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Full text of "Vector analysis and the theory of relativity"

PU, 

M 

VECTOR ANALYSIS 

AND THE 

THEORY OF RELATIVITY 



BY 



FRANCIS D. MURNAGHAN, M.A. (N.U.I.), PH.D. 
Associate Professor of Applied Mathematics, Johns Hopkins University. 



546G58 



BALTIMORE 

THE JOHNS HOPKINS PRESS 
1922 



JL\\t ^jnuitay ~i tinting Company 

lETTEIPIESS AND OFFSET 



One of the most striking effects of the publication of Ein- 
stein's papers on generalized relativity and of the discussions 
which arose in connection with the subsequent astronomical 
observations was to make students of physics renew their study 
of mathematics. At first they attempted to learn simply the 
technique, but soon there was a demand to understand more; 
real mathematical insight was sought. Unfortunately there 
were no books available, not even papers. 

Dr. Murnaghan's little book is a most successful attempt to 
supply what is a definite need. Every physicist can read it with 
profit. He will learn the meaning of a vector for the first time. 
He will learn methods which are available for every field of 
mathematical physics. He will see which of the processes used 
by Einstein and others are strictly mathematical and which are 
physical. Every chapter is illuminating, and the treatment of 
the subject is that of a student of mathematics and is not de- 
veloped ad hoc. The extension of surface and line integrals is 
most interesting for physicists and the discussion of the space 
relations in a four-dimensional geometry is one most needed. 
This is specially true concerning the case of point-symmetry 
which forms the basis of Einstein's formulae for gravitation as 
applied to the solar system. 

I feel personally that I owe to this book a great debt. I have 
read it with care and shall read it again. It has given me a 
definiteness of understanding which I never had before, and a 
vision of a field of knowledge which before was remote. 

JOSEPH S. AMES. 
JOHNS HOPKINS UNIVERSITY, 
June 1, 1921. 



in 



CONTENTS 

PAGE 

Introduction 1 

CHAPTER ONE 
THE TENSOR CONCEPT 

Spreads in Space of n Dimensions 4 

Integral over a Spread of One Dimension 6 

Integral over a Spread of Two or More Dimensions 7 

Transformation of Coordinates 11 

Covariant Tensors of Arbitrary Rank 15 

Contravariant Tensors 16 

Mixed Tensors 17 

Invariants 18 

CHAPTER TWO 

THE ALGEBRA OF TENSORS 

The Rule of Linear Combination 21 

The Rule of Interchange of Order of Components 22 

The Simple Tensor Product 24 

The Outer Product of Two Tensors of Rank One 25 

The Rule of Composition or Inner Multiplication 25 

Converse of the Rule of Composition 27 

Applications of the Four Rules 29 

Stokes' Generalised Lemma 33 

Examples 34 

The Curl of a Covariant Tensor of Rank One 34 

Integral of an "Exact Differential" 35 

Maxwell's Electromagnetic Potential 37 

Lorentz's Retarded Potential 38 

v 



VI CONTENTS 

PAGE 

CHAPTER THREE 

THE METRICAL CONCEPT 

The Metrical Idea in Geometry 40 

The Reciprocal Quadratic Differential Form 41 

The Transformation of the Determinant of the Form 43 

The Invariant of Space-Content 46 

The Divergence of a Contra variant Tensor of Rank One ... 47 

The Magnitude of a Covariant Tensor of Rank One 48 

The First and Second Differential Parameters 48 

General Orthogonal Coordinates 49 

The Special or Restricted Vector Analysis .' 50 

Four- Vectors and Six- Vectors 51 

Reciprocal Relationship between Alternating Tensors 52 

Reciprocal Six- Vectors 53 

CHAPTER FOUR 

THE RESOLUTION or TENSORS 

The Unit Direction Tensor 54 

Angle between Two Curves 55 

Coordinate Lines 56 

Orthogonal Coordinates 57 

Resolution of a Covariant Tensor of Rank One 57 

Coordinate Spreads of n 1 Dimensions 58 

The Normal Direction-Tensor to a Spread F_i 59 

The Resolution of a Contra variant Tensor of Rank One ... 63 

Application to General Orthogonal Coordinates 64 

Oblique Cartesian Coordinates 65 

Genesis of the Term "Tensor" 66 

General Statement of Green's Fundamental Lemma 67 

Normal and Directional Derivatives 68 

The Direction of a Covariant Tensor of Rank One 68 

The Invariant Element of Content of a Spread V^-i 70 

The Mixed Differential Parameter . . 70 



CONTENTS Vll 

PAGE 

Uniqueness Theorems in Mathematical Physics ........... 71 

Application to Maxwell's Equations ...................... 72 

The Electromagnetic Covariant Tensor-Potential ......... 73 

The Current Contravariant Tensor ...................... 74 

Maxwell's Equation in General Curvilinear Coordinates. ... 76 

The Constitutive Relation B = pH ...................... 77 



CHAPTER FIVE 

INTEGRAL INVARIANTS AND MOVING CIRCUITS 

Definition of an Integral Invariant ...................... 79 

Relative Integral Invariants ............................ 80 

General Criterion of Invariance ......................... 81 

Faraday's Law for a Moving Circuit .................... 82 

The Mechanical-Force Covariant Tensor ................. 85 

CHAPTER SIX 

THE ABSOLUTE DIFFERENTIAL CALCULUS 

The Calculus of Variations ............................. 86 

Geodesies of a Metrical Space ........................... 88 

The Christoffel Three-Index Symbols .................... 89 

Covariant Differentiation .............................. 90 

Applications .......................................... 94 

The Riemann Four-Index Symbols ...................... 95 

Einstein's Covariant Gravitational Tensor ............... 95 

Gaussian Curvature ................................... 95 

Definition of Euclidean Space ........................... 99 

Riemann's Definition of Curvature ...................... 100 

The Differential Character of the Definitions ............. 101 

CHAPTER SEVEN 

PROBLEMS IN RELATIVITY 

The Einstein Concept of a Physical Space ................ 102 

The Single Gravitating Center (Statical) ................. 103 



Viu CONTENTS 

PAGE 

Hypotheses of Symmetry 105 

The Einstein-Schwarzschild Metrical Form 109 

Einstein's Law of Inertia 110 

Modification of the Newtonian Law of Gravitation 113 

The Motion of Mercury's Perihelion 118 

The Law of Light-Propagation 118 

Minimal Geodesies 119 

The Fermat-Huyghens' Principle of Least Time 121 

Deviation of a Ray of Light which Grazes the Sun 125 



PREFACE. 

This monograph is the outcome of a short course of lectures 
delivered, during the summer of 1920, to members of the graduate 
department of mathematics of The Johns Hopkins University. 
Considerations of space have made it somewhat condensed in 
form, but it is hoped that the mode of presentation is sufficiently 
novel to avoid some of the difficulties of the subject. It is our 
opinion that it is to the physicist, rather than to the mathe- 
matician, that we must look for the conquest of the secrets of 
nature and so it is to the physicist that this little book is 
addressed. The progress in both subjects during the last half 
century has been so remarkable that we cannot hope for investi- 
gators like Kelvin and Helmholtz who are equally masters of 
either. But this makes it, all the more, the pleasure and duty 
of the mathematician to adapt his powerful methods to the 
needs of the physicist and especially to explain these methods 
in a manner intelligible to any one well grounded in Algebra 
and Calculus. 

The rapid increase in the number of text books in mathematics 
has created a problem of selection. We have tried to confine 
our references to a few good treatises which should be accessible 
to every student of mathematics. 

Ch. V should be omitted on a first reading. In fact it is 
quite independent of the rest of the book and will be of interest 
mainly to students of Hydrodynamics and Theoretical Elec- 
tricity. There are several paragraphs in Ch. IV which may be 
passed over by those interested mainly in the application of the 
theory to the problems of relativity. For these we may be 
permitted to suggest, before taking up the subject matter of 
Chap. VII, a reference to an essay "The Quest of the Absolute" 

IX 



X PREFACE 

which appeared in the Scientific American Monthly, March 
(1921), and was reprinted in the book "Relativity and Gravita- 
tion," * Munn & Co. (1921). It may be useful to add the well- 
known advice of the French physicist, Arago "When in 
difficulty, read on." 

The manuscript of the book was sent to the printer in June, 
1921, and its delay in publication has been due to difficulties in 
the printing business. In the meantime several important papers 
bearing on the Theory of Relativity have appeared; it will be 
sufficient to refer the reader to some significant notes by PainlevS 
in the Comptes Rendus of this year (1922). We are under a 
debt of gratitude to Dr. J. S. Ames for valuable advice and en- 
livening interest. And, in conclusion, we must thank the officials 
of The Johns Hopkins Press for their painstaking care in this 

rather difficult piece of printing. 

F. D. M. 
OMAGH, IRELAND. 
June, 1922. 

* Edited by J. Malcolm Bird. 



VECTOR ANALYSIS AND THE THEORY OF 
RELATIVITY 



INTRODUCTION 

Vector Analysis owes its origin to the German mathematicians 
Mobius* and Grassmann f and their contemporary Sir William 
Hamilton.! Since its introduction it has had a rather checkered 
career and it is only within comparatively recent times that it 
has become an integral part of any course in Theoretical Physics. 
It is well known that the subject was regarded with disfavor 
by many able physicists, among whom Sir William Thomson, 
afterwards Lord Kelvin, was probably the most prominent. 
The reason for this is, in our opinion, not hard to seek. Grass- 
mann, who undoubtedly had a much clearer conception of the 
generality and power of his methods than most of his followers, 
expounded the subject in a very abstract manner in order not 
to lose this generality. Naturally enough his writings attracted 
little attention and when, some forty years later, Heaviside and 
others were earnestly trying to popularize the method they 
swung to the other extreme and, in attempting to give an 
intuitive definition of what a vector is, failed to convey a clear 
and comprehensive idea. Roughly speaking their definition was 

* Mobius, A. F., Der barycentrische Calcul (1827). Werke, Bd. 1, Leipzig 
(1885). 

f Grassmann, H., Ausdehnungslehre (1844). Werke,- Bd. 1, Leipzig (1894). 
Grassmann was particularly interested in the operations he could perform 
upon his " vectors " and not in the transformations of the components of 
these which occur when a change of " basis " or coordinate system is made. 
In this respect the point of view of his work will be found very different from 
that adopted here. 

J Hamilton, W., Elements of Quaternions. Dublin Univ. Press (1899). 

Heaviside, 0., Electromagnetic Theory, Vol. 1, Ch. 3. London (1893). 

1 



2 VECTOR ANALYSIS AND RELATIVITY 

that " a vector is a quantity which, in addition to the quality 
of having magnitude, has that of direction." The fault with 
this definition is, of course, that it fails to explain just what is 
meant by " having direction." That this idea requires ex- 
planation is clear when we realize that the simple operation of 
rotating a body around a definite line through a definite angle 
which, a priori, " has direction " in the same sense that an 
angular velocity has is not a vector whilst an angular velocity 
is. Then, again, endless trouble arises when vectors are intro- 
duced in a manner making it difficult to see their " direction " 
and even today some of the better text-books on the subject 
speak of " symbolic vectors " such as gradient, curl, etc., as 
if they are in any way different from other vectors. In 1901 
Ricci and Levi-Civita* published an account of their investiga- 
tions of " The Absolute Differential Calculus " a kind of dif- 
ferentiation of vectors. This paper was written in a very con- 
densed form and did not at once attract the notice of students 
of Theoretical Physics. It was only in 1916 when Einsteinf 
called attention to the usefulness of the results in that paper 
that it received adequate recognition. However it seems to be 
the common opinion that the methods there dealt with (and 
often referred to as the " mathematics of relativity ") are 
extremely difficult. It is the purpose of this account to lessen 
this difficulty by treating several points in a more elementary 
and natural manner. For example, in an interesting introduc- 
tion to their paper, Ricci and Levi-Civita point out, as an instance 
of the power of their methods, that they can obtain easily, 
by means of their absolute differentiation, the transformation 
of Laplace's differential operator A 2 which in Cartesian co- 
ordinates takes the form 

A ^l+^l + il 
~ dx*'*' dy*^ dz* 

*Ricd, 0., and Levi-Civita, T. M&hode* de Cdcul diff4rentiel absolu. 
Math. Annalen, Bd. 54, p. 125 (1901). 

] Einstein, A., Die Grundlage der allgemeinen Relativitdtstheorie. Annalen 
der Physik, Bd. 49, p. 169 (1916). 



VECTOR ANALYSIS AND RELATIVITY 3 

into any curvilinear coordinates whatsoever. This trans- 
formation was first obtained by Jacobi,* and, while expressing 
admiration for the ingenuity of his method, they justly remark 
that it is not perfectly satisfactory for the reason that it brings 
in ideas those of the Calculus of Variations foreign to the 
nature of the problem. Now by a method due to Beltramif it 
happens that this very transformation can be obtained by 
Vector Analysis without any knowledge of absolute differentia- 
tion; the apparently fortuitous and happy disappearance from 
the final result of the troublesome three index symbols of that 
part of the subject is thus explained. In addition we hope to 
make it clear that the methods of the " Mathematics of Rela- 
tivity " are applicable to, and necessary for, Theoretical Physics 
in general and will abide even if the Theory of Relativity has to 
take its place with the rejected physical theories of the past. 

* Jacobi, C. G., Werke, Bd. 2, p. 191. Berlin (1882). 
t Beltrami, Ricerche di analisi applicata alia geometria. Giornale di mate- 
matiche (1864), p. 365. 



CHAPTER I 

1. Every student of physics knows the important role played 
by line, surface and volume integrals in that subject. For 
example, the scalar magnitude work is the line integral of the 
rector magnitude force and this will suggest a simple mode of 
defining a vector. As, however, we shall wish to apply our 
results in part to gravitational spaces it is desirable at the 
outset to state as clearly as possible what we mean by the various 
terms employed. 

Space. By this term is meant a continuous* arrangement or 
set of points; a point being merely a group of n ordered real 
numbers. In our applications n is either 1, 2, 3, or 4 and the 
space is said to be of one, two, three, or four dimensions respec- 
tively. The ordered group of numbers we denote by z (1) , z <2) , 
, z (n) , and call the coordinates of the point they define. 
Nothing need be said for the present as to what the coordinates 
actually signify. A space defined in this way is a very abstract 
mathematical idea and to distinguish it from a more concrete 
idea of space in which, in addition to the above, we have a funda- 
mental concept called length, we may, where necessary, call the 
latter a metrical space and the former a non-metrical space. 
We use the symbol S n to indicate our space, metrical or not, 
of n dimensions. 

SPREADS IN S 

It is possible to choose from the points of S n an arrangement 
or set of points such that any one point is determined by the 
value of a single variable.- Thus if, instead of being perfectly 
independent, the n coordinates x (1 \ , a; (n) are all functions 

Continuity is assumed as an aid to mathematical treatment. In certain 
modern theories preference is given to a discontinuous or discrete set of points. 

4 



THE TENSOR CONCEPT 



of a single independent variable, or parameter, u\ 



(*= 1,2, ...,n) 

the point x is said to trace a curve or spread of one dimension 
as MI varies continuously from the value MI to MI (I) . The points 
corresponding to the values MI = MI and MI = MI (I) are called 
the end points of the curve and if they coincide, i.e., if all cor- 
responding coordinates are equal the curve is said to be closed. 
A spread of two dimensions in S n is similarly defined by 

X = X \U\, Uz) \8 = 1, " * *, Tl) 

where MI and Uz are independent parameters. Here we have 
two degrees of freedom because we can vary the point x by 
varying either MI or M 2 . It is necessary, however, that the func- 
tions a;' (MI, Uz) should be distinct functions of the parameters 
MI, Uz; the criterion for this being that not all the Jacobian 
determinants 



a ( 



d (MI, 



du\ 



dx (t *> 



i \ 

sz = 1, , n) 



should vanish identically. If this were to happen, we would not 
have two degrees of freedom but only one and the points would 
lie on a curve and not on a proper spread of two dimensions. 

Similarly by a spread of p dimensions in S n (p ^ n) we 
mean the locus of points x with p degrees of freedom; 



where not all the Jacobian determinants 



= 1, , n 



s= 



6 VECTOR ANALYSIS AND RELATIVITY 

vanish identically. This we denote by F p (the corresponding 
French term being variete*) and we shall suppose all our V v 
to be " smooth "; by this we mean that all the partial deriva- 
tives 



\m= 

are continuous. This restriction is not really necessary but is 
made to avoid accessory difficulties. 

INTEGRAL OVER A SPREAD OF ONE DIMENSION FI* 

Consider an ordered set of n arbitrary continuous functions 
Xi, ", X n of the coordinates x (l \ , x (n) . (For brevity sake 
we shall hereafter use the phrase " functions of position. ") 
The numerical value assigned to the label r in the symbol X r 
tells which one of the components Xi, , X n , which are ordered 
or arranged in this sequence, we are discussing. Now for any 
curve FI given by 

*<> = 
form the differentials 



and then form the sum Xidx + X^dx ---- h X n dx (n) which 
is, by definition, identically the same as 



If in each of the functions X, of position we replace the co- 
ordinates x w , , x (n) by their values on the curve FI 



X t -z becomes a function of ui, F(UI) let us say, and we may 



* Reference should be made to the classical paper by H. Poincart, " Sur 
les r6sidus des int^grales doubles," Acta Math. (9), p. 321 (1887). 



THE TENSOR CONCEPT 7 

evaluate the definite integral J^ (l \F(ui)dui. This is called the 
integral of the ordered set of n functions of position (X\, , X n ) 
over the curve. If, now, we change the parameter u\ to some 
other parameter i by means of the equation u\ = Ui(vi) the 
points on the curve are given by a; ( * ) = a; (8) (wi) = X M (VI) say 
(* = 1, , n) and it is conceivable that the value of the integral 
might depend not only on the curve but on the parameter used 
in specifying the curve. However this is not the case since 



/" l(1) /iO> f 

F(i)rfia I 

J,o Jujo I =i 



and 

7 / n \ *l U 1)] I / n I * 1 f}lL] 1 



This independence, on the part of the integral, of the accidental 
parameter used in describing the curve allows us to speak of the 
integral as attached to the curve and the symbol J'^",=iX,,dx ( ^ 
is used since it contains no reference to the parameter u. 

In what follows we shall adopt the convention that when a 
literal label occurs twice in a term summation with respect to 
that label over the values 1, , n is implied. Thus our line 
integral may be conveniently written 

/i = SX*dx" 

Such a label has been called by Eddington a dummy label (or 
symbol) of summation. We prefer to adopt the term " umbral " 
used by Sylvester in a similar connection; the word signifying 
that the symbol has merely a shadow-like significance disappear- 
ing, as it does, when the implied summation is performed. 

