pervasive adjectives and adjectival particles
We gain great simplicity of explanation, without loss
of any essential considerations by confining our con
sideration of events to routes. These routes are of course
not true events, but merely ideal limits with only one
dimensional extension remaining.
* I borrow the term historical from Prof. C. D. Broad.
II]
THE RELATEDNESS OF NATURE
31
32 THE RELATEDNESS OF NATURE [CH
A factor will be said to be an adjective pervading a
route when it is an adjective of every stretch of the
route. Such a factor will be called a pervasive adjective,
or uniform object. I think without being very certain
that true pervasive adjectives are only to be found
qualifying historical routes ; but that pervasive pseudo-
adjectives also qualify spatial routes. The essential
difference between time and space finds its illustration in
the difference between these two different types of route.
As an illustration of pervasive adjectives, consider a
mass-particle m. The enduring existence of this particle
marks out a historical route amid the structure of events.
In fact the mass-particle is merely a pervasive adjective
of that route, since it is an adjective qualifying in the
same sense every stretch of that route. But here a
further explanation is necessary. The mass-particle as
a pervasive adjective is a universal and has lost its
concrete individuality.
Another mass-particle of the same mass pervading
another historical route is the same pervasive adjective
also qualifying every stretch of that other route. It
follows that the separate concrete individualities of the
two mass-particles arise from the separate individualities
of their two historical routes. Thus a concrete mass-
particle is the fusion of a pervasive adjective with the
individuality of a historical route. We say that a mass-
particle is situated at each event -particle of its historical
route. I will call a pervasive adjective as qualifying a
particular historical route an adjectival particle. The
principle underlying the conception of an adjectival
particle is that the individual embodiment of character
always involves process and that this process is here
represented by the historical route.
ii] THE KELATEDNESS OF NATUKE 33
Spatial routes cannot be pervaded by mass- particles.
Thus if a mass-particle of the same mass be situated at
each event-particle of a spatial route, that route is not
pervaded by the one adjective which is the same uni
versal for each of the concrete mass-particles. In fact a
stretch of the spatial route is qualified by quite a differ
ent adjective, which represents the sum of the masses
situated at the event-particles of the stretch. Accord
ingly spatial routes and historical routes function quite
differently in respect to the adjective mass w, and thus
illustrate the difference between the spread of space and
the lapse of time.
There are however pseudo-adjectives which do pervade
spatial routes. Consider a sense-object, such as the
colour red. It is not a true adjective of its situation,
since there is always a necessary reference to a per
cipient event. But for the one observer conscious of the
natural relations of that percipient event, who is pre
supposing this reference to his bodily life as a condition
for appearance, the colour red is an adjective of its
situation. But any part of its situation is also red,
neither more nor less so than the whole enduring patch
of red. Thus red pervades its situation. However I have
already argued at length that sense-objects are not true
adjectives. They simulate adjectives for an observer
who in his intellectual analysis of the circumstances
forgets to mention himself. Accordingly they may be
called pervasive pseudo-adjectives.
The common material objects of perception, such as
chairs, stones, planets, trees, etc., are adjectival bodies
pervading the historical events which they qualify. In
so expressing myself, I have gone beyond the ideal
simplicity of a route, and the terms pervasion and
w. R. 3
34 THE RELATEDNESS OF NATURE [CH
historical event require, strictly speaking, a more
elaborate explanation than I have yet given. In this
lecture however it is unnecessary to undertake the task,
and I need only refer to my Principles of Natural
Knowledge where the requisite definitions are given in
connection with uniform objects.
(iii) The atomic field of an adjectival particle
Science has been driven to have recourse to more
precisely delimited adjectival objects than these ad
jectival objects of perception. The standard types of
such objects are mass-particles and electrons; and we
will fall back on our ideal simplicities by conceiving
them as adjectival particles defined, as above, for the
ideal simplification of historical routes.
Now the essence of an adjectival object, whether it
be the unprecise object of perception or the more precise
object of science, is that it reduces the contingency of
nature. It is an adjective of events which to some extent
conditions the possibilities of apparent sense-objects.
It must be admitted that it is itself a contingent
adjective. But owing to the simplicity of the relation
of an adjective to its qualified substance, it involves a
simpler contingency than the contingency of the complex
relationships of sense-objects. In other words we are
limiting contingency by the fixed conditions which are
the laws of nature.
It is evident therefore that a scientific object must
qualify future events. For otherwise the future contin
gency is unaffected by it. In this a scientific object
differs decisively from a sense-object viewed as a pseudo-
adjective. A sense-object qualifies events in the present.
ii] THE RELATEDNESS OF NATURE 35
It is confined to a spatial region with the minimum of
historicity requisite for the duration of the present.
Whereas the scientific object qualifies a region extending
from the present into the future. Thus the seemingly
contingent play of the senses is controlled by the
conditions introduced by its dependence upon the
qualification of events introduced by scientific objects.
A scientific object qualifies the future in two ways,
(a) by its permanence and (b) by its field.
Let us take the permanence first. The permanence
of an adjectival particle lets us know that there will be
some historical route pervaded by that particle. It does
not in itself tell us more than that some pervaded route
will stretch into the future from the situation in the
present. The permanence of the unique particle is
nothing else than the continuity of the unique historical
route, and its pervasion by the adjectival particle.
The further laws of physical science represent the
further conditions which determine, or partially deter
mine, the particular historical routes pervaded by these
adjectival particles. The most simple expression of such
a law consists in associating an atomic field with each
adjectival particle as situated in each one of the event-
particles of ifys pervaded route. Again this association
of the field represents another eruption of contingency,
but also again this contingency is of a simple defined
type. The field of an adjectival particle m at a situation
P is a limited region stretching from P into P s futurity.
This region is qualified by an adjective dependent upon
m and P only. For this simple type of law, the only
limited region which can satisfy this demand is the three-
dimensional boundary region between P s co-present
and P s kinematic future. I have called this region P s
32
36 THE RELATEDNESS OF NATURE [CH
causal future. Accordingly the field of m at P must be
P s causal future. Expressing this statement in terms
of one consistent meaning for time and its associated-
permanent space, we first note that P consists of a point
S P at a time t p , and m situated in P means m at the
point S P at the time t r . The causal future of P means
those points S B , at times t B [i.e. those event-particles
such as B], reached by a physical character due to m, and
starting from S P at time t p and arriving at S B at time
t B9 and travelling with the critical maximum velocity c.
Experiment shows that this critical maximum velocity
is a near approximation to the velocity of light in vacuo,
but its definition in no way depends upon any reference
to light. Thus the adjectival character of the field of
m Sit P consists in the correlated physical characters
of the different event-particles of the field. The whole
conception is practically the familiar one of tubes
of force, with one exception. A tube of force is con
ceived statically as a simultaneous character stretching
through space. This statical conception destroys the
true individuality of a tube by piecing together frag
ments of different tubes. As we pass along a tube
radiating from S P we keep to the same tube by allowing
for the lapse of time required by the velocity c.
The peculiar correlation of adjectives attaching to the
various event-particles of the field of m at P will depend
upon the particular contingent law which science con
jectures to be the true expression of m s physical status.
There are, also, less simple laws of nature for which
the influence of the contingent configurations of other
adjectival particles will be essential factors. Such laws
will in general involve the deflection of the field of m at
P from P s causal future into P s kinematic future. The
ii] THE RELATEDNESS OF NATURE 37
region will be dependent upon the fields of the other
relevant adjectival particles. It is evident that with
such laws we are rapidly drifting towards the difficulty
of having to know everything before knowing any
thing.
I will call such fields obstructed fields. Differential
equations help us here. But even their aid would be
unavailing unless we could approximate from the first
assumption of unobstructed fields for the adjectival
particles producing the obstruction. In this way the
influence of gravitation upon the electromagnetic field
can be calculated and vice versa.
This account of the status of scientific objects com
pletely changes the status of the ether; from that
presumed in nineteenth century science. In the classical
doctrine the ether is the shy agent behind the veil : in
the account given here the ether is exactly the apparent
world, neither more nor less. The apparent world dis
closes itself to us as the ingression of sense-objects amid
events. In this statement the term ingression is used
for the complex relationship of those abstract elements
of the world, such as sense-objects, which are devoid
of becomingness and extension, to those other more
concrete elements (events) which retain becomingness
and extension. But a bare event is a mere abstraction.
Events are disclosed as involved in this relationship of
ingression. This disclosure is our perceptual vision of
the apparent world. We now ask on behalf of science
whether we cannot simplify the regulative principles
discerned in this apparent world by treating events as
something more than relata in the relationship of in
gression. Cannot we discern true Aristotelian qualities
as attaching to the events ? Is not each event something
38 THE RELATEDNESS OF NATURE [CH
in itself, apart from its status as a mere relatum
in the relationship of ingression \ The apparent world
itself gives an answer, partially in the affirmative.
Chairs, tables, and perceptual objects generally, have
lost the complexity of ingression, and appear as the
required Aristotelian adjectives of some events. Their
appearance involves that borderline where sense-aware
ness is fusing with thought. It is difficult to make any
account of them precise. In fact, for the purpose of
science they suffer from incurable vagueness. But they
mark the focal centres to be used as the radiating
centres for an exact account of true Aristotelian adjec
tives without any of those qualifications here referred
to as * vagueness. The events of the apparent world
as thus qualified by the exact adjectives of science are
what we call the ether. Accordingly in my previous
work, The Principles of Natural Knowledge, I have
phrased it in this way, that the older ether of stuff
is here supplanted by an * ether of events.
This line of thought, supplanting stuff* by events,
and conceiving events as involving process and exten
sion and contingent qualities and as primarily relata in
the relationship of ingression, is a recurrence to Des
cartes views with a difference. Descartes, like the
rest of the world at that time, completely dissociated
space and time. He assigned extension to space, and
process to time. It is true that time involves extension
of some sort, but that does not seem to have coloured
his philosophy. Now according to Descartes extension
is an abstract from the more concrete concept of stuff.
He, like the rest of the world, considers stuff as being
separable from the concept of process, so that stuff
fully realises itself at an instant, without duration.
ii] THE RELATEDNESS OF NATURE 39
Space is thus a property of stuff, and accordingly follows
stuff in being essentially dissociated from time. He there
fore deduces that space is an essential timeless plenum.
It is merely an abstract from the concrete world of
appearance at an instant. If there be no stuff to appear,
there can be no space.
Now re-write this Cartesian account of space, sub
stituting events (which retain process ) for stuff
(which has lost process ). You then return to my ac
count of space-time, as an abstract from events which
are the ultimate repositories of the varied individualities
in nature. But space as pure extension, dissociated from
process, and time as pure serial process, are correlative
abstractions which can be made in different ways, each
way representing a real property of nature. In this
manner the alternative spaces and the alternative times,
which have already been mentioned, are seen to be
justifiable conceptions, according to the account of the
immediate deliverances of awareness here given, provided
that our experience can be thereby explained.
Mere deductive logic, whether you clothe it in mathe
matical symbols and phraseology or whether you enlarge
its scope into a more general symbolic technique, can
never take the place of clear relevant initial concepts
of the meaning of your symbols, and among symbols
I include words. If you are dealing with nature, your
meanings must directly relate to the immediate facts of
observation. We have to analyse first the most general
characteristics of things observed, and then the more
casual contingent occurrences. There can be no true
physical science which looks first to mathematics for the
provision of a conceptual model. Such a procedure is to
repeat the errors of the logicians of the middle-ages.
CHAPTER III
EQUALITY
THE criticism of the meanings of simple obvious state
ments assumes especial importance when any large
reorganisation of current ideas is in progress. The up
heaval produced by the Einstein doctrine of relativity
is a case in point. It demands a careful scrutiny of the
fundamental ideas of physical science in general and of
mathematical physics in particular. I propose therefore
in this lecture to take one of the simplest mathematical
notions which we all come across when we start mathe
matics in our early school life and to ask what it
means.
The example I have chosen is the notion of equality.
There is hardly a page or a paragraph of any mathe
matical book which does not employ this idea. It
appears in geometry in the more specialised form of
congruence.
If I am not mistaken, clear notions on equality are
of decisive importance for the sound reconstruction of
mathematical physics. Congruence is a more special
term than equality, being confined to mean the quanti
tative equality of geometrical elements. Equality is also
closely allied to the idea of quantity; but here again
I think that equality touches the more general ideas.
The consideration of quantity necessarily introduces
that of measurement. In fact the scope of a discussion
on quantity may be defined by the question, How is
measurement possible? Lastly, equality has an obvious
affinity with identity. Some philosophers in considering
CH m] EQUALITY 41
the foundations of mathematics would draw no distinc
tion between the two. In certain usages of equality
this may be the case. But it cannot be the whole truth.
For if it were, the greater part of mathematics would
consist of a reiteration of the tautologous statement
that a thing is itself. We are interested in equality
because diversity has crept in.
In fact a discussion of equality embraces in its
scope congruence, quantity, measurement, identity and
diversity. The importance of equality was discovered
by the Greeks. We all know Euclid s axiom, Things
that are equal to the same thing are also equal to one
another (TO, TO) avrq) Lcra /cat aAA^Xoi? tcn\v icra). This
axiom deserves its fame, in that it is one of the first
efforts to clarify thought by an accurate statement of
premises habitually assumed. It is the most conspicuous
example of the decisive trend of Greek thought towards
rigid accuracy in detailed expression, to which we owe
our modern philosophy, our modern science, and the
creeds of the Christian Church. But grateful as we are
to the Greeks for this axiom and for the whole state of
mind which it indicates, we cannot withdraw it from
philosophic scrutiny. The whole import of the axiom
depends on the meaning of the word tcro?, equal. What
do we mean when we say that one thing is equal to
another? Suppose we explain by stating that equal
means equal in magnitude, that is to say, the things
are quantities of the same magnitude. But what is a
quantity? If we define it as having the property of
being measurable in terms of a unit, we are thrown
back upon the equality of different examples of the same
unit. It is evident that we are in danger of soothing
ourselves with a vicious circle whereby equality is
42 EQUALITY [CH
explained by reference to quantity and quantity by
reference to equality.
Let us first drop the special notion of quantitative
equality and consider the most general significance of
that notion. The relation of equality denotes a possible
diversity of things related but an identity of character
qualifying them. It is convenient for technical facility
in the arrangement of deductive trains of reasoning to
allow that a thing is equal to itself, so that equality
includes identity as a special case. But this is a mere
matter of arbitrary definition.
The important use of equality is when there is
diversity of things related and identity of character.
This identity of character must not be mere identity
of the complete characters. For in that case, by the
principle of the identity of indiscernables, the equal
things would be necessarily identical.
Accordingly when we write
A=B
we are referring implicitly to some character and asserting
that A and B both possess it. The assertion of equality
is therefore generally couched in a highly elliptical form
since the expression of the character in question is
often omitted. This is a source of most of the confused
thinking which haunts discussion on this subject. Let
us remedy our notation so as to rid it of its misleading
ellipticity. Let (c l} c 2 , . . . , c n ) denote a class of characters
c lt c a , ... , c n , such as colour for example.
Then we write
A=B-*(c l9 c a , ..., c n )
to mean that A and B both possess the same character
out of the set (c lt c 2 , ... , c n ); and we write
tfa.c,, ...,c n )
m] EQUALITY 43
to mean that different characters out of the set apply
to A and B respectively. Our notation still has the
defect of implying that the class of characters is a finite
or at least an enumerable class. Let us therefore take
y to represent this class, so that
means that the same member of the class y qualifies
both A and B\ and
means that one member of y qualifies A and that another
member of y qualifies B. I will call y the qualifying
class.
It is now evident that
A=B-*y
and Z?=(7->y
implies that A = (7>y.
This is evidently a general rendering of Euclid s first
axiom.
But we are not yet at the end of our discussion. In
the first place, we cannot yet prove that
A=B-*y
and A^B-*y
are incompatible with each other. For we have not yet
excluded the case that more than one character of the
set y may attach either to A or to B or to both. For
example if ^ and c 2 are members of y, both attaching to
A, but only c x attaching to B, then both
A=B-*y
and
Accordingly we must re-define the meaning of our
44 EQUALITY [CH
symbols by introducing the additional limitation that
A=B-*y
and A^B-^y
both mean that A and B each possess one and only one
character of the class y. It is well to note that the two
propositions represented by these symbolic statements
are only contraries to each other. For though they
cannot both be true, they will both be false if either A
or B does not possess any character out of the qualifying
class y. For example if A does not possess any such
quality or if it possesses two such qualities, then
A=A->y
and A =j= A -> y
are both false. This example also illustrates the sharp
distinction between equality and mere identity.
In this most general sense of equality, the notion of
matching/ in the sense in which colours match, might
with advantage replace equality, so that we should
interpret
A=B-+y and A =f B -> y
as meaning respectively
A matches B in respect to the qualities y
and
A does not match B in respect to the qualities y!
This verbal statement in its common meaning pre
supposes our three conditions :
(i) that A and B each possess one of the qualities y,
(ii) that neither^ nor B possesses more than one such
quality,
(iii) that A and B possess the same one of the
m] EQUALITY 45
qualities y, and (in the second case) that A and B do
not possess the same one of the qualities.
The set of entities such as A and B possessing one and
only one of the qualities of the class y will be said to
form the qualified class for y, and we have already
named y the qualifying class.
Congruence. Congruence is a subspecies of the general
type of the equality relation. Let us start with the
simplest example and consider a one-dimensional space.
The points of this space are terms interconnected by a
relation which arranges them in serial order with the
ordinary continuity of the Dedekindian type. The
points may be connected by other relations which sort
them out in other ways ; but when we say that they
form a one-dimensional space, we are thinking of one
definite relation which produces the continuous serial
order, both ways infinite.
Now in the particularising of the equality relation so
as to produce a congruence relation for this space, we
first demand that, if y be the qualifying class, the class
qualified by y must be composed of all the finite
stretches of the space. Thus the terms A, B, etc. in
the previous explanation of equality are now stretches
of the serial space, and every finite stretch belongs to
the qualified class. It will be convenient to confine
attention to those stretches which include their two
end-points. Let two stretches which do not overlap,
except that they have one end-point in common, be
called adjoined stretches, or stretches adjoined at that
end-point.
Now the conditions which have to be fulfilled in order
that this type of equality may reckon as a congruence
are:
46 EQUALITY [CB
(i) If A be any stretch and p any point, there ar^
two stretches P 1 and P 2 adjoined at p, such that
and A = P 2 -> y.
In other words, from a given point p stretches of ai
assigned length can be measured in either direction.
(ii) If P and Q are two stretches, and P contains Q,
then
In other words, the whole is unequal to its part.
(iii) If P and Q be two stretches, and P be composed
of the adjoint stretches P x and P 2 , and Q of the adjoint
stretches Q l and Q z , and furthermore if
and P 2 = Q z -> y,
then P =Q > y.
In other words, if equals be added to equals the wholes
are equal.
(iv) If the first clause of the hypothesis of (iii) hold,
and furthermore if
P= Q -y
and P 1 =Q 1 -+ y,
then P n =Q 2 > y.
In other words, if equals be taken from equals the
remainders are equal.
(v) The axiom that the whole is greater than its
part suffers from the difficulty that we have not defined
what we mean by greater than. Our condition (ii)
states that the whole is unequal to its part. But the
idea of greater than really follows from the condition
which we wish to express. I think that the missing
condition is best stated thus :
in] EQUALITY 47
, Let A and B be two stretches of which one contains
,he other, so that either A contains B or B contains A,
md let H and K be two other stretches with the same
property in regard to each other.
Also let
Then if H contains K, it also follows that A contains
B. The point of this condition is that we exclude the
crosswise equality in which A is congruent to a part of
H and H to a part of A.
Then the idea of any stretch P being greater than
any stretch Q must be defined to mean that there is a
stretch H containing a part K such that
Thus the verbal form, the whole is greater than its
part, becomes a mere tautology. The true point being
first our condition (ii) that the whole is unequal to any
of its parts, and our condition (v) which excludes the
crosswise equality of wholes to parts.