2. INTEGRAL 7 2 OVER A SPREAD F 2 OF TWO DIMENSIONS 

Consider a set of n 2 ordered functions of position (to indicate 
which we use two labels si, Sz) 

X 8l , tt (ti t *2 = If *>) 



8 VECTOR ANALYSIS AND RELATIVITY 

The numerical values assigned to *i and * 2 tell which one of the 
set of n 2 functions we wish to discuss. It is convenient to think 
of the functions as arranged in a square or " checkerboard " 
with n rows and n columns; then Si may indicate the row and 
*2 the column. K 2 is specified by means of two parameters 
u\, uz through the equations x (l) = x (t) (u\, Wj). Substitute these 
expressions for the coordinates in the functions X^ and con- 
sider the definite double integral 

T r ( v dz (tl) dx<" > \ , , , u i i u i \ 

7 2 = j { A,... -- - J duidut (*i and $ 2 umbral labels) 
\ dui duz ) 

extended over the values of u\, ui which specify the points of F 2 . 
This integral will depend for its value not only on the spread F 2 
but on the parameters u\, iiz used to specify it unless the set 
X tl) , f is alternating, i.e., X tl , ,, = X H , ,, which implies the 
identical vanishing of the n functions ATi, \; X n , n and the 
arithmetical equality in pairs of the remaining n 2 n so that 
there are but n(n l)/2 distinct functions in the set. Grouping 
together the functions of each pair we have 



, 

0(Ui, Uz) 

where now the umbral symbols do not take independently all 
values from 1 to n but only those for which the numerical value 
of $1 is less than that of * 2 . If a change of parameters is made by 
means of the equations 

Ui = Ui(Vi, Vi) 



where u\ and uz are distinct functions of v\ and r 2 the coordinates 
are given by equations 

x (.) = x (.)( Ul> uj = f (.)(,, lf ) (, = 1, . . ., n ) 

and the value of 7 2 when the TI, r 2 are used as parameters is 



THE TENSOR CONCEPT 
which, by the rule for multiplying Jacobians, 



r [ v d(x*<> x (s *>) } d(ui, 

= f \ X, ltt ~b -- r ^ 

d(ui, uz) j dfa, 



and this by the formula for the change of variables in a double 
integral 

__ f 



d(ui, uz) 

Starting, then, with an alternating set of functions of position 
XM we can form an integral, (over any F 2 ), which depends in no 
way on the parameters chosen to specify it. To avoid all refer- 
ence to the accidental parameters we write 7 2 in the abbreviated 
form f{X tl . tl d(x (ai \ z ( ">)} (ft < $ 2 ). We adopt this in pref- 
erence to the customary notation ^[X^dx^dx^} (si < 5 2 ) 
since no product of differentials, such as will occur later when 
we use quadratic differential forms, is implied. 

In an exactly similar way an integral I p over a spread V p of 
p dimensions (p ^ n) is defined.* By an alternating set of 
functions X tli ,, ..., Sp of position we mean that a single inter- 
change of two of the labels merely changes the sign of the func- 
tion. If, then, two of these labels are the same the function 
must be identically zero. Then 



is a definite multiple integral of order p extended over the values 
of MI, , Up which specify the points of V p . We write 

/*-.'/ 

where, in the summation with respect to the umbral symbols, 
ft, s z , , s p , si < s z < - < Sp. To emphasize the fact that 

* When p = n it is customary to use the phrase region of S n in preference 
to spread of n dimensions in S n . 



10 VECTOR ANALYSIS AND RELATIVITY 

IP does not depend in any way on the parameters u\, , u 
it will be written 



IP = fX tl , -.., . f d(* (tl \ 

Examples, n = 4 x 1 = x, x (2) = y, x (3) = z, x (4) = 
Zi = X, X z = Y, etc. 

<fy -f Zdz + r<&) 
, z) + Z 3 . id(z, x) + Xud(x, y) + X l4 d(x, t) 

+ X z <d(y, t) + W(z, 
utfL(x, y, z) + Xiud(x, y, + ^134^(2;, 2, 

+ Xtud(y, 2, 
i, z, 3. *d(x, y, z, t) 

Here in 7 2 we may write Xs\d(z, x) instead of X\, *d(x, z) since 
X 3 i = Xn and d(z, x) = d(x, z) 

As a concrete example of 7 2 we may take the case of a moving 
curve in ordinary Euclidean space of three dimensions, the curve 
being allowed to change in a continuous manner as it moves. 
Here x, y, z may be rectangular Cartesian coordinates and t 
may denote the Newtonian time. u\ is any parameter which 
serves to locate the points of the curve at any definite time 
t = to and Uz may well be taken = t. Then the equations of 
our Fa are 



and the parameter curves u^ = constant are the various positions 
of the moving curve, whilst the curves u\ = constant are the 
paths of definite points on the initial position of the moving curve. 
Denote dx/dt by x and we have 

d(s, >) - 

d(x, 3s - dmdi - .dujt 

i, dui 



THE TENSOR CONCEPT 11 

(It may not be superfluous to point out that it is essential to the 
argument that u\ and Uz should be independent variables. Thus 
in the present example u\ could not stand for the arc distance 
from an end point of the moving curve if the curve deforms as it 
mows although it could conveniently stand for the initial arc 
distance.) Our 7 2 may here be written 



dz 1 

-r- 0X34 +X&X X^y) \ duidt 
dui j 

showing it in the form of a time integral of a certain line integral 
taken over the moving curve. Before proceeding to define the 
idea of vector quantities it is necessary to make one remark of a 
physical nature. We have written expressions of the type 

(s an umbral symbol) 



and regarded the separate terms of these expressions Xidx (r >, - -, 
etc., as mere numbers. To actually perform the indicated sum- 
mations it is necessary, when we apply our methods to physics, 
that the separate terms in a summation should be of the same 
kind, i.e., have the same dimensions. Thus if the coordinates 
#(1) . . . X M are a li o f the same kind the coefficients 



occurring in the various integrals must all have the same di- 
mensions. 

3. TRANSFORMATION OF COORDINATES 

It has already been seen that if the various line integrals 
under discussion are to have values independent of the choice of 
parameters (MI, , u p ) care must be taken that the n p functions 
of position X^, .... 8p which form the coefficients of the I p should 



s 



12 VECTOR ANALYSIS AND RELATIVITY 

be alternating. Let us now see what happens to these coefficients 
when we change, for some reason, the coordinates x w , , x (n) 
used to specify the points of the V p . The formulae of transforma- 
tion are given by n equations 



the functions z (t) being supposed distinct so that the Jacobian of 
the transformation 



does not vanish identically. These equations may be regarded 
in two ways. First the y (t) may each denote the same idea as 
the corresponding a: (f) and then we have a correspondence set up 
between a point y and some, in general different, point x. 
Secondly the symbols y (t) may have a meaning quite distinct 
from the symbols x (l) and then we have a correspondence 
between one set of coordinates y (t) of a point and another set 
of coordinates x (t) of the same paint. It is the second way of 
looking at the matter that interests us and we speak then of a 
transformation of coordinates. (From the first point of view we 
would have a point correspondence.) Since the functions x (t) are 
distinct we can, in general, solve the equations* and obtain 



As an example take n = 3 and let z (1) , z (2) , x (3) be rectangular 
Cartesian coordinates and (y w , y (2) , y (3) ) space polar coordinates 
in ordinary Euclidean space of three dimensions. 



j.d)2 

sin 



(2)2 



v ' 
_ Z (2) 

y ~ ** u ff(i) 

Cf. Gawtat-Hedrick, Mathematical Analysis, Vol. 1, Ch. 2, or Wilson, 
E. B., Advanced Calculus. 



THE TENSOR CONCEPT 13 

In order to have a uniform transformation of coordinates so 
that to a given set of numbers y (l) , y\ y (3) there may correspond 
but one set z (1) , x (2) , z (3) and conversely it is frequently neces- 
sary to restrict the range of values of one or the other set. Thus in 
the example chosen we puty (l) > 0; < y (2) < TT; < y (3) <2ir. 
If now in 

/i = J*X a dx (t) (s an umbral symbol) 
we substitute 

(s = 1, .-,) 



dx (t) 
X s becomes X(y l , , y n ) say, and dx^ = du\ becomes 

OUi 

(Q x () Qy(r) \ 
rr - I dui (r an umbral symbol) 
oy (r > dui ) 

and so Ji becomes 

( dx (t) dy^ r) \ 
X g -TT-. -* ) dui (r, s both umbral symbols) 
dy (r) duij 



where Y is defined by the equation 

Y r s X, (r = 1, , n; s umbral) 

We shall from this on drop the bar notation above the X, which 
indicates that the substitution x w = x (9) (y (l \ , 2/ (n) ) has been 
carried out. It will always be clear when this is supposed done. 
For an 7 2 we have 



d(x^\ x ( '^ (si < * 2 ) (si, s 2 umbral) 

(d(x 91 x' 1 ) ] 
^i* -^r r \du\dui (s\ < * 2 ) by definition 
0(Ui, Uz) ) 

(dx ( ' l) dx ( ' 2) 1 
X*H T~ -\duiduz 
dui duz J 

since the functions X SlSt form an alternating set. 



14 VECTOR ANALYSIS AND RELATIVITY 

Now 

dxM dx^ dyM 

1ST ; ' efzra^ (ri umbral) 

so that 



dx (l *> __ dx (tl) dx (> *> dy (r *> dy (rt) 



dui ditz dy (ri) dy (r *> dui duz 

(ri and r 2 both umbral symbols) 
Hence if we define 



, , n 

(Sl and * 2 umbral) 
7 2 takes the form 

r fv dy^ay 
f \ YW -% 

I di dw 

Now 

dar (tl) dar^^ 
^. n = ^. ^ ^ ^5 (by definition) 

dx (lt} dx ( ' l) 

= X, t , ,, a~~?T) (^ a mere interchange of the letters 

dy dy l standing for the umbral symbols 
$1 and * 2 ) 

dx (tl) dx ( '*> (since X tl , , t is alternating by defi- 

* li> dyM dyfi) nition) 
= F r ,. r, (by definition) 

Accordingly the set of functions Y fl , r, of position, defined as 
above, is also alternating and we may write 



Generalizing we may write I p in the form 



(fit - - , TP umbral) and r\ < r 2 < < r p 

where the coefficients 7 r , ..... r p form an alternating set of n p 
functions of position defined by the equations 

f 

(...,*, umbral symbols) 



THE TENSOR CONCEPT 15 

Accordingly, then, if an integral over a curve, or more generally 
a spread of dimensions p, is to have a value independent of the 
coordinates the coefficients are completely determined in every 
system of coordinates once they are known in any particular 
system of coordinates. The coefficients in a line integral form 
as we shall see later a set of functions which " have direction " 
in Heaviside's sense and so might be called a vector. As, how- 
ever, the term vector is derived from a geometrical interpretation 
of the idea which loses to a great extent its significance when we 
apply our ideas to spaces of arbitrary metrical character the 
name has been changed and the coefficients of a line integral are 
said to form, taken as a group, a Tensor of the first rank of which 
the coefficients are the ordered components* To distinguish 
between this definition and another of similar character this 
Tensor is said to be covariant. More generally the coefficients 
of an Ip, n p in number, are said to form a covariant tensor of 
rank p of which the separate coefficients X 8l ..... 8p are the ordered 
components. Knowing the values of the components X Sl ..... if 
of a covariant tensor in any suitable system of coordinates x (t) 
the components in any other set y (s) are furnished by the equa- 
tions 



Although not of such physical importance it is convenient to 
extend the idea of Tensor to an arbitrary set of functions of 
position X Slt ..., 8p which follow the same law of correspondence, 
when a transformation of coordinates is made, as the alternating 
set above. If we do this it is merely the alternating covariant 
Tensors which arise as coefficients in integrals over geometric 
figures. The reason for the correspondence between the com- 

* The term Tensor was used by Gibbs in another sense in his lectures (see 
his Vector Analysis, Chap. V, edited by Wilson, E. B.) and also with the same 
meaning as that given here by Voigt, W., " Die fundamentalen Eigen- 
schaften der Krystalle," Leipzig (1898). Cf. Ch. IV, 4, infra. 



16 VECTOR ANALYSIS AND RELATIVITY 

ponents in different systems of a Tensor in the general non- 
alternating case would remain to be explained. 

4. INTRODUCTION OF CONTRAVARIANT TENSORS 
In the expression 

h = SX.dx M = f W> (* umbral) 

the quantities by which the components X, of the covariant 
tensor of rank one are multiplied have a law of correspondence 
defined by the equations 

as 



Similarly in the integral 

/, - y 



the factors X", Y rt which multiply the components X rt , Y rt 
respectively of the alternating covariant tensor of rank two 
have a law of correspondence given by the equations 

(bv definition) 

' duidv * (T " " umbral symbol9) 

(by definition) 

and so in general for an integral over a spread of p dimensions 
(p < n). These factors, regarded as a whole, are said to form 
a contravariant Tensor of the first, second, , pth rank as the 
case may be. The sets introduced in this way are not, as in the 
case of the covariant tensors, alternating. Even though the 
correspondence between the two sets of functions of position 



THE TENSOR CONCEPT 17 

X' 1 '* '" *' and y* 1 " may not arise in the above manner the 
set is said to form a contravariant tensor of rank p if the corre- 
spondence between the ordered components is defined by the 
equations 



(f ..., 



The labels which serve to oHer the components are written 
above in the case of contravariant and below in the case of co- 
variant Tensors. The following remark may be useful in aiding 
the beginner to remember easily the important equations defining 
the correspondence. The umbral symbols are always attached 

to the x coordinates on the right. When the labels are , [ 

on the left the y coordinates are f ow \ on the right. 
Thus 



(si, Sz umbral) 
oy v " oy^'*' 

whilst 



By an obvious and useful extension we can now introduce mixed 
Tensors partly covariant and partly contravariant in nature. 
Thus the set of n 3 functions of position X r r \ Tt form a mixed 
tensor of rank three, covariant of rank two and contravariant of 
rank one, if the correspondence between the two sets of ordered 
components is defined by the equations 



Now when we recall that the x coordinates are perfectly 
arbitrary as also are the y's it becomes apparent that it must be 
possible to interchange the x and y coordinates in the equations 



18 VECTOR ANALYSIS AND RELATIVITY 

defining the correspondence. Thus, to give a concrete example, 
it must be possible to derive from the r? equations 



r t = 



which serve to define a covariant tensor of rank 2, the equations 

(*i, 82 umbral) 



In fact 



all umbral) 



is umbral ) is = by the rule for 



composite differentiation and this, on account of the mutual 
independence of the x coordinates, is = unless t\ = r\ in which 



case 



it = ll 



To conclude these definitions it will be sufficient to state that 
a single function of position may be regarded as a tensor of rank 
zero if its value (not its formal expression) is the same in all sets 
of coordinates. No labels are here required to order the com- 
ponents and the equation defining the correspondence is simply 

Y=X 

Such a function of position is also called an invariant or absolute 
(or in the text-books on vector analysis a scalar) quantity. The 
reason for regarding this as a tensor (of either kind) of rank zero 
will become apparent from a study of the rules of operation with 
tensors. 

Example. Consider the formulae of transformation from rec- 
tangular Cartesian to space polar coordinates ( 3). 



THE TENSOR CONCEPT 19 

Here 



= sin y (2) cos y (3) ; - = + y (1) cos y (2) cos !/ (3) ; 



) 

= (V sin /< 2 > sin 



etc., and we obtain 

,, daP> , Y , 

Yl = Xl dy W 

= (Xi sin y (2) cos y (3) + X 2 sin ?/ (2 > sin y (3) + X 3 cos 2/ (2) ) 



cos cos y z cos y sn y - 3 sn 

C7v -rr C7*C -inr C7X 



= y w [ Xi sin i/ (2) sin y (3) + JJT 2 sin y (z) cos 

the X's on the right hand side being supposed expressed in terms 
of the i/'s. If then we denote by R, 0, $ the resolved parts of 
the vector X\, Xz, X 3 (the theory of the resolution of tensors 
will be dealt with later but we may anticipate here) along the 
three polar coordinate directions at any point 



r; 3 = 2/ sn s r sn 

For a contravariant tensor of rank one we have 



yd) = Vd) __ i_ v(2) __ L 



sin y ^ C os i/< 3 > + X sin < 2 > sin 

4. 
^ 



V(3) Vd) _L -F(2) __ L V(3) 

^ 



cosy (3) + X cosy (2 > sin y< 3 > - Z<*> sin 

fy 



20 VECTOR ANALYSIS AND RELATIVITY 

1 



sn 



(- X 1 sin 7/ (3) + X* cos 



where the Jf 's on the right are supposed expressed in terms of the 
y's. Call the resolved parts of (X w , Z (2) , Z (3) ) along the polar 
coordinate directions R, 6, $ as before and we have 

yd) == R. y( 



r ' r sin 

* A general result of which this is a special case is given in Chapter IV. 



CHAPTER II 
THE ALGEBRA OF TENSORS 

1. ELEMENTARY RULES FOR DERIVING AND OPERATING WITH 

TENSORS 

(a) The Ride of Linear Combination 

( n i \ 
) and 

q = 0, 1, / 

X%'.". r is another tensor of the same kind then the set of 
n iH- functions of position found by adding components of like 
order (that is with all corresponding labels, both upper and 
lower, having the same numerical values each to each) forms a 
tensor of the same kind as X and X which is called the sum of 
X and X. By the phrase " of the same kind " we imply 
not only that X and X must have the same rank both as to 
covariant and contravariant character, but that corresponding 
components have the same dimensions. The proof of the state- 
ment is immediate for from the equations 



Q:::^all umbral) 

and a similar one obtained by writing a bar over Y and X we 
obtain by addition 



which is the mathematical formulation of the statement that 
X + X is a tensor of the same kind as both X and X. 

If we multiply the equations written above, which express the 
tensor character of X^'." r ,^ by an invariant function of position 

21 



22 VECTOR ANALYSIS AND RELATIVITY 

(possibly a constant) m we have that mX is a tensor of the same 
character as X. Combining this with the previous definition 
of a sum, repeatedly applied if necessary, we have what is known 
as a linear combination of Tensors 



where the l\, Jj, are either mere numbers or scalar (invariant) 
functions. The separate members of this linear combination must 
be of the same kind. If, as a special case, /2 is a negative number 
lz = - 1 say and li = + 1 then X + (- Z 1 ) is written X - X 1 
and in this way subtraction is defined. A tensor all of whose 
components are zero is said to be the zero tensor. (It is im- 
portant to notice that the property of having all the components 
zero is an absolute one; i.e., it is independent of the particular 
choice of coordinates in terms of which the components are 
expressed. This follows at once from the equations defining 
the correspondence between the ordered components in different 
systems of coordinates. The General Principle of Relativity 
merely says that all physical laws may be expressed each by the 
vanishing of a certain tensor. This satisfies the necessary de- 
mand that the content of a physical law must be independent of 
the coordinates used to express it mathematically. The fixing 
of the number of dimensions n as 4 rather than 3 and the inter- 
pretation of the physical significance of the coordinates are the 
difficult parts of the theory of relativity; the demand that all 
physical laws express the equality of tensors has nothing to do 
with these and must be granted by everyone. Here we regard 
an invariant as a tensor of zero rank.) Since the idea of a linear 
combination of tensors is reducible to a linear combination of 
the corresponding components it follows that the order of the 
separate members in a linear combination is unimportant. 

2. (6) The Rule of Interchange of Order of Components. 

A specific example will show most briefly and clearly what is 
meant by this rule. Consider the co variant tensor X Tl r t of the 



THE ALGEBRA OF TENSORS 



23 



second rank. The components have a definite order which may 
be conveniently specified by a square arrangement. 



Xln 



If now we rearrange the n. 2 functions amongst the n. 2 small squares 
in such a way that the rows and columns are interchanged, 
then this same interchange of rows and columns will take place 
in the square for any other coordinate system y. We denote 
the new ordered set by a bar thus 

~X r ..= X 8 , r (r,8= 1,2, ,) 

From X r , we obtain Y ra by means of the equations of corre- 
spondence and we wish to show that Y ri = Y tr where the Y ra are 
obtained from the X r by the same equations of correspondence. 
All we have done is to rearrange the order of summation on 
the right hand side of the equations of correspondence and the 
formal proof is very easy. 



V = 



= V 

* sr 



by definition (p and a umbral) 



from definition of X 



(from equations of correspondence). 



Combining this rule with rule (a) we derive some important 
results. Thus starting with X whose components are X r , we 
derive X whose components are X rs = X sr and then the differ- 
ence X X whose components are X rg X rs = X rt X iT ' 
This new tensor is alternating and an important example of this 
type will be given to exemplify the next rule. 



24 VECTOR ANALYSIS AND RELATIVITY 

3. (c) The Ride of the Simple Product. 