The theory of numerical measurement depends upon
three additional conditions which can be conveniently
preceded by some definitions. Let a sequence of n
successively adjoined stretches A L9 A. 2 , ... , A n , which is
such that
A p = A q ->y, [p, 2 = 1,2, ...,?i]
be called a stretch sequence for y. Let each individual
stretch of the sequence be called a component stretch
of the sequence, and let the stretch which is composed
of all the stretches of the sequence be called the
resultant stretch of the sequence.
48 EQUALITY [CH
Furthermore if c be the member of y which charac
terises each component stretch of the sequence of n
stretches, let nc be the symbol for the member of y
which characterises the resultant stretch of the sequence.
Also if c f be an alternative symbol for nc, let - c f be an
n
alternative symbol for c.
The three conditions are :
(vi) If A be any stretch and n be any integer, then
a stretch sequence for y can be found composed of n
members such that A is its resultant.
(vii) If A and B be any two coterminous stretches,
and A be part of B, then we can find an integer n such
that there exists a stretch sequence for y of n terms
such that A is its first term and B is part of the re
sultant of the sequence.
(viii) If A be any stretch and n any integer, then A is
a member in any assigned ordinal position of two stretch
sequences for y of n terms, the two sequences running
in opposed directions.
The condition (vii) is the axiom of Archimedes.
It is evident that we may conceive y as the class of
magnitudes and the stretches as the class of concrete
quantities. The difference between a magnitude and a
concrete quantity is the difference between the length,
called a yard, and the particular concrete instance which
is in the custody of the Warden of the Standards.
It is not necessary to plunge further into the exact
analysis of the theory of extensive quantity. The dis
cussion has been carried far enough to make it evident
that the qualifying class y, which is the class of magni
tudes, is simply a class of qualities which happen to be
sorted out among the qualified class (which in the above
example was a class of stretches) in such a way that,
in] EQUALITY 49
when one member of y has been taken as the standard
of reference, the unit, all the other members of y can
be described in terms of it by means of real numbers.
But a quality which belongs to the set y is in itself in
no way otherwise distinguished from any other quality
of things. Quantity arises from a distribution of
qualities which in a certain definite way has regard to
the peculiar fact that in certain cases two extended
spatio-temporal elements together form a third such
element. In fact the qualifying qualities are distri
buted among extended things with a certain regard to
their property of extension. Also it is evident that
two stretches A and B which are equal for one quali
fying class y may be unequal for another qualifying
class y.
If we apply this doctrine to the classical theory of
space and time, we find, following Sophus Lie s analysis,
that there are an indefinite number of qualifying classes
y, y, y", etc., which for the case of three-dimensional
space generate relations of congruence among spatial
elements, and that each such set of congruence relations
is inconsistent with any other such set.
For the case of time the opposite trouble arises. Tune
in itself, according to the classical theory, presents us
with no qualifying class at all on which a theory of
congruence can be founded.
This breakdown of the uniqueness of congruence for
space and of its very existence for time is to be con
trasted with the fact that mankind does in truth agree
on a congruence system for space and on a congruence
system for time which are founded on the direct evidence
of its senses. We ask, why this pathetic trust in the yard-
measure and the clock? The truth is that we have observed
something which the classical theory does not explain.
W. R. 4
50 EQUALITY [CH
It is important to understand exactly where the diffi
culty lies. It is often wrongly conceived as depending
on the inexactness of all measurements in regard to very
small quantities. According to our methods of observa
tion we may be correct to a hundredth, or a thousandth,
or a millionth of an inch. But there is always a margin
left over within which we cannot measure. However
this character of inexactness is not the difficulty in
question.
Let us suppose that our measurements can be ideally
exact; it will be still the case that if one man uses one
qualifying class y and the other man uses another
qualifying class S, and if they both admit the standard
yard kept in the exchequer chambers to be their unit
of measurement, they will disagree as to what other
distances places should be judged to be equal to that
standard distance in the exchequer chambers. Nor need
their disagreement be of a negligible character. For
example, the man who uses the qualifying class y might
be in agreement with the rest of us, who are also using
y, and the other man who uses S might also be a well-
trained accurate observer. But in his measurement the
distance from York to Edinburgh might come out at
exactly one yard.
But no one, who is not otherwise known to be a
lunatic, is apt to make such a foolish mistake.
The conclusion is that when we cease to think of
mere abstract mathematics and proceed to measure in
the realm of nature, we choose our qualifying class y
for some reason in addition to the mere fact that the
various characters included in y are sorted among
stretches so as to satisfy the conditions for congruence
which I have jotted down above.
When we say that two stretches match in respect to
in] EQUALITY 51
length, what do we mean? Furthermore we have got
to include time. When two lapses of time match in
respect to duration, what do we mean? We have seen
that measurement presupposes matching, so it is of no
use to hope to explain matching by measurement.
We have got to dismiss from our minds all considera
tions of number and measurement and quantity, and
simply concentrate attention on what we mean by
matching in length.
It is an entirely different and subsequent considera
tion as to whether length in this sense of the term is a
class of qualities which is sorted out to stretches in
accordance with the congruence conditions.
Our physical space therefore must already have a
structure and the matching must refer to some qualifying
class of qualities inherent in this structure. The only
possible structure is that of planes and straight lines,
such that stretches of straight lines can be conceived
as composed of points arranged in order.
An additional factor of structure can be that of
ordinary Euclidean parallelism. By this I mean that
through any point outside a plane there is one and only
one plane which does not intersect a given plane. You
will observe that I have had to adopt what is termed
Playfair s axiom for the definition of parallels. It is the
only one which does not introduce some presupposition
of congruence, either of length or angles. I draw your
attention to the absolute necessity of defining our
structure without the presupposition of congruence. If
we fail in this respect our argument will be involved in
a vicious circle.
With this definition of parallels it is now very easy
to get some way in the explanation of what we mean
4-2
52
EQUALITY
[CH
by stretches matching in length. For since our structure
includes parallels, it also includes parallelograms. Ac
cordingly we can agree that the opposite sides of
parallelograms match in length. It is then easy enough
to show that we have a complete system of congruence
for any one system of parallel stretches in space. This
means that if there are any two stretches either on the
same straight line or on parallel straight lines, we have
a definitely determined numerical ratio of the length of
one to the length of the other.
But we cannot go further and compare the lengths
of two stretches which are not parallel, unless we in
troduce some additional principle for the matching of
lengths.
We can find this additional principle provided that
we can define a right-angle without any appeal to the
idea of congruence or equality. For let us anticipate
such a definition independent of congruence.
8 D C
Let D be the midpoint of the stretch BC, and draw
DA perpendicular to BC. Then our additional principle
of matching shall be that AB is equal to AC. In this
way we can compare the lengths of stretches which are
not parallel, and the whole theory of congruence in space
is established.
in] EQUALITY 53
But as yet we have not gone any way towards finding
any theory for the congruence of lapses of time. Accord
ingly if we are to explain how it is that in our observa
tion of nature we all agree in our systems of space and
time congruence, we have to explain what we mean by
planes, and by right-angles, and how we should match
lapses of time. We can omit straight lines from this
catalogue, since they can be defined as the intersections
of planes. We shall however have to explain how the
points on straight lines come to be arranged in order.
When we are conscious of nature, what is it that we
really observe? The obvious answer is that we perceive
various material bodies, such as chairs, bricks, trees.
We can touch them, see them and hear them. As I
write I can hear the birds singing in a Berkshire garden
in early spring.
In conformity with this answer, it is now fashionable
and indeed almost universal to say that our notions of
space merely arise from our endeavours to express the
relations of these bodies to each other. I am sorry to
appear pigheaded ; but, though I am nearly in a minority
of one, I believe this answer to be entirely wrong. I will
explain my reasons.
Are these material bodies really the ultimate data of
perception, incapable of further analysis?
If they are, I at once surrender. But I submit that
plainly they have not this ultimate character. My allu
sion to the birds singing was made not because I felt
poetical, but to warn you that we were being led into
a difficulty. What I immediately heard was the song.
The birds only enter perception as a correlation of more
ultimate immediate data of perception, among which
for my consciousness their song is dominant.
V
54 \ \EQUALITY [CH
Material bodies only enter my consciousness as a
representation of a certain coherence of the sense-objects
such as colours, sounds and touches. But these sense-
objects at once proclaim themselves to be adjectives
(pseudo-adjectives, according to the previous chapter)
of events. It is not mere red that we see, but a red
patch in a definite place enduring through a definite
time. The red is an adjective of the red time and place.
Thus nature appears to us as the continuous passage of
instantaneous three-dimensional spatial spreads, the
temporal passage adding a fourth dimension. Thus
nature is stratified by time. In fact passage in time is of
the essence of nature, and a body is merely the coherence
of adjectives qualifying the same route through the four-
dimensional space-time of events.
But as the result of modern observations we have to
admit that there are an indefinite number of such modes
of time stratification.
However, this admission at once yields an expla
nation of the meaning of the instantaneous spatial
extension of nature. For it explains this extension as
merely the exhibition of the different ways in which
simultaneous occurrences function in regard to other
time-systems.
I mean that occurrences which are simultaneous for
one time-system appear as spread out in three dimen
sions because they function diversely for other time-
systems. The extended space of one time-system
is merely the expression of properties of other time-
systems.
According to this doctrine, a moment of time is
nothing else than an instantaneous spread of nature.
Thus let t lt t 2 , t 3 be three moments of time according to
Ill]
EQUALITY
55
one time-system, and let T lt T 2 , T 3 be three moments
of time according to another time-system. The inter
sections of pairs of moments in diverse time-systems
are planes in each instantaneous three-dimensional space.
In the diagram each continuous line accordingly sym
bolises a three-dimensional space; and the intersections
of continuous lines, such as A or B or (7, symbolise
ft fe
planes. Thus ^ and T l are each a three-dimensional
space, and A is a plane in either space.
Parallelism is the reflection into an instantaneous
space of one time-system of the property of moments of
some other system. Thus A and B are parallel planes
in t lt since T l and jT 2 are moments of the same system
which is not the system to which ^ belongs.
56 EQUALITY [OH
But when we talk of space we are not usually thinking
of the instantaneous fact of immediate perception. We
are thinking of an enduring scheme of extension within
which all these instantaneous facts are fitted. It follows
that we ought to be able to find a meaning for the idea
of a permanent space in connection with each time-
system.
This conception must arise from our immediate obser
vations of motion and rest. Both rest and motion have
no meaning in connection with one mere instantaneous
space. In such a space everything is where it is and there
is an instantaneous end to it ; to be succeeded by another
instantaneous space. But motion and rest at once warn
us that our perception involves something more.
The instantaneous moment is merely an ideal limit
of perception. Have you ever endeavoured to capture
the instantaneous present? It eludes you, because in
truth there is no such entity among the crude facts of
our experience. Our present experience is an enduring
fact within which we discriminate a passage of nature.
Now within this enduring fact we observe rest and
motion. A body at rest in the space of our observation
is tracing out a certain historical route intersecting the
moments of our time-system in a sequence of instan
taneous points. This route is what we mean by a point
of the permanent space of our time-system. Thus each
time-system has its own space with its own points, and
these permanent points are loci of instantaneous points.
The paradoxes of relativity arise from the fact that
we have not noticed that when we change our time-
system we change the meaning of time, the meaning of
space and the meaning of points of space (conceived as
permanent).
in] EQUALITY 57
Now the route of a small body at rest in the space of
a time-system, that is to say, a point of that time-system,
has a certain symmetry in respect to the successive in
stantaneous spaces of that system, which is expressed for
us by the perception of lack of change of position. This
symmetry is the basis of the definition of rectangularity.
If the body be at rest in the space of the time-system
t, it is moving in a straight line in the space of another
time-system T. This permanent straight line intersects
any moment of T, say T 19 in an instantaneous straight
line ^ (say). Then /> is perpendicular to the series of
instantaneous parallel planes in which the moments of
system t intersect T^. In other words the planes to
which motion is perpendicular are the planes of inter
section with the moments of that time-system for whose
space that motion would be represented as rest.
We have thus defined both parallelism and perpen
dicularity without reference to congruence, but in terms
of immediate data of perception. Furthermore, the
parallelism of the moments of one time-system enables
us to extend parallelism to time as also expressing the
relation to each other of permanent points of the same
time-system. It thus follows that we now possess a
structure in terms of which congruence can be defined.
This means that there will be a class of qualities y one
and only one of which attaches to any stretch on a
straight line or on a point, such that matching in respect
to this quality is what we mean by congruence.
The thesis that I have been maintaining is that
measurement presupposes a perception of matching in
quality. Accordingly in examining the meaning of any
particular kind of measurement we have to ask, What
is the quality that matches?
58 EQUALITY [CH
Furthermore, in applying this doctrine to measure
ments in space and time, I have maintained that the
things whose qualities match are events. In other words,
I maintained that it is events that are congruent, and
that spatial congruence and temporal congruence are
merely special instances of this fundamental congruence.
In conformity with this doctrine I also maintain that space
and time are merely the exhibition of relations between
events.
The usual opinion, or at any rate the more usual mode
of expression, is that space and time are relations between
the material objects implicated in events. It is difficult
to understand how time can be a relation between two
permanent objects. Also with the modern assimilation
of time and space, this difficulty in respect to time also
attaches to space. Furthermore, I hold that these per
manent objects are nothing else than adjectives of
events. It follows that a yard measure is merely a
device for making evident the spatial congruence of the
events in which it is implicated.
The divergence between the two points of view as to
space-time, that is to say, as to whether it exhibits re
lations between events or relations between objects in
events is really of the utmost importance in the stage
of physical science. If it be a relatedness between events,
it has the character of a systematic uniform relatedness
between events which is independent of the contingent
adjectives of events. In this case we must reject Ein
stein s view of a heterogeneity in space-time. But if
space-time be a relatedness between objects, it shares
in the contingency of objects, and may be expected to
acquire a heterogeneity from the contingent character
of objects. I cannot understand what meaning can be
in] EQUALITY 59
assigned to the distance of the sun from Sirius if the very
nature of space depends upon casual intervening objects
which we know nothing about. Unless we start with
some knowledge of a systematically related structure
of space-time we are dependent upon the contingent
relations of bodies which we have not examined and
cannot prejudge.
Furthermore, how time is to be got from the relations
of permanent bodies completely puzzles me. And yet
the moderns assimilate time with space. I have never
seen even the beginning of an explanation of the meaning
of the usual phraseology.
I have already reiterated, that measurement presup
poses a structure yielding definite stretches which,
in some sense inherent in the structure, match each
other; and I have explained the type of structure which
is formed by our space-time.
The essence of this structure is that it is stratified
in many different ways by different time-systems. This
is a very peculiar idea which is the product of the
speculations of the last fifteen years or so. We owe the
whole conception notably to Einstein. I do not agree
with his way of handling his discovery. But I have no
doubt as to its general correctness. It is at first sight
somewhat of a shock to think that other beings may
slice nature into time-sections in a different way to
what we do. In fact we have differences even among
ourselves which luckily are quite imperceptible. How
ever if we allow this possibility we not only explain
many modern delicate experiments, but we also obtain
explanations of what we mean by the spatial extension
in three dimensions, and by planes and straight lines,
and parallels and right-angles. We also obtain a definite
60 EQUALITY [CH m
meaning for the matching which is the basis of our con
gruence. The explanation is too sweeping to be put aside.
Our whole geometry is merely the expression of the
ways in which different events are implicated in different
time-systems.
I have also hitherto omitted to point out that all
order in space is merely the expression of order in time.
For a series of parallel planes in the space of our time is
merely the series of intersections with a series of moments
of another time-system. Thus the order of the parallel
planes is merely the time-order of the moments of this
other system.
I must stop now. We started from the simplest idea
which meets every child at the beginning of his or her
schooldays. I mean the idea of equality.
We asked what it meant. We have then been led on
and on, till we have found ourselves plunged in the
abstruse modern speculations concerning the character of
the Universe. They are not really very difficult. I call
them abstruse because they deal with questions which
we do not ordinarily think about. It is therefore a strain
on our imaginations to follow the line of thought. But
when we have once allowed the possibility of different
meanings for time in nature, the argument is a straight
forward deduction of the consequences.
CHAPTER TV
SOME PRINCIPLES OF PHYSICAL SCIENCE
IT is my ambition in this lecture to discuss some general
principles of mathematical physics, and to illustrate
them by their application to the problem of the gravi
tational field. In a sense such a discussion should form
the first chapter of the science, but it is that first chapter
which is studied last.
The Apparent World. It would be easy to quote an
imposing array of authorities, almost a consensus of
authorities, in support of the thesis that the subject
matter of physical science is composed of things observed
by the senses. Such things are sights, sounds, touches,
bodily feelings, shapes, distances, and their mutual rela
tions. I will call the whole assemblage of them the
apparent world. Natural science is therefore the study
of the interconnections of the things forming the ap
parent world.
This profession of the motive of science seems however
in sharp contradiction to its actual achievement. The
molecular theory, the wave theory of light, and finally
the electromagnetic theory of things in general have, as
it seems, set up for scientific investigation a society of
entities, such as ether, molecules, and electrons, which
are intrinsically incapable of direct observation. When
Sir Ernest Rutherford at Cambridge knocks a molecule
to pieces, he does not see a molecule or an electron.
What he observes is a flash of light. There is at most
a parallelism between his observation and the conjectural
molecular catastrophe.
62 SOME PRINCIPLES OF PHYSICAL SCIENCE [OH
I suggest to you that, unless we are careful in our
formulation of principles, the outcome of this train of
thought is apt to be unsatisfactory and very misleading
to scientific imagination. The apparent world becomes
an individual psychological reaction to the stimulus of
an entirely disparate interplay of electrons and ether.
The whole of it is in the same boat. There is no prin
ciple by which we can assign for some of it any indepen
dence of individual psychology superior to that of the
remainder.
On this theory we must entirely separate psychological
time, space, external perceptions, and bodily feelings
from the scientific world of molecular interaction. This
strange world of science dwells apart like the gods of
Epicurus, except that it has the peculiar property of
inducing our minds to play upon us the familiar antics
of our senses.
If we are to avoid this unfortunate bifurcation, we
must construe our knowledge of the apparent world as
being an individual experience of something which is
more than personal. Nature is thus a totality including
individual experiences, so that we must reject the dis
tinction between nature as it really is and experiences
of it which are purely psychological. Our experiences
of the apparent world are nature itself.
Two-fold Cognisance. We have a two-fold cognisance
of nature, and I will name the two factors of this ex
perience cognisance by adjective and cognisance by
relatedness.
Think of yourself as saying, There is a red patch.
You are affirming redness of something, and you are
primarily conscious of that something because of its
redness. In other words, the redness exhibits to you the
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 63
something which is red. This is cognisance by adjective,
red being the adjective. But your experience has gone
further than mere cognisance by adjective. Your know
ledge is not merely of something which is red. The
patch is there and it endures while you are observing it.
Thus you are cognisant of it as having spatio-temporal
position, and by this we mean a certain type of related-
ness to the rest of nature which is thereby involved in
our particular experience. This knowledge of nature
arising from its interconnectedness by spatio-temporal
relations is cognisance by relatedness.
For example, the physiological account of the function
of the brain as determining the conditions of external
perception presupposes that the events of the brain
signify the totality of contemporaneous space. Again
the disclosure of space behind the looking-glass as
qualified by images situated in it exhibits the fact that
the events in front of the glass are significant of con
temporaneous space behind it. Also we know that there
is space inside the closed cupboard.
Nature is an abstraction from something more con
crete than itself which must also include imagination,
thought, and emotion. This abstraction is characterised
by the systematic coherency of its interconnections
disclosed in cognisance by relatedness. Thus the sub
stances of nature which have the cognised adjectives as
their qualities are also the things in nature connected
by the cognised relatedness. Nature is delimited as the
field of this closed system of related things. Accordingly
the ultimate facts of nature are events, and the essence
of cognisance by relatedness is the ability to specify the
event by time and by place. Dreams are ruled out by
their inability to pass this test.
64 SOME PRINCIPLES OF PHYSICAL SCIENCE [CH
But an event can be specified in this way without its
being the subject of direct cognisance by adjective. For
example we can exactly specify a time and a place on
the further surface of the moon, but we should very much
like to know what is happening there. There is however
a certain fullness in the dual cognisance both by adjec
tive and by relatedness. I will use the term perception
for this full experience.
Mere cognisance by relatedness is essentially know
ledge of an event merely by its spatio-temporal relations
to other events which are perceived and thus form a
framework of what is fully experienced. In this sense
there is no cognisance by relatedness without perception.
It is not the case that the analysis of the adjectives
of appearance attached to the events within any limited
field of nature carries with it any certain knowledge of
adjectives attached to other events in the rest of nature,
or indeed of other such adjectives attached to those
same events. I will refer to this fact by the phrase, the
contingency of appearance.
On the other hand, though the character of time and
space is not in any sense a priori, the essential related-
ness of any perceived field of events to all other events
requires that this relatedness of all events should con
form to the ascertained disclosure derived from the
limited field. For we can only know that distant events
are spatio- temporally connected with the events imme
diately perceived by knowing what these relations are.
In other words, these relations must possess a systematic
uniformity in order that we may know of nature as
extending beyond isolated cases subjected to the direct
examination of individual perception. I will refer to
this fact by the phrase, the uniform significance of events.
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 65
Thus the constitutive character of nature is expressed
by the contingency of appearance and the uniform
significance of events. These laws express characters of
nature disclosed respectively in cognisance by adjective
and cognisance by relatedness. This doctrine leads to
the rejection of Einstein s interpretation of his formulae,
as expressing a casual heterogeneity of spatio-temporal
warping, dependent upon contingent adjectives.
~The case of the yard-measure illustrates my meaning.
It is a contingent adjective of the events where it is
situated. Its spatio-temporal properties are entirely
derived from the events which it qualifies. For example,
its use depends on the recognition of simultaneity, so
that we shall not observe its two ends at widely different
times. But simultaneity concerns events. Also the
mere self-identity of the yard-measure does not suffice
for its use, since we also admit the continued identity
of objects which shrink or expand. The yard-measure
is merely a device for making evident obscure relations
between those events in which it appears.
If congruence merely meant relations between con
tingent adjectives of appearance, there would be no
measurement of spatial distance or of temporal lapse
without knowledge of actual intervening appearances,
and no meaning for such distance in the absence of these
adjectives. For example, the distance of the star Sirius
would be a phrase without meaning.
You will have observed that in this doctrine of cog
nisance by relatedness I am merely taking the old belief
that we know of unbounded time and of unbounded
space and am adapting it to my inversion which gives
the supremacy to events and reduces time and space to
mere relations between them.
W. R. 5
66 SOME PRINCIPLES OF PHYSICAL SCIENCE [CH
The Doctrine of Time. It follows from my refusal to
bifurcate nature into individual experience and external
cause that we must reject the distinction between
psychological time which is personal and impersonal
time as it is in nature. Two conclusions follow, of which
the one is conservative, and indeed almost reactionary,
and the other is paradoxical.
The conservative conclusion is that in cognisance by
relatedness the apparent world is disclosed as stratified
into a succession of strata which are subordinate totali
ties of immediate experience. Each short duration of
time is merely a total slab of nature disclosed as a totality
in cognisance by relatedness, and for any individual ex
perience partially disclosed in cognisance by adjective.
There can be no other meaning for time, if we admit
the position from which my argument has started. I will
state the doctrine in this way, Time is a stratification
of nature. Adherence to this doctrine is today the mark
of a reactionary. I accept the term with the qualifi
cation that it is reaction to the admission of obvious
fact.
We now pass to the other conclusion which is para
doxical. The assumption of the uniqueness of the tem
poral stratification of nature has slipped into human
thought. Certainly in each individual experience such
uniqueness must be granted. But confessedly each
individual experience is partial, and we cannot safely
reason from partial experience to the limitation of the
variety of nature. Accordingly the uniqueness of time
succession for each of us does not guarantee its consis
tency for all.
At this point I put by urgent metaphysical questions
concerning any supposed distinction between past,
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 67
present, and future as to the character of their existence.
Also I need not recall to your minds the reasons, based
upon refined observations, for assuming the existence in
nature of alternative time-systems entailing alternative
systems of stratification.
I think that no one can study the evidence in its
detail without becoming convinced that we are in the
presence of one of the most profound reorganisations of
scientific and philosophic thought. But so many con
siderations are raised, so diverse in character, that we
are not justified in accepting blindfold the formulation
of principles which guided Einstein to his formulae.
You will have observed that for reasons which I have
briefly indicated, I maintain the old-fashioned belief in
the fundamental character of simultaneity. But I adapt
it to the novel outlook by the qualification that the
meaning of simultaneity may be different in different
individual experiences. Furthermore, since I start from
the principle that what is apparent in individual ex
perience is a fact of nature, it follows that there are in
nature alternative systems of stratification involving
different meanings for time and different meanings for
space. Accordingly two events which may be simul
taneous in one instantaneous space for one mode of strati
fication may not be simultaneous in an alternative mode.
Time and Space. The homogeneity of time with space
arises from their common share in the more fundamental
quality of extension which is a quality belonging exclu
sively to events. By extension I mean that quality in
virtue of which one event may be part of another or two
events may have a common part. Nature is a continuum
of events so that any two events are both parts of some
larger event.
52
68 SOME PRINCIPLES OF PHYSICAL SCIENCE [CH
The heterogeneity of time from space arises from the
difference in the character of passage in time from that
of passage in space. Passage is the same as significance,
and by significance I mean that quality of an event
which arises from its spatio-temporal relationships to
other events.
For the sake of simplicity I will speak of events whose
dimensions are ideally restricted. I will call them
* event-particles. Also we may conceive of an event
restricted except in one dimension. Such an event may
be termed a route or path, where I am now thinking of
a route of transition through the continuum of nature.
A route may evidently be also conceived as a linear
chain of event -particles. But its essential unity is
thereby lost. A spatial route is a route which lies
entirely in one instantaneous space. A historical route
is such that no two of its event -particles are simultaneous
according to any time-system. Along such a route there
is a definite antecedence and subsequence in time which
is independent of alternative time-systems.
Thus the distinction of time from space, which I have
just asserted, consists in the fact that passage along a
spatial route has a different character from passage along
a historical route. For proof of this fact think of a spatial
route which has a material particle situated in each of
its event-particles. We pronounce at once that all these
material particles are different, because no material
particle can be in two places at the same time. But if
a historical route is in like case and the material particles
be of like character even with some differences, we
equally pronounce them to be the same material particle
at successive stages of its existence. This difference of
judgment can only arise from the distinction in the
characters of spatial and temporal passage.
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 69
It only strengthens this argument when we remember
that the events are the ultimate substance of nature
and that the apparent material particle is an adjective
of appearance which qualifies them. For the unique
type of individuality possessed by the emergence of the
same adjective throughout the historical route must
be due to the special peculiarity of the route. I will
recur to this question later when I define adjectival
particles.
Time-Systems. According to the view which I am
urging on you a moment of time is to be identified with
an instantaneous spread of the apparent world. The
relations of interconnection within this moment form a
momentary three-dimensional space. Such a space is an
abstraction from the full-bloodedness of a moment of
time which includes all that is apparent in that space. A
time-system is a sequence of non-intersecting moments
including all nature forwards and backwards. I call the
moments of such a consistent system * parallel/ because
all parallelism is derived from their mutual relations
and from their intersections with the parallel moments
of other time-systems.
I am also assuming on rather slight evidence that
moments of different time-systems always intersect.
This hypothesis is the simplest and I know of no pheno
mena that would be explained by its denial. The result
is to introduce the peculiar properties of Euclidean
parallelism.
One advantage of the admission of alternative time-
systems is that they afford explanations of the notion
of position and of the notion of evenly lying loci, such
as planes and straight lines. However, I will not in
this lecture enter into a detailed examination of the
origins of geometry.
70 SOME PRINCIPLES OF PHYSICAL SCIENCE [CH
Permanent Space. The momentary spaces of a time-
system are matters of direct observation, at least when
we construe momentary in an approximate sense. They
must be discriminated from the permanent space of that
time- system.
Best and motion are ultimate data of observation, and
permanent space is the way of expressing the connec
tions of these data. The ultimate elements of perman
ent space are therefore somewhat elaborate. It will be
sufficient for my immediate purpose in this lecture to
exhibit the meaning to be ascribed to a point of the
permanent space of a time-system.
Consider observations wedded to a single .temporal
mode of stratification. Some apparent bodies will be
observed to be in motion and others at rest. The his
torical route forming the successive situations of an
apparent particle at rest for such an observer is a point
in the permanent space which corresponds to that time-
system. For an observer who is wedded to another
time-system the same apparent particle will be moving
with uniform velocity. Accordingly the point of the
space of the former time-system cannot be a point of the
space of the latter time-system, since to be at rest
occupying one point in the space of one time-system is
to be moving through a succession of points in the space
of another time-system. A permanent point is thus
highly complex and only serves for one particular
specification of the meaning of space and time. Each
event-particle will occur at one point in each permanent
space, and is thus the vertex of a pencil of points, one
point for each time-system.
Each point intersects any moment, of whatever time-
system, in just one event-particle. There is thus a
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 71
point- wise correlation between the event-particles of any
momentary space and the points of the permanent space
of any time-system. This correlation explains the na
turalness with which observation of momentary spaces
is expressed in terms of permanent space so as to gain
the facile representation of the phenomena of rest and
motion, which can have no existence for a single momen
tary space.
This general theory of the grounds in nature for
geometry and time is consistent with a rigid relativity
whereby space and time are simply expressions for a
certain observed ordering of events. Also it is essential
to note that the spatial relations between apparent
bodies only arise mediately through their implication
in events. It is essential to adopt this view if we are
to admit any assimilation of space and time.
The Physical Field. We now pass to the considera
tion of the status of the physical field of natural science.
The scope of the contingency of appearance is limited,
and the conditions of limitation are what we term the
laws of nature.
They are expressed by assuming that the apparent
adjectives of the past indicate a certain distribution of
character throughout events extending from the past
into the future. It is further assumed that this hypo
thetical distribution of character in its turn expresses
the possibilities of adjectives of appearance attachable
to the future events. Thus the regulation of future ad
jectives of appearance by past adjectives of appearance
is expressed by this intermediate distribution of cha
racter, indicated by the past and indicating the future.
I call this intermediate distribution of character the
physical field. The true expression of the physical
72 SOME PRINCIPLES OF PHYSICAL SCIENCE [OH
field is always to some extent a matter of conjecture.
The only guarantee for correctness is the pragmatic test
that the theory works.
The physical field is not the cause of perception nor
is it the object perceived. The search for a cause of
perception raises a problem which is probably meaning
less and certainly insoluble. The physical field is merely
that character of nature which expresses the relatedness
between the apparent adjectives of the past and the
apparent adjectives of the future. It therefore shares
in the contingency of appearance, and accordingly can
not affect spatio-temporal relations.
Atomicity. Luckily the physical field is atomic, so far
as concerns our approximate measurements. By this
I mean that we can discriminate in the four-dimensional
continuum certain regions or events, such that each
exhibits a physical character which is entirely indepen
dent of the physical characters of other events or of the
other physical characters of that event. This physical
character requires the whole region for its complete
exposition. Thus atomicity implies two properties, one
is the breakdown of relativity in that the atomic
character is independent of the physical characters per
vading the rest of nature, and the other is that we
cannot completely exhibit this character without the
whole corresponding region.
This physical atomic character is the only case in
which the Aristotelian idea of an attribute of a sub
stance holds without grave qualification, at least so far
as the realm of nature is concerned. Furthermore,
atomicity is a property which is capable of more or
less complete realisation. Failure to attain complete
atomicity is illustrated when one aspect of the physical
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 73
field modifies another aspect of it, for example, when
the physical field of mass modifies that of electro-
magnetism.
Observe that the practical atomicity of the physical
and apparent characters is essential for the intelligibility
of the apparent world to a finite mind with only partial
perception. Without atomicity we could not isolate our
problems; every statement would require a detailed
expression of all the facts of nature. It has always been
a reproach to those philosophers who emphasize the
systematic relatedness of reality that they make truth
impossible for us by requiring a knowledge of all as a
condition for a knowledge of any. In the account of
nature which I have just given you this objection is met
in two ways: In so far as nature is systematically
related, it is a system of uniform relatedness; and in
the second place, intelligibility is preserved amid the
contingency of appearance by the breakdown of related-
ness which is involved in atomicity.
This breakdown of relatedness in the expression of
the laws of nature is reflected into observation by our
perception of material objects. Such an object is more
than its colour, is more than its touch, and is more than
our feeling of its resistance to push. The object, taken
throughout its history, is a permanent factor conditioning
adjectives of appearance, and it is a factor which is
largely independent of its relatedness to other contingent
facts. It is the endeavour to make precise this aspect
of a perceived material object which has led to the
atomicity of modern science.
Thus it is not true to say, without qualification, that
the physical field is not perceived. We do recognise
permanences in the relatedness of things sensed, per-
74 SOME PKINCIPLES OF PHYSICAL SCIENCE [CH
manences which are largely disconnected. The physical
field is the endeavour to express precisely these per
ceived permanences as atomic characters of events.
Adjectival Particles. The discussion of these recog
nised permanences is reduced to an ideal simplicity by
the introduction of adjectival particles, by which I mean
the ideally small perceived bodies and the elementary
physical particles.
I have already stated that an adjectival particle
receives its enduring individuality from the individuality
of its historical route. Let me now give a more precise
statement of my meaning : An adjectival particle is the
adjective attached to the separate event-particles of a
historical route by virtue of the fact that some one and
the same adjective attaches to every stretch of the route.
It is the outcome of the transference to the individual
event-particles of a common property of all the stretches.
Accordingly the unique individuality of the particle
is nothing else than the fusion of the continued sameness
of the adjective with the concrete individuality of the
historical route. We must not think of an adjectival
particle as moving through its route. We will say that
it pervades its route, and that it is situated at each
event -particle of the route, and that it moves in an
orbit in each permanent space.
It follows from this conception of the meaning of an
adjectival particle that the expression of its properties
should require the consideration of stretches of its route.
In order, even now, to attain ideal simplicity we proceed
to the limit of making all such stretches infinitesimally
small. A stretch of a historical route, as thus employed
in the process of proceeding to a limit, will be called a
kinematic element. A kinematic element is equivalent
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 75
to both the position and the velocity of an adjectival
particle in any permanent space at any time.
Mass -Particles. A mass-particle is an adjectival par
ticle. It follows that for some limited purposes we can
treat it as being situated in an event-particle, but that
for the final purpose of enunciating the laws of nature
we must conceive it as pervading a stretch of its his
torical route.
Consider [cf. figure, p. 31] first the former conception
of a mass-particle m as situated at an event-particle
which we will call P. The physical field due to m at P
has to stretch away into the future. It is to be a limited
atomic field with a foot in two camps, for it represents
the property of the future as embodied in the past. It
may therefore, so far as it is completely atomic, be ex
pected to consist of that region within the future from P
which has peculiar affinities with the region co-present
with P.
Now what I call the kinematic future from P is the
region traversed by the pencil of permanent points
which has P as vertex, considering only the portions of
those points which stream into the future from P. It
will be remembered that there is one such point for
each time-system. Again the region co-present with P
is the region reached by the moments containing P. It
will be remembered that each moment is an instan
taneous three-dimensional space, and that there is one
such moment for each time-system. Both these regions,
the kinematic future from P and the region co-present
with P, are four-dimensional. The ordered geometry of
the four-dimensional continuum shows that the boundary
region which separates the two is a three-dimensional
region which belongs to neither. This three-dimensional
76 SOME PRINCIPLES OF PHYSICAL SCIENCE [CH
region will be called the causal future from P. It has
all the properties that we want for an atomic region
completely defined by P and for its delimitation not
dependent upon any contingent characters of the rest
of nature.
The atomic physical field of the mass-particle at P is
P s causal future together with P itself. We will call
P the origin of the field. The physical character of this
field as a whole is what is meant by the mass-particle
at P. This is merely Faraday s conception of the tubes
of force as constituting the physical particle, with the
modification that the tubes in the act of streaming
through space also stream through time. Conceived
under the guise of time and permanent space the mass-
particle is a transmission of physical character along its
lines of force with a definite finite velocity.
Metrical Formulae. A few mathematical formulae
are now necessary for my argument. The assumption,
adopted as the simplest representation of observed facts,
that the permanent space of each time-system is
Euclidean, leads to the formulae of the special theory
of relativity. There is however this difference that the
critical velocity c has no reference to light, and merely
expresses the fact that a lapse of time and a stretch of
spatial route can be congruent to each other.
Define the quantities
OV, |> = 1,2, 3, 4J|
by <=1, |> = 1, 2, 3] | (1).
o>/= -c 2
Let a rectangular Cartesian system of coordinates in
the permanent space of the x time-system be (x lt x 2) x s )
and let the lapse of x-time since zero time be x 4 . Thus
(x lt x 2 , x s , x 4 ) are the four coordinates of an event-
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 77
particle, which we will name X. Also in the l y time-
system, we denote analogously a permanent point by
the Cartesian coordinates (y lt y 3 , y a ) and a lapse of
y-time by y,. Let (y, 9 y^y^ y,} and (x l9 x z , x 3 , x 4 ) denote
the same event-particle.
Then [cf. The Principles of Natural Knowledge, Ch.
xin] the relations between the two systems of coordi
nates, the *x system and the y system, are of the form
^(^-^) = 2^ a w a x a , |> = 1, 2, 3, 4]... (2),
a
where the symbol S means summation for a = 1, 2, 3, 4
successively, and the I s are constants satisfying the
conditions 2^ = 0, 0*j8]l
-1, [a-fflj
These conditions entail analogous formulae for the con
verse transformation from ?/ to #.
It follows that, if the coordinates of another event-
particle, named P, be (p lt p z , p 3 , p 4 ) in the x system
and (q lt q,, q 3 , q,) in the y system,
Let r (X ) and r^ be respectively the x-distance and the
^/-distance between X and P. Then this invariant for
X and P can be expressed indifferently either by
IA\
or by
Then
(i) X and P are co-present, if
cPfa-jpjr-ffr^O,
(ii) P is kinematically antecedent to JT, if
X 4 >j9 4 , and C 2 (x 4 -p 4 ) 2 -r (:r) 2 >0,
(iii) X lies in the causal future from P, if
78 SOME PRINCIPLES OF PHYSICAL SCIENCE [CH
Routes of Adjectival Particles. Let the mass-particle
M be situated at X and the mass-particle m be situated
at P, and let X and P f be event-particles respectively
neighbouring to X and P on the historical routes of M
and m in the four-dimensional continuum of nature.
Let their coordinates be respectively
(x^ + dx^ ...) and (p^ + dp^ ...), [ju = l, 2, 3, 4].
These are accordingly infinitesimal invariants dG M ~ and
dG n ?, respectively expressing a spatio-temporal property
of the kinematic elements XX and PP r . This property
depends on the existence of the whole bundle of diverse
time-systems without special emphasis on any one of them.
These invariants [cf. equation (4)] are expressed by
Let the route of M be expressed by assuming x lt x z , x 3
to be appropriate functions of o? 4J and the route of m by
assuming p lt p, p s to be appropriate functions of p^.
Thus, always in reference to these assumptions, we write
Also we put
and
v m *
and
Impetus. In order to exhibit the character of the
physical field due to a mass-particle we must consider
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 79
it as pervading a kinematic element, which has the
advantage over an event-particle of retaining the quality
of historic passage. A loss of spatial dimensions is com
paratively immaterial, though it probably represents a
simplification beyond anything which obtains in nature.
In expressing the physical field due to ra we must
therefore consider the kinematic element PP r of its
route. Also we must take any arbitrary element XX ,
and consider how its qualifications as a possible kine
matic element of the route of Mare affected by the fact
that m pervades the element PP .
Each kinematic element, such as XX , having X as
initial starting-point will have certain physical charac
ters. The assemblage of quantities defining these
physical characters for this pencil of elements constitutes
the physical field at X. The two such characters which
we need consider, as qualifying XX for pervasion by
M, are its potential mass impetus and its potential
electromagnetic impetus.