Consider any two tensors not necessarily of the same kind or 
rank. Let us form the product of each component of the first 
into each component of the second and arrange the products in a 
definite order. The set of products will form a tensor whose 
rank is the sum of the ranks of the original tensors. Again it 
will suffice to show how the proof runs in a special example. 
Let the two tensors be X Tt and X Tt and denote by the symbol 
X'lr\ the product X ri r t -X' 1 '*. (Here r\, r 2 , *i, s z have definite 
numerical values so that X^'\, defined in this way, is a single 
function out of a group of n 4 obtained by giving r\, r 2 , *i, s% 
each all values from 1 to n in turn.) We have to show that the 
group of n 4 functions X? t really form, as the notation implies, 
a tensor of rank four covariant of rank two and contravariant 
of rank two. To do this we have 



Yr['rl a F rir , . y-ii by definition of 



(Pit P2 frb 02 umbral) 



dx(pl) dx(n) d y (t 



by definition of X% 
which proves the statement. 

It is quite apparent that X^ is not the same as X%% so that 
the order of the factors in this kind of a product is important. 
Multiplication of tensors is not in general commutative. This 
remains true even when both the factors are of the same kind and 
rank. Consider the simplest case where we have two tensors 
X and X both covariant of rank one. Then the product X- X 
is_a tensor X r = X T .X,_covariant of rank two whilst the product 
X'X is a tensor X rt = X r >X t . 



THE ALGEBRA OF TENSORS 25 

The difference X ra X ra is again a covariant tensor of rank 
two which is alternating since X r = X tr . Since alternating 
tensors have a more immediate physical significance than non- 
alternating tensors it is natural to expect that this difference 
should be more important than either of the direct products 
X rs or Xra- It is what Grassmann called the outer product of 
the two tensors X, X in contrast to another kind of product which 
he calls " inner " and which we now proceed to discuss. 

4. (d) The Rule of Composition or Inner Multiplication. 

Let us first consider a simple mixed tensor of rank two X ri r * 
for which the equations of correspondence are 



Y ri r * = ^V 2 -r (*i and *2 umbral symbols) 



If now we make r 2 = n = r (say) and use r as an umbral symbol 
we get 



The remarkable simplification on the right hand side is due to 
the results from composite differentiation 

dy (r) = dz (<l) 



dy (r) 

= if s z 4= Si and = 1 if s 2 = si 

In this way we can form from a given tensor a tensor of lower 
rank (in this case an invariant). 

The proof in the general case is of the same character. 

Consider the mixed tensor X^'.^mi"'^ which is, as the 
labels indicate, covariant of rank p + / and contravariant of 
rank p + <? so that the equations of correspondence are 



where -^rr stands for -^7-, - v and so for the others. 
dx ( " ox ( p ' 



26 VECTOR ANALYSIS AND RELATIVITY 

If now we make p\ = r\, p% = Tt p p = r p and use r\ r p 
as umbral symbols of summation, -^ ^ on the right hand 
side becomes 

(TI T p umbral) 



1 fr^ 1 Cr^ 

oy v ' ox ( ' 
and successive applications of the results 



dy (T 



gives us that 



-r = Unless t\ = T\ 

ri) 

= 1 if *! = n 



= U SS ^ = 
= 1 if <i = ri, 



, t p = r 



so that 



>, Ml " 



r, -. r, 



(r, ra, * all umbral) 



giving the result that (^rj'."^^'.'.'/"!,) is a tensor, co variant of 
rank / and contravariant of rank q. If q = 0, / = we have the 
result that 

X r r \"'. r r f r is an invariant (fi r p umbral) 

explaining why we regard an invariant as a tensor of zero rank. 
If now we have two tensors not both entirely covariant or 
contravariant and take their simple product we have a mixed 
tensor to which we may apply the method here described and 
obtain a tensor of lower rank. This is called composition or 
inner multiplication of the two_tensors. Thus starting with 
X r and X' we obtain X, T = X r -X $ and then making r = * (i.e., 
picking the n diagonal elements or components of the tensor Xf 
of rank two)^and summing with respect to * we derive an in- 
variant X t 'X' which is the invariant inner product of the two 



THE ALGEBRA OF TENSORS 27 

tensors. (To obtain an inner product the tensors must be 
of different character one covariant, the other contravariant.) 
Similarly from the two tensors of rank two X rir * and X tlSt we 
first obtain the mixed tensor of rank 4 



X 1* == 

and from this the scalar or invariant function of position 

X'lrl = X rir *'X ri r t (n, r 2 umbral symbols) 

Notice that in these cases the order of the factors is not im- 
portant the same invariant results if we change the order. 

5. (e) Converse of Rule of Composition. 

Again, for the sake of simplicity, let us explain this for a 
special case. We consider a set of n functions of position X r 
which has such a law of correspondence between components 
in different coordinate systems that for any contravariant tensor 
X r of rank one whatsoever the summation X r X r is invariant 
(r umbral). Then we shall prove that the set X r actually form, 
as the notation implies, a covariant tensor of rank one. 

We have 

Y r - F = X t -X w (by hypothesis) 



= Xt'Y 1 -r :-. (since X r is contravariant of rank one) 
dy M 

We now take as a special illustration of the tensor X T that one, 
which, in the y system of coordinates, has all its components = 
save one which is = 1, e.g., Y* = if * 4= r whilst Y r = 1. 
This choice of X is permissible since we make the hypothesis 
that X is any tensor we wish to choose. And we have 



proving on assigning, in turn, to the label r the numerical values 
1, , n, the statement made. (It is apparent that instead of 



28 VECTOR ANALYSIS AND RELATIVITY 

taking X T as perfectly arbitrary it is the same thing to say that 
X (T) shall be any one of the n tensors which in some particular 
system of coordinates have each all but one of their coordinates 
= 0, the remaining one being =1.) As another example of 
this converse let us suppose that the n 2 functions X r * have such 
a law of transformation that the summation X r * X tt is a covariant 
tensor of rank two (* umbral) where X tt is an arbitrary covariant 
tensor of rank two; we have to prove that the n 2 functions of 
position X/ actually form, as the notation implies, a mixed tensor 
contra variant of rank 1 and covariant of rank 1. 
We have 



Y r 'Y. t - CX/JW by hypothesis 



- 

' z 

(since X is covariant of rank 2) 

Now as our arbitrary tensor X let us choose that one for which 
Fjm = unless both I = s and m = t 

Qy(t) Q x (r) 

Y, t = 1 and using ^- } . _ = 1 ( T umbral) 
we obtain 

d 7/) fob) 

r '' mZ 'M*ij (*,P umbral) 

proving the statement. The essence of the proof is that the 
multiplying tensor must be an arbitrary one. In concluding 
these remarks on the elementary rules of tensor algebra it may 
not be superfluous to remark that although, for example, the 
product X T = X r * 'X t t is a definite tensor we do not introduce 
the idea of quotient X r , -f- X T *. The reason for this is, of course, 
that there is no unique quotient; there are many tensors X,t 
which when multiplied by a given tensor X r * in this way will 
yield a given tensor X r - In the algebra of tensors it is possible 
to have a product (inner) of two non-zero tensors equal to zero. 



THE ALGEBRA OF TENSORS 29 

6. Applications of the Four Rules of Tensor Algebra. 

The most useful applications of these rules will be found by 
returning to a consideration of the integrals which served to 
introduce us to the tensor idea. It will be remembered that a 
curve V\ is either open and has two end points as boundary or 
else is closed and has no boundaries; a spread Vz of two dimen- 
sions is either open and bounded by one or more closed curves or 
closed and without boundaries. In general a spread Vp+\ of 
p + 1 dimensions (p < n 1) is either open and bounded by 
one or more closed spreads V p of p dimensions or else closed and 
without boundaries. When the spread V p +i is open there is an 
important theorem giving the value of an arbitrary integral I p 
extended over the closed boundaries V p in terms of a certain 
connected integral extended over the open Vp+i bounded by V p . 
The simplest case is when p = 1 in which case an integral over 
a closed curve is shown to be equivalent to a certain integral 
extended over any surface or spread of two dimensions Vz 
bounded by the curve V\. This case was discussed by Stokes 
for ordinary space of 3 dimensions and the general theorem is 
known as " Stokes' generalized Lemma."* It will be noticed 
that the theorem is a non-metrical one as we have not yet had 
occasion to say anything about the metrical character of the 
space S n containing the spreads V p . We shall prove the theorem 
when p = 2 as this will suffice to show the details in the general 
case. 

Here the equations of the open V$ are 



and the boundaries will be specified by one or more relations on 
the parameters u\, Uz, u 3 . If there are several distinct boundaries 
Vz we may connect them by auxiliary surfaces Vz so as to form 
one complete boundary. The parts of the 7 2 over this complete 
boundary coming from the auxiliary surfaces will cancel (each 
* H. Poincart, loc. cit. 



30 VECTOR ANALYSIS AND RELATIVITY 

auxiliary connecting surface may be replaced by two, infinites- 
imally close, surfaces and it is the integrals over these pairs of 
surfaces that cancel each other in the limit as the surfaces are 
made to approach each other indefinitely). The relation between 
the parameters on the boundary may be 

3 = <t>(ui, ut, u 3 ) = 

and we introduce two other functions v\ and 2 of u\, it*, u 9 
such that 01, 02, 03 are distinct functions, and change over to 
0i, 02, 03 as parameters. We shall suppose the parameters such 
that the equations giving the coordinates x are uniform both 
ways. Not only does an assigned set of parameters give a 
unique point x but to a point x there corresponds but one set of 
parameters 0. 

Accordingly the surfaces 03 = const, cannot intersect each other 
and they form a set of closed level surfaces filling up the initial 
open Fj. On each of these closed level surfaces we shall have 
the level curves 0i = const., 0j = const., and we suppose the 
functions 0i, 03 of Ui, u*, M so chosen that these level curves 
are closed. 

Now consider the integral 

/, = fX^ tl d(x (tl) , x (t *>) (si, 8 Z umbral and *i < * 2 ) 

extended over the boundary 03 = 0. If, instead of integrating 
over 03 = 0, we take it over any of the level surfaces 03 = constant 
it will take on different values depending on this constant and 
to indicate this we write 



?= rx 

* d0id0 



+ 



Ar<'d d r () ] 

/tj vJ, VJ, I 

-= I 

001 002 



THE ALGEBRA OF TENSORS 31 

(It is only necessary to differentiate the integrand since the 
limits of the integral are independent of 3 ). Now if F is any 
function of position (not merely of the parameters)* on a closed 

dF 

curve with parameter v the integral J* - dv taken round the 

dv 

closed curve is necessarily zero. For it is the difference of the 
values of F at the coincident end points of the curve. If, in 
particular, we take as F the function 



F = X Sl t t ($1, 82 umbral) 

and integrate round the closed curve v t = constant we get 

8 i 8 J I a a a ' a a a I 
\ 001003 002 003 001002 / 



_ vi = 
dvz J 



and integrating this with respect to 02 over the surface 3 = con- 
stant we have 



f\X gltt {^. >_|_te^d*a: -\ 

= 



Similarly on taking 



00 



f)F 
and integrating f , dvidvz over the closed surface v 3 = const. 

dvz 

* The distinction implied here should be clearly grasped. If the equations 
of the curve are 



xi = a cos v 
x* = a sin v 

F must be periodic in v with period 2ir. 



32 VECTOR ANALYSIS AND RELATIVITY 

we get 

r\X ( d * x(>t) dx( ' l) 

[ ''''Vdfladfla dvi 303 



0t>2 003 Ofli 

Now add these two equations together and note that 

(*i *2 umbral) 

because the terms in the summation cancel out in pairs owing to 
the alternating character of X tlH the factor multiplying X, ltt 
in the summation being obviously unaltered by an interchange 
of the symbols *i and s z . We find that 



* 1 * 1 



4- 



d * x(tt) dx( ' l) \ 
dvydvz dv\ J 



so that 

dh_ r \dX^dx^dx^ 



dt>3 dvi dfy dv\ 



Now the X^ H are functions of position, i.e., of the coordinates x 
so that 

air s v ?_f.l 

(* 8 umbral) 



The second term in dlz/dv^ we shall slightly modify by a change 
in the umbral symbols. Thus 

-i /..\ -> r.,\ 

(si, s z , s 3 all umbral) 



dx ( ' l) 
dX^ 
dx (tl) 



THE ALGEBRA OF TENSORS 33 

so that we can write 

= J < - ~~ - " 7~"v i dcidvz 

On writing 

Y = 
A ii 



and integrating the expression for dlz/dv 3 with respect to 8 
we find 

' 

dvidvzdv* (si, s z , s 3 umbral) 



since the set of functions X, ltt t defined as above is obviously 
alternating (on account of the fact that X rt is an alternating set). 
The limits for v$ are 3 = and v 3 = some constant for which 
/2 = since the corresponding F 2 is either a point or a spread 
traced twice on opposite sides. Let the integration be such 
that 3 = is the upper limit and we have 

/ 2 = fX, lH d(x (t i>x {t *>) (*i<*2> over boundary 

= fXw t d(x^x^x^) (Si < * 2 < * 3 ) over the F 3 .* 
In general from 

I p = 



over a closed boundary we derive as equivalent to I p an 

/+! 

where 



* It will be observed that placing the + sign before / on the left makes 
= the upper boun 
from the open spread 



t> 3 = the upper bound of the integral f -^ dvs. Thus v t is increasing away 



34 VECTOR ANALYSIS AND RELATIVITY 

It is usual to preserve a cyclic arrangement of suffixes for the 
X's and then, on account of the alternating character of the X's, 
we have 






the upper signs being used when p is even and the lower when p 
is odd. Since I p is by hypothesis invariant so is Ip+i because 
IP+I = Ip and accordingly the coefficients X Sl ... Vl form an 
alternating covariant tensor of rank p -f- 1 [seen either directly 
as when tensors were introduced or as a case of the converse of 

rule (d), the set of functions -r dvi dvp+i form- 

001 OVp+i 

ing an arbitrary contra variant tensor of rank p + 1]. In this 
way we can derive from any alternating covariant tensor, by a 
species of differentiation, a covariant tensor of higher rank. , 

EXAMPLES. 

p = 1. From any covariant tensor X T of rank one we derive 
an alternating covariant tensor of rank two 

X - dXr - dXt 

It is the negative of this tensor that is called the curl of the 
vector X in the earlier vector analysis. It is rather important 
to notice that this, and the other tensors of this paragraph, have 
no reference to the metrical character of the fundamental space 
S n . The derivation of them by the methods of the Absolute 
Differential Calculus introduces, therefore, extraneous and un- 
necessary ideas. 

p = 2. From an alternating covariant tensor of rank two 
X rt we derive the alternating covariant tensor of rank three 

f\ *y \ TT A T^ 

Y _ O**-Tt i OA. t t , OA.tr 

" rtt ^ s 



THE ALGEBRA OF TENSORS 35 

If n = 3 there is only one such function and in the usual analysis 
it is called the divergence of X rs . We shall have to modify this 
slightly for the general tensor analysis. It is interesting to 
notice that if we take as X r the tensor of the previous example 



we find Xrst 0. It is easily seen that this happens in general. 
If we derive X tl ... 9p from X tl ... g^ in this way then the 
X tl ... v , derived from X tl ... 8p is = 0. When the X Sl ... v , 
derived from X Sl ... 8p is = we have that Ip+i = and so I p 
(extended, of course, over any closed spread of p dimensions) 
is = 0. In this case I p is said to be the integral of an exact 
differential. It can then be proved that the value of I p over 
any open V p is equal to the value of a certain integral Ip-i over 
the closed boundary of this V p * 

"If 

IP - SX^ ... tp d(xW X<P>) (i<a,... < 8 P ) 

(an \ 71 ? 

-i ) - n - r T7 
p + !/ n p lip + 1! 

partial differential equations 

Y = n 

Atl2 ... Ip+l W 

The theorem stated is that these are the necessary and sufficient conditions 
that there exist ( _ i ) functions of position X n ... .^j satisfying the ( ) 
partial differential equations 



That the conditions are necessary is an immediate result of a direct substitution 
of the left hand side of the equation just written for X tl ... , in the equation 
of definition 



To prove the sufficiency an appeal is made to the principle of mathematical 
induction. Let us, for definiteness, take p = 2. Then we shall prove the 
statement that if the theorem is true for a particular value of n it is true 
for the next greater integer value n + 1. Granting this, for the moment, we 



36 VECTOR ANALYSIS AND RELATIVITY 

p = n 1. This is the next and last case if n = 4. For 
an arbitrary value of n it is second in importance only to the 
first case p I. In order to avoid having to write out separately 

observe that the theorem is true for n 2. (In this case there are no in- 
tegrability conditions necessary; on account of the alternating character of 
the Tensor X^^ whose vanishing expresses these conditions, it is neces- 
sarily s 0.) We have two unknowns X\ and Xt satisfying the single differen- 
tial equation 



and a particular solution is found by assuming that neither A"i nor X t involves 
z ( . Then X l may be any function of z> and X t = - f* (1> Xitdx (1 \ the 
lower limit being any constant i n> . In the integration z (1) is regarded as a 
constant. Hence by the induction lemma the theorem is true for n = 3 
and then for n 4 and so for every integer n. 

To prove the induction lemma let us seek for a solution of the equations 

y dX r dXt f 

rt * az^> ~ az^> (r 

where the unknown X n s 0. We have then 

AY 
X "-+i) <r-l,...,n-l) 

whence 

X T - +X rn dx<*> +X r (r = 1, -, n - 1) 



where x ( * ) is a constant; X T is any function of x (1 >, , z*""" and in the 
integration z a) , , z ( *~ x) are constants. The remaining equations 

xr dXr dX t . . 

X " tow ~ Sw (r < - 1, -, n - 1) 

give on substituting these values 

C*dX n f^dX n dX r dX. 

JU ^^ ~ JU a^)^ ^ a^ ~ ^o 

dx r . d~x t ax. 

<-> T ( > ^> 



1 v- a y- 

Z r . X,. + T-TTT T-T;: where X r , is the function X r , when x (B) is 
oz v *' aZ v ' 



put - z ( 



THE ALGEBRA OF TENSORS 37 

the cases corresponding to n even and n odd we shall adopt the 
first form for X tl ... , ,. 



Hence we have the 



( o ) 



dX. 

dx< r > 



with n 1 unknowns X T and involving n 1 independent variables 
x (1) , -, x (n-1) . Also we have ( ) integrability equations X rtt ^0 

found by putting x (n) = Xo (n) in 

X Tlt = (r < s <t = 1, , n - 1) 

Hence if we can solve these equations, i.e., if our hypothesis is true for n 1, 
we can solve the original equations which are identical in form but involve 
one more independent variable x (n) . The particular case of this theorem 
corresponding to n = 4, p = 2, tells us that Maxwell's equations 

_ j a 5 _ 

curl E -\ --- ^r = div B = (in the usual notation) 

C at 

imply the existence of the electromagnetic potential (A t , A v , A,, c<t>) 
which is as in the general case when p = 2 a covariant tensor of rank one 
such that 



B - curl A; E - - grad ^ - - 

C at 

For further details cf. Physical Review, N. S., Vol. 17, p. 83 (1921). 

It is apparent that there is a great degree of arbitrariness allowed in the 
determination of the functions X tl ... , _ t ; in fact we may add to any solution 
any alternating covariant tensor of rank p 1 whose integral over any closed 
spread V p -\ of p 1 dimensions is zero. For example we may add to the 
electromagnetic potential any gradient, of a function of position; that is 
if (A,, A y , A t , c<t>) is any determination of the electromagnetic potential, 
so is 



A dF 
A V ~T 7~ 



, where F is an arbitrary function of x, y, z, t. 

Of 

- 
idF 



38 VECTOR ANALYSIS AND RELATIVITY 

Here p + 1 = n and there is only one distinct function X tl ... , 
on account of the alternating character of this set. Let us choose 
this one as A'i ... and our formula is 



_ dX v ... 



n-l -1, 2 n-t n 



Now there are only n distinct functions X, t ... ,_, and it will be 
possible, and convenient, to indicate these by means of a single 
label. Thus we write 



n-2, n 

n- 3, n 1, n = 



where we are careful to put parentheses round the symbols (X r ) 
to indicate that they are not the components of a covariant 
tensor of rank one. 