The potential mass impetus along XX will be written
JdJ 2 , and the potential electromagnetic impetus will
be written dF. If the mass of the particle M be also
denoted by M, and its electric charge, in electrostatic
units, by E, then the realised mass impetus due to
pervasion of XX by M will be
and the realised electromagnetic impetus, due to the
same pervasion, will be
c~ l EdF.
The total impetus along XX realised by its pervasion
by M is
(9).
80 SOME PKINCIPLES OF PHYSICAL SCIENCE [CH
Summing along the route of M between the assigned
event-particles A to B, we obtain the realised impetus
along this route which is symbolised by
A
If this total impetus is to be finite, it is evident that
JdJ* and dF must be homogeneous functions of du iy
du z , du 3 , du of the first degree, where (u 19 u^ u 3 , u 4 )
are any generalised coordinates of X. Thus, guided
empirically by the ascertained character of dynamical
equations and of the electromagnetic field, we can assume
Thus \\jy\\ is a symmetric covariant tensor of the second
order and ||jF*J?|| is a covariant tensor of the first order.
The elements of these tensors are functions of the co
ordinates of X, that is, of (u 19 u 2 ,u 3) u 4 ). These tensors
define the physical field at X so far as inertial and electro
magnetic properties are concerned.
Hence, writing as above
for differentiation along the route of M, -= is a function
of u 1} u. 2 , u 3 , and of u 19 u 2 , u 3 , u 4 . We now assume that
the actual route of M satisfies the condition that the
realised impetus is stationary between A and B for
small variations of route. We thus obtain the equations
of motion
, [M =1,2, a] ...(ii).
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 81
Expression for the Gravitational Field. I will now
confine myself to the proper determination of dJ 2 , as
affected by the existence of other mass-particles m, m ,
etc., in other routes. In expressing the conditions re
straining the contingency of appearance it is necessary
that we have recourse to that aspect of nature which is
independent of this contingency. The only such aspect
is that arising from spatio-temporal properties. Also
dG M 2 and dG^ are the invariants expressing the quan
titative aspect of the historical passage of the elements
XX and PP .
Again in considering the physical character of XX
as affected by m in its route, we must select that
kinematic element PP f of m s route which is causally
correlated with XX . By this I mean that PP f has a
point-wise correlation with XX such that X is in the
causal future from P and X is in the causal future
from P . With this correlation the physical character
of PP is already determined when XX occurs.
This assumption of causal correlation is mathematically
expressed by the relation
x,-p, = r (x] jc .................. (12)
between corresponding event-particles on XX and PP .
The main empirical facts of gravitation are expressed
by the assumption that
......... (13),
where 2 means the summation for all mass-particles
m
such as m in kinematic elements such as PP 1 , causally
correlated to XX , and ^ m expresses the gravitational
law of fading intensity. The factor 2/c 2 is inserted so
that, when the main intensity is empirically adjusted
w. R. 6
82 SOME PRINCIPLES OF PHYSICAL SCIENCE [CH
to give the main inverse square law of gravitation, V m
may be the analogue of the familiar gravitational poten
tial at X due to m. It is easy to prove [cf. Part III] that,
apart from any assumption of causal correlation between
J^andP,
^mM^4-p 4 )-n}
has an invariant value for all sets of rectangular Car
tesian coordinates in all time-systems. Also with the
causal correlation between PP r and XX which we are
assuming, this invariant expression reduces to
& m {r (x] - m }
Accordingly, guided by our knowledge of the Newtonian
law of gravitation, we assume
v - _ r^ _ (14: )
"Q. {-"
where y is the familiar constant of gravitation so as to
produce the scale of intensity of the main inverse square
Newtonian term.
If we write
then in an empty region M* satisfies
We might, if we had preferred to do so, have started
from the differential equation as the only invariant form
of linear differential equation of the second order, and
then deduced the above solution for ^ m as the only in
variant solution for a single point-wise discontinuity.
The procedure of thought which I have adopted seems to
me to be better suited to throw into relief the funda
mental ideas concerning nature.
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 83
Comparison with Einstein s Law. In the formula
corresponds to Einstein s proper timers. By identi
fying the potential mass impetus of a kinematic element
with a spatio-temporal measurement Einstein, in my
opinion, leaves the whole antecedent theory of measure
ment in confusion, when it is confronted with the actual
conditions of our perceptual knowledge. The potential
impetus shares in the contingency of appearance. It
therefore follows that measurement on his theory lacks
systematic uniformity and requires a knowledge of the
actual contingent physical field before it is possible.
For example, we could not say how far the image of a
luminous object lies behind a looking-glass without
knowing what is actually behind that looking-glass.
The above formula, assumed for dJ 2 , also differs from
Einstein s. In his procedure the e/ s are conditioned by
making them satisfy the contracted Reimann-Christoffel
tensor equations. He obtains a solution of these equa
tions for a single point-singularity under the assumption
that the gravitational field is permanent for the coordi
nates adopted so that no elements of the array ||/^|| are
functions of the time in the system of coordinates
adopted. This limitation rules out any application of this
solution to cases like that of the moon s motion, where
the sun and earth evidently cannot both produce gravita
tional fields permanent for the same system of coordi
nates. My formula, given above, applies generally to all
such cases. It is a matter for investigation whether the
small terms depending on the motions thereby introduced
into the gravitational formulae produce effects which
62
84 SOME PKINCIPLES OF PHYSICAL SCIENCE [CH
are verified in observation as recorded in the discrepancies
of the moon s tables. I have traced some theoretical
effects of these terms of the order of magnitude of one
or two seconds of arc with periods of the order of a
month or a year, but I have not yet succeeded in hitting
on a term of a period long enough to aggregate an ob
servable effect, having regard to the state of the moon s
tables. We want periods of about 250 years.
If the above formula gives results which are discre
pant with observation, it would be quite possible with
my general theory of nature to adopt Einstein s formula,
based upon his differential equations, for the determina
tion of the gravitational field. They have however, as
initial assumptions, the disadvantage of being difficult
to solve and not linear. But it is purely a matter for
experiment to decide which formula gives the small
corrections which are observed in nature. So far as
matters stand at present both formulae give the motion
of Mercury s perihelion, my formula gives a possible shift
of the spectral lines dependent upon the structure of the
molecule and on the interplay of the gravitational and
electromagnetic fields, and lastly, assuming a well-known
modification of Maxwell s equations giving such an
interplay, the famous eclipse results follow^.
Alternative Laws of Gravitation. Perhaps neither of
the above formulae will survive further tests of other
delicate observations. In this event we are not at the
end of our resources. There are, in addition to Einstein s,
yet two other sets of tensor differential equations which
on the theory of nature explained in this lecture satisfy
all the general requirements. These requirements are,
(i) to have no arbitrary reference to any one particular
* In Part II the Limb Effect and the doubling or trebling of the
spectral lines are also deduced.
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 85
time-system, and (ii) to give the Newtonian term of the
inverse square law, and (iii) to yield the small correc
tions which explain various residual results which
cannot be deduced as effects of the main Newtonian
law.
The possibility of other such laws, expressed in sets
of differential equations other than Einstein s, arises
from the fact that on my theory there is a relevant
fact of nature which is absent on Einstein s theory.
This fact is the whole bundle of alternative time strati
fications arising from the uniform significance of events.
It is expressed, without emphasis on any one such
time-system, by the Galilean tensor |(?jj?||. This tensor
is defined by the property that, when expressed in
terms of rectangular Cartesian coordinates (# x. 2J x 3 , x 4 )
for any time-system x,
, .
Thus we have on hand two tensors, the above Galilean
tensor and the tensor of the gravitational field which is
ira-
In order to formulate the differential equations in
volving the gravitational laws we shall require the
three-index symbols of the first and second types for
both the tensors || J" M J and 1 6r M J. They will be written" 3 "
for the symbols of the first type, and
J>*, X}<> and G{^\} (u)
for the symbols of the second type. Also the associate
contravariant tensors are written || J&H and || 6rg5||, and
the determinant \J ( $ is symbolised by J (u \
* Cf. Part II, Chapter v, equation (8), and Chapter vi, equa
tion (13).
86 SOME PKINCIPLES OF PHYSICAL SCIENCE [CH
(i) Einstein s Law is
(19).
The two other laws which involve differential equations
depend upon making the proper substitutions for the
mixed tensor M T r\ \\
in the following tensor equations
^^log{- <?<*>}*]
-22[J5T
= o,
- ...(20).
(ii) In this law the mixed tensor
equation is to stand for
(iii) In this law the mixed tensor
tion above is to stand for
of the above
of the equa
- 22
G
where || T*$) \\ is some contravariant tensor arising
from some quality of the electromagnetic field. This
law is suited to express the interaction (if any) of the
electromagnetic field on the gravitational field.
If the equations of laws (ii) and (iii) be referred to
rectangular Cartesian coordinates, they become
(ii) 2^ 2
and
= 0, [,u,v=l,2,3,4] (21),
( Hi ) ??;wT
= 0, [>, *- 1, 2, 3, 4] (22).
iv] SOME PRINCIPLES OF PHYSICAL SCIENCE 87
(iv) The fourth law has already been considered. It
can be expressed in the integral form
-.(23),
where the kinematic element corresponding to dG m * is
causally correlated to that corresponding to dG M z .
According to this law the fundamental character of
inertial properties is derived from their intimate con
nection with the abstract measure of uniform process
in the spatio-temporal field. Thus JdG^ and JdG m 2
are these abstract measures of spatio-temporal process
in the elements XX and PP f of the tracks of M
and m respectively. The inertial physical field modifies
this abstract measure of process into the more concrete
potential impetus Jd>P, and full concreteness, so far
as it is ascribable to nature, is obtained in the realised
impetus M JdJ\
Rotation. In conclusion I will for one moment draw
your attention to rotation. The effects of rotation are
among the most widespread phenomena of the apparent
won exemplified in the most gigantic nebulae and in
the Hiinutest molecules. The most obvious fact about
rotational effects are their apparent disconnections from
outlying phenomena. Rotation is the stronghold of
those who believe that in some sense there is an abso
lute space to provide a framework of dynamical axes.
Newton cited it in support of this doctrine. The Ein
stein theory in explaining gravitation has made rotation
an entire mystery. Is the earth s relation to the stars
the reason why it bulges at the equator ? Are we to
understand that if there were a larger proportion of
run-away stars, the earth s polar and equatorial axes
88 SOME PRINCIPLES OF PHYSICAL SCIENCE [CHIV
would be equal, and that the nebulae would lose their
spiral form, and that the influence of the earth s rotation
on meteorology would cease ? Is it the influence of the
stars which prevents the earth from falling into the sun ?
The theory of space and time given in this lecture, with
its fundamental insistence on the bundle of time-systems
with their permanent spaces, provides the necessary
dynamical axes and thus accounts for these fundamental
phenomena. I hold this fact to be a strong argument
in its favour, based entirely on the direct results of
experience.
Conclusion. The course of my argument has led
me generally to couple my allusions to Einstein with
some criticism. But that does not in any way repre
sent my attitude towards him. My whole course of
thought presupposes the magnificent stroke of genius
by which Einstein and Minkowski assimilated time and
space. It also presupposes the general method of seek
ing tensor or invariant relations as general expressions
for the laws of the physical field, a method due to
Einstein. But the worst homage we can pay to genius
is to accept uncritically formulations of truths which
we owe to it.
PART II
PHYSICAL APPLICATIONS
CHAPTER V
THE EQUATIONS OF MOTION
THE equations of motion of a mass-particle (M) are
[cf. Chapter iv, equation (11)]
^L^_JL^ =0 , o-i,2,3]...(i),
where [cf. Chapter iv, equation (9)]
dl= MJdJ* + c~ l EdF (2).
We write
dJ* = dG M * 2 2g ( $ du^du, (3),
^ du p du^
Then the equations of motion can be written
M *-(-i<?r
/ 5 \
& p> [ft = 1,2, 3]. ..(6),
p
where (u iy u Z9 u s , u^) are any generalised coordinates of
the situation of M.
If (x lt x z , x 3 , x 4 ) are Cartesian coordinates for the
spatio-temporal system #, these equations become
d Mx^ _^_^p,^^^^r nn . ^(x)
x f , 0=1,2,3] (7),
where
92 THE EQUATIONS OF MOTION [CHV
and F is written for T (x] . We write 2 for summation
p
for p = 1, 2, 3, excluding p = 4. Then the terms
are called the pure centrifugal gravitational terms,
the terms
are called the composite centrifugal gravitational
terms, and the term
is the pure gravitational term. Also
is the electric force [electrostatic units] and
(rF (x} rF (x} ^F (x} )
\cr 2 3 , cr 31 , CJT 12 /
is the magnetic force.
It is convenient to note for future reference that
[cf. Chapter iv, equations (5) and (7)]
x , .
m = c*ci m -*d P ;r
Also if c(xt-p<) = r (x)t
fir
then dp, = dx,-^ ............... (10).
CHAPTER VI
ON THE FORMULA FOR dJ 2
WE adopt the formula [cf. Chapter iv, equation (17)]
where c (x 4 p 4 ) = r .................. (2),
and m is a typical member of the attracting particles,
situated at (p lt p, p. 3 , p 4 ), and r stands for r (x) in
Chapter iv.
Then [cf. equations (3), (9) and (10) of Chapter v]
tta (x) dx dx - 2 2 ym dG 2
M,,, ctx aju v z, 7~ u ***
2ym
Thus
, dr\
4 "
...(3).
dr dr
2ym dr /dr . r
- su- O* 4 ]
...(4).
Also
L/ ^* J
> 4 1 dr r Sr
> C 9 ^M
...(5).
94 ON THE FORMULA FOR dJ* [OH
Also
The Potentials. It is convenient to express the com
ponents of || J^ v |t in terms of various potential functions
which have either a tensor or an invariant character
for transformations between space-time systems. We
will limit our statements to rectangular Cartesian
coordinates.
(i) The General Potential. This is symbolised by 3>,
where
Vp^ ...(7).
o m t*-
Here, as elsewhere, it is to be noticed that (x lt x 2 , x 3 , # 4 )
lies in the causal future of (p lt p^ p 3 , p 4 ) where m is
situated ; so that
c(x,-p.) = r.
This condition always holds unless a special exception
is made.
<l> is invariant. For
is invariant. Also, dropping for the moment the causal
relation between
(a^, a? 8f a?,, x,) and (p lt p t , p, p<),
l m {c(Xt-pt)-} and n w -
vi] ON THE FORMULA FOR dJ* 95
are invariant. Hence
is invariant. Hence, replacing the causal relation after
differentiation, we immediately find that
OJ 2 (* -pjp, - \ <V (r - f ) 2 lVP,
is invariant. The in variance of 3> immediately follows.
(ii) The Tensor Potential. This is a covariant tensor
of the first order, symbolised by
where ^ = _ 2 ............... (8) .
m d* (r )
The tensor property follows from the fact that
.(*-)
is invariant, and that
II A. V^ II
is a covariant tensor.
We note that =
(iii) The First Associate Potential. This is symbolised
by A, where
A-S^^r-S) ............ (10).
Ttl C
It is obvious that A is invariant.
(iv) The Second Associate Potential. This is sym
bolised by B, where
......... (11).
It is obvious that B is invariant.
96 ON THE FORMULA FOR dJ* ICH
Then, neglecting terms involving c~ 3 as a factor,
= (c 2 + 6<S>) dx? - (l+ - 2 , ] ^Jdx* - 82 ^ p dx p dx,
...(12).
It is now easy to transform to any other pure spatial
coordinates in the x -space by noting that
is a vector in the os-space, and that
2 z dx = dB.
P dx p
and that, in any coordinate system (u 19 u Z9 u 2
is a covariant tensor of the second order, whatever may
be the coordinates (u lt u a) u. 3 , u). Here^ (r{^, p} (u) is
the ChristofFel three-index symbol defined by
0fcu % p}-Sflfo0[/u % a]W (13).
Furthermore, to our order of approximation [i.e. neg
lecting terms involving c~ 3 ], the terms involving B will
disappear from the equations of motion. Accordingly
from this source no terms arise in these equations which
involve c" 1 . We shall also show that no terms of this
order of magnitude arise from <I>, since these terms
disappear from the approximate expression for <.
The Contemporary Positions. It is often more con
venient to express the formulae in terms of the positions
of the attracting particles in the x-space contemporary
with the event-particle (x 19 x 2 , x 39 x 4 ). Let this con
temporary position of the particle m be (q 19 q 2 , q 3 , x 4 ).
* The definition of 7gJ and of Gfi^ is given in Chapter x, equation
(3), below, and that of G[pv, p] (u} in Chapter v, equation (8), above.
vi] ON THE FORMULA FOR dJ* 97
We have to assume that r/c is a small time. Then
Let R be the x-distance between (a;,, a; 2 , a;,) and (q lt q,, q 3 ),
and
1 -
.(15).
Then [cf. equation (14)]
. -.(16).
We write
for suiBficient approximations to the various potential
functions, neglecting terms involving c~ 3 .
Then
.-(18),
m C
W. R.
.(19).
7
98 ON THE FOKMULA FOR dJ* [CH
We note that V (x) M = - 2 2 ............... (20),
C "
s 2 s 2
v +
It easily follows [cf. equation (20)] that, if the attract
ing matter be a uniform sphere in the x-space of mass
M and of radius a, then at a distance R from the
centre of the sphere and for points outside the sphere
[i.e. R^a]
(21),
(22),
(23):
and for points inside the sphere [i.e. R < a]
Associated Space. We now introduce new co
ordinates (X lf X z , X s , X 4 ) which are not in general pure
spatio-temporal coordinates (unless the attracting Jbodies
be at rest in the os-space), but are closely associated
with the Cartesian coordinates (x lt x z ,x z , x 4 ). We write
1 ;) Jo
We can then by an easy transformation deduce
dJ" = (c 2 + 64>) dX; -
&
j -(27).
t + 42 - dX f dX t
VI]
ON THE FORMULA FOR dJ*
It is to be noticed that (X lt X z , X 3 , JQare transformed to
(Y 19 Y 2 , Y 3 , F 4 ) cogrediently with the transformation of
(x lf x 2 , x 3 , # 4 ) to (y lt 2/2, t/3, y 4 ). Hence we can conceive
that (X lt X^ X 3 , X 4 ) and (Y lt F 2 , F 3 , F 4 ) are two sets
of rectangular coordinates to an event-particle in an
Associated Space-Time Continuum.
Then corresponding to a path in the #-space tra
versed with velocity (x lt x. 2 , x 3 ), there is a path in the
associate space traversed with velocity (X lt X,, X^),
where
Y -^
~~
= 1,2, 3, 4].
Also we write
and
Now put
,(28).
A= 1 +
2 ym
<?. R
(29).
The equations of motion [cf. equation (6), Chapter v,
and equation (27) of this chapter] now become [for
/*=!, 2, 3]
1 + JT
- 3*)
where
= _
...(30),
.(31),
7-2
100 ON THE FORMULA FOR dJ* [CH vi
and (since || F^ || is a co variant tensor)
1 rP /4
F^^F^+^J-^F^^- (32).
c 2 a> a dx p dx a
We note that throughout the small terms we can
neglect the distinction between the true and the
associate continua.
CHAPTER VII
PERMANENT GRAVITATIONAL FIELDS
WHEN the attracting masses (m 15 m 2 , ...) are per
manently at rest in the x-space, we obtain those par
ticular cases of gravitational action for which Einstein s
general equations of condition have been solved.
We now have
T,=^;=O, 0=1,2,3]
X, = x t
Hence [cf. equation (27) of Chapter vi]
a - = (c 2 -2^ 4 ) dX?
and [cf. equation (12) of Chapter vi]
= (c 2 - 2 4 ) dx t * - l + < S
Thus the equations of motion become
A.
dX t
....(2).