Maxwell availed himself of this arbitrariness and chose F so that div A a 
whence 

dF d*F . 3*F 



yielding, from the theory of the Newtonian Potential, 
Fas _L ydivA 

4 r 

The usual procedure with modern writers is to choose F so that 

div A 

The equation determining F is now 



whence 



F 
from the theory of the retarded potential. 



/YdivA- !) 

I V Cdt/ t _r 
1 / ?_dT 

4/ r 



THE ALGEBRA OF TENSORS 39 

Then we have 

Xi n = . ,; (s an umbral label) 

ox (t) 

Although the (X a ) do not form a covariant tensor of rank one 
they are very closely related to a contravariant tensor of rank one. 
In fact there is a reciprocal relationship between an alternating 
covariant tensor of any rank r and an allied contravariant alter- 
nating tensor of rank n r. It is a special case of this reciprocity 
stressed so much by Grassmann in his Ausdehnungslehre that 
gives the dual relationship of point and plane, line and line in 
analytic projective geometry and it is from the terminology of 
that subject that the terms " covariant " and " contravariant " 
are taken. In order to bring out this reciprocal relationship in 
the clearest manner we must make a digression and discuss what 
are meant by " metrical properties " of space. 



CHAPTER III 

1. INTRODUCTION OF THE METRICAL IDEA INTO OUR GEOMETRY* 
Let us consider a curve V\ specified by the equations 

X M = z<*>() (*= 1, -,*,) 

The quadratic differential form 

g rt dx (r) dx (t) (r, s umbral) 

where the g r are functions of position, will be invariant provided 
that these functions form a covariant tensor of rank 2. (This 
is a consequence of our rule (d), Ch. 2, 4, and its converse since 
the set of n* functions 

/fr(r) 



du du 

form a contravariant tensor of rank two.) Accordingly the g rt 
being of this kind the integral 



du 



du du 

has a value independent of the choice of coordinates x; it is called 
the length of the curve V\ from the point specified by UQ to that 
specified by u'. If the upper limit u' is regarded as variable 
and written, therefore, without the prime S is a function of this 
upper limit u and its differential is given by 

(<fo) 2 = g rt dx (r) dx (t) (r, s umbral) 

where the positive radical is taken on extracting the square root. 
It will be convenient to agree that, in some particular set of co- 
ordinates x, we arrange matters so that g T = g r ', this can always 

* The most satisfactory presentation of the general idea of a metrical space 
is that given in Bianchi, L., Lezioni di Geometria Differenziale, Vol. 1, 152. 

40 



THE METRICAL CONCEPT 41 



be done by rewriting any two terms, g z zdx (z) dx ( * > 
for example, of the summation which do not satisfy this require- 
ment in the form %(g 23 + g 32 )dx (2) dx^ + %(g n + 
The equations defining the covariant correspondence 



where 

then show that 



since *,- 



We inay express this result by saying that the property of any 
special tensor of being symmetric is an absolute one just as is 
the property of being alternating. 

2. RECIPROCAL FORM FOR (dsf 
Consider the n linear differential forms 

r = g rt dx (t) (s umbral; r = 1, , n) 

We can solve these for the differentials dx M in terms of the n 
quantities r as follows. (Note that the r form, as the notation 
indicates, a covariant tensor of rank 1 from our rule (d) of com- 
position or inner multiplication.) Let us denote the cofactor of 
any element g rs in the expansion of the determinant 

ii <7i2 gin 



71 ' * ' <7nn 

by (G r ), observing in passing that (G>) == (G tr ). The parenthe- 
ses indicate that the (G>) do not form a tensor. From the 



42 VECTOR ANALYSIS AND RELATIVITY 

definition of a cofactor the summation 

0r.(G>m) = g when m = s (r umbral) 

= when m ^ s 

We shall now introduce the hypothesis that our metrical space 
is such that g does not vanish identically (it will be presently seen 
that this is an absolute property) and for all points where g is 
not zero we have 

(C ^ 

0r^ ^ 1 when m = * , , lx 

g (r umbral) 

= when m 4= s 

Write 0* m = (Gim)/g and let us justify the notation by showing 
that the g lm form a contravariant tensor of rank two. From our 
definition it is symmetrical and so we have in addition to 

g r g rm =1 if m = s 
= if m 4= s 

the equivalent equations 

0.r0 mr =1 if m = s 
= if m ^ s 

These relations suggest that we multiply the equations of defini- 
tion 

r = g ri dx^ 

by g rm and use r as an umbral symbol. We obtain then 



(r, a umbral) 
= dx (m) from our relations just written 
Accordingly 

(<fc) 2 = gi m dx w dx = gi m g lr r'g m 't. (I, m, r, s umbraD 
= g rl rk, (r, s umbral) 

since gi m g* r = unless m = r when it = 1. 

The r form, by rule (c), Ch. 2, 3, an arbitrary contravariant 

tensor of rank 2 and (<&) 2 being, by hypothesis, invariant, the 



THE METRICAL CONCEPT 43 

converse of rule (d), Ch. 2, 5, gives us the result that the g rt 
form a contravariant (symmetrical) tensor of rank 2. When we 
write 

g"te t (r, s umbral) 



it is said to be written in the reciprocal form. We could start with 
this form and write 



and solving these obtain 

. = 9. 
and then find 

(ds) z = g r 



3. If now we have two determinants a = \a rs \, b = \b rs \ 
each of order n (the notation implying that a rs is the element in 
the rth row and sth column of the determinant a) it is well 
known that the product of the determinants a and b may be 
written as a determinant c* of which the elements c rs are defined 
by 

c r = airbis (I an umbral symbol) 



This kind of a product is said to be taken by multiplying columns 
of a into columns of b. 

We can, with the aid of this rule, easily see how the determinant 
g behaves when we change our coordinates x to some other 
suitable coordinates y. We get a determinant / of which the 
r, sth element is 

frs = gin 



x 

Here -T-. may be conveniently denoted by (/&) since it is the 
(r) 



dx (l) 

-T 

dy ( 

I, rth element of the Jacobian determinant J of the transformation 
from x to y coordinates 

* Cf. Bdcher, M., Introduction to Higher Algebra, Chap. 2, Macmillan (1915). 



44 



VECTOR ANALYSIS AND RELATIVITY 



,7 = 



and then 



,<!> 



dy ( 



,(n) 



/ # x (l) \ Q x (m] 

is the mrth element of the product gJ so that I 0j m ^-rr ) -^-r^ 



is the r*th element of the product of the determinants gJ by J. 
Hence / = g J 2 . 

This important formula shows us that if g ^ neither will 
/ s= unless J = in which case the y's would not be suitable 
coordinates. / can be zero at points where 0=t=0if/ = 0at 
those points; such points would be singular points of the system 
of coordinates and the quantities f rt would not be defined for 
them. 

EXAMPLE 

In Euclidean space of 3 dimensions with rectangular Cartesian 
coordinates x w x w x w we write 



so that 0n = 022 = 033 = 1> 0i2 = 0i3 = 023 = 0. In space polar 
coordinates we find 

fii = 1 /2z = (y (1) ) 2 /33 = y (l) * sin 2 y (2 > 

/12 = /13 = /23 = 0. 

Here 0=1 

/ = /U/22/33 = J 2 

so that 



___1. /22__ = __ . 

fu / lf (1)l> 






1 



THE METRICAL CONCEPT 45 

and 



In fact 1 = dx w , etc. There are no singular points in the x 
coordinates but there are in the y system; those for which J = 0, 
i.e., 

yO)giny-0 

These are the points on the polar axis 

y< = r = 0; i/ (2) = 6 = or ir 

4. If now ui " ' Un are any independent parameters in terms 
of which it is convenient to specify both the x and y coordinates 
we have, by definition of the symbol, 



and a similar equation for d(x (l) - z (n) ) so that 



i u ) 
If we multiply the determinants - - - r^ and . . \ n - ^7 



together and note that 

dx^ du 



= itt^r 

we find that their product is unity and so we can write the 
quotient 



w * x- 

(m umbral) 



46 VECTOR ANALYSIS AND RELATIVITY 



as above 



33 Jgjf since / = gJ*. 
Accordingly 

Vf (%< . 



so that this expression is an invariant. In view of the fact that 
it depends on the fundamental quadratic differential form (fo) 2 
it is called a metrical invariant. 

Let us consider an integral over a region of the fundamental 
space S n , fX\ ... n d(x (l) - x (n) ). Here X\ ... n is the single 
distinct function of an arbitrary alternating covariant tensor of 
rank n. Since the integrand is invariant and since V</ d(x w 
x (n) ) is invariant it follows by division that X\ ... n -5- V^ is an 
invariant. As an application of Stokes' Lemma we have already 
seen that if 

(Ai) = ( l) n ^L2 ." n = A n 2 n 1 ' ' ' (A n ) = -^1 n 1 

(where X tl ... .^ is any alternating covariant tensor of rank 
n 1) then 

V d/TT\ /Ll\ 

X i ... n = T-77j (A) (* umbral) 

is the coefficient of an integral over a region of S n . We see 

1 a 

therefore that p-r-ri (-^) is an invariant. 



We shall now investigate the nature of the n functions (X t ). 
Under a transformation of coordinates from x to y we find, for 
example, 

(V \=v Y dx(tl) dx( '"~* 

""""- i ay< 1 >" ' ' dyt- 

(! *_i umbral) 
a(a;Ci) s<'-i>) 
l """- 1(1 > ( - 1 > 



(owing to alternating character of X^ ... ..,) 



THE METRICAL CONCEPT 



47 



In general 



And, accordingly, if we denote the cofactor of ^-^ in the expan- 
sion of J by (J r ) we have 

(F n ) == (J n )(X.) (s umbral) 

(Y r ) S (J. r )(X.) 

If we solve the n equations 

= -^-7-) =1 if s = r (p umbral) 
= it s ^ r r = 1 n 
for^- ( -j we find 

jdy (p) _ (Jgp) 

so that we may write 



or 



, n 
umbral) 



V/ Jg 9*** 

showing that ^^ is a contravariant tensor of rank one. We 

may then put (X t } = ^g X* and our previous result takes the 

1 d 
form that -7= (V0 X") is an invariant; X 9 being any contra- 

\gdx (8 ' 

variant tensor of rank one. This metrical invariant is known 
as the divergence of the contravariant tensor. 

5. SPECIAL RESULTS 

If u(x m - - - x (n) ) is any invariant function of position the rule 
of differentiation 

du du dx (9) f , 1N 

T~T^ = TT^ -^~77\ ( s umbral) 



48 VECTOR ANALYSIS AND RELATIVITY 

0tt 

tells us that the n functions X t = r-n form a covariant tensor 

dx { *' 

of rank one; this is known as the tensor gradient. If X r is any 
covariant tensor of rank one its simple product by itself or 
" square " is a covariant tensor of rank two, X r = X T X t . 
Hence by rule (d), Ch. 2, 4, 

(f'X T X is an invariant (r, s umbral) 

This is called the square of the magnitude of the tensor. In 
particular the square of the tensor gradient is the invariant 

A . du du , , lx 

AlWHEE/ o^>dx^> (r,* umbral) 

This is known as the " first differential parameter of u." Similarly 
the magnitude of the square of a contravariant tensor of rank 1 
is the invariant g^X^ X^ . 
Again 

9 "<^ = X' (r umbral) 

oar 

is contravariant of rank one (rule (d)). Hence 

7r JIT) ( ^S 9" Q-^) ) is & n invariant (r, * umbral) 



by the result of the preceding paragraph. It is written 
and is known as the " second differential parameter."* In 
ordinary space of three dimensions in which the s's are rec- 
tangular Cartesian coordinates 

g n - if r =t= * 
= 1 if r = * 

and g rt = /'; Vjj = 1 so that A 2 w takes the form 
d z u . d*u . d z u 

I i /o^o I 



dx (l)2 

* Larmor, J. t Transactions Cambridge Phil. Soc., Vol. 14, p. 121 (1885), 
obtains this transformation in the case n = 3 by the application of the Calculus 
of Variations. 



THE METRICAL CONCEPT 49 

When we change over to any " curvilinear " coordinates y 
WP Viavft i in Her thft form 



we have under the form 



the expression of this magnitude in a form suited to the new 
coordinates. 

6. GENERAL ORTHOGONAL COORDINATES 

Whenever we have, in any space, coordinates x such that the 
expression (ds) z involves only square terms, i.e., g r , = if s ^ r, 
the coordinates are said to be orthogonal (for a reason to be 
explained later). It is usual to write, in this case, 

I i 

accordingly 



so that 


1 
11 1 


i 

9nn ~ ft 
1 


*~W A 2 2 An 2 ' 

11 Z, 2 ~n Z, 5 
9 == 1 ' * ' 9 "* 

The square of the gradient is 
/ du \ 2 

A , J, 2 f vu> \ _1 


2 ( dll \ 2 


whilst the 

A 2 w = AiA 


t-A\(A; '*'l 1 /i\ I | 

\oa: a) / 
quantity 

-JL( d ( hl 


du W 


^lax^VA.--- 


Andz'V 1 

d ( An du 



The reader should write out the explicit formulae for space polar 
and cylindrical coordinates in ordinary space of three dimensions. 



50 VECTOR ANALYSIS AND RELATIVITY 

7. THE SPECIAL OR RESTRICTED VECTOR ANALYSIS 

In the form given to the theory by Heaviside and others only 
those coordinates x or y were considered in which the fundamental 
metrical form is 

d** = (<&i (1) ) 2 + ---- h (<& (n) ) 2 s (dy w )* + ---- h (rfy 00 ) 8 

These coordinates we call rectangular or orthogonal Cartesian 
coordinates and the space we call Euclidean. It is true that 
use was made of Stokes' Lemma to find expressions for important 
invariants as A 2 w in other than orthogonal Cartesian coordinates 
but no attempt was made to define the components of a vector 
in these coordinates. Now when we restrict ourselves to that 
subgroup (of all the continuous transformations) which carries us 
from one set of orthogonal Cartesian coordinates to another 
the distinction between covariant and contravariant tensors com- 
pletely disappears. The transformations are necessarily of the 
linear type 

x (r) = My w (s umbral, r = 1 n) 

where the a's are constants. Since here /= g = !,/*= 1* 
and so the equations just written have a unique solution for the 
y's. To get this most conveniently note that dx (r) = (a rt )dy (t) 
and squaring and adding we have 

(a r ,)(o r j) = t ^ s (r umbral) 

= 1 t = s 

Hence multiplying the equations for x by a rt and using t 
as an umbral symbol we find 

(a r )z (r) = (a r )(a r .)y<-> (r, * umbral) 



Accordingly the equations of correspondence defining covariant 
* We shall consider only direct transformations; those for which J = + 1. 



THE METRICAL CONCEPT 51 

and contra variant tensors are, for this restricted set of trans- 
formations, identical. Again denoting by (A rs ) the cof actor 
of (a rt ) in the expansion of the determinant J we have by the 
usual method that 



and since the solution is unique we must have (a r( ) = (A r t)* 
Hence since g = 1 we have that the n distinct components of an 
alternating tensor of rank n 1 form a tensor of rank one. It 
is for this reason that when n 3 it was found necessary to 
discuss but one kind of tensor that of the first rank which was 
called a vector.^ Still some writers felt a distinction between the 
two kinds; that of the first rank they called polar and the 
alternating tensor of the second kind, whose three distinct com- 
ponents form a tensor of the first kind, they called axial. Thus 
a velocity or gradient are polar vectors (the first being properly 
contravariant, the latter covariant) whilst a curl or a vector 
product are axial vectors. 

When, in the mathematical discussion of the Special Rela- 
tivity Theory, it was found convenient to make n = 4 [the trans- 
formations (Lorentz) being still those of the linear orthogonal 
type], a new kind of tensor or vector is introduced. Here it is 
the alternating tensor of the third rank which, when we consider 
merely its four distinct components, is equivalent, from its 
definition and the properties of the transformation, to a tensor 
of the first rank or " four-vector." There remains the alter- 
nating tensor of the second rank and the six distinct components 
of this were known, for want of a better name, as a six-vector. 
As an example of the general theory we have that 

dX (g) . 
(a) the divergence of a four-vector , . is an invariant. 

OX 

(s umbral) 

* This is merely a special case of the previous result that J -~-^ = (/r). 
f Until a consideration of non-alternating tensors became desirable. 



52 



VECTOR ANALYSIS AND RELATIVITY 



(6) From any six-vector X Tt we may derive a four-vector 
(really an alternating tensor of the third rank) 

Y _ dX I dXt i dXtr 

~ (t) (r) (>) 



It is this four-vector that was written lor X n in honor of 
Lorentz. 

8. GENERALIZATION OF THE RECIPROCAL RELATIONSHIP be- 
tween an alternating tensor of rank r and one of opposite kind 
of rank n r from the case r = I already treated to a general 
value of r. 

We have already seen that" 



where J is the determinant 



of the transformation 



and (J. p ) is the cofactor, in the expansion of J, of the element 

dx (t) 
(j lp ) = ^-r:* of this determinant. 



Hence 



J* 



'in 



V, 



Now the determinant of the minors of J is well known to be 
equivalent to the product of J by the determinant of order n 2 
obtained by erasing the *ith and *jth rows and the nth and f2th 
columns of J affected with its proper sign (the determinant of 



order n 2 is the cofactor of 



/(ri) 



dy (rt) 



in the Laplacian ex- 



THE METRICAL CONCEPT 63 

pansion of / in terms of two row determinants from the *ith 
and s 2 th rows and the nth and r 2 th rows). Hence we have the 
result that the n(n l)/2 distinct components of an alternating 
covariant tensor of rank n 2 when divided by V<7 form the 
distinct components of an alternating contravariant tensor of 

rank two. And so in, general. Similarly the ( j distinct com- 
ponents of an alternating contravariant tensor of rank n r 
when multiplied by V<7 form an alternating covariant tensor of 
rank r. 

Example. Take n = 4, r = 2 and consider the linear orthog- 
onal transformations of the Special Relativity Theory. Here 

Z M = Z<; Zi, = X 42 ; Z 14 = X* 
Z 23 = Z"; Z 24 = Z; Z 3 4 = X 12 

The two tensors or six vectors X rt and X rl were said to be 
reciprocal* 

* Cf. Cunningham, E., The Principle of Relativity, Ch. 8, Camb. Univ. 
Press (1914). 



CHAPTER IV 
1. GEOMETRICAL INTERPRETATION OF THE COMPONENTS OF A 

TENSOR 

DEFINITIONS 

(a) Direction of a curve at any point on it. 
At any point u on the curve V\ specified by the equations 

z<> = x<>(w) (, = 1, ..., w ) 

whose length s from a fixed point UQ is defined by the integral 



: f 

/m 



j 
du du 

we may form the n quantities 



j j -J 
as du du 



/ _ -i \ 

V' A > > n ) 



We exclude from consideration here the " minimal " curves 
along which ds = 0. Since X r = dx (r) is a contravariant tensor 
of rank one and ds is an invariant we have that the n quantities 
l (r) form a contravariant tensor of rank one which we call the 
" direction " tensor of the curve at the point in question. The 
n components we call direction coefficients. The equation of 
definition 

= g rt dx (T) dx (t) (r, s umbral) 



shows us that g rt fi r) l (t) = 1 so that a knowledge of the mutual 
ratios of the direction coefficients suffices to determine their 
magnitudes (save for an indefiniteness as to sign). Otherwise 
expressed the magnitude of the direction tensor is unity. Fixing 
the indefiniteness as to sign by a particular choice is said to fix 

54 



THE RESOLUTION OF TENSORS 55 

a sense of direction on the curve and the curve may be then said 
to be directed. 