CHAPTER VIII
APPARENT MASS AND THE SPECTRAL SHIFT
IN the first place consider the vibration of some in
ternal part of a molecule. Let M be its mass and V its
undisturbed velocity. Then in the absence of gravitation
and in the presence of gravitation
But in either case Mt is the effective mass. Accord
ingly, assuming that the electromagnetic forces which
bind together the molecule are unaltered by the presence
of the gravitational field, the period of vibration is
lengthened in this field from
T to
where ST/T-^v; .................. (3).
But the electromagnetic forces will be affected by the
field. Accordingly, it requires some knowledge of the
structure of the molecule to be certain what the shift
(if any) of the spectral lines should be.
For example, assuming the electromagnetic laws con
sidered under Chapters x and xui below and assuming
that the cohesive forces of an atom depend on the
statical distribution of electric charges, the presence of
CH vm] APPARENT MASS AND SPECTRAL SHIFT 103
the gravitational potential will (on the average) change
any such cohesive force from F to
according to the formula (10*1) of Chapter xm below.
Thus the shift would now become
*,V; .................. (4).
The whole question is discussed in detail in Chapters
xm, xiv, and xv below.
CHAPTER IX
PLANETARY MOTION
LET the sun be the only gravitating body and let it
be permanently at rest in the cc-space at the spatial
origin of the coordinate-system (x 19 x z , x s ). The corre
sponding polar coordinates are (r 19 r a , r 3 ). Then
, A = ymr^ ............ (1).
Hence *, = a>l - , [>= 1, 2, 3]... (2).
We then put
............ (3),
and evidently (JBj, r 2 , r s ) are the polar coordinates in
the associate space of the point (X 19 X z , X 3 ).
Consider uniplanar motion of a planet in the plane
r f = 0.
The equations of motion become
O-0 ..................... (5).
Thus ^R^r t = h ..................... (6).
Then transferring to r 2 as independent variable and
putting
=1/^ ..................... (7),
we find
CHIX] PLANETARY MOTION 105
But, to our approximation,
y ra V
4 = yww- j ............... (9).
c/
Thus the equation becomes
Now K is the constant energy of the orbit on the sup
position that c is infinite. Hence, to our approximation,
c~*K is constant.
We now put
u = r i (l+ecos0) ............ (11),
where
6 = (l-k)r z + a ............... (12),
and immediately deduce
4 \ .
(lo)
2 7 2
A 2 c 2 /r c 2
7
and A=
This value for & is Einstein s result.
The path of the planet in the as-space is
(15).
l l
It is to be noticed that we have not assumed that e is
small.
It is evident that Kepler s second law receives a slight
modification, since
hb- 1 .................. (16).
CHAPTER X
ELECTROMAGNETIC EQUATIONS
WE have to consider a modification of the Maxwell-
Lorentz Equations which will exhibit an influence of
the gravitational field on the electromagnetic field.
The electric and magnetic forces in the #-space are
expressed by the skew symmetric tensor
II*U
where
Hence it follows that, if
A
then
Thus one set of four out of the Maxwell-Lorentz
equations is identically satisfied. We now choose the
remaining set of four in a form which exhibits a gravi
tational influence. Let || </ M " || denote the contravariant
tensor conjugate to || J^ v ||, defined by
= 1, |> = v]J
We then define the skew contravariant tensor
!()
Then since || (JF is skew,
is a contravariant tensor.
CHX] ELECTROMAGNETIC EQUATIONS 107
Also let p (x) be the electric density in the #-space, so
that
is the Contra variant Electric Motion Tensor. Then the
second set of the remaining four of the Maxwell-Lorentz
equations is
? l-(jy) S = ^*,, 0=1,2,3,4]...(5).
If the circumstances are such that the gravitational
elements [i.e. J$~\ can be taken as constant throughout
the region of the electromagnetic field, then [cf. eqn.
(3) above] we can write these equations in the co variant
form
O J/X*) A
ratf-T A tTTy) . w^) f\ _
L A -
CHAPTER XI
GRAVITATION AND LIGHT WAVES
THE wave-lengths of light waves are short compared
with the linear dimensions of any region within which
the gravitational elements vary. Also it is possible to
assign regions such that the gravitational elements are
constant within them and yet large enough to contain
areas of wave-fronts of linear dimensions large com
pared to the wave-lengths. Also any lengths expressive
of the curvatures of the wave-fronts may be assumed
to be large compared to the linear dimensions of such
regions, at least in the application considered below.
It follows that the characteristic equations for light
waves in uncharged space are [cf. equation (6) of
Chapter x]
In the corresponding coordinates of the associate con
tinuum these equations become
Also with our assumption as to the constancy of the
e/ s, X^ is a linear function of x lt x. 2) x 3 , # 4 . Hence a
plane wave in the x- space is a plane wave in the associate
JT-space.
Now assume
K
Then our conditions give
........................ (4),
CHXI] GRAVITATION AND LIGHT WAVES 109
...(5).
We derive therefore a determination of VF l + F 4 as a
multiple of the small quantity
and the equation
V*J-2VJ> + J& = ............ (6)
for the determination of V.
Assume that the gravitating bodies are permanently
at rest in the x-space. Then [cf. equation (ri) of
Chapter vn and equation (3) of Chapter x]
(X)
Also, measuring along the normal (in JT-space) to the
plane wave,
Hence the equation for V becomes
dJ* = ..................... (9).
Thus Einstein s assumption is proved for the normal
advance of very short electromagnetic waves, such as the
light waves, considered as advancing in associate space.
This result holds for any short waves for which the
radii of curvature of the wave-fronts are large compared
to the wave-lengths. For then a small area of wave-
front can be treated as plane.
Now consider a ray from a fixed point P to a fixed point
110 GRAVITATION AND LIGHT WAVES [OH
Q in the x-space. By Huygheiis principle its course is
given by making the time T to be stationary for small
variations of the path between these points. Now by
comparison with the associate space, since the gravita
tional field is permanent, X 4 and x 4 are identical. Thus
P p
But V is given by
=0 ...... (11).
Now V is nearly equal to c. Hence to our approximation
Thus
P
Thus, keeping PJandjQ fixed,
(14).
Thus the associate path of the ray in the associate space
is obtained by assuming the associate space to be filled
with a medium of refractive index
Also in the particular case when the gravitational
field is due to the sun, the refractive index is
h c 2 ^
and since A is a function of i\ only, the polar coordinates
(r 19 r a , r 3 ) in the #-space correspond to the polar co
ordinates (R, r 2 , r 3 ) in the JT-space. Hence the angle
subtended at the sun by the two points at infinity on
the ray in the x-space is equal to the analogous angle
xi] GRAVITATION AND LIGHT WAVES 111
subtended at the sun by the two points at infinity on
the associate ray in the JT-space. Thus Einstein s result
as to the deviation immediately follows.
Furthermore, it follows from the expression for T, that
no modification of interference fringes can arise, due to
the terms in dJ 2 involving B, by the use of an apparatus
by which alternative rays for light, originating from
the same source at P, are sent along alternative paths
from P to Q, since
[1 CQ I O \ ~~I other path
\ }+-^\dS\ ...(15).
Cjp\ C* V Jonepath
If ^ 4 be constant along the paths, this equation be
comes other path 1 / 2 \ other path , .
I 1 Jonepath 0\ l *tf 4 ) L J one path
Now on the surface of the earth, if the axes of co
ordinates be fixed relatively to the surface and the axis
of x 3 be vertically upwards, we have [cf. equations (22)
and (26) of Chapter vi]
-
*- 5 c 2
where g is the gravitational acceleration and a is the
earth s radius.
It follows that S = s.
Accordingly, if the alternative geometrical paths of a
divided ray be of equal geometrical length,
p rrrt other path _ ^
L J one path
Thus in any experiment of the Michelson-Morley type,
the earth s gravitational field will produce no modifica
tion of the interference fringes. The null result of the
Michelson-Morley experiment is therefore fully explained.
CHAPTER XII
TEMPERATURE EFFECTS ON GRAVITATIONAL
FORCES
ASSUME that the attracting body is at rest, except
that its separate molecules have a velocity of agitation
of which the mean square is w 2 . Let Av stand for
Average value of. Consider
AvdJ\
We use the formula of Chapter vi, equation (12), for
dJ 2 and the values of the potentials given in equations (7)
to (1 1) of that chapter. Then R refers to the permanent
position of a molecule, neglecting its agitation due
to temperature.
Write
m R
and note that
Then we easily find
AvdJ* = jc 2 - 2 (l + U -\ ^1 ^;
A
...(3).
Thus the gravitational potential requires the co
efficient
due to the temperature of the attracting body.
CHXII] TEMPERATURE AND GRAVITATION 113
The coefficient due to the temperature of the attracted
body is complicated by the change of apparent mass due
to the velocity of agitation and by the possible effect
of this velocity on the electromagnetic forces. Accord
ingly the special circumstances must be known before
any calculation can be applied.
Returning to the consideration of the correction for
the temperature of the attracting body, let 6r, and u? be
its gravitational attraction at a given point and the
mean square of its molecular velocities when its absolute
temperature is T 19 and G and u^ be analogous quan
tities when its absolute temperature is T . Then
< = a7;, M 2 = aT (4),
where a is some constant depending on the physical
constitution of the body. For example, for water vapour
at
<=5x 10 9 [cm. 2 /sec. 2 ], T = 273 (5).
Hence a= 1 83 x 10 7 (approx.) (6).
Hence, putting
a = a xl0 7 (8),
we find
Thus the temperature correction due to ordinary
molecular velocities is inappreciable. It may be the
case however that we should consider the velocities
within each molecule, particularly within the nucleus
of an atom. It may well be that such velocities attain
to a sensible fraction of the velocity of light. The
temperature correction might then be appreciable,
w. B. 8
CHAPTER XIII
THE ELECTROSTATIC POTENTIAL AND SPECTRAL
SHIFT
SINCE the gravitational field is permanent [cf. equation
(1-2) of Chapter vn]
= (c 2 - 2^ 4 ) dx? -
\ /*
We also assume (as in Chapter xi) that in the region
considered the gravitational field is practically uniform,
so that the spatial rate of variation of g^ v is negligible.
This amounts to neglecting gravitational accelerations
but retaining gravitational potentials or quasi-potentials.
Wewrite
so that "^ and A^ [/x, v = 1, 2, 3] are constants through
out the region considered.
Neglecting terms involving c~ 4 as a factor, we find
...(4),
(5),
(6).
CHXIII] ELECTROSTATIC POTENTIAL 115
Hence putting X = 4 in equation (6) of Chapter x, we
find that in a steady electromagnetic field and a per
manent gravitational field, the equation for the electro
static potential F becomes
Hence the solution for a point-charge e at the origin is
(8),
where ?\ is the distance of the point (x lt x. 2 , x 3 ).
Now consider a number of molecules in the region
each forming an isolated electrical system. Let each
molecule have an axis-system (x lt x 2 , a? 3 ) at the centre
of its nucleus, and let each such axis-system have the
same relation to its electrical configuration as any
other such axis-system has. But the molecules are
orientated in every possible manner with respect to the
gravitating field.
Hence if e be the charge at the centre of the nucleus
which can be conceived as keeping the molecule together,
and e be the charge of any part of the molecule
whose vibration is being considered, it follows that on
the average the cohesive radial force is
But V 2
c
} ............ (9).
Hence the average cohesive force on that element of
the molecule is
82
116 ELECTROSTATIC POTENTIAL [CHXIII
But the apparent mass of the element due to the
gravitational potential is
where M is its apparent mass in the absence of the
field. Hence, if T be the period of vibration of the
element in the absence of the gravitational field and
T-}- Sj T in its presence,
?^-L* (11)
T - 6c ^" .-111J.
Einstein s formula for the shift of the spectral lines is
For observational purposes the two formulae are in
distinguishable.
CHAPTER XIV
THE LIMB EFFECT
LET a> = a M r lf [>=1,2,3] ...... (1).
Then [cf. equation (8) of Chapter xiu]
a.} ............ (2).
Here (a 1? a 2 , a 3 ) are the direct ion- cosines of the vector
from the origin to (x lt x 2 , x. A ). Thus the radial force is
Now consider the internal vibration of a molecule
which radiates light of period T (in a non-gravitational
field) as capable of being represented as the vibration
of a variable electric Hertzian doublet with this period.
Let (a lf a 2 , a 3 ) be the direction-cosines of the axis of
the doublet. Then owing to the gravitational field the
electric force which controls the vibration of the doublet
is changed by the presence of the factor
M v **" M
Let T become T+8 T owing to the joint effect of
this factor and of the change in the apparent mass of the
electrons forming the doublet (due to the gravitational
field). Then
8 T
Let there be a large number of electrons forming the
atmosphere of a star (say, the sun). Let the observer
118 THE LIMB EFFECT [OH
be at a great distance along the axis of x^ Put
c&! = cos Oj, a 2 = sin Oj cos a 2 , a, = sin a : sin a 2 . . . (5).
Now doublets radiate light unequally in different
directions. The intensity (measured by the energy
radiated) varies as the square of the sine of the colatitude
of the direction, the latitude being reckoned from the
equatorial plane of the doublet. Thus the intensity of
the light from the doublet in direction (a,, a. 2 , a s ) sent
to the observer varies as sin 2 a x . Also the average change
of period (8T) of the light sent to the observer with
colatitude a, (as reckoned from the equatorial planes of
the doublets) is given by
1 p-S T
2. -T
i.e. by
S T 1 3^
...(6).
sin 2 a x cos 2
Now the light from the molecules for which a x is
nearly 90 will be the brightest, both because of the
factor sinX in the intensity, and because the equa
torial belt of angular space of breadth d^ is greater
than the belts of the same angular breadth as c^
approaches zero. Hence the shift of the spectral lines
will approach that given by taking a x = ?r/2. This con
clusion is reinforced by the discussion of the next
chapter on permanent directions of vibration at least
so far as relates to the centre or the edge of the sun s
disc. Thus
xiv] THE LIMB EFFECT 119
Now let en be the radius of the sun, and let the centre
of the sun be the point
( a cos {} a cos /3 2 , a cos /? 8 ),
so that at the point of the sun s surface from which the
light is taken the direction-cosines of the upward
vertical are
(cos ft, cos&, cos/3 3 ).
Let R be the distance from the molecule at (x lt x z , x 3 )
to the centre of the sun. Then after differentiating we
can put a for R, and zero for x 19 x z , x 3 .
Then A=(&+y?) ............... (8),
where <TJ <-- ..................... (9).
It follows from equation (22) of Chapter vi that 77
would be exactly % if the sun were homogeneous. But
it is probably considerably smaller.
Then ^ a = = {2, + (l-3,)8in A} ...(10).
Thus = I* 4 {1+^ + 1(1- 3,) sin A} ...(11).
This formula exhibits a Limb Effect. For if the light
comes from the centre of the sun, then
and = (l+i,) ............ (12);
and if the light comes from the edge of the disc, then
(13).
120 THE LIMB EFFECT [CHXIV
Hence, as we proceed from the sun s centre to its
rim, there is a shift of spectral lines towards the red,
defined by
Thus, if we take 77 = ^
(which is probably not far from the truth), we find
& iv T 7
It is unnecessary to point out the roughness of the
assumptions, particularly the conception of the molecule
as a vibrating doublet emitting light. But the investi
gation does suffice to show that our general assumptions
do require the existence of a limb effect of the same
order and sign as that actually observed.
CHAPTER XV
PERMANENT DIRECTIONS OF VIBRATION AND
THE DOUBLING EFFECT
CONSIDER a vibrating element of a molecule of charge
e , the charge of the central nucleus being e. Let
(PU P 2 , P 3 ) be the mechanical force on the element due
to the electrostatic attraction of the nucleus. Then [cf.
equation (8) of Chapter xm]
> ee x * r
*~ r 3 1
2ee
= 1,2,3]
...(1).
Let this element of the molecule be that element whose
radial vibration in direction (a lt a. 2 , a 3 ) constitutes the
variable doublet to which the radiation of the light is
due. This direction of vibration cannot be permanent
unless the force (P 15 P 2 , P 3 ) is in the direction (a l9 a 2 , a 3 ),
when
x. = W.> [5=1,2,3] (2).
Hence for permanence we require
V 4 n rr* n [~e 1 9 3~l f l\
2* Si^Ctp OC a s , [_.S 1, <6, OJ...^O^.
/*
Consider a molecule in the atmosphere of the sun as
in the previous chapters. Then [cf. equations (8) and
(10) of Chapter xi v]
1
<r "
1
= ~ - ? cos cos , . -
...(4).
122 THE DOUBLING EFFECT [CH
Hence
a M = ^-~ ^ 4 cos fa S a M cos fa
+ p* 4 a s .-(5).
Hence
a M cos
or
Thus a permanent direction of vibration must be either
normal or tangential to the gravitational level surface.
Accordingly, in the gaseous mass of molecules forming
the atmosphere of the sun there will be an excess of
molecules with their vibrations either normal to the
level surface or in one of the directions tangential to
the level surface.
First consider the vibrations normal to the level
surface, and as in the previous chapter let the observer
be on the axis of x l at a great distance. Then for these
vibrations we should put
a, = cos&, [*=!, 2, 3] ...... (7).
Thus [cf. equation (4) of Chapter xiv]
t .
-y*. .............. ...................... <>
Thus these molecules yield a constant shift of the
spectral lines all over the sun s disc. But the intensity
of the light due to them varies as sin 2 /^ . Accordingly,
xv]
THE DOUBLING EFFECT
123
they should yield faint lines from the centre of the disc
and comparatively strong lines from its edge.
Secondly, consider molecules vibrating tangentially to
the sun s gravitational level surface. No generality is
lost by taking the axes of x 2 and x 3 , so that the sun s
diameter through the point of the disc considered is in
the plane x^. In this case
I- -(9).
Also the level surface at the point contains the axis
of x 3 .
x*
Then we can put
! = sin 9 sin /3 X = sin 8 cos ft. 2 *
a z = sin cos &
a 3 = cos
Hence equation (4) of Chapter xiv becomes
...(10).
T
- \ sin a (A u cos 2 ft - 2A n cos ft cos & + A* cos 2 ft)
^^33 cos 2 sin cos (A^ cos ft ^4 31 cos ft).
124 THE DOUBLING EFFECT [OH
By the use of equations (4) of this chapter, this
reduces to
T ~ 2c* 4
To consider the comparative brightness of light from
these molecules taken at different points [i.e. for
different values of ft] note that [cf. equations (10) of
this chapter]
sin 2 a, = I - a, 2 = 1 - sin 2 sin 2 ft ...... (12).
Hence the total light from this type of molecule is
brightest at the centre of the disc [ft = 0], since for
every value of
sin 2 ^=1 ..................... (13).
The brightness falls off as we pass towards the edge
/ 7T^
of the disc, and finally at the edge ( ft = -
2
sin 2 a x = cos 2 6
Also the average value for sin 2 c^ at any point is
l-isin 2 ft (15).
It will also be noticed that the larger angular area
of an equatorial belt of angular breadth 28^ over a
polar cap of angular diameter 28^ gives the tangential
molecules another advantage in brightness over those
of the former type vibrating normally to the level
surface.
To sum up the discussion on the shift of the spectral
lines contained in this and the two preceding chapters :
The molecules in the sun s atmosphere can be sorted
into three groups, (i) a group of molecules uniformly
pointing in all directions, (ii) a group of molecules
pointing normally to the gravitational level, and (iii) a
xv] THE DOUBLING EFFECT 125
group of molecules pointing tangentially to the gravi
tational level. A molecule is said to point in the
direction of the equivalent electric doublet whose varia
tion generates the light waves.
The relative brightness of the light from these three
groups changes as we pass from the centre to the edge
of the sun s disc. It is constant for group (i), it varies
as sin 2 ft for group (ii), and it varies as (l ^ sin 2 ft) for
group (iii).
The spectral shift for group (i) is on one hypothesis
[cf. Chapter xm]
!_*
6c 4)
and on another hypothesis [cf. equation (11) of Chap
ter xiv] it approximates to
where 77 is probably not greater than ^Q.
The spectral shift for group (ii) is
3 ~ 2r J ,
~~2d r ^ 4
The spectral shift for group (iii) is
2 _2<p
2c 2
Accordingly, in light derived from the sun, or a star,
or a nebula in (1) a general shift of the spectral lines to
the red may be expected ; (2) since groups (ii) and (iii)
change in relative importance as we pass from the
centre to the edge of the disc, and since the shift due
to group (i) also changes, so a shift of spectral lines
towards the red (the limb effect) may be expected ;
(3) in the case of the sun or a nebula some evidence of
126 THE DOUBLING EFFECT [CH xv
a doubling or even a trebling of the spectral lines may
be expected.
It is also to be noticed that the number of vibrations
of a doublet emitting light from the visible spectrum
during a mean free path of the molecule is of the order
10 4 , even allowing for the fact that the velocity of the
molecule is largely due to a high temperature. Accord
ingly, within each mean free path there is time for the
vibrations to settle down into one of their permanent
directions.
Finally, we note that when it shall be possible to
measure with reasonable accuracy the spectral shifts of
light from the stars and the nebulae, we obtain a
numerical determination for the mass divided by the
radius of the body concerned. Hence if either the
mass or the radius be known, the other can be found.
CHAPTER XVI
STEADY ELECTROMAGNETIC FIELDS
THE equations (6) of Chapter x for a steady electro
magnetic field become
y<y, 0=1, 2, 3]...(i).
Consider a region where there is no current and let
F^ |>, v= 1, 2, 3, 4] be the value of F^ which is the
first approximation when the gravitational influence is
neglected.
We use equations (3) to (6) of Chapter xm and put
We also take 1 as the magnetic potential for the
approximate magnetic force (cF Z3 , cF 3l , cF^), so that
(cFj, cFJ, cFJ)= -gradft ......... (3).
The equations to determine the magnetic force
(cF,, cF 3l , cF K )
now become
curl (Si, S,, S,) = Q ............... (4),
where
an 2/dBdF dB3F 4 \ /cx
Si^cFv-VSfA+z- +-U-^r-5-^r (5),
dx v C \dx., dx 2 dx. 2 dx 3 1
with analogous meanings for and >S 3 .
128 STEADY ELECTROMAGNETIC FIELDS [CH
Hence the second approximation gives
cF, = cFJ - 2 {A u cF a + A a cFJ + A a cFj}
- ...(7),
- 2 { A cFJ + A x cF 3 ? + A.cFj}
_2M 35 ....
In the first place we note that a steady electric force
(Fu, F^,FU) in a permanent gravitational field produces
the magnetic force
where \_H.R f ~\ stands for the vector product of the two
vectors R and R f .
Accordingly, the magnetic force is perpendicular to
the electric force which produces it and to the vector
grad B.
Consider a field on the surface of the earth. Let a be
the earth s radius and let the axis of x l be the upward
vertical. Then at the origin (which is on the earth s
surface)
grad = (^ 4 , 0, o) (9),
where [cf. equation (23) of Chapter vi]
076 ... <e<l.
Here e would attain its lower limit if the earth were
uniform throughout. We shall assume
e = 88 (10)
xvi] STEADY ELECTROMAGNETIC FIELDS 129
as a sufficient approximation in the actual circum
stances.
Hence the magnetic force produced by the electric
force
a, Fsina, 0)
is ^.(0,0, -sin a),
c/
i.e. is the horizontal force
2ega ^ .
-~ Fsma,
c
perpendicular to the vertical plane containing the
electric force and proportional to the sine of the angle
which the electric force makes with the vertical. Here
g denotes the ordinary gravitational acceleration.
Accordingly, a given electric force produces the
greatest magnetic effect when it is horizontal. But in
any case the magnetic force produced is extremely
small, being about
r2xlO~ 9 x^sin a (gausses),
where F is the measure of the electric force in electro
static units.
The corresponding effect on the surface of the sun
would be about
3*8xlO~ B x^sinct (gausses).
This effect is the only effect I have found which depends
on the existence of B. Accordingly, an experiment of
sufficient accuracy to detect the magnetic force, if it
exists, would be of great interest as forming a crucial
experiment to test the formula for dJ 2 here adopted.
A steady magnetic field is also modified by the
presence of the gravitational field.
W.R. 9
130
STEADY ELECTROMAGNETIC FIELDS
[CH
For example, consider a current / (electromagnetic
measure) along the axis of x lt and let R be the distance
of (x lt x. 2 , x. d ) from this axis. Then
,(11).
R
Let the wire (i.e. the axis of x t ) make an angle ft with
the upward vertical, and let the axis of x 2 lie in the
vertical plane through the wire. Also let the plane
through the wire and the point (x lt x 2 , x 3 ) make an angle
< with this vertical plane through the wire. Then we
find [cf. equations (4) of Chapter xv]
AU = - (1 - 817) sin ft cos ft . -
A = Q
Hence [cf. equation (6) above]
...(12).
c~ K
Thus there is a small magnetic force parallel to the wire
which is equal to
(l-3i))ga . n0 21
- 12 - - cos (j> sin 2ft x -p-
G s\i
at distance R from the wire.
This force vanishes if the wire be vertical or hori
zontal and is greatest when the wire is inclined at an
angle of 45 to the vertical. Also it is greatest in the
vertical plane through the wire, and vanishes in the
plane through the wire perpendicular to this vertical
plane.
xvi] STEADY ELECTROMAGNETIC FIELDS 131
Thus its greatest value at a distance R from the wire
& ^ , and cj> = or TT\ is
(l-3rj)ga 2/ /
jT ~ x R (gausses).
Hence, taking 77 = ^ , its greatest value at distance R
from the wire is about
27
x 10 9 x - (gausses).
92
CHAPTER XVII
THE MOON S MOTION
ASTRONOMICAL tables, which depend on observations
made at all times of the year, must finally register
spatio-temporal elements in terms of the space -time
which is the rest-system of the sun. We must therefore
distinguish between relative motion and difference
motion in respect to a given space -time. Thus the
relative motion of the moon with respect to the earth
is the motion of the moon in the earth s rest-system at
the moment of observation. But the difference motion
of the moon from the earth in the sun s rest-space is the
vector excess of the motion of the moon over that of the
earth reckoned in the sun s rest-space. On the classical
theory of a unique space and unique time difference
motion and relative motion were identical. We have to
treat them as distinct with distinct formulae. It is
evident that astronomical tables for the moon concern
the difference motion of the moon from the earth with
respect to the sun s rest-system of space-time.
Let (x l} x 2 , x 3 , x) be the coordinates of the moon at
the time x 4 with respect to rectangular coordinate
axes in the sun s rest-space with the sun as origin.
Let the contemporary [i.e. at the same sun-time # 4 ]
coordinates of the earth be (q l , q 2 , q 3 ). We now take the
earth as a moving origin in the x- space and obtain
the difference coordinates for the moon, referred to
moving axes parallel to the fixed axes (y lt y 2 , y a ), where
&-*,-&, |> = l,2,3] (1);
also the difference coordinates for the sun referred to
CH xvii] THE MOON S MOTION 133
the earth as moving origin are ( q lt <? 2 , <? 3 ), where
the three positions in the sun s rest-space for the sun,
earth and moon are contemporary at the time x t .
The difference velocity of the moon from the earth is
therefore (y l9 y 9 ,y 3 ).
Let v be the magnitude of the velocity of the moon
in the sun s rest-space, and let U be the velocity of the
earth in the same space. Also let F be the magnitude
of the difference velocity of the moon from the earth.
Then
v*= F 2 + Z7 2 + 22 ?/ M ^ (2).
Also we write
s x >
Assume a uniplanar motion in the plane of the
ecliptic, so that
* 3 = 0, 2 3 = 0, 2/3 = ............... (4).
Let (R s , r a ) be the polar coordinates of the earth
relatively to the sun, and let (R 19 R 3 ) be the polar
difference-coordinates of the moon from the earth. Thus
y l = R, cos ^ 3 , y t = R, sin
134 THE MOON S MOTION [CH
Also let r a be the distance of the moon from the
sun, so that
r* = R* + ^ + 2^11, cos Pa ......... (6),
where p* = R*-r* .............................. (7).
We consider R 8 and r 3 to be given functions of the
time Xi, and take J?, and R 3 as the coordinates to be
determined in terms of the time by means of the
equations of motion. Two propositions (A and B) are
easily proved, of which particular cases are important :
Prop. (A)
The important particular case of this proposition is
AJL^-JL^o (9)
dx.dR.r, dR,r,
Prop. ()
jLJ^( r r) L( r ri ) = ......... (10).
The important particular case is
"
Let (_p 15 _p 2 , _p 3 , jp 4 ) be the coordinates of the ante
cedent position of the earth causally correlated to the
moon s position at time x 4 , referred to the fixed axes
origin at the sun.
Let r e be the distance between the moon and this
antecedent position. Also put
t* -L . i * / \ .
-i
.(12).
XVIl]
THE MOON S MOTION
135
Then the potential impetus of the moon s route through
the space-time manifold is [cf. equations (1) and (3)
of Chapter vi] determined by
dJ* = (c 2 - v*) dx? - 8 l - - dx-
where m s and m e are the masses of the sun and earth
respectively.
We now put
Thus
Also
Hence
+ terms due to planetary attraction (14).
(15).
(17).
c" i\
where
+ ~^T L+ cR,
...(18),
We put
w l = R s cos p 3 + R s r 3 sin
2 = R s r 3 cos p. A R s sin
^^(R,- R s r*) cos p 3 + (R s r 3 + 2R s r 3 ) sin p 3
,r 3 ) cos yo 3 - (R 3 - R s r?) sin /> 3 j
....(20).
136 THE MOON S MOTION [CH xvii
Then w^w^a^, a. 2 are functions of R A and x 4 only. Also
(21).
Thus
TT ym e r-f. & -,
e = -^n\^ K ^^^ U
......... (22).
There are now [cf. equation (6) of Chapter v] two
equations of motion of the type
(23).
It follows from the special cases of Props. (A) and
(B), that the two terms involving c~ l as a factor dis
appear in both equations. Hence these equations both
take the form [ft = 1, 3],
d az az d d
dx^R^ <?r l
The terms on the left-hand side of these equations
are those introduced by ordinary lunar theory ; the
terms on the right-hand side are the new small cor
rections introduced by the formulae of relativity of the
form here adopted. I have not succeeded in eliciting
any terms which, in the present state of the Lunar
Tables, can be made the subject of comparison with
observation. The investigation will therefore not be
pursued further.
PART III
ELEMENTARY THEORY OF TENSORS
CHAPTER XVIII
FUNDAMENTAL NOTIONS
1. Coordinates. The mutual relations to each
other of event-particles can be determined by charac
terising each event-particle by four measurements of
four assigned types respectively. These four measure
ments are called the coordinates of the event-particle ;
and the four types of measurement must be such that
(i) each type assigns to each event-particle one and
only one coordinate of that type, and (ii) each set of
four coordinates (as ordered in that assignment to
types) characterises one and only one event-particle.
Four given types of measurement with these properties
are called a coordinate-system.
A coordinate-system will be called pure if one of
the coordinates be the time of some given space -time
system x and the other three coordinates be spatial-
quantities of the space of the same system x" A co
ordinate-system which is not pure is called mixed.
If (u 19 u. 2 , u 9 , u 4 ) be the coordinates of an event-particle
in a pure system, then it will be adopted as a conven
tion that (u 19 u Z9 u a ) represent the spatial coordinates
of a point in the space of the space-time system to
which the coordinates refer and u* represents the time
in the same space-time system. Thus the event-particle
(u l9 u 2 , u 3 , u 4 ) happens at the time u 4 and at the point
(u l9 u 2 , u 3 ) in the corresponding space.
If (u^ u z , u 3 , u^) and (x l9 x 2 , x S9 x 4 ) be the coordi
nates of the same event-particles according to different
140 FUNDAMENTAL NOTIONS [OH
coordinate-systems, the w-system and the ^-system
respectively, then there will be four equations of trans
formation
^=/M(I x -2> x s> x ^ [/*=!> 2 > 3, 4] ...(1).
These four equations can be solved so as to give
*> = ^K> u *> u *> *0> [> =1 > 2 > 3 > 4 1 (I l).
If both systems are pure in the same space-time
system, then
u, = x,,
and (u lt u z , u s ) and (x 19 x z , x 3 ) are different spatial co
ordinates of the same point in the space of that space-
time system.
2. Scalar Characters and Invariant Expressions.
Consider the measurement of some physical quantity
arising in the physical field at an event-particle, such
as the gravitational potential according to some definite
meaning of that term. Its measure (so far as the defini
tion of meaning is kept unchanged) must be indepen
dent of coordinate-systems. But its law of distribution
throughout the various event-particles of space-time
will be expressible as a function of the coordinates of
the event-particles under consideration.
Such a physical character is called a scalar quantity.
We must distinguish between a scalar quantity
and an invariant formula expressing that quantity.
When a formula in terms of coordinates of relevant
event-particles is such that it gives the same value for
the scalar quantity whatever coordinate-system be
employed, it is called an invariant formula. There
may also be formulae which are only invariant for a
limited set of systems of coordinates, derivable one
from the other by transformations forming a group (in
xvm] FUNDAMENTAL NOTIONS 141
the mathematical sense of that term). In this limited
case we have l group invariance/
When we can conceive a scalar character in such a
way that it has no special or peculiar relation of any
sort to one coordinate-system of a group which it has
not to any other system of that group, it follows that
there must be some group-invariant formula for the
scalar character which is limited to that group of
systems of coordinates.
3. Physical Characters of the First Order. A scalar
character is a character of zero order.
A character of event-particles is of the first order
when given any coordinate -system (u lt u z , u 3 , u^) it is
expressible by an array of four quantities (functions of
the coordinates of the event-particles in question) such
that each quantity is specially related to one of the
types of coordinate measurement. These four quantities
are called the components of that character for that
coordinate-system.
For example, let (u lt u 39 U B , u 4 ) be a pure coordinate-
system, and let a region of the it-space be filled with a
continuously moving substance. Let the motion of the
substance at (u lt u. 2 , u 3 ) at the time u be represented by
Thus the array (u lf u t9 u 3 , 1) represents a character of the
first order which is descriptive of the motion of the fluid.
Again, let <X> be the gravitational potential at the
event-particle (u l9 u 2 , u 3 , w 4 ). Then the gradient
represents a character of the first order.
142 FUNDAMENTAL NOTIONS [CH
Let (x 19 x z , x 3 , x 4 )be the coordinates in the coordinate-
system x of the same event-particle as denoted by
(u ly u Z9 HS, Ut). Then (x lt x 2J x a> 1) is a character of the
first order descriptive of the motion by reference to the
coordinate-system x. We are at once brought to the
consideration of the relations to each other of these two
distinct descriptions of the same fact of motion by
means of (u lt u z , u 3 , 1) and (x lt x 2) x 3 , 1) respectively.
The relations between the two will be peculiarly
simple (and therefore important) if the components of
one character (say the M-character) are expressible as
linear functions of the components of the other character
(the x-character), where the coefficients may be functions
of the coordinates of the event-particles in question
which are purely determined by the general relations
of the two coordinate -systems in question and are
independent of the particular values of the components.
Thus if (7V 10 , T* (u \ T, T 4 (u) ) be a first order description
of some fact in coordinate-system u, and
be a description of the same fact in coordinate-system
x, the desired linear relation is
where the coefficients Z Ma [//,, a = 1, 2, 3, 4] are expressible
in terms of the equations of transformation between
6 u and x without any reference to the particular
values of
4. Tensors of the First Order. Furthermore, we
pass from the two assigned coordinate-systems u and
1 x to the consideration of a group of systems (as in
the case of invariance), if the determination of the
xvm] FUNDAMENTAL NOTIONS H3
coefficients [i.e. Z M J can be fixed by a general rule which
is identical for any two systems of the group.
A first order character as thus described in any
coordinate-system of a group is called a Group-Tensor
for that group.
If the general rule for the formation of coefficients
in the linear relation between the components hold for
all pairs of coordinate-systems whatsoever, then the
character as thus described in all coordinate-systems is
called a General Tensor or more simply a * Tensor.
It is a tensor of the first order.
It is obvious that in the case of a group-tensor or
a general tensor the rule for the formation of the
coefficients in the linear equations giving the com
ponents of the character for one coordinate-system in
terms of the components for another coordinate-system
must be such that the transformations of components
from one system to another form a group. For there is
to be only one description of the character in each co
ordinate-system. Accordingly, if u, v, x be three
coordinate-systems and J" a tensor character, then the
transformation of
(T, T?\ T 3 (w) , T) to (T*\ T?\ T*\ I")
and then of
(T, ZV", Tf\ r 4 w ) to (T, T*\ T, T^)
must give the same components in system x as the
direct transformation from system u to system x.
In future we will write
|| 7"" ||
for the array of the components of a character in
system ( u.
144 FUNDAMENTAL NOTIONS [CH
5. Covariant and Contravariant First -Order
Tensors. A tensor may refer to many event-particles.
Suppose that one of these with a peculiar definite
relation to the character in question is picked out and
termed the dominant event-particle of the character.
Let || T (u) || be the tensor in system u* and || T (x) \\ be
the tensor in system x, J and let (u lt u 2 , u 3 , u^) and
(x l9 x 2) x 3 , x) be the coordinates of the dominant event-
particle of the character expressed by the tensor.
The tensor is covariant if its components in any
system u are related to its components in any
system e x* by
n^Fff^, [/*=!, 2, 3,4]...(3).
In the case of contravariant tensors it is convenient
to adopt an alternative notation (T (u) \ T (u) \ T (u) \ T (u f) for
the components in any system u 9 shortened into || T (u) ||
when the whole array is to be mentioned.
Then a tensor is contravariant if its components in
any system * u are related to its components in any
system x by
It is easy to prove the group property of the co
variant and contravariant modes of transformation by
the use of the equations
. , (5-1)-
= 1, [p = V_ _
If the tensor property is restricted to a group of
xvin] FUNDAMENTAL NOTIONS 145
systems of coordinates, we obtain covariant group-
tensors or contravariant group-tensors as the case
may be.
6. Characters and Tensors of Higher Orders.
A physical character is of the nth order when, in any
coordinate system u, it is expressible by an array of
4 M quantities (functions of the coordinates of the event-
particles in question) so that each component of the
array is specially related to one permutation of the
types of coordinate measurement, the types being taken
n together in each permutation and repetitions of type
being allowed.
Thus a character of the 2nd order will require the
array
T, [^=1,2,3,4].
For example, those seven components (out of the
whole sixteen) which involve the coordinate-type u^ are
/TT() /TT(U) /rr() rp(u) rp(u) /TT(W) /TF(w)
* 11 J * 12 > * 21 > * 13 -* 31 > * 14 J - 1 41
A character of the 3rd order will involve 64 com
ponents, and of the 4th order 256 components.
The same general explanations, mutatis mutandis,
apply as in the case of characters of the 1st order.
The covariant tensor transformation (for the 2nd
order) is
and the contravariant tensor transformation (for the
2nd order) is
OU
and analogously for characters of other orders.
w. B. 10
146 FUNDAMENTAL NOTIONS [OH
But mixed tensors now appear in which both co-
variant and contra variant qualities are involved.
For example, a mixed tensor of the 2nd order [repre
sented by the notation T*(u) for coordinate-system ^ ]
is transformed by the rule
x) d ^ d ^ ............ (7),
and analogously for higher orders.
7. Tensor-Invariance of Formulae. The tensor
description of a physical character must not be confused
with the tensor-invariance for mathematical formulae.
If the array || T^ M \\ be an array of formulae involving
the ^-coordinates (u 19 u 2 , u s , u 4 ) as arguments, then
these formulae have tensor-covariance if HJ^H, as
obtained from || T u) || by the co variant rule of trans
formation, are expressible by the same formulae in
volving (x 19 x. 2 , x 3 , x 4 ) as || TJ U] \\ are expressed by the
use of (u l} u 2 , u 3 , u 4 ). Also similarly for tensor-contra-
variance.
Thus tensor-invariance (as this property will be
termed) implies the persistence of the same formulae
after transference from one coordinate-system to another
by means of the appropriate tensor formula (covariant
or contra variant).