2. (6) Metrical Definition of Angle 

Consider two curves with a common point and let their direc- 
tion tensors at this point be l (r) and m (r) . The simple product 
X rt = l (r) m (l) is contra variant of rank two (Rule (c), Ch. 2) and 
so the expression g rt l (r) m (t) is invariant (r, s umbral; Rule 
(d), Ch. 2). This we call the -cosine of the angle between the 
two curves (directed) at the point. If the quadratic differential 
form defining (ds)* is supposed to be definite, i.e., if it is supposed 
that (ds) cannot be zero, for real values of the variables z (r) 
and dx (r) save in the trivial case when all the dx (r) = 0, it can 
easily be shown that the angle defined in this way is always real 
for real curves. Let us write instead of dx (r) the expression 
XZ (r) + M m(r) an d thus form the quadratic expression in X and n 



This is not to vanish for real values of X, /* save when X = 0, n = 
(we suppose the quantities Z (r) and m (r) all real and the two direc- 
tions as distinct). Using 

g ri l (r) lM = 1 = g n m (r) m M 
we have that 

X 2 + 2X M cos 6 -f M 2 = 

must have complex roots when regarded as an equation in 
X : M- Hence 1 cos 2 6 > so that the angle as defined above 
is always real for real directions under the assumption that (ds) 
cannot vanish on a real curve. It must be remembered however 
that this assumption is not always made, e.g., in Relativity 
Theory. 

When cos 6 = the curves are said to be orthogonal or at 
right angles at the point in question. 



56 VECTOR ANALYSIS AND RELATIVITY 

EXAMPLES 

In ordinary space with the z's as rectangular Cartesian co- 
ordinates we have the usual expression 

cos 6 = Z<ro< + J< 2 >ro< 2 > + J< s >m<*> 

where (J (1> , J (2) , J (I) ), (m (1) , m (2) , m (3) ) are the direction cosines of 
the two curves. If now we use any " curvilinear " coordinates 
(y w , y (2) , 3/ (l) ) the angle between two curves is 



COS 6 = Jrt 



In particular if we use orthogonal coordinates 



Thus for a curve in polar coordinates r, 0, 



It will now be clear why those coordinates in terms of which 
(ds)* has no product terms are said to be orthogonal. 
For 

QyJ Q x (m) 

fr, = 9im^-M^-7:r (from its co variant character) 
dy (r > 



If now all the coordinates y but one, y (r) say, are kept constant 
we have a curve whose equations, in the x coordinates, may be 
conveniently specified by means of the parameter y (r) 

X M = z(->(yM) ( S = 1, ...,n) 

Through each point y there pass n curves of this kind which we 
shall call the n coordinate lines y through that point. On the 
rth of these coordinate lines the direction tensor is 



ds 



THE RESOLUTION OF TENSORS 57 

and so the vanishing of the component / r states that the co- 
ordinate lines y (r) and y (t) are orthogonal. Hence if (ds) z does 
not contain any product terms the coordinate lines are everywhere, 
all mutually orthogonal and so the coordinates are said to be 
orthogonal. In ordinary space, i.e., where the a;'s are rectangular 
Cartesian coordinates and where the y's are orthogonal co- 
ordinates, 

/^p 
Ll Zr 



and 

/n = 
so that 



- 



\dy (1 >J 

(du (l) Y 
y \ 
dx") 



a result which is sometimes useful in the calculation of the 
coefficients /n, /22, /ss of the form (ds) 2 in the curvilinear 
coordinates y. 

3. RESOLUTION OF TENSORS 

If we consider any covariant tensor X r of rank one and take the 
inner product of this into a direction tensor l (r) we derive the 
invariant Xrl (r) (r umbral; Rule (d)). This we call the 
resolved part of the co variant tensor along the direction / (r) . 
Let us now make a transformation of coordinates from x to y 
and consider the coordinate line y (>) . The n components of the 
direction tensor for this curve are proportional to 

. 

To determine the actual values of these components we must 
divide through by the positive square root of 



yy (/,m umbral) 

and this is equivalent to 



58 VECTOR ANALYSIS AND RELATIVITY 

The equations defining the covariant correspondence for a 
tensor of the first rank are 

Ft I *^~ 1 *J f 1 1 TY\ w\f*Q 1 ^ 

J -^'^TTn " ~ L > n > r umorai; 



= V/H times the resolved part of the tensor X r along the co- 
ordinate direction y w 

EXAMPLE 

Space polar coordinates y in ordinary space of three dimensions. 
The x are rectangular Cartesian coordinates. Denoting the 
resolved parts of the covariant tensor X in the directions 
2/ (1) > y (2) > y (3) by R, 0, $ respectively we have since /n = 1 ; / 2 2 = r 2 ; 
/ M = r 2 sin 2 

FI = R; Yz = r0; y, = r sin 0<t>. 

The three distinct components of the alternating covariant tensor 
of rank two, curl X, in polar coordinates are 



dR o . . f.,.. 
-- (r sm 6$) 
d<t> dr 



Similarly for cylindrical coordinates p, <f>, z where f\\ = 1; 
/22 = p 2 ; /as = 1 if we denote the resolved parts of X along the 
three coordinate directions by R, $, Z we have Y\ = R; 
F 2 = p$; Y 3 = Z and the components of the curl are at once 
written down. 

Resolution of Contravariant Tensors. 

To define what is meant by this we require, not as before the 
coordinate lines y (r) along each of which all the coordinates y but 
one, y (r \ are constant, but the coordinate spreads F_i along each 



THE RESOLUTION OF TENSORS 59 

of which all the variables but one, y (r) say, vary. The parameters 
HI - ?/ n _i may here be very conveniently chosen to be the 
coordinates y\ y n themselves omitting y (r) , and then 7/ (r) is a 
constant (a particular function of u\ u n -i). Now, in general, 
when we have a V n -\ specified by equations 

X M = x W( Uli ... f Un _^ ( s = i, . . ., n) 

we obtain on the spread, through each point, n 1 parameter 
lines by letting in turn each parameter vary, keeping all the rest 
fixed. Any one of these, u r varying, say, has at the point in 
question a direction tensor whose components are proportional to 

. 

Let us look for a direction orthogonal at once to the n 1 
directions of these parameter curves. Such a direction tensor 
has components n (1) n (n) say and is said to be normal to the 
spread V n -\ at the point in question. To express the required 
orthogonality we have n 1 equations 

dx (m) 

gi m n (l) - = (/, m umbral; r = 1 n) 
du r 

homogeneous in the n (1) n (n) and thus serving to determine 
then* mutual ratios. To actually solve divide across by one of 
the unknowns n (n) say and we have n 1 linear, non-homogene- 
ous equations for the (n 1) unknowns 

(1 > < n - * 



* The algebra following here is somewhat complicated and so it may be 
desirable to derive the expressions for the components of the normal direction 
tensor to the spread y (n) as follows. Working with the coordinates y the n 1 
parameter curves y (>) varying (s = 1, , !) have their direction coef- 
ficients proportional to 

(1, 0)1 

(0, 1,0 0)f 

(0, 1, 0) 



60 VECTOR ANALYSIS AND RELATIVITY 

The determinant of the coefficients has as the element in the 
rth row and *th column 

(m umbral; r, s = 1, , n 1) 



du r 

This determinant is therefore the product of the two matrices 
9\\ <7i2 * ' ffi* 



du\ du\ 



0*-i, i <7-i, 

each of n 1 rows and n columns. It is well known that this 
product can also be written as the sum of products of all corre- 

respectively. The n 1 equations expressing that n (r) is orthogonal to these 
n 1 directions are 

/, r n (r > =0 (t - 1, -, n - 1; r umbral) 
Hence the ratios 

the actual values being these divided by 



[one must be warned against thinking that "^ ( 1, , n) are contra- 

variant. When a change of coordinates from y to x is made the spread 
y(> = const, does not become x ( "> = const.] If now we wish to use x co- 
ordinates, the normal direction tensor, being contravariant of rank one, has 
components proportional to 



nW "fjjfi (r - 1, -, n; umbral) 

ay" dy<> dx r 
aSnSSfr a , umbral) 

-9**l (I umbral) 

If y<) = F(xO) f ., x (>) ) we have that the normal direction tensor to the spread 
has its components proportional to g lr r- the result required. 



THE RESOLUTION OF TENSORS 61 

spending determinants of order n 1 that can be formed from 
each matrix. Let us write for brevity 

2 n\ 

( T } = ( 1^i 



and the determinant of the coefficients becomes 

(.)(/.) (sumbral) 

which s may be written g g nt (J s }. The numerators of the fractions 
furnishing vi v n -i are dealt with in the same way and we have 



(Since the (J a ) are really the n distinct components of an alter- 
nating contravariant tensor of rank n 1 we know that 
X (J a ) V^ is a covariant tensor of rank one verifying the 
contravariant character of the n (r) (Rule (d)))- If all the (J) 
vanish the point is said to be a singular point of the spread and 
the determination of n (r) becomes impossible. 

Let us now apply these generalities to the spread F n _i given 
by a single equation 

F(*< *<>) = 

connecting the coordinates x. We may solve for one of the 
coordinates, ar (n) say, in terms of the others x w x (n-1) and 
these others we use as the n 1 independent parameters of the 
spread: 



are then the equations, in parametric form, of the spread F_I. 
Our matrix 

dz (r) 

(r = 1, -^, n- s = 1, -, n - 1) 



62 VECTOR ANALYSIS AND RELATIVITY 

is now 

1 

1 



and so 







dx(n> 



But, on differentiating the equation V(x w 
spread F_i we obtain 



so that 
whence 



: (J) : : (J.) = 



dV ^ dV 

' dx M 

dV dV 



x (n) ) = of our 



_ 

'dx 



In particular, if the spread Vn-\ has, in the y coordinates, the 
equation y (r) = const., we have for its normal direction tensor 



d) 



... j n (n) = 



The actual magnitudes of these components are found by dividing 



THE RESOLUTION OF TENSORS 63 

through by the positive square root of 

rh/ r > rhyW 

<"""" few '"'Ilk (r <>< umbral) 

Qy(r) Qy(r) 

which expression is = </ m ' T - 0* not umbral) 



If now we have a contravariant tensor X (r) of rank one it is 
meaningless to call J5T (r) / (r) the resolved part of the tensor in the 
direction I for the simple reason that this expression is not 
invariant but takes on different values in different systems of 
coordinates. However, we may first form the co variant tensor 

X t = g.rX^ (r umbral. Rule (d)} 

This tensor is said to be reciprocal to the contravariant tensor 
X (r) with respect to the fundamental metrical quadratic differ- 
ential form and its resolved part in any direction we call the 
resolved part of the contravariant tenser in that direction. Thus, 
for example, the res^lvcJ part of the contravariant tensor X r 
in the direction normal to the coordinate spread y (r) = constant is 

(s, p, t umbral) 



yto 



Hence any component Y (r) of a contravariant tensor of rank 
one is the product by V/ rr of the resolved part of the contravariant 
tensor normal to the coordinate spread y (r) = constant. It is 
now apparent that to deal with covariant and contravariant 
tensors of the first rank we require the coordinate lines through 
each point and the normals to the coordinate spreads through 
that point. When the coordinates are orthogonal, and only then, 



64 VECTOR ANALYSIS AND RELATIVITY 

these lines and normals coincide and a great simplification is due 
to this fact. This explains why orthogonal coordinates have 
been used, almost to the point of excluding all others, in the 
investigations of Theoretical Physics. 

4. EXAMPLE (a) 

Space polar coordinates. These being orthogonal the normals 
to the spreads r = const., d = const., < = constant are the 
coordinate lines r, 6, <f> respectively and, if we denote the resolved 
parts of the contravariant tensor X (t) in these directions by 
R, 0, $ the three components are 

y(i) _ p. y(2) _ . y(s) _ *** 

1 /I * A m M A 

r r sm 9 

In general for orthogonal coordinates y with 

we have/"" = l/f rr and if, as usual, we write f rr = I/hS we have 
/ = (hf A, 2 ... An 2 )- 1 and /" = h r * 

Here F (1 > = Ai(fli) 7 (n) = hn(Rn) where we denote by 
(Ri) - - (RJ the resolved parts of the contravariant tensor along 
the coordinate directions 1,2, , n respectively. The divergence 
of the contravariant tensor 



takes the form 

d 



h 



Thus, for space polar coordinates, the divergence is 



THE RESOLUTION OF TENSORS 65 

and for cylindrical 

p [ dp d<t> dz ] 

Example (b) 

In order to illustrate the distinction between covariant and 
contravariant tensors we now consider oblique Cartesian coordi- 
nates y so that 



where the constants X, JLI, ? are the cosines of the angles between 
the oblique directed axes. Here 



1 v /x 
y 1 X 
M X 1 



= square of volume of unit parallelepiped with 
its edges along the three axes. 



i.e., Vf = sin X cos 0i = sin n cos B z = sin v cos 3 where 0i is the 
angle between the coordinate line y\ and the normal n\ to the 
coordinate plane y\ = const, with similar definitions for 62 and 6 9 . 
Hence 






= sec 0u =sec0 2 ; = sec 3 



If we have any vector whose components in rectangular Cartesian 
coordinates (a: (1) , (2) , x (3) ) are -X"i, Zj, ^3 this vector may be 
regarded as either a covariant or contravariant tensor, i.e., 
X\ = X w ; Xz = X (2) ; Xz = X (3) and if we denote the resolved 
parts of this vector along the coordinate lines y by (X^, X^, XiJ 
and along the normals of the coordinate planes y by (X ni , X nt , 
X nt ) we have 



^ = X^; YZ ^ X^', YS = Xi t 
= Vf 1 X ni = Z ni sec ^; F = X^ sec 2 ; T 3 = X nt sec 3 



66 VECTOR ANALYSIS AND RELATIVITY 

Hence (Yi, F 2 , Y 3 ) are the resolved parts of the vector along the 
three coordinate lines whilst (Y w , Y (Z) , y (3) ) are the components 
of the vector along these same directions. The tensors Y r and 
Y r are reciprocal with respect to the differential form (ds)*, i.e., 

Y l = y< + V Y<*> + /iY (3 >, etc. 



Let us now consider the contravariant tensor whose components 
are 

y (1 > = Pl y<i> ; y< 2 > = P2 y (2 >; y< 3 > s P3 y< 3 > 

where P i, P j, PS are scalar or invariant numbers; we find for the 
components in the rectangular coordinate system x 



yd> = yd) i __ i_ y(2) rz __ t- 



= Pl Z + Z2 + jy + etc 

~ Pl \ dx^ dx^ S&Bjdj^ 

or 

X r = P /Z (* umbral) 

where 



r== , , 

Pl m (t) P * (t) 



Now r TTT is a contravariant tensor and -. is a covariant tensor 
dy ( dx (t) 

if we regard the y's as fixed and consider merely transformations 
on the x's so that P /, being the sum of three mixed tensors, is 
actually, as the notation implies, a mixed tensor of rank two. 
It was in this geometrical way that Voigt introduced the idea 
which he called a tensor. The mixed tensor P / is completely 
specified by the three directions y and the scalar numbers p\, pz, 
p 3 . If the mixed tensor is to be symmetric for every choice of 
Pit P 2, P s we must have 



These equations lead to the conclusion that the " axes " y of 



THE RESOLUTION OF TENSORS 67 

the tensor are mutually at right angles and so such a tensor was 
called symmetric. 

In order to study the behavior of the vector X as X changes 
direction, keeping its magnitude unaltered, we may solve the 
equations for X and obtain 

X r = TT/Z- (s umbral) 

where from the geometrical construction TT/ is a mixed tensor 
with the same axes as p but 

?TI = , etc., 
Pi 

so that 

r _ ldx (r) dy (l) ldx (r) dy Idx^dy 
~ m M + ^^ ^ < 3 > ' 



Then squaring and adding the equations for X r we find that X 
traces an ellipsoid, called the first tensor ellipsoid. 

For a symmetric tensor the directions y are orthogonal so 
that YI = F 1 , etc. A simple example of a symmetric tensor is 
furnished by the uniform stretching of a medium along three 
mutually perpendicular directions successively. It was from 
this example that Voigt originally took the name " Tensor." 
Reference may be made to any treatise on the Theory of Elas- 
ticity for an amplification of the remarks of this paragraph. 

5. GENERAL FORM OF GREEN'S FUNDAMENTAL LEMMA 

Starting with any invariant function of position F(z (1) z (n) ) 
we have seen how to form its covariant tensor gradient 

dV 



the square of whose magnitude is the first differential parameter 
of F 



68 VECTOR ANALYSIS AND RELATIVITY 

Now the normal direction tensor to F(x (1) a; (n) ) = const, has 
components whose ratios are 

n< : : : <"> = f : . . . : g*" (s umbral) 



the actual magnitudes of these being found on division through 
by the positive square root of AiF. Hence the resolved part of 
the covariant tensor gradient along the normal is 

i - 
^^ (M umbral) 



and this is = VAiF.* This we shall call the normal derivative 

cf V and denote by the symbol 

i 

gradient along any direction I is 



dV 

cf V and denote by the symbol The resolved part of the 

on 



dV 

This we denote by -?r and call the directional derivative of V 
ol 

along the direction 1. The angle 8 between n and I is given by 

1 dV 

cos 6 = gJPn - = f-PV" ('. , * umbral) 



az (r) 

Hence 



showing that the maximum directional derivative is that along 
the normal. (In general, if we say that any covariant tensor X T 
has a direction specified by the reciprocal contravariant tensor 



X' = fX r (r umbral) 

* If we define the " direction " of any covariant tensor of rank one as 
that of its reciprocal contravariant tensor we may say that the gradient of 
any invariant function of position has a direction normal to it. 



THE RESOLUTION OF TENSORS 69 

the resolved part of X r along any direction / is the product of the 
magnitude of the tensor into the cosine of the angle between I 
and the direction of the tensor.) 
The contravariant tensor reciprocal to the gradient of V is 



Accordingly, on multiplying each of these expressions by V0, 
we derive the n distinct components of an alternating covariant 
tensor of rank n I (cf . Ch. 3, 4) and so we can form the 
integral /_! 

I~-i s f 



over any spread of re 1 dimensions given by 
the symbol (J r ) denoting as before 

(- l)n-r^ 

The normal contravariant tensor to the spread of re 1 dimen- 
sions has, as has been shown, components proportional to 

g rt (J t ) (r = 1, , re; s umbral) 

the actual magnitudes being found by dividing through by the 
positive square root of 

gimg l '(J*)g mt (Jt) (I, m, s, t umbral) 

= 0"(^)(7*) (s, t umbral) 

dV dV 

Hence /-(J.) z-^y = product of V '(,/.) (J)by fa ^ direc- 



tional derivative V normal to the spread V n -i over which 
is being extended. Hence we may write 



70 VECTOR ANALYSIS AND RELATIVITY 

where by dV n -.\ we mean the invariant V^ m< (J m )(t/ t ) du\ 
dun-\. (That this is invariant follows from rule (d) since V<7 (Jr) 
is a covariant tensor of rank one (cf. Ch. 3, 7).) 
Applying Stokes' Lemma to /_! we have 



where the integral 7 n _i is extended over any Vn-i which is closed 
and the integral / on the right is extended over any region of 
space V n bounded by Fn-i. Here 



and dV n is the invariant V<7 d(x m z (n) ). 

dV 

If, instead of the contra variant tensor X r = g rt ^-7-5 > we start 

* 

out with 



where U is an invariant function of position we find 



On interchanging the functions U, V and subtracting we have 



which is the usual form of Green's Lemma. The previous 
equation may be written 



where A(C7, F) is the invariant mixed differential parameter 

(r, S umbral) 



THE RESOLUTION OF TENSORS 71 

In particular, if the invariant functions U, V are identical we 
have 



- 
on 

The last identity is the basis of various uniqueness theorems of 

dJJ 

Theoretical Physics. If we know the values of U or - over a 

dn 

closed V n -i as well as the values of A 2 I7 throughout the region 
bounded by Vn-i the function U is unique, save possibly to an 
unimportant additive constant. For, applying the last identity 
to the function W = TJ\ U-t where U\ and C7 2 satisfy the above 
conditions, we have 

-dVn = 



Now under the hypothesis that 



is a definite form we see that &\W is one signed and vanishes only 

dW 
when all r^are zero. Hence since fAiW-dV n = we must 

dx (r) 

dW 
have rr = throughout the region of integration (r = 1 n). 

OX V ' 

Therefore, W is a constant and if the values of U are assigned 
W = Ui - U z = 

on the boundary and so W = or Ui = Uz- 

The whole argument depends on the definiteness of (ds) z . 
Suppose we wish to apply the theorem to solutions of the wave 
equation 



dx z dy* 
Here we have 



72 VECTOR ANALYSIS AND RELATIVITY 

and so 



and the theorem cannot be applied since AiF can vanish without 
implying the vanishing of all the derivatives. 

6. APPLICATION TO MAXWELL'S EQUATIONS 
One of the most interesting applications of the algebra of 
tensors is the discussion of Maxwell's Electromagnetic Equations. 
These consist of two sets, which in the symbols of restricted 
vector analysis and the units employed by Heaviside are 



= j; div D = p 
c at 



(6) ~ + curl E = 0; div B = 

C at 

D is the electric displacement, H the magnetic force, and j the 
current vector; B is the magnetic induction, E the electric force 
and p is the volume density of electrification. We take n = 4 
and as coordinates, in the above form, 

x w = x . x w = y . x m = 2 . x (4) = t 

If we assume that 

XM = B x ; Xti = B v \ X\i = B,', Xu = cE x ; Xu = cE y ; 

X u - cE. 

are the six distinct components of an alternating covariant tensor 
of rank two, the four equations (6) express that 

V _ dX\i , 6X23 i dXii _ dB t , dB x . 3B V .. D 
A ui = ---- h -z -- H -r = -z- + ^ + -r-^ = div B = 0, 
02 ox ay az ox oy 

Y &Xi* i dXu_^&Xn _ dB t , / dE v dE x \ 
A 114 = ZT- T -3 -- T -r = -rr + C I = - = 0, 

dt dx dy dt \ dx dy ) 

Y = d^n i dXu , 8X41 _ 6B V / d.E, dJ x \ _ ft 

A 1M a . ~r ~T ' -r i C I --- - I U, 

dt dx dz dt \dx dz J 

Y _ dXtt . dXu . dX& _ dB x i (dE B dE v \ n 

A M4 = -r^- + -5 h -3 =-sr + c(-r -- ? =0 

d< dy dz dt \ dy dz J 



THE RESOLUTION OF TENSORS 73 

In other words the integral 



* SBxd(y, z) + Byd(z, x) + B,d(x, y) + c^(;r, 

, + C^(2, 



is the integral of an exact differential its value when extended 
over any closed spread Vz is identically zero. Hence its value 
when extended over any open spread Vz can be expressed as a 
line integral J*X r dx (r) round its boundary. On writing 

X\ = A x ', Xz = A v \ Xa = A t \ Xi = c0 

we have 

72=7!== f(c<l>dt A,j(ly A^dy 

and an application of Stokes' Lemma tells us that 

_ dX T dX t 



or 



7? = Y = 2 _ ^-^3 = dA f _ dA y 

~ ~""~'' 



dA x dA, n _ dA v dA 



dz dx ' dx dy 

^r -- Y -dXi dX 4 _ dA x d< 

c&x = A 14 = 



dt dx 

The covariant tensor of the first rank (A x , A v , A t , c<f>) is the 
" electromagnetic covariant tensor potential " of which the 
first three components form Maxwell's vector potential, < being 
his scalar potential. 

Similarly, if we assume that ( D x , D v , D g , cH x , cH v , cH t ) 
are the six distinct functions of an alternating covariant tensor 
X rt of rank two the equations (a) say that 



74 VECTOR ANALYSIS AND RELATIVITY 

and we have Iz = Iz where 



7 2 = fcH^x, + cHJ(y, t) + cH4(z, <) - D x d(y, 2) 

- Dyd(z, x) D4(x, y), 
h = fcj,d(y, z, f) + cjJL(z, x, f) + cj t d(x, y, t) pd(x, y, 2) 



7 2 being taken over any closed spread F 2 of two dimensions and 

7 3 being taken over the open F 3 bounded by V?. Accordingly 

(jx, jv> j*> ~~ P/ c ) are tne f ur distinct functions of an alternating 

_ (j^ r ) 

tensor of rank three and so, on writing c(X\) = X 234, etc., r^- 



form a contra variant tensor of rank one (Ch. 3, 7). From its 
definition we know that its divergence is zero. This tensor we 
may call the current contravariant tensor and write 

fn _ Ji . _ ~ P 

u -- p , o -- p 

V0 c^g 

Let us now apply these methods to the problem of writing 
Maxwell's equations in a form suitable for work with curvilinear 
coordinates y w , y (y) , y w in space of three dimensions the time t 
not entering into the transformation. The equations connecting 
the x and y coordinates are of the type 



and denoting tensor components in the new coordinate system 
by primes we have 



12 
the terms in //i, H 2 , HZ vanishing since 



/ TT \ / _ 

(H l} ' " 



THE RESOLUTION OF TENSORS 75 

\^ > "^ ' -L. fll \ \^ ' ^ ' _1_ / Z7 N ^ > ^ / 



the terms in (Di) (D 2 ) (Z) 3 ) vanishing since 

te^> 



Hence in the three-dimensional space with coordinate systems 
(ar (1) , *' 2) , o; (3) ) and (?/ (1) , i/ (2) , y (3) ) the variable / being regarded 
merely as a parameter which does not enter into 

(ds) 2 = g ra dx^dx^ = frJyVdyM (r, 5=1,2, 3) 
the three quantities (Di) (D z ) (#3) are the three distinct members 
of an alternating covariant tensor of rank two. Hence ^ = X r 

iff 

/ n \ _ 

is a contra variant tensor of rank one; similarlv ^ ^- = X r is a 

V0 _ 

contravariant tensor of rank one whilst E r = X r and H r = X T 
are covariant tensors of rank one. We derive by our rule (d) 
of composition the invariants 

(ED) . (EB) . (HP) . (HB) 
V? V<7 Vsr V^ 

where as in the usual vector notation 

(ED) = EiDi + E Z D Z + E 3 D 3 

and similarly for the others. 

Dividing Maxwell's equations, as usually written, across by 
V<7 we obtain 



- z- + -curl r (10 = & (r = 1, 2, 3) 
cdt V0 

(where C r = -^is the contravariant current vector). 
V? 



76 VECTOR ANALYSIS AND RELATIVITY 

div Z = p 

where p is the invariant charge density and similarly from the 
second set 

+ i|l' + J = ciirl r OE)-0 
cdt V0 

div X r = 

Denoting, then, as usual resolved parts along the coordinate lines 
by subscripts (li, h, /j) and along the normals to the coordinate 
surfaces by the subscripts (n\, n$, n 3 ) we have the three equations 



The equation div X T = p becomes 



^ + sps ( ^ ^ I 



(by D, is meant the resolved part of the contravariant tensor 
Z)/V? along the direction ni). 

The equations (6) are similar and are simplified by the fact 
that there C,, *,, C w ,, p are all zero.* 

* When the coordinates y are orthogonal 

(<fc) - -, (dy<) 4- ^ (<*>)* + ^ (rfy<) f 

/ - ...... ; /" - hi*, etc., and Maxwell's equations become since ni - li, etc. 

i "i "a 



and two similar equations together with 



Cf. jRienumn-W6r, Die PartieUen Differentialgleichungen der Mathemat- 
ischen Physik, Bd. 2, p. 312 (Vieweg & Sohn) (1919). 



THE RESOLUTION OF TENSORS 77 

EXAMPLE 
In space polar coordinates Maxwell's equations are 



c dt r 2 sin 6 



cdt r sin 



^^4^ lj- r (r sin 6D r ) + i (r sin 0D,) + ^ (rD.) I = p 

It is particularly to be noticed that Maxwell's Equations are 
essentially of a non-metrical character. No real simplification 
is introduced by the hypothesis that the fundamental space is 
of the ordinary Euclidean character. Another point to which 
attention should be directed is the difference in character of the 
tensors B and H or of D and E. A relation of the familiar type 



H, the coefficient of permeability, being supposed invariant is 
not the proper mode of statement of a physical law if we under- 
stand by B\, 5 2 , BZ the three components of the tensor B. The 
true statement of the law is 



where by (B)i we mean the resolved part of the contra variant 
tensor CB)/V<7 along the direction I and by (H)i we mean the 
resolved part of the covariant tensor H along the same direction. 
Thus any constitutive equation of this type is an allowable state- 
ment of a physical law not because it is a tensor equation (since 
it is not such), but because it is an equality between invariant 
magnitudes or a scalar equation. The true tensor equation is 
found by equating the covariant tensor i*H to the covariant 
tensor reciprocal to the contravariant tensor 



CHAPTER V 

1. CONNECTION OF TENSOR ALGEBRA WITH INTEGRAL INVA- 
RIANTS AND APPLICATION TO THE STATEMENT OF 
FARADAY'S LAW OF MOVING CIRCUITS* 

Suppose for example we have a curve V\ whose equations 
z<> = x ^(u, T) (= 1, ..-,n) 

involve a parameter T. This curve may be said, adopting the 
language of dynamics, to move and trace out a Vi whose equa- 
tions are those given above, the parameters being u and T. 
Any one of the curves T = constant will then be a position of the 
moving curve. We shall suppose that the values of u serve to 
identify the various points on the moving curve; thus if u 
denotes the distance along the initial position of the moving 
curve from a certain fixed point, or origin, the curves V\ obtained 
by taking u = constant (wo) in the equations 

x (.) = a.(.)( W) T ) (5 = 1, ..-,n) 

are the path curves of the definite point on the curve V\ which 
initially was at the distance w from the origin on V\. It will 
fix our ideas to Consider V\ as made up of particles of a fluid; 
then the curves V\ are the paths of the various material particles 
of V\. It is well to insist, at the outset, on the point that the 
parameters u and r are independent. Thus if the moving curve 
V\ were rigid, u could be taken as the arc distance along V\ at 

An elementary presentation of the theory of Integral Invariants is given 
by Goursat, E., in two papers: 

(a) Sur les invariants intgraux. Journal de Mathematiques, 6 e se>ie, 
t. IV (1908), p. 331. 

(6) Sur quelques points de la the'orie des invariants intdgraux. Journal de 
mathe'matiques, 7* s6rie, t. 1 (1915), p. 241. 

78 



INTEGRAL INVARIANTS AND MOVING CIRCUITS 79 

any time r; if, however, as in the case of the curve made up of 
material fluid particles, V\ is not rigid, u may only be taken as 
the initial identifying arc distance; otherwise u would vary with 
T. Let us now consider an integral I\ = J*X r dx (r) extended 
over V\ and ask the conditions that l\ should be the same for 
all the curves V\, i.e., that I\ should not vary with T. If this is 
so, /i is said to be an integral invariant. 

Now /i is in general a function of r defined by 

X"' / rM r >\ 
lx r ~^\du (rumbral) 

the limits U Q and u' being, however, since u and T are independent, 
quite independent of T. Hence 



dr dr \ du 

The coefficients X r are functions of position and therefore involve 
r indirectly; it is somewhat more general to contemplate the 
possibility that they may involve r, not only in this indirect 
manner but also directly. Then for any one of the coefficients 
X r we have 



dr dx (s ' dr 

It is now convenient to denote the contravariant tensor of rank 

dx (r) 

one by the symbol X r and to use the result 

or 

J -\ _.<V^ 1 *\ __(V\ * VM ft VM 3 -.(*} 

f umbral) 



dr du du dr du dx (8) du 
and we have 

dh = r (dXrdx" d_dx^\ , , ,, 

dr~~ f dr"du~ VXr dr~d^ dU 



dX ' \ dx(r) -u 

^)-^r + 'dx^-d^ 

(r, s umbral) 



80 VECTOR ANALYSIS AND RELATIVITY 



dr dx (t) dz (r) J du 

(on modifying suitably the umbral symbols) 

Hence if dli/dr is to vanish identically for all curves V\ we must 
have 

dX r . v dX r , v dX'"' n / 1 U 1\ 

-dt +X dx" +X 'dx"^ Q (rl,---,n, umbral). 

Sometimes it is only necessary that /i should be unchanged for 
all closed curves V\\ in this case 1\ is said to be a relative integral 
invariant. To find the conditions for this we use Stokes* Lemma 
to replace the I\ over a closed curve by an 7 2 over an open F 
and then find the conditions that 7 2 should be an (absolute) 
integral invariant. 

The analysis necessary to find the conditions that an 



extended over a V v (moving) whose equations are 

. . Up> T ) ( 1 - n) 



should be an absolute invariant is identical with that given for 
the simplest case p = 1. Let us write as before 



dr ' 
and denote by the symbol F the derivative 

dF oF , dF v( m \ , i t\ 

T" s F~ + F7S (* umbral) 

dr dr dx (t > 

where F is any function of position which may also involve r 
explicitly. Then 



=f- [X tl .... f d(x^ . . **>)} (!< *,< umbral) 



INTEGRAL INVARIANTS AND MOVING CIRCUITS 81 

since the limits of integration with respect to the variables u are 
independent of r. This we write 



' dr 
and availing ourselves of the relation 



dr du r du r da; (m) du r 

we arrive at the conditions expressed in the form that 



- J- Y __ 

.,.... ^,...^^^- A ,.. f-fafij- 

dX (m) 
+ ^IP-I^^V " ( umbral) 

An especially simple case is that in which p = n. Here there is 
a single condition 

^ ,, fdXV\ . , n 

X i... ft + Ai... n ( ^ j (r umbral) 

Since Xi... n is the single distinct member of an alternating co- 
variant tensor of rank n 

*i.. = TjgU 

where U is an invariant function of position and writing out 



., _ \... n , y, l... n 

dr 6aP> 

our condition that J'U-dVn should be an integral invariant may 
be written in the form 



or on dividing out by V<7> which does not involve r explicitly, 

f)TT 

- -- f- div (UX^) where as usual the divergence of the contra- 

OT 



82 VECTOR ANALYSIS AND RELATIVITY 

1 a 

variant tensor of rank one UX T is the invariant -= 77-: (V0 UX*"). 

V<7 dx (r) 

In this form the invariance of the condition for an integral 
invariant is apparent. If we are considering a moving charged 
material body where p is the density of charge, the total charge 
J*pdV n remaining constant gives us that 



where X (r) is the contravariant velocity tensor of rank one. 

Faraday's Law for a Moving Circuit. 
We have seen that 



the integral in each case being taken over the position of the 
moving curve at time T. The expressions 

dX r i -v.dXr i - 



must accordingly form a covariant tensor of rank one. In fact 
we ma write this as 



dX r . Yt \dX T dX t 
"~dr" \dx" dx" 



when the covariant character is apparent by rule (d*), Ch. 2, 4, 
since 

dX r _ dX( t ) _ y 

dx" dx" " 

is covariant of rank two and X t X (t) is invariant. 

Let us now write down the expression for dl^/dr where /a is 
any surface integral and transform the coefficients as above so 
as to make evident their tensor character. 



INTEGRAL INVARIANTS AND MOVING CIRCUITS 83 
Writing 

/2 

we get 



where 

~Y dX r sj_ Yt-^ T8 i Y X m - Y vX 



the integrals being in each case extended over the position of the 
moving spread or surface V% at time T. We may write 

~Y _ dXra I -rA I vX T8 , dX a t dX.tr 

: ~" 



where we have availed ourselves of the alternating character of 
X TS . The covariant character of X ra then follows from rule (d). 
We shall apply this result to the surface integral 



n = 3 

so that (Z>i), (Da), (D 3 ) are the three distinct members of an 
alternating covariant tensor of rank two. Hence Z) (r) = (D r )/V<7 
is a contravariant tensor of rank one. The covariant tensor of 
rank one whose curl appears in the expression for X ra is 

X rm X m (m umbral) 

so that its first component is 



It accordingly appears as that derived from the outer product 
of the velocity contravariant tensor and the displacement contra- 
variant tensor. 



84 VECTOR ANALYSIS AND RELATIVITY 

The expression 



If now we assume as Maxwell's equations for the moving material 
medium 

|-() = ccurlff-(j); divZK= p 
at 

where (j) is the alternating co variant current tensor of rank two, 
so that (j)Hg is the contra variant current tensor of rank one C r , 
we have for X r the equations 

X a = V ~ 



etc. 

Using Stokes' Lemma to transform the surface integral of the 
part in face brackets into a line integral as well as that involving 
curl H in dD^jdt we find 



The integrand in the surface integral on the right is found by 
writing r, s, t in cyclic order and summing the terms corresponding 
tor = 1, 2, 3 respectively. (The line integral is to be taken over 
the boundary of the moving surface.) The contravariant tensor 
pX (r) is called the convection current. In exactly the same way 
we obtain, on making a similar assumption as to what Maxwell's 
equations should be for moving media, 



INTEGRAL INVARIANTS AND MOVING CIRCUITS 85 

there being now, however, no surface integral on the right-hand 
side. Accordingly the covariant tensor 

E r + - V0(^ (8) # (0 - Z ( '>5 (8 >) (r = 1, 2, 3; r, s, t cyclic) 
c 

is taken as the effective electric intensity along the moving curve; 
its line integral being called the effective electromotive force 
round the curve. (X (r) is the contravariant velocity tensor.) 
On multiplication by charge this tensor gives the mechanical force 
tensor. 

Example. In space polar coordinates the mechanical force 
tensor per element of length on a moving curve with linear density 
a is 

II 11 

{ E T + - r 2 sm 0(0 fl J5,* v^Bg) - r -. - } crds 
c r sm 6 J 

I rE e + - r 2 sin 0(v+B r - v r BJ }-~ - 1 ads 
c r sin J 

(r sin &EA + - r 2 sin 6(v r B e v g Br) - \ ads 
c r\ 

where E r , B T , v r are the resolved parts of E, B, X along the direc- 
tion r and so on. To get the resolved parts of the mechanical 
force along the three coordinate directions multiply these by 1, 

- , : - respectively and we obtain the well-known formula 
r r sm 

F = E+[vB] 
c 

In the general case when the coordinates y\, y%, y$ are not orthog- 
onal the three resolved parts of the mechanical intensity (covar- 
iant) tensor along the coordinate lines y\, y%, z/ 3 respectively are 



TJT _ mr JTT | -* IJJ J T TTJ TJ I . 

where v ni v nt v nt denote the resolved parts of the velocity along 
the normals to the coordinate surfaces y\ const., yz = const., 
3/3 = const., respectively. 



CHAPTER VI 

1. THE TENSOR OR ABSOLUTE DIFFERENTIAL CALCULUS 

Since the Calculus of Variations deals with properties of curves 
and surfaces without making any particular reference to the 
special coordinates used in describing the curves there must be 
a close relationship between that subject and that which we 
are discussing. It is this absolute or tensor character of the 
calculus of variations that has urged writers on Theoretical 
Physics to express the laws of physics, as far as possible, in the 
language of the Calculus of Variations. However, this subject 
has been placed on a clear and firm basis only within the past few 
decades and so it may be well to discuss one of its simpler prob- 
lems the more so as the solution of this problem is involved in 
the statement of Einstein's fundamental law of Inertia in the 
Theory of Relativity. 

Let us consider a curve V\, in space S n of n dimensions, given 
by the equations 

*<> = X M (U) ( = 1 n) 

and in connection with this curve a function, not merely of 
position, but of the coordinates x and their derivatives 



du 
The integral I\ over the curve V\ where 



has a value depending on the curve V\ as well as on the particular 
function. The problem we wish to discuss is: What, if any, are 
the curves V\ making, for a given function F, I\ a minimum, all 
the curves V\ being supposed to have the same end points. 