For example, if A be any function of the position
of the event-particle (u lt u 2 , u s , u t ), then the array
has tensor-covariance. For
cL4 =s cL4.a^ ,
dUp dx a du lli " *\ /
Again let dN be any homogeneous rational integral
xvm] FUNDAMENTAL NOTIONS 147
function of du^ du z , du 3 , du^ of the first degree, and thus
analogously expressible in any coordinate-system. Also
write
idN
where u lt u 3 , u 3 represent the definite velocity of a
substance at (u iy u 9 , u 3 ) at time w 4 . Then
(N (u} u lt Wu,, N (u} u 3 , N M ) ......... (8-1)
has tensor-contravariance, since
It is evident that the formulae expressing a law of
nature which is not known to have any particular
relation to the coordinate-systems in question should
have tensor-in variance.
102
CHAPTER XIX
ELEMENTARY PROPERTIES
8. Test for Tensor Property. If an array character
possess the tensor property (covariant, contra variant,
or mixed) for transference from one given coordinate-
system to every other coordinate-system, then it
possesses it in general, namely for transference from
any system to any other system. For let t u t be the
given coordinate-system and let * p and q be any
other coordinate-systems. As an example consider the
mixed tensor || $/ ||. Then by hypothesis
(9-1).
O ^
Multiply equation (9 1) by ~ ^ and sum for p
cu a cq^j.
and cr. [Note that in future this type of operation
will be described as operating with 2S ~ -*. !
p * du a dq v
Then [cf. equations (5) and (5*1)]
CH xix] ELEMENTARY PROPERTIES 149
Hence substituting in equation (9) for S a ft (u)
This proves the required property.
9. Sum of Tensors. If || S \\ and || T\\ are two tensors
of the same order and type, then || S+ T\\ and || S- T\\
are tensors of that same order and type. The proof is
obvious.
Again, if every component of a tensor vanishes in one
coordinate-system, the same property holds in every
coordinate- system.
It therefore follows that if || S|| and || T\\ are tensors
of the same order and type and their corresponding
components are equal in one coordinate-system, they are
equal in every coordinate-system. This is the principle
of tensor-equations.
10. Product of Tensors. Let || S \\ be a tensor of the
rath order and || T\\ a tensor of the nth order. Form a
new array of the (m + n)th order whose components are
the products of any component of \\S\\ with any com
ponent of || T\\. This new array is a tensor with the
covariant and contravariant affixes of both tensors.
As an example, let || S^ \\ be a covariant tensor of the
1st order and || T v p \\ be a mixed tensor of the 2nd order.
We have then to prove that || S^Tf \\ is a mixed tensor
of the 3rd order for which p, and v are the covariant
affixes and p is the contravariant affix.
150 ELEMENTARY PROPERTIES [CH
For S = SS<%,
Hence S u (u} T v p (u)--
V a J3 y QU^ OU V CX y
This proves the proposition. An analogous proof holds
for any other types of tensors.
11. Representation of a Tensor as a Sum of
Products.
Case (i). If || T^ v \\ be any covariant tensor of the
2nd order, then four pairs of covariant tensors of the
1st order can be found, namely
||-4 M || and ||-4/||, ||J? M || and ||J?/||,
||C; || and ||(7;||, ||Zy and ||Z>/||,
such that
T^ApAJ + B^BS+CpCr + D^DJ ...(10).
For by sections 9 and 1 the left-hand side is a tensor
of the right order and type. Hence, again by section 9,
we have only to choose A, A , etc., so that in one co
ordinate-system the components of the composite system
are equal to the components of T.
Consider the coordinate-system l u. In thisjsystem let
A^T^, 0=1,2,3,4],
Then ^ = A^Al + B& + .. . , O = L 2,;3, 4].
Also let _# = T^, O = 1, 2, 3, 4],
xix] ELEMENTARY PROPERTIES 151
Then T^ = A^Aj + B^BJ + . . . , [> = 1 , 2, 3, 4].
Also treat T^ and T^ in the same way, so that
C^T^ [,.= 1,2,3,4]
and D;=ZU [ ].
Hence in the u system
T^= A f A. +&+.. . + ...,
and hence the equality holds in every system.
This proof will hold equally well for contra variant or
mixed terms of the 2nd order.
Case (ii). A tensor of any type and of the 3rd order
can be exhibited as a sum of products of four pairs of
tensors, one tensor in each pair being of the 2nd order
and one of the 1st order.
For example, consider the mixed tensor || T* v \\.
We can find four pairs of tensors,
Ap, and A p , B^ and B p , C^ and C p , D^ and D p ,
such that
TI=A^A>+B^B>+... + ......... (11).
For as before we have merely to obtain the equality
in one coordinate-system. Now take in coordinate-
system Uy
ATl, 0,^=1,2,3,4],
=Tl, O,v=l,2,3,4],
=Tl, O,v=l,2,3,4],
152 ELEMENTARY PROPERTIES [OH xix
#M,= Z, [>,!> = 1,2, 3, 4],
The theorem is now proved.
It is obvious that this mode of representation can be
proved successively for tensors of any order or type.
CHAPTER XX
THE PROCESS OF RESTRICTION
1 2. Definition of Restriction. Let S;;; p (u) be a tensor
of any order, with p as a covariant affix, and otherwise
of any type. Let T"- p (u) be a tensor of any order, with
p as a contravariant affix and otherwise of any type.
Then the array ^(u) T;-(u)
P
will be proved to be a tensor. It will be called a
restricted product of the two tensors \\S\\ and || T\\.
The order of the restricted product will obviously be two
less than the sum of the orders of || S\\ and || T\\.
In the proof we will take the two tensors S^ p (u) and
!<), but it will be obvious that the steps of the proof
are absolutely general.
We have S
T Se u v " ? Uf > vp
t u \ r _ - 2i T r -
l) dv^ du y 3u 8 P dv p du e
Hence
*S V M T x *
Z<O D \ U I J. f v \
p
Hence [cf. equation (5 1) of Chapter xvm]
This proves the required tensor property, and an analo
gous proof is obviously applicable to all analogous cases.
13. Multiple Restriction. The analogous process of
restriction can be applied for two or more pairs of con
trasted indices [i.e. one index covariant and the other
contravariant]. The multiply restricted product thus
obtained will still be a tensor. If there be n processes
of restrictive summation and m be the sum of the
154 THE PKOCESS OF KESTKICTION [OH
orders of the two tensors \\S\\ and \\T\\, the order of
the multiply restricted tensor will be m 2n.
To prove the tensor property, take as an example
Now in equation (ll l) above put
X=S/A = <JT,
and sum for cr. Then [cf. equation (5*1)]
JJ (H-2).
14. Invariant Products. If the two tensors subject
to restriction are of the same order n, and there be n-fold
restriction, the order of the restricted product is zero,
so that it is an invariant scalar quantity. For example,
and Z^TZ^ZS^TM (12 1),
ft a
and 22 S/ (v) Tf (v) = 22 S a ft (u) Tf (u)
(12-2).
15. The Tensor || /||. Let //be defined in reference
to a given coordinate-system u by
-J/=o,
~ wi ^y^ (13).
Consider the mixed tensor || 1* (v) \\ whose com
ponents in the system u are equal to those of the
array ||//||. Then
T vl \^ TpdU a dV y
= V g ^q dv >>
= ^/.".... a (13-1).
xx] THE PROCESS OF RESTRICTION 155
Hence // has the same relation to every coordinate-
system as that which it has to coordinate-system u.
16. Restriction of a Single Mixed Tensor. It
follows from the theory of restriction that
is a tensor, i.e.
is a tensor. Accordingly the mixed tensor of the 3rd order
II Tl (u) ||
has been restricted into a tensor of the 1st order. This
proof is obviously quite general. For example, if || Tj ||
be a mixed tensor of the 2nd order, then
is invariant.
17. Argument from Products [Restricted or Un
restricted] to the Tensor Property.
This argument is best shown by a series of examples :
Case (i). If || T^ || be an array of components, of the
1st order, defined for every coordinate-system, and if,
whatever Ist-order contra variant tensor [| S* \\ may be,
we have the invariance of
S T () fift
2<J. p (tt) ,
then || T^ \\ is a covariant tensor. For by hypothesis
v T (v) (> ? (u)a
and fife
Hence S T - 2 T . p l S ( u} = 0.
P OM a J
156 THE PROCESS OF RESTRICTION [OH
But the tensor || <S" || is arbitrary. Now make four suc
cessive choices
8^=1, S W 2 = 0, S w = 0, ,,< = 0,
and S M l = 0, S M =1, S M S = 0, S M =0,
and S M > = 0, S M 2 =0, S w *=l, S^=0,
and ^., = 0, ,,= 0, S M = 0, S w =l,
and substitute successively in the above equation. We
obtain
np () _ 5* np (v) v^p
> 8.
Hence operating with S*, we obtain
Thus[cf. section 8] the required tensor property is proved.
Secondly, it is evident that if the arbitrary tensor
|| Sp || had been co variant, then || T* || would have been
contravariant.
Case (ii). If || T^ \\ be an array of components, of the
2nd order, defined for every coordinate-system, and
if, whatever Ist-order contravariant tensor || S v \\ may
be, the restricted product
is a covariant tensor of the 1st order, then || T^ v || is a
co variant tensor of the 2nd order. For by hypothesis
anrl ^ 7* v) S p V TV T (u) ^ a l f *
sr^ M =|[|r /( .s ( .j^-.
Hence by substituting from the former into the latter
equation
xx] THE PROCESS OF RESTRICTION 157
Thus, as in Case (i), by suitable choices for || >S P ||J the
tensor property is proved.
An analogous theorem holds in which invariance and
contravariance are interchanged, or in which the array
|| T\\ is proved to be a mixed tensor.
Case (iii). If || T^ v \\ be an array of components, of
the 2nd order, defined for every coordinate-system, and
if, whatever 2nd-order contra variant tensor || M " || may
be, the restricted product
|| 2 Z^ |l
is a mixed tensor of the 2nd order, then the array
|| T^ || is a co variant tensor of the 2nd order.
For by hypothesis
/..v
Hence operating with 2^*, we deduce
p vv
Also by hypothesis
)U a nfiyV,,
^ - -^ Mw) ^
p |_a Y
Thus
Hence, as above, the tensor property follows. Analogous
theorems follow for suitable interchanges of the covariant
and invariant types.
158 THE PROCESS OF RESTRICTION [OH
Case (iv). If || T^ v \\ be an array of components, of any
order, denned for every coordinate-system, and if, what
ever Ist-order co variant tensor \\S^\\ may be, the product
||$ A jT MJ/ || is a co variant tensor of the 3rd order, then
|| T^ || is a co variant tensor of the 2nd order.
For by hypothesis
Also by hypothesis
Hence from both equations
[rr,(v] yy rwi(u) <^/3 ^y O (v) _ rv
/ "f7 >l *8^8
Hence, by suitable choices for the arbitrary tensor
|| S K (V) ||, the tensor property for || T ( ^ || is proved.
Analogous theorems can be proved for any suitable
interchanges of covariance and contra variance of type.
General Theorem. If the product, restricted (multiply
or singly) or unrestricted, of an array, of any order
and defined for every coordinate-system, with every
arbitrary tensor of any one definite type and order be
a tensor [of suitable type and order] , then the array is
a tensor [of suitable type and order].
It is evident that the types of proof given above for
the four special cases can be adapted for every case of
this general theorem.
1 8 . Differential Forms . Since
it follows that
\\dv J|, and || dv dv v ||, and || dv k dv^dv v ||, etc.,
xx] THE PROCESS OF RESTRICTION 159
are contravariant tensors of the 1st, 2nd, 3rd, etc.,
orders respectively. V
Hence if || F^ \\ be a co variant tensor
is invariant. We adopt the notation
dF^ZF^du^ ............... (14).
Then dF is a differential form of the 1st order.
Similarly if || S^ v || be a co variant tensor
is invariant. We adopt the notation
dS^HS^du^du, ............ (15).
M v
Then dS 2 is a differential form of the 2nd order.
CHAPTER XXI
TENSORS OF THE SECOND ORDER
IT is proposed in this chapter to bring together some
of the simpler notations and theorems relating to tensors
of the second order.
19. Symmetric Tensors. The covariant and contra-
variant tensors || S^ \\ and || T* v \\ are respectively called
symmetric if in every measure-system
and ,, ...... (16).
If a tensor is symmetric in one measure-system, it is
symmetric in every measure-system.
For in measure-system u let
S = S%, [/*,!, = 1,2, 3, 4].
Then
^
dv
1(V)
The theorem holds for contravariant tensors with suit
able interchanges of the covariant and contravariant
types.
We notice that in the case of the differential form
dS 2 in the preceding section there is no loss of
generality in considering the tensor ||$ MJ/ || to be
symmetrical.
CHXXI] TENSORS OF THE SECOND ORDER 161
20. Skew Tensors. The covariant and contravariant
tensors || S^ \\ and || T v || are respectively called skew*
if in every measure-system
8^ + 8^ = 0, and 7^+7^ = ...... (17).
If a tensor is skew in one measure-system, it is skew in
every measure-system.
For in measure-system u 1 let
o() . O(M) A
& + 0, M =
Then
(M) 4- .Sf (M)> l a
-+a__
An analogous proof holds for || T*"^.
It is evident that for skew tensors
5^ = 0, T^ = ............ (17-1).
21. The Determinants. If ||SJ| and ||^
respectively covariant and contravariant tensors of the
second order, the symbols S (u} and T (u} represent the
determinants formed by the components as elements,
so that
(18),
and y M = det. T\
It at once follows from the law of the multiplication of
determinants that
nd T T . p (>.... P.. P
T w =7 >- x |a (,,, ,,
A tensor is called special if its determinant vanishes.
It is evident from the above equation that if a tensor
w. R. u
162 TENSORS OF THE SECOND ORDER [CH
be special in one coordinate-system, it is special in every
coordinate-system.
Since we are considering a four-dimensional manifold,
a skew tensor is not necessarily special. But in a three-
dimensional manifold every skew tensor would be special.
If || M^ || be a mixed tensor, the symbol M will denote
the determinant formed by the components as elements,
so that M=det. M;(x)\ ............ (19 2).
It is unnecessary to denote the coordinate-system * x
in the symbol for the determinant since the value of the
determinant is the same in all coordinate-systems, that
is to say, the determinant is an invariant. For if M f be
the value in system *y and M in system e x,
x
d (x 19 x 99 x 3 , 05 4 ) 9 (y 1 , ?/ 2 , 2/3, y,)
= M ....................................... (19-3).
22. Associate Tensors. Let || S^ \\ and || T 1 " \\ be a
pair of tensors of the second order, one covariant and
the other contravariant, such that in the coordinate-
system U* ^ T
then the analogous property holds for every coordinate-
system.
For
ay /3
ay
xxi] TENSORS OF THE SECOND ORDER 163
A pair of tensors with this property are called Associate
Tensors.
If either of the two tensors be not special, it has one
and only one associate tensor which is also not special.
In the sequel, unless it is otherwise expressly stated,
in dealing with associate tensors we shall always assume
that we are considering non-special tensors.
Thus the associate of the tensor associated with a
given tensor is the original given tensor. The associate
of any tensor || S^ \\ will be denoted by || >S M " ||, and
conversely. Also with the above notation,
of fi^ in S (U] ]--S M (20 1),
and S<2 = [cofactor of Sft in S ( ^+S M (20 2),
and S (u) S (u] =l.
Associate tensors enable us to solve tensor equations
of the form
p ; , Zr = jC r ,[,,= !, 2 , 3,4] ...... (21).
For operating with S $JJ* , we find
I 2 /;*::: = 2 ssic,
i-e- X:: = is^D^.
Analogously we can solve
ZStiX;:; f = D::; [>=i, 2, 3, 4]...(2i-i).
The theory of associate tensors applies also to mixed
tensors. For, exactly as above, if || /S || and || T* \\ be a
pair of mixed tensors of the second order, such that in
one coordinate-system
25;7;- = /; .................. (22),
ft
then the property holds for every coordinate-system.
Also all the analogous theorems hold.
11 2
164 TENSORS OF THE SECOND ORDER [CH
The associate of a non-special mixed tensor || Sf \\
will be written || Sf\\.
Thus Iia*ll = imi ............... (22-1).
Either both or neither of a pair of associate tensors
(invariant or covariant) are symmetrical, and either
both or neither are skew.
For if Hfiyi be symmetrical, the cofactor of S ( ^ in S (u]
is equal to the cofactor of S in S (u \ and analogously
for contravariance.
Also if || S^ || be skew, the cofactor of SJJ in S (u] is
equal to the negative of the cofactor of S ( ^ in S (u \ and
analogously for contravariance.
23. Derivative Tensors. By means of a pair of
tensors of the second order we can derive tensors of
various types which are called tensors derivative from
the given tensor.
For example, let || S^ \\ and || S* v \\ be one pair of as
sociate tensors and || T^ v || and || T* v || be another pair.
Then the derivative tensors
2 S 2*
f
and \\tST\\ea>.d\\lS flt T"
P P
are derivative tensors of mixed type. If one of the
tensors ||$ Ml/ || or || J^J be symmetrical these four tensors
coalesce into two tensors, and if both be symmetrical
they coalesce into one tensor.
Similarly || ^ T T v * S p(r \\ ............ (23 1)
is a contra variant derivative tensor. There are four
derivative tensors of this sort which coalesce into three
if || S^ || be symmetrical, and into one if || T^ v \\ be
symmetrical.
xxi] TENSORS OF THE SECOND ORDER 165
Again \\ttTT n S~\\ (23-2)
P *
is a covariant derivative tensor, with other analogous
tensors of the same sort.
Finally there are analogous sets of derivative tensors
in which || S^ \\ and || T^ v \\ have interchanged roles.
CHAPTER XXII
THE GALILEAN TENSORS
24. Galilean Tensors. We have already [cf.
Chapter iv of Part I] defined the symbols
,
Define the Galilean tensor [i.e. || 6r{w)>||] of the co
ordinate-system u by
Then in any other coordinate-system v,
(24 . 2) .
We will always assume that in any coordinate-system
f v the coordinate-type which is to play the part of
the exceptional axis for the Galilean tensor is to be
assigned the subscript 4.
With this convention, the condition that the co
ordinate-systems u and ( v* have the same Galilean
tensor is
\
]
dVp dv^
Operate on 2 ^ with 2 *
> v
OH xxn] THE GALILEAN TENSORS 167
Then, from the two equations above, we obtain
i.e. a*-^. = u*-aL [u, =1,2,3, 4]... (25-1).
av, a^
25. Galilean Differential Forms. The differential
form arising from this Galilean tensor is
It must be remembered that this particular Galilean
differential form has the Galilean property for the group
of coordinate-systems, such as % which are connected
with the coordinate-system ( u by sets of equations of
the type of equations (25 1) above. Call such a group
of coordinate-systems a Galilean group. It is evident
that a Galilean group is defined by any one of the
coordinate-systems which belong to it, since each such
system belongs to one and only one such group.
26. The Linear Equations of Transformation.
Let a track in the manifold be defined by considering
(u lt u 2) u 3 ) as appropriate functions of u\ and with this
(dui du 2 du\
supposition let (u lt u,, u,) stand [or ^, ^, ^J .
We now seek the condition that
/
along a track between any given pair of event-particles
A and B of the manifold, may have a stationary value.
This is given by the adaptation to this case of equations
168 THE GALILEAN TENSORS m [OH
(7) of Part II Chapter v. Since the coordinate-system
t u t is a member of the relevant Galilean group, these
equations reduce to
/
c 2
i ck n. xy n, I 7) r.. i o o~l ^9 I 7\
J-.c. u/^ u/^ t^ 4 f Up , |ju/ i , zjj o j ^ / y,
where a^ and 6^ are constants.
If the coordinate-system v be another member of
the same Galilean group, the same track, from A and B
and stationary, must be expressible in the form
0=1,2,3] (27-1),
where c p and d p are constants.
Hence the equations of transformation relating any
pair of coordinate-systems u and v belonging to the
same Galilean group must be of the linear form
M (iV-e M ) = 2k a w ft ............ (28),
where e^ and l^ [//,, a = 1, 2, 3, 4] are constants.
Furthermore, from equations (2 5 ) above [interchanging
u and v in their application],
......... <->
Also we can conceive equations (28) to be solved in
the form
.K-/J-2^^n ......... (28-2).
Hence .9%
^a7 ^ ^
^7?
and 0)^^ = 0)^1^.
Thus [cf. equations (25*1)]
^ *
xxn] ; THE GALILEAN TENSORS 169
27. Cartesian Group. Thus the Cartesian co
ordinate-systems of Chapter I v of Part I are a particular
Galilean group of coordinate-systems [such as the system
*# ] which have a peculiar spatio-temporal significance
in the four-dimensional continuum of nature. When
we are discussing the Galilean tensor of this group, we
symbolise it by || 6? MI/ || in place of the longer || 0{&}p,||.
We will call this Galilean group of coordinate-systems
the Cartesian group, and the corresponding Galilean
tensor is (in case of doubt) called the Cartesian Galilean
tensor. Furthermore, in discussing Galilean tensors we
will habitually consider in illustration the Cartesian
group and its Galilean tensor. But the theorems are
quite general and hold for any Galilean group.
28. Associate Galilean Tensors and Galilean
Derivative Tensors. Let the coordinate-system x
belong to the Cartesian group. Then the associate of
the Galilean tensor is 6r M ", where
I
(29).
[>=1,2,3,4]
By means of the Galilean tensor and its associate tensor
derivative tensors are found from any given pair of
associate tensors, || M J and US 1 *"!!, which are called the
6 Galilean derivatives from || S^ \\ or || S" v \\.
Thus the Galilean derivatives of mixed type are
|| 2 tf^ || and \\ZG^\\ (30).
The components in the coordinate-system x of the
former tensor \ji the covariant affix] are
S !2. b*, v = i, 2, 3, 4],
170 THE GALILEAN TENSORS [CH
i.e. - 5J3, = 1, 2, 3, 4 ; v = 1, 2, 3]]
and |SJ2, 0=1,2.3,4]
(30-1).
The components in the coordinate-system * x of the
latter tensor \JL the covariant affix] are
&%Sy [>,"=1,2, 3, 4],
Le - -Sg;, [>=1,2,3,4; ft=l,2,3]
and c*S, [v=l,2, 3, 4]
The Galilean derivative of contravariant type is
/~Y i/cr O 1 1 / Q 1 \
ur Op,|| (^olj.
[ (30-2).
The components in the coordinate-system x of this
tensor are
GKGS%> o,"=i,2,3,4],
i.e. S2, O." =1 .2.3] ......... (31-1)
and -\S%, O =1 > 2 3 : i- = 4]...
C
and -\S [ :l, O = 4; "=l,2,3]
C
and 1S, O = 4; - = 4]
The components of this contravariant Galilean derivative
are linear functions of the components of || S^ ||.
The Galilean derivative of covariant type is
^5" || ............... (32).
The components in the coordinate-system l x j of. this
tensor are
, 0^=1.2,3,4],
xxn] THE GALILEAN TENSORS 171
i.e. Sft, |>,* = 1,2, 3] ............ (32-1)
and -c 2 S$, [> = 1,2,3; i/=4] ...... (322)
and -c Sg, [> = 4 ; i/=l,2,3] ...... (32 3)
and c 4 S*>, [> = 4; i/ = 4] ......... (32 4).
The components of this covariant Galilean derivative
are linear functions of the components of H^H.
Finally the Galilean invariants are
SSG^Sp, and SSG^S" ......... (33).
p <r p <r
Thus in any coordinate-system x of the Cartesian
group there are the two group invariants
......... (33-1)
and S+S+S-c"S ......... (332).
29. Galilean Derivative Tensors of the First Order.
Let \\Fn\\ be a covariant tensor of the first order, then
its Galilean derivative is the contravariant tensor
l|2(?-^|| .................. (34).
The components in the Cartesian coordinate-system x
of this tensor are
GSF?, = 1,2,3,4] ...... (34-i),
i.e. -F?, O=l,2,3] ............ (34-2)
and -,F ( :\ = 4] .................. (34 3).
The Galilean invariant is
?SG "F f F v ............... (34-4).
P <r
Thus in any coordinate-system x of the Cartesian
group there is the group invariant
- -(34-5).
c
172 THE GALILEAN TENSORS [OH xxn
Again let || F* || be a contra variant tensor of the
first order, then its Galilean derivative is the co-
variant tensor
IISG^II (35).
The components in the Cartesian coordinate-system
# of this tensor are
<?/%,[> =1,2, 3,4] (35-1),
i-e. -/%, [>=1, 2,3] (35-2)
and c 2 ^,, [> = 4] (35-3).
The Galilean invariant is
ttG^F F* (36).
p a-
Thus in any coordinate-system e x of the Cartesian
group there is the group invariant
TO + TO + TO -c-TO ...(36-1).
CHAPTER XXIII
THE DIFFERENTIATION OF TENSOR COMPONENTS
30. The Christoffel Three-Index Symbols. Let
|| H^ v || be any symmetric covariant tensor. The Chris-
toffel Three-Index Symbol of the first kind is defined by
[X, /t,v=l,2, 8, 4]
The Christoffel Three-Index Symbol of the second
kind is defined by
Hfav, \} ( *> = 2HH[jtr, />]"" ...... (37-1).
Then
2 H Hv cr"" = 2S H H
] w ......... (37-2).
It is evident that
T ] ......... (38),
and J7{fi* 9 X} M J7{pm X} M ...... (38-1).
In general neither H\_p,v, X] (u) nor H{^v y \} (u] is a
tensor, though we shall prove that they are group
tensors for any Galilean group.
31. Differentiation of Determinants of Tensors.
Let the covariant tensor || S^ \\ be infinitesimally in
creased to || Sp V + 8S^ V ||, and in consequence let S (u} be
come 8S (U \ Then
(U) = SS {cofactor of S$} x 8^?
............... (39).
mi tfO OWN? * eHF****? /3Q 1\
Ihus - = o w SSoJ5-r- (oy ij.
8^ A ** " GU K
174 DIFFERENTIATION OF [CH
Analogously, if the contravariant tensor || T^ } || be
infinitesimally increased to || T^ + 8T^ } \\, and in con
sequence T (u) increases to T (U} + &T (U) , then
ST M =T M ^^xBTs: } ......... (40),
d T 3 T* v
U u
Now recur to the symmetric covariant tensor
Then
p a-
_u
9u f "
Now the interchange of the symbols p and cr does
not affect the value of
Hence [cf. equation (39*1)]
32. The Standard Formulae. There are certain
standard formulae which are the foundation of the
theory of the differentiation of tensor components.
We consider the symmetric covariant tensor || -H^||.
Now H-VHy>%&.
3 >du a d
and = 2--5 .
xxin] TENSOR COMPONENTS 175
Then, remembering that || II ^ \\ is symmetric,
y -\ d * U * 9 M
,, "*" 9^9*;,, 9^ J
xfy 81;,
Hence interchanging X, p, v cyclically, and in the
former of the two summations also interchanging a, /3, y
cyclically, but in the latter retaining /3 and y in their
original functions, we find
/ <yi/ a v <.
C ^\2
()
and
y
Hence by combining these three equations
This formula relates the three-index symbols of the first
type, as expressed in different coordinate-systems.
Operate on this formula with
We consider separately the effect of this operation on
each of the three terms of the above formula,
176 DIFFERENTIATION OF [ CH
And
, a]"" ^ ^ 22 HJJ ^ |L.
J (v)
,3 y
And
p vv\
Thus, transposing terms, the formula becomes
^|i- = 2^jy{^,pf ) -22^{^ r ,f ) ^^...(43).
dv ov v p cv P y dv dv
This is the standard formula for - ~ .
Finally operate with
and transpose terms. We obtain (putting a for e)
(44).
This is the standard formula relating the three-index
symbols of the second type, as expressed in different
coordinate- systems.
Now let || K^ v || be another symmetric covariant tensor
of the second order. Then we at once prove that
\\H{^,\r-K{^ v ,\r\\ (45)
is a mixed tensor of the third order, for which X is the
sole contravariant affix.
xxm] TENSOR COMPONENTS 177
For, it at once follows from the formula above that
(45-1).
This proves the theorem.
33. Covariant Tensors of the First Order. Let
|| T^ || be a covariant tensor of the first order, and let
|| H^ v || be any symmetric tensor of the second order.
Then
and
Hence
dv
Now use formula (43), and remember that
We deduce
ft y
Thus
IN
(46)
is a covariant tensor of the second order.
Interchanging /x and v, and subtracting, we find that
dv,, dv.
,(47)
is a covariant tensor of the second order.
w. R.
12
178 DIFFERENTIATION OF [CH
When || Tp \\ is a given co variant tensor of the first
order, we shall use T^ v to mean
dv
Thus, with this meaning, || T ( ^ v || is a skew co variant
tensor of the second order. Also identically
34. Contravariant Tensors of the First Order. Let
|| S* || be any contra variant tensor of the first order.
Then
, 3 . du B d
and = 2) ^ ^
8^ P 8^8^
Thus
Hence [cf. equation (43)]
/3 e
Now in the second term on the right-hand side inter
change a and e, and in the third term note that
and
xxm] TENSOR COMPONENTS
Hence, rearranging terms,
179
Thus
..(48)
is a mixed tensor of the second order, ft being the
covariant affix. It will be noticed that the differenti
ation adds the covariant affix to the original tensor.
Since we have a mixed tensor we can apply the
process of restriction, identifying a and ft. Hence
[cf. equation (41) and changing e to a]
is invariant.
35. An Example. Let A be any scalar function of
dA
the position of an event-particle. Then
du a
is a co
variant tensor. Hence, using the Galilean tensor for
the Cartesian group of coordinate-systems,
is a contravariant tensor. Hence [cf. formula (48*1) of
section 34]
is invariant.
In any Cartesian coordinate- system
reduces to /a* 4 &A d*A\ 1
(49-1)
this invariant
..(49-2).
122
180
DIFFEKENTIATION OF
[CH
Thus we have transformed this fundamental ex
pression to any coordinates.
HI
dv..\
Again substituting the covariant tensor
T (V J\\ in formula (46) of section 33, we deduce that
(50)
for
(topdv, P dv p
is a covariant tensor of the second order.
36. Tensors of the Second Order. Any tensor of
the second order can be expressed as a sum of products
of pairs of tensors of the first order [cf. section 11].
Thus if || S^ || be a covariant tensor of the second
order, we can write
H3J-P4AII.
where || A^ \\ and || B v ||, etc., are covariant tensors of
the first order.
Thus
f() rdd (u) aR (M) ~i
v = s M B (u) | A (u) v
^ [_ du^ " du^ J
[lr\ A M
(^-f*
Hence [cf. formula (46)], since the sums of products
of tensors are tensors,
are
du.
-S
...(51)
is a covariant tensor of the third order, since it is equal
to such a tensor.
xxm] TENSOR COMPONENTS 181
Let || T^ || be a contravariant tensor of the second
order. We can write
|| 2^11= || 2 ^5 ||,
where || A" \\ and H^H, etc., are contravariant tensors
of the first order.
Thus
Hence [cf. formula (48)], since the sums of products
of tensors are tensors,
du^ p
is a mixed tensor of the third order, X being the sole
covariant affix.
Identifying X and v and summing, we obtain by this
process of restriction the tensor
(52-1),
which is a contravariant tensor of the first order.
Mixed tensors of the second order can be dealt with
by exactly the same method as that applied to covariant
and contravariant tensors in this article, and by the use
of the same formulae (46) and (48). If || L ^u) \\ be a
182 DIFFEKENTIATION OF [CH
mixed tensor of the second order we deduce the tensor
which is a mixed tensor of the third order in which v
is the sole contra variant affix.
Identifying X and v and summing, we find by this
process of restriction the tensor
( M )_L log {_#<">}* ...(531),
which is a contra variant tensor of the first order.
37. Tensors of the Third Order. These are dealt
with by the same method as those of the second order,
by the use of the formulae obtained in sections 33, 34
and 36. The only such tensor which we need explicitly
consider is a mixed tensor of the third order with only
one contra variant affix. Let || K v \\ be such a tensor.
We can write this tensor in the form
II je* II -US ^5* II,
where || A^ v \\ is a covariant tensor of the second order
and || B K || is a contra variant tensor of the first order,
and so on for the other pairs of tensors.
Hence by the use of formulae (51) and (48) we deduce
that
> />n] |
(54)
is a mixed tensor of the fourth order in which X is the
sole contra variant affix.
XXIII]
TENSOR COMPONENTS
183
Identifying X and TT and summing, we obtain by tbis
process of restriction the tensor
H
(55).
which is a covariant tensor of the second order.
CHAPTER XXIV
SOME IMPORTANT TENSORS
38. The Riemann-Christoffel Tensor. Consider the
Tensor Differentiation of the co variant second- order
tensor
which is obtained as formula (46) in section 33. We
use formula (51) of section 36, substituting the given
tensor for || S^ ||. We deduce, after arranging the terms,
the covariant tensor
du t
...(56).
Now interchange X and v in this tensor and subtract
the latter from the former tensor. We obtain the tensor
Hence by section 17,
-H{pX, W y*-/-H{fv, Tr} (u) + 2H{w, Trf >
...(57)
CH xxiv] SOME IMPORTANT TENSORS 185
is a mixed tensor of the fourth order, in which TT is the
only contravariant affix. This is the Riemanri-Christoffel
Tensor.
Now identify v and TT and sum. Then, as the result
of this process of restriction, we obtain the covariant
second-order tensor
du p duju,,
P
du ff
i
>} (n} H{pp, o-} (M) (58).
This is the restricted Riemann-Christoffel Tensor. It is
a symmetric covariant second-order tensor.
39. The Linear Gravitational Tensor. In formula
(45) of section 32, we have proved that, if || H^ v \\ and
|| J^ v || are any symmetric covariant second-order tensors,
||/{/AI>, X} (w) - H{JJLV, X} (w) || (59)
is a third-order mixed tensor in which X is the sole
contravariant index.
Hence operating with
we find that
\\Jfrr, ir]" -2 JSV, X} w || -..(59-1)
A.
is a covariant third-order tensor.
Again operating on this latter tensor with
^ T/p**
*r-"w*>
and interchanging the indices, viz. putting X for p, p for
TT, and cr for X, we find that
rfU (59-2)
is a third-order mixed tensor in which X is the sole
contravariant index.
186 SOME IMPORTANT TENSORS [OH
Now we replace || H^ v \\ by the Galilean tensor || G^ ||,
and obtain the tensor
lSQ%rb*,pf*- S&G%JG{p l , t *r\\ (59-3).
This tensor is linear in the components of ||/^||. In
any coordinate-system x of the Cartesian group, this
tensor reduces to
A
(59 4).
If in formula (55) of section 37, we replace || H^ v \\ by
|| 6r MV || and replace || K v || by the above tensor, we
obtain the tensor utilised in the law (ii) of gravitation
mentioned in Chapter iv of Part I. In the coordinate-
system x this law of gravitation becomes
?^a!/[> t X r ) = [/*."= 1.2, 3, 4] (60)
where there is no attracting matter.
It is evident that || H^ || is introduced in the above
reasoning disconnectedly on two distinct occasions,
namely in formula (59), and in the operation SJ?^*.
There is no logical necessity that || H^ \\ should be
the same on each of these occasions, still less that it
should be the Galilean tensor. Accordingly this is an
opportunity of framing other laws of gravitation in which
tensors characteristic of other fields of force are intro
duced instead of || G^ v \\ on one or more of these three
occasions. In this way, the influence (if any) of these
fields on the gravitational field may be represented.
40. Cyclic Reduction. The Cyclic Eeduction of the
third-order array || A^ v || is the array
\\A^ V + A^ K + A V ^\\ (61).
This reduced array will be symbolised by
11 (61-1).
xxiv] SOME IMPORTANT TENSORS 187
This definition and the symbolism will be applied to
arrays of any order. Thus
UCycL^JIHI^ + ^JI ...... (61-2),
and
|| Cycl. A^ || = || A^ + A^+A^ + A n ^ v \\ (6 1 3).
The cyclic reduction of a covariant, or contravariant,
tensor of any order is a tensor of the same order and
type as the given tensor. The permanence of the order
is obvious; we have only to prove the tensor property.
Consider a covariant tensor of the third order. Then
\u.v
* ft
Now permute (X, /x, v) and (a, /3, y) each cyclically.
Then T(V) _ T(U] dupdv^du,
^" X ~tf7 ^dv^dvydv,
and analogously for T^. Hence
) a p
This proves the required proposition, and an analo
gous proof evidently holds for contra variance, and for
tensors of any order.
Now replace the covariant tensor || T^ v \\ by
-
+
The preceding theorem on reduction, applied to this
case, tells us that
[r)S (v
S
(62)
is a covariant third-order tensor.
188
SOME IMPOKTANT TENSORS
[CH
Hence if ||flL|| be a skew tensor, then
+
.
r
ass;
dv
(62-1)
is a covariant third-order tensor.
41. Some Cartesian Group Tensors. We first note
that if || $ M || and || T* || be Cartesian group tensors (co
variant and contravariant) of the first order, then [cf.
section 29]
and
.(63)
are Cartesian group tensors (contravariant and co
variant) of the first order. Furthermore if || S^ v \\ and
|| T* v || be Cartesian group tensors (covariant and con
travariant) of the second order, then [cf. section 28]
" and I !,>, T*- 1 1 (63-1)
s a
are Cartesian group tensors (contravariant and covari
ant) of the second order. And so on for higher orders.
Also in the case of the two Cartesian tensors (co
variant and contravariant) of the first order
0).
and
(63-2)
are Cartesian invariants. And in the case of the two
Cartesian tensors of the second order
2 --^S^ and
O),
,(63 3)
are Cartesian invariants.
Let (x 19 x,,x SJ x) and (Pi,jp 2 ,_p 3 ,_p 4 ) be the coordinates
of any two event-particles referred to the same co
ordinate-system c x of the Cartesian group. Then
||^ pj| and || dx^ \\ and ||cpj! ...(64)
are contravariant Cartesian tensors of the first order.
xxiv] SOME IMPORTANT TENSORS 189
Thus
2a> M a (se M -p M ) 3 and Zu^dx,* and Sov d/y (64 1)
are Cartesian invariants. Put
r = JV(x f - P J> ............... (65).
Then ^(x f -p f ) 1 = i>-<f(x t -p^f ...(65 1).
Also put
*"-; * -!; ^=^2, 3, 4] (65-2),
*"=*, +*, +#, vj=p;-+p;+p; (05-3),
= -2
^
-4),
(65-5);
then from above
Qrdxi and n^dp 4 ............... (66)
are Cartesian invariants.
Hence differentiating the Cartesian tensor ||# M p^ \\
with respect to # 4 , we deduce that
Hn^JI and IK^VU ......... (67)
are Cartesian tensors (contra variant and co variant). Also
differentiating the Cartesian invariant r 2 c a (# 4 p$
with respect to p 4 , we deduce that
n m {c(x 4 - Pl )-? m } ............ (68)
is a cartesian invariant.
Again differentiating the Cartesian tensor || 1 M x^ ||
with respect to # 4 , we deduce that
and
(69)
are Cartesian tensors (contravariant and covariant).
190 SOME IMPORTANT TENSORS [OH xxiv
Again differentiating the Cartesian invariant
^mMaV-^O-fm}
with respect to x we find that
a^ji-is ^&J (70)
is a Cartesian invariant.
Also differentiating the same Cartesian invariant with
respect to p\ , we find that
i <V [c (x t - Pt ) - m ] S &jj, - 1V2 (a;, - jv) JV (7 1 )
is a Cartesian invariant.
Also from either of the tensors of formula (69) we
find that
IV [S V + ^ IV (1 + ^ 2 ) (f*,^) 2 ] (72)
is a Cartesian invariant.
CAMBRIDGE: PRINTED BY J. B. PEACE, M.A., AT THE UNIVERSITY PRESS
QC Whitehead, Alfred North
6 The principle of
W57 relativity
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