86 



THE ABSOLUTE DIFFERENTIAL CALCULUS 87 

To answer this question we consider a new curve V\(a) given by 

the equations 

x (s) = x (s) (u, a) (a = 1 TO) 

where a is quite independent of u. We suppose this parameter 
a. to be such that when a = 0, Vi(a) makes /i a minimum. 
Vi(a) is now completely determined by the equations just written 
when a is given and so I\ is a function of (a) which may, we sup- 
pose, be expanded by Taylor's Theorem in the form 

(dT\ o? (d*I\ 

I (a) = 7(0) + a( ) +r^(^2J +" 
\da/ a=0 l-2\daV tt=0 

This is written 

I (a) = 7(0) + 57 + 6 2 7 + 

and 5/ is called the first variation of the integral. If 7(0) is to be 
a minimum it is necessary (although not always sufficient) that 
67 = for otherwise A7 = I(a) 7(0) would change sign with 
a when a is sufficiently small. Now the limits of the integral 
for 7i are fixed and so to find dl/da we have merely to differen- 
tiate the integrand F with respect to a. F involves a, not 
directly, but indirectly through the coordinates x and their 
derivatives x'. 
Thus 

dF dF dx<* . dF dx^' 

= J-T-. -- h - 777-5 (* umbral) 

da dx (s) da Q X (Y da 

and therefore 

dF 



a7 = p' / dF dx< , 
da J uo \jM* da 



, 

a uo a dx^ a 

Now 

dx^' ^ a 2 a; ( ' ) = d_ ^ dx^ 
da dadu du da 

so that, on integration by parts, 



6F dx M ' , dF 



dx (t) 



"' C 


J,, 



'dx (9) d dF 



da 



du 



88 VECTOR ANALYSIS AND RELATIVITY 

Since the end points of the curve are fixed, dx (t) /da = at the 
limits of integration and so the integrated part vanishes and, 
collecting terms, we have 

dF d dF 



dl r' 

= 
da J uo 



, 
du (s umbral) 



If ( ) is to be zero for all possible varied curves F(a) it is 

\da:/ a= o 
evidently sufficient and can be shown to be necessary that all the 

/ ftf /) flV \ 

coefficients ( ^-77, 5 - r ) in this integral should vanish 
\dx (t > dud x ()'J a= o 



(s= 1, "-,n). 

These n expressions are the components of a covariant tensor 
of rank one where now, however, the term is used in a wider 
sense than hitherto. F is now not merely a function of the 
coordinates x but of their derivatives x' . From 

x (.r)' = d _j^ y w' tfumbral) 

we have 



, , n 
(r umbral) 



showing that -- - f = X r is a covariant tensor of rank one. 
dx (r) 

Suppose we wish to find the geodesies of our metrical space S n . 
These are the curves for which the first variation of the length 
integral is zero. 



so that 

dF dF 



. 
dx (T) 

dF dx 



the gim being functions of position. We shall find it convenient 



THE ABSOLUTE DIFFERENTIAL CALCULUS 89 

to take as parameter u the arc distance s along the sought-for 
geodesic.* Then when we put a = after the differentiations 
F = 1, from the definition of arc distance s, and we have 

(dF\ _ 1 

~ 



(dx w \ 
- } so that X (r) ss (i 00 ) is the 
as / a , 

unit contravariant direction tensor along the sought-for geodesic. 
Also 

\ =1.9/7, (r< m >^ 

~' / 9 ^ tm( - x > 

/a=0 ^ 



and our equations are 

(l) (m) _ <L_ 

ds 






(r, w umbral; t = 1 w) 

Multiply through by g pt and use < as an umbral symbol so as to 
obtain the n components of a contravariant tensor of rank one 



Qfftm _ J ^rm\ Q , . 

w 2ft?J 



pt (r)+m 



It is now convenient to introduce the Christoffel three-index 
symbols of the first and second kinds defined as follows: 



(6) {rs, t] = {sr, t] = g^[rs, p] 

* However, this rules out those minimal geodesies along which s is constant. 



90 VECTOR ANALYSIS AND RELATIVITY 

which equations imply 

gtq{rs, t] = gt q g tp [rs, p] (t, p umbral) , 

s [rs, q] 
Equations (a) give 

[rs, t] + [rt, s] = ~ 
Then we may write 

x (r) x m (dffg _ l^\ (f> m umbra ,) 



t, m] + [rm, t] - %[rt, m] - %[tm, r]] 

t, m] + [rm, t] - %[mt, r]} 
= x (r) x (m) [rm, t] 

since an interchange of the umbral symbols r, m in the last 
three-index symbol leaves the summation unaltered. 

Accordingly, on using the definition (6), the differential equa- 
tions of the geodesies are 

x (p) -f {rm, p}x r x m = (p = 1 n) 

From their derivation we know that these equations are contra- 
variant of rank one. We proceed now to obtain a general rule 
which makes the tensor character of equations of this type 
apparent on inspection. 

2. THE FORMULAE FOR COVARIANT DIFFERENTIATION 
From the covariant character of the g rt we have 

(/,m umbral) 



. dfr._ ^d^dx^dx^dx^ ( BW 

QyW Q X (n) Qy(T) Qy( t ) Qy(t) -T 9 lm \Qy(r)Qy 



H" 



) \ 

, n I (I, m, n umbral) 
y (i) ) 



* f \ , , ^ , 
dy (r) dy (t) dy ( 

where in the differentiation we have remembered that gi m is a 



THE ABSOLUTE DIFFERENTIAL CALCULUS 91 

function of the y's only indirectly through the ar's. We easily 
obtain two other similar equations by merely interchanging 
(r, and (*, t) in turn. We are careful to so distribute the 
umbral symbols I, m, n as to facilitate combination of the three 
equations obtained in this way. Thus 



. ( d z x 
9lm \dy^ 



dx (l) dy (r) dy (t) 

. 



dy (t) dy (r) dy (t) 



Now adding the first two of the equations and subtracting the 
third we have, on writing 



etc 



v _ dx(l) dx(m} dx(n) & x dx(m) 

ln > m l)M t^ gim " 



[I, m, n umbral] 
Now 



from its covariant character (p, q umbral) 

dx (m) , dy (p) 



To remove the coeflScient of , .^ ... multiply across by f s n ... 

dy (r) oy (t) oy (lc) 

and make s and k umbral when we get 



, m 



1 dy (k) 
from the relation (contravariant) 



,i umbral) 



92 VECTOR ANALYSIS AND RELATIVITY 

Finally 



from which on interchanging the role of the x and y coordinates 
we have 



_ = lrt M_ i /n , r 

W ' ' ' Jl 



Suppose now we have a covariant tensor of rank one X r so that 

Q x (r) 

F. = ^^-(7) ( r umbral) 

Then 



O i t __ Y v 



1*1 /.\ \ ( ]\ f\ { m\ 1 *\ \ r 

H-r\~> rtT^*'rlT^^' I ri X 

[ s t f k]' {lm, r\ [+ 



on altering suitably the umbral symbols Zm to rp. These 
equations state that 



is a covariant tensor of rank two. Consider now a contra variant 
tensor of rank one so that 

p " *f (r umbral) 

Then 



M 



dy (t) ~ Q X (P) Qy(t) Q x (r) 

, , 

(r)^ (t) {TP ' CJ 



. n m ,i' 
' ! 55 



THE ABSOLUTE DIFFERENTIAL CALCULUS 93 



These equations state that 



is a mixed tensor of rank two. 

These tensors of rank two are called the covariant derivatives 
of the covariant and contra variant tensors X r and X r respectively. 
Similar analysis can be carried out to obtain the covariant 
derivative of a tensor of any rank and character. To make this 
perfectly clear let us take the case of a mixed tensor X, T of rank 
two: 

yr= Y 



dy (t) ~ dx (l) dy w dy (t) dx (p) 




whence 



expressing that -^-^ X k p {ql, k} + X q k {kl, p\ is a mixed 

tensor of rank three being covariant of rank two and contra- 



94 VECTOR ANALYSIS AND RELATIVITY 

variant of rank one. In general, the covariant derivative of 



It will be noticed that + signs go with the contravariant symbols 
and negative with the covariant. Also the new label s is always 
second in the three-index symbols; the umbral label is first if 
taken from the contravariant and third if taken from the co- 
variant indices. 

3. APPLICATIONS OF THE RULE OF COVARIANT DIFFERENTIATION 

(a) Riemann's four-index symbols and Einstein's Gravitational 

Tensor 

From any covariant tensor X r we obtain as its covariant 
derivative 

X T = r-^jj Xk{rs, k} (k umbral) 

and as its second covariant derivative 

dra v 
I AT v , , 



X r t = 



Y 

" 



From this by the elementary rule (6), Ch. 2, 2, of tensor 
algebra we derive a new covariant tensor X rtt = X rt9 and the 
difference of these is a covariant tensor of rank 3 by rule (a), 
Ch. 2, 1; i.e., 



THE ABSOLUTE DIFFERENTIAL CALCULUS 95 

whence 

r a % 

X nt = X k \ {rt, k} - {rs, k} + {ps, k} {rt, p} 



- {pt, k} [rs, p} 



the terms involving the derivatives of the X r cancelling com- 
pletely out. Now Xk is an arbitrary covariant tensor of rank 
one and so by the rule (e), Ch. 2, 5 the converse of the rule 
(d) of composition 

a a 

faM^t' ^ ~fa&{ rs ' k ^ + lP s > k }{ rt >P\ ~ {pt,k}{rs,p} 

= Y & 
-A- rat 

is a mixed tensor of rank four of the type indicated by the 
positions of the labels. 

If we write k = t and use t as an umbral symbol we derive by 
rule (d) Einstein's gravitational covariant tensor of rank two 

Gr ^ 



The mixed tensor X rs t k is usually denoted by the symbol {rk, ts] 
and is known as the Riemann four-index symbol of the second 
kind. From it we obtain by the rule of composition the co- 
variant tensor of rank four 

[rj, **] = gjkXrst* = gjk(rk, ts} (k umbral) 

which is known as the Riemann four-index symbol of the first 
kind. From Einstein's tensor of rank two we obtain the in- 
variant 

G = g r 'G rs (r, s umbral) 

which has been called the Gaussian or total curvature of the space. 
This name is given since G is regarded as a generalization of the 
expression given by Gauss for the curvature of a surface (i.e., 



96 VECTOR ANALYSIS AND RELATIVITY 

n = 2). The term curvature is widely used in the literature of 
Relativity and so it may be well, in order to avoid a possible 
confusion of ideas on the subject, to discuss briefly what is 
meant by the curvature of a metrical space. To do this it is 
necessary to say a few words about the four-index symbols. 
We have, by definition, 

[a a 

ipgff***) ~fa&{ ps > k \ 

+ {pr, t}{ts,k} - {ps,t}{tr,k}'] 

Recalling that 

\P r > ?] = 9*k{pr, k} 
we have 

a a /)/7 

~ 



= fa& [pr, g]~ ipr,k\ ([qs, k] -f [ks, q]) 
from definition of [qa, k] so that on operating similarly with 



and wfang g q k{ts,k\ = [ts, q] we find 




a a 

[pq, rs] 3= \pr, q] - [ps, q] - {pr, k\[qs, k] 



, k] 

(the terms {pr, t}[ts, q] and {pr, k}[ks, q] cancel since t and k 
are merely umbral symbols) . Finally, in terms of the three-index 
symbols of the first kind, 

Iwwl s ^w^ d-^)b'd 

+ 9 kj ([p*, j\[qr, k] - [pr, j][qs, k]) (k, j umbral) 
Writing out, in the first two terms of this expression, the values 
of the symbols, e.g., 

,ird a d ~] 

[pr> q] " = 2 [d^ 9rq + d^ 9pq ~ e& 9pr \ 



THE ABSOLUTE DIFFERENTIAL CALCULUS 97 

we find 

, = i r a 2 a 2 a 2 

m ' n - 



a 2 



~\ 

r) 



From this formula it is apparent that 

(a) An interchange of the indices or labels p, q merely changes 
the sign of the symbol. 

\pq, rs] + [qp, rs] = 
(6) Similarly 

[pq, rs] + [pq, sr] = 

(c) A complete reversal of the order of the labels does not 
alter. the symbol [pq, rs] = [sr, pq]. This depends on the sym- 
metry relations g ki = g ik . 

(d) If we keep the first label fixed and permute the other three 
cyclically we get 3 symbols whose sum is identically zero, i. e., 

[pq, rs] + [pr, sq] + [ps, qr] = 

The number of non-vanishing symbols which are linearly distinct 
now follows. If p = q or r = s the symbol vanishes on account 
of (a) and (6). The number of choices for the first pair (p, q) is 

HZ = - - and similarly for the second pair (r, s). However 
& 

relation (c) shows us that we do not get n 2 2 symbols by combining 
the two choices but 

n 2 2 ^2(^2 1) = ^2(^2 + 1) 



The relation (d) will still further reduce the number of linearly 
distinct symbols. When the indices or labels p, q, r, s have 
numerical values which are not all distinct the relation (d) merely 
reduces to a combination of the relations (a), (6), (c). There are 
therefore n(n T)(n 2)(ra 3) new relations in (d). How- 
ever since there are three letters q, r, s permuted cyclically, each 



98 VECTOR ANALYSIS AND RELATIVITY 

relation will occur three times. Each of the relations (a), (6), (c) 

, n(n l)(n - 2)(n - 3) , . , . . 

reduces the number -- - - - - which remains in 

o 

half and so there are 

n 4 = n(n l)(n 2)(n 3) -5- 24 distinct relations (d). 
There are accordingly but 



distinct Riemann four-index symbols. For n = 2 there is out 
one which we may write [12, 12], When we change the coordi- 
nates from x to y we have 

, 
[12, 12], = (pq, ] 



(from covariant character) 

Since there is but one distinct symbol [pq, rs] it will factor out 
on the right and we get (since there are but four of the symbols 
which do not vanish) 

[12, 12] y = [12, 12] -J* 

d(z (1) z (2) ) 
where J is the Jacobian , ' - We have already seen that 



[12 12] 
/ = gJ 2 and on division we obtain the invariant K = - 

9 

It is this invariant which Gauss called the total curvature of the 
space of two dimensions under discussion. 
In order to compare this with the invariant 

g r 'G rt (r, s umbral; n = 2) 

we have 

G rt = {rt, ts} (t umbral) 



{since if p = I or t = 1, [Iptl] = by relations (a) and (6),} 
= - <7n[12, 12] -r g 



THE ABSOLUTE DIFFERENTIAL CALCULUS 99 

from definition of </ 22 , 

= - gii-K 
Similarly 

G u = 12 [12, 12] = - gn-K = - g n .K 

Gn = <7 21 [21> 21] = giz-K = gzi'K 

from relation (c), 

22 = n [21, 12] = -022-tf 
so that 

/<? = - Kfg n = -2K 
since 

- 2tf (r umbral) 



For a space in which, in some particular coordinate system x, 
the coefficients g rs are constants all the three-index symbols 
[pr, s] and in consequence all the symbols {pr, s\ and also the 
four-index symbols [pq, rs] and {pq, rs} = 0. On account of the 
tensor character of these latter symbols we know that the Rie- 
mann tensors [pq, rs] {pq, rs} will be zero no matter what the 
coordinates are. Conversely the vanishing of the tensor [pq, rs] 
expresses the fact that it is possible to find coordinates y such 
that the f r defined by the equations 

Q X (l) #("*) 

t~ m **yfiipS (/,m umbral) 

shall be constants. We may now apply the well-known method 
of reduction of a quadratic expression to a sum of squares (as in 
the determination of normal vibrations in dynamics where the 
expression for the kinetic energy is reduced to a sum of square 
terms) ; the transformations on the y's are linear in this operation 
and we finally get 



(If we restrict ourselves to real transformations there may be some 
negative squares; thus in the relativity theory there are three 
and one + term.) A space of this character is said to be Euclidean 



100 VECTOR ANALYSIS AND RELATIVITY 

and the y's are called orthogonal Cartesian coordinates. Rie- 
mann defines curvature by means of his tensor [pq, rs]. When 
this tensor vanishes the curvature of the space is said to be zero 
so that Euclidean space is one of Zero Riemann Curvature and 
conversely. If the ratio of the component [pq, rs] of the curvature 



tensor to the two-rowed determinant 



9p 



is the same for 



all values of p, q, r, s, Riemann says the space is of constant 
curvature; otherwise the curvature will be different for different 
orientations at a point:* Gauss' total curvature, on the other 
hand, has a numerical value at each point in space and has 
nothing to do with the different orientations at that point. We 
may sum up by saying that a gravitational space is, at points 
free from matter, non-Euclidean, i.e., it has a Riemann curvature 
but its Gaussian curvature is zero. 

It may be well to call attention to the fact that the definition 

* The differential equatic ns of the non-minimal geodesies of any space are 

<Px" . ( lm\ rfz> dx<> 

-df- + i r S'dr~dT " (f " *' " *l1"*D 

a being the arc length along the geodesic. It is known that the solutions i (r) 
of these equations are completely determined by the values of x (r> and 3 

for a particular value of , = let us say. This is stated geometrically by 
saying that through any point in space there passes a unique geodesic with a 
given direction. If, now, through a definite point we construct the geodesies 
with the distinct directions (r) and > <r) respectively (r 1, -, n) and con- 
sider the family of geodesies through the point in question obtained by assign- 
ing to each a direction tensor whose rth component is proportional to 
\%(r) _|_ ^w an( j then letting the ratio X : M vary, we obtain a geodesic spread 
Vt of two dimensions which at the point in question has the orientation deter- 
mined by the two directions and j through the point. It is the curvature of 
this geodesic Vi that Riemann calls the curvature of the space relative to the 
orientation determined by ( and ij. There is a remarkable theorem due to 
Schur (Math. Anualen, Bd. 27. p. 563, 1880) which says that if at every 
point the Riemann curvature of space is independent of the orientation the 
curvature at all points is the same. Such a space is, then, properly called 
a space of constant curvature. 



THE ABSOLUTE DIFFERENTIAL CALCULUS 101 

of Euclidean space given above is a " differential " definition; 
spaces which are Euclidean according to this definition do not 
necessarily satisfy the postulate that one can proceed indefinitely 
in a given direction without coming back to the starting point. 
The simplest example is the well-known one of a cylinder of unit 
radius. In this case n = 2, y m = <f>, the longitudinal angle 
measured in radians, and 7/ (2) = z, the distance measured parallel 
to the axes of the cylinder: 

(<fe) 2 s (<Z0) + (dzY = W>) 2 + W 2) ) 2 

If the cylinder is cut along a generator and developed on a plane 
it will cover a strip of breadth 2r on the plane. If we take 
rectangular Cartesian axes in the plane, with the x (l) axis parallel 
to the strip, points whose z (2) = < differ by 2ir correspond to a 
unique point in the strip (that one with the same x w ) and to a 
unique point on the cylinder. Hence there are an infinity of 
straight lines (i.e., geodesies) joining any two points (with 
different z's) on the cylinder. They develop into the oo l straight 
lines joining the points 

(1 >, x + 2nir) (n = 1, 2, ) 



on the plane. It is evident that speculations as to the " finite- 
ness " or " infiniteness " of a space based on its differential 
characteristics must be regarded with distrust. 



CHAPTER VII 

1. In this final chapter we shall treat in a brief way, as an 
application of the preceding analysis, the classical problems of 
Relativity. As in other applications of the methods of mathe- 
matical analysis to problems in physics the first, and here the 
most serious, difficulty is that of giving a physical significance 
to the coordinates. All systems of coordinates are, without 
doubt, equally valid for the statement of the laws of physics 
but not all are equally convenient. It is reasonable to suppose 
that for a given observer of phenomena a certain coordinate 
system may have a direct and simple relationship to the measure- 
ments he makes; such a coordinate system is called a natural 
system for that observer. It is necessary to define this natural 
system and to find by experience, or otherwise,* how the natural 
systems of different observers are related. This has been well 
done in the special or " Restricted Relativity Theory " but in 
the more general theory, which we propose to discuss here, 
much remains to be done in this part of the subject. In what 
follows we shall consider (a) the problem of determining the 
metrical character of the space-time continuum round a single 
gravitating center and (6) in consequence of the results of (a) 
the nature of the paths of a material particle and of a light ray 
in a gravitational field. We shall, following Einstein, make the 
fundamental assumption that the space which has a physical 
meaning or reality, i.e., with reference to which the laws of 
physics must have the tensor form (cf. Ch. 2, 1), is one of four 
dimensions (commonly referred to as the Space-Time continuum). 

* The relationship between the different systems may be arrived at by 
making various hypotheses whose truth or falsity must then be tested in the 
light of experience. 

102 



PROBLEMS IN RELATIVITY 103 

2. THE METRICAL SPACE ATTACHED TO A SINGLE GRAVITATING 

CENTER 

We assume that for an observer attached to the gravitating 
center one of the four coordinates, x (4) say, of his natural system 
is such that the coefficients gu, gzt, #34 of the quadratic differential 
form for (ds) 2 vanish identically whilst those remaining are 
independent of z (4) ; z (4) is said to be a tune coordinate and the 
field is said to be statical. Accordingly 



(ds)* = 044(<fo (4) ) 2 + g lm dxWdx (I, m = 1, 2, 3 uinbral) 



Now in any space of three dimensions we can always find orthog- 
onal coordinate systems; for, writing the metrical (d$) 2 in its 
reciprocal form (ds) 2 = / ra 7? r T7, we have merely three equations 
f r * == -Q (r =J= *) or explicitly 



a i m 

9 



to determine the three unknown functions y of x so that the 
coordinate curves y may be orthogonal. There is no lack of 
generality, then, in writing (ds) z for the statical field in the 
orthogonal form 



where we have dropped the double labeling as unnecessary. (In 
general it is impossible to find orthogonal coordinates in space of 
four dimensions since there are now six differential equations 
f r> =0 (r ^ s) for the four unknown functions y and these 
equations are not always consistent.) We must now go through 
the details of evaluating Einstein's gravitational tensor (cf. Ch. 
6, 3) for an orthogonal space. 

The relations g rs = 0; g rt = if r 4= s make matters com- 
paratively simple. We shall use r, s, t to denote distinct numer- 
ical values of the labels. Then 



104 VECTOR ANALYSIS AND RELATIVITY 



H(^> + i^-IfsH by<Mnition 

{rs,t} = ^[rs, k] ss g'^rs, t] = Q (k umbral) 



, r\ = {sr, r\ = /*[-, *1 - rr , r] = 



being the only umbral label here) 



similarly and 

{rr, 3} = - 9~J17 

The Riemann four-index symbol of the second kind (cf. Ch. 6, 
3) is defined by 

t \ ^ t \ I iif 



[pa, I] {/r, q] (I umbral) 

and those components vanish identically, for an orthogonal 
coordinate system, where the pq, rs are distinct; [{pr, 1} vanishes 
unless / = p or r in which case {/, q\ vanishes]. To evaluate 
the remaining symbols write r = q without, for the present, 
using q as an umbral symbol 

1 dQ a \ . 1 d(j a dg 
T ~r 



dg p dg q 1 dg, 



4g t g q dx (p) dx (l) 

The formulae from this on take a simpler form if we use the 
symbols H defined by g r z= H r 2 ', thus 

1 d * H * 1 dH p dH q 



PROBLEMS IN RELATIVITY 105 

Similarly we find 
{pq,qp} = - 



1 d*H q .H p d*H p H p dHpdH q 



. **.p I 1 dHp dHq 
+ 



. 
*" 



1 dHp 



where r and s are the two labels different from p and q. The 
components of the Einstein tensor are now found by summing 
with respect to q. It will be recalled that [pp, rs] = (p, r, s 
any values distinct or not, cf. Ch. 6, 3). Hence 

{pp, rs} = g pk [pk, rs] = g pp [pp, rs] = (k umbral) 

Similarly {pq, ss] = 0, so that in forming Gu, for example, 
we need merely write 

12 = {13,32}+ {14,42} 
whilst 

<?us {12,21} + {13,31}+ {14,41} 

It will be observed that differentiation with respect to x (p) and 
x (>) occurs in every term of {pq, qs} and so the absence of the time 
coordinate z (4) from the coefficients makes GU, GU, G& all iden- 
tically zero. 

We shall now make the following hypotheses of symmetry (a) 
we shall suppose that the coordinate lines z (1) are geodesies of the 
space (all passing through the gravitating center). The equa- 
tions of the non-singular geodesies have been found to be (Ch. 6, 

D 

x (r) + [lm, r}x (I) x (m) = (r = 1, -, 4; /, m umbral) 

where dots denote differentiations with respect to the arc distance 
which we take as our coordinate x (1) . Writing 

4(8) = = z (3 > = i (4) , xi = 1 



106 VECTOR ANALYSIS AND RELATIVITY 

(since x (2) , x (8) , x (4) are constant along the coordinate lines x (I) ) 
we find {11, r} = which from the values given for this symbol 
yields g\ = constant. The constant is in fact unity since, 
by hypothesis, ds = dx m along the curves x (2) = const., 
(3) = const., x (4) = const. It is apparent that it is sufficient 
that ^i be a function of x (1) alone for we may make a change of 
variable x (1) = x (1) (y (1) ) leaving the other coordinates unaltered; 
the argument shows conversely that if g\ is a function of x (1) 
alone the coordinate lines x (1) are geodesies, the arc length along 
them being given by * = J V<7i dx (l \ 

(6) x (2) and x (a) are directional coordinates serving to locate a 
point on the geodesic surface x (1) = const., x (4) = constant. We 
shall suppose that the arc differential on this surface (which 
may conveniently be called a geodesic sphere) cannot involve the 
" longitude " coordinate x (>) nor can the arc differential along a 
given " meridian " x (8) = constant depend on the " latitude " 
coordinate x (2) . Hence g t is a function of x (1) alone whilst </ 3 is 
a function of Xi and x> alone. 

(c) 04 does not involve the directional coordinates x (2) and x (3) 
and so is a function of x (1) alone. 

Accordingly, then, x (8) does not appear in the expression for 
(<fo) 2 and so, in addition to Gu = 0, Gu = 0, (734 = we have 
GU = 0, (?23 = 0. We must wiite down the five equations 
Gu = 0, Gu = 0, G n = 0, 33 = 0, Gu = 0. The fact that H 4 
is a function of x (1) alone and HI = 1 (x (1) being the arc distance 
along the geodesic curves x (1) ) gives {14, 42} = and from 
(? = {13,32} = Owe get 



which gives, on integration with respect to x (1) , 

1 dH 

-=5- s-ja = a function independent of Xi (A) 

.a 2 ox' ' 



PROBLEMS IN RELATIVITY 107 

n = yields 



(?44 = gives 



t g 3ff 4 [ 1 dH 2 3 _ Q 

* 



which on integration with respect to z (1) gives 

independent of * (C) 



f 

Eliminating -^ between (C) and 04) we get H z * -r-^ independ- 
2 ax (1) 



ent, of x (1) . Since it cannot involve any variable but x (1> we 
have 

H<?Hi a constant a, let us say; (C") 

primes denoting differentiations with respect to x m . 

f)H dd> 

Again from (-4) -^ = F 2 X a function of a; (2) = H 2 ^-^ say 

where ^> is a function of z (2) alone. Then H 3 = ^^ + / where 

1 f&i] 

/ is a function of z (1) alone. Now (5) shows that - is a 

- 2 



function of z (1) alone so that its derivative with respect to 
vanishes. Evaluating this derivative we find 



= 



We can now proceed in various ways; either make < a constant 
or fHz' f'Hz (of which the second factor is the derivative) a 



constant giving / = const. X Htj TTT^ We choose the latter 

HZ 

alternative and make the constant zero so that / = giving 
H 3 = H 2 <j> where <J> is a function of z (2) alone. 



108 VECTOR ANALYSIS AND RELATIVITY 

<p is determined by means of the equation Gzz = 0. This gives 

- 2 HW = (D) 
#4 

On substituting H 3 = Hrf in (D) we find that - - - is equal 

' 



to a function of x (1) alone; but from its form and the definition 
of it cannot involve a: (1) and so must be a constant. This 
constant may, by a proper choice of unit for ar (2) , be put either 
1 or zero. We choose the first alternative and find, by suitably 
choosing the origin of measurement for z (2) , < = sin x (2) . 
33 = gives 

1 dH, 1 dH< 

-- --- ---- } = V 



and on substituting <f> = sin x (2) , ^3 = Hrf, both (D) and 
yield the same equation 

HW + HM { jjJjL' + jjj I . i 

(B) gives 



On difiFerentiating (CO and eliminating H^H^" we find 

//2"/l4 = Hi'Ht 

which gives on integration 

#' = /3T 4 
where is an arbitrary constant. 

Eliminating # 4 between (CO and (C") we have tf," 
which on integration gives 



where 7 is an arbitrary constant. 



PROBLEMS IN RELATIVITY 109 

Putting # 4 = (Ht')fP in (#') we have 

2# 2 # 2 " + (// 2 ') 2 = 1 

so that 1 = 2y giving 7 = 1 and hence finally H 2 is determined 
by the differential equation 



and then 

# 2 2 <fo< 2 > 2 + sin 2 



It is usual to change the coordinate z (1) , leaving the others 
unaltered. We write x (1) = x w (y w ) where y (l) = H 2 . 



and we have 



This is the form chosen by Einstein (that it is only one of many 
is evident from its derivation). If aft = it reduces to the 
well-known Euclidean form where y (l) = r, z (2) = 6, x (3) = $ are 
space polar coordinates. It remains to attach some physical 
significance to the constant a/3 and to take up the problem (6) 
stated at the beginning of this chapter. In order to conform 
to the usual notation we write henceforth y w = ir; z (2) = 6; 
x = <j>; z (4) = t where i 2 = - 1. 

Choosing the unit of z (4) or t so that /3 2 = + 1 and writing ia 
= a we have 

"' 



= - { ( 1 - 



sn 



110 VECTOR ANALYSIS AND RELATIVITY 

3. DETERMINATION OF THE PATH OF A FREELY MOVING 

PARTICLE 

A physical law of inertia is postulated to the effect that a 
freely moving material particle in a gravitational field mil follow 
the non-minimal geodesic lines of the four-dimensional space time 
continuum which, for the single gravitating center, has the 
metrical geometry characterized by the form given above by 



A second postulate is that rays of light follow the minimal 
geodesies those for which ds = 0. In the ordinary Euclidean 
space these lines are imaginary, i.e., have points with imaginary 
coordinates but the occurrence of the negative signs in the ex- 
pression for (ds)* gives real minimal lines in our problem. For 
example, the light rays directed towards or away from the cen- 
ter, those for which 6 and < are constant, are characterized by 
the equation 

.(,.5)^., or j- 

In order, then, to solve the problem of the free motion of a 
material particle we have merely to determine the non-minimal 
geodesies whose equations are 

x (f) + {lm, p}x w x (m) = (cf. Ch. 6. 1) 

the dots denoting differentiations with respect to the arc length 
along the geodesic. For an orthogonal space of four dimensions 
these simplify to four equations of the type 

x (1) + {11, l}z< 1)2 + {22, l}(z<) + {33, l}(z (8) ) z 

+ {44, l}(z>)* + 2{12, 1 }*>*< + 2(13, l)z (1) z< 8) 

+ 2{14, l}z (1) z> = 

However we need use only three of these equations, replacing the 
fourth by g r x (r) x (t) = 1 which is easily seen to be a consequence 
of the differential equations 

z (r > + {lm, rJzWxO-o = 



PROBLEMS IN RELATIVITY 111 

(if we multiply these by r and use r as an umbral symbol to 
obtain 

and then avail ourselves of the definition of the symbols 
[lm, s]. (Ch. 6, 1.) j- (grtXrXs) is found to be zero). In our 

problem it is convenient to omit the first of the four equations, 
the other three simplifying, on using the values for the three- 
index symbols given (Ch. 7, 2), to 

* 1_ /,v 2 ^0J , J[_d02 .a _ n ,,, 

202 vv 02 C/7" 



where 0i = ( 1 ) ; 

02 = 7-2; 

03 = r 2 sin 2 6; 



To these we have to add the first integral 

gii* + g*P + gw* + g*i* = 1 (D) 

Equations (5) and (C) are immediately integrable giving 

gzb constant = h say (B 1 ) 

and 

gd = constant = + C say 

or on substituting the values of gz and 4 

r 2 sin 2 0^ = h; (l-~\i-C 
Equation (^4) may be written 

(r 2 ^ - r 2 sin cos ^(<^ 2 = 



We now proceed to eliminate the parameter s and find a relation 
connecting 6 and 0. Assuming that < 4 s (0 = constant is a 
special case which is susceptible to the analysis given below on a 



112 VECTOR ANALYSIS AND RELATIVITY 

mere interchange of and tp) we have 

6- 
so that 



On substituting the value of </> from (B) we have 
j 



= {2rr'6' + rtf"}^ 2 - r 2 0' - r'< 2 + 2 cot 00'< 2 

where we denote differentiations with respect to the new inde- 
pendent variable <f> by primes. Equating this to r 2 sin 8 cos 0(<) 2 
and dividing out by r 2 ^ 2 we obtain 

0" - 2 cot 0(0') 2 = sin 6 cos 
If now we choose our directional coordinate so that initially 



We see that 0" = and then on differentiating the above equa- 
tion with respect to <f>, B'" = and so for all the other derivatives, 
i.e., is a constant as <f> varies. Otherwise expressed the general 
integral of the equation for as a function of <p is found by 
writing z = cot yielding 2" + z = to be cot = L cos (<f> -f M) 
where L and M are arbitrary constants. We choose our initial 

1P 

conditions as above so that L = giving = - . Putting in this 

value for we find 

1*4 =h (BO 



(0") 

\ r / 

and from (D) 



PROBLEMS IN RELATIVITY 113 

Just as in the usual Newtonian treatment of planetary motion, 
it is convenient to write r = l/u and to again use <p as the 
independent variable. We have 

r = u/u z = u'<j>[u? = hu' (from J3') 

and then (Z)') yields, on making use of (C"), 

nz 1 Onu 

(u'Y + <u? = 2au* + g- + -- () 

Now, in the Newtonian treatment, the equation giving the path 
of a particle under a central force is 

u" + u = F/tfu? 

where F is the acceleration towards the center and h=r z - 

dt 

is the constant of areas. Instead of this we have on differen- 
tiating the equation (E) just obtained 

u" + u = 3au z + ?L 
/r 

so that we may, in a general manner, express Einstein's modifica- 
tion of the Newtonian law of gravitation by saying that there is 
superimposed to the inverse square law attraction an inverse 
fourth power attraction, the relative strength of the attracting 
masses being as 1 : 3h z . It remains to determine, at any rate 
approximately, the nature and magnitude of the constants a, h 
and C which arose in the integration of our differential equations. 
For large values of r, and therefore small values of u, the New- 
tonian law is a first approximation and so neglecting the term in 
u z in the equation for u", a = F/u z = jura; ju being the gravita- 
tional constant and m the mass of the sun. Hence if we choose 
our unit of mass so that n=l,a = m, where now m is what is known 
as the gravitational mass of the attracting center (notice that we 
have identified, for small values of u, our r and < with the usual 
polar coordinates of Euclidean geometry). The velocity of light 

directly towards the attracting center is 1 and accordingly 

f 



114 VECTOR ANALYSIS AND RELATIVITY 

our unit of time is such that for small values of u the velocity of 
light is unity; i.e., if the unit of length be 1 cm., the unit of 
time employed is 1/c seconds where c = 3.10 10 . In the theory 
of relativity there is no absolute distinction between space and 
time and so we refer to our time unit as one centimeter (1 cm. 
being the distance traversed by light in one time unit). It is 
to be observed that in Newtonian mechanics gravitational mass 
m has dimensions L*T~ Z so that if L and T have the same dimen- 
sions a = m has the dimensions of a length. The equation 

n i m 

"+- p 

of the Newtonian theory yields 

u - j z = P cos (<f> - ) 

where P and fa are arbitrary constants of integration. 
Comparing this with the polar equation of a conic 
lu = \-\-e cos <t> (I = semi-latus rectum, e = eccentricity) 

h* 

we have = I = A(l e-) where A is the semi-major axis. 
m 

If T is the period of revolution 
2 X 



whence 

m = W{A(\ - r) = 



where o> is the angular velocity of the planet. This gives for the 
sun m = 1.47 kilometers or 1.47.10 6 cms. For the planets then 
m/r is a small quantity of the order 10~*. In order to determine 
the constant C we differentiate 

m ,- ... 

u = rj (1 + e cos 0) 

and find 

(') + if = j(l + 2e cos* + e 2 ) = ?gu - g(l - **) 



PROBLEMS IN RELATIVITY 115 

and comparing this with the equation (E) we have 



It is to be observed that the values of m, C and h obtained in 
this way are found from the Newtonian theory and so are to be 
regarded as first approximations. In particular we have iden- 

tified the h of (J3') with r 2 -j- so that we have written -^ -^ 

at ds at 

Accurately 

(fromC") 



ds dt ds dt 
But 

M z 



= d- -V 

\ A/ 



so that neglecting quantities of the order 10~ 8 

d<p d<p 
ds dt 

Substituting the expressions just obtained in (E) we have to 
integrate the first order differential equation 

f^Y = 2mu* -u?+ 2mu/h* - m 2 (l - 



This equation defines u as an elliptic function of <f>; or inversely 
<f> as an elliptic integral. It simplifies the algebra somewhat to 
write mu v and to put ra 2 /A 2 = a. We have already seen 

that ra 2 /F = - - so that if e is not very nearly equal to 



unity a is a small quantity of the same order of magnitude as m/A 
or 10~ 8 . Our equation is now 



(//7)\ 2 
M = 2tf - 



Now the discriminant of the literal cubic 

tti* 2 + O2 + 3 = 



116 VECTOR ANALYSIS AND RELATIVITY 

is 



4ai 3 a 3 27a 2 a 3 2 

For the cubic on the right-hand side of the equation giving (dv/d<fi)* 
this is 

8o 3 (l - Oe 2 ) - 108a<(l - e 2 ) 2 . 



On account of the small magnitude of a this is positive, the first 
term being the dominant one. Hence the cubic has three real 
roots which we denote, in descending order of magnitude, by 
i, tfc, *> When a = the roots are , 0, 0, and so we try first 
= ka and find k = (1 e) or (1 + e) and then secondly 
i + fra and find k = 2. Hence, to a first approximation, 
the three roots of the cubic giving (dv/dpy are t> 3 = a(l e); 
t = a(l + ); t>i = ^ 2a. Further since (dv/d<f>)* cannot be 
negative in the problem t must lie between t> 3 and 0j or between 
d and + oo . As r does not tend to zero v does not tend to oo 
and hence t> lies between Vt and 0j. We have 



"I 

*/, 



dv 



The variable t> oscillates between the values r 3 and t>j; at these 
values dv(d<f> = 0, so that v has an extreme value; as v passes 
through the value t>j retracing its values both dv and the radical 
change signs so that <f> steadily increases. The change in <p 
between two successive extreme values of v, i.e., between peri- 
helion and aphelion of the planet, is 

p dt 

&<f>= I 

Jn 



It is convenient to make a simple linear transformation of the 
variable of integration. Write v = a + 62 and determine the 
coefficients a and 6 of the transformation so that to the roots r 
and 2 of the cubic will correspond values and 1 of 2 respectively. 
The values are a = r 3 ; 6 = v t v t and then the third root v\ 



PROBLEMS IN RELATIVITY 117 

goes over into 2 = ^ where k 2 = - - The cubic 2(v Vi) 

(v Vz)(v vz) transforms into 26 3 2(1 2) [j- 2 2) 

so that 

k C 1 dz 



This simplifies considerably on writing 2 = sin 2 6 when in fact 

2k C' 12 d6 



Now 

jo 02 ^3 2ae 



(to a first approximation) is a small quantity of the same order 
of magnitude as a; hence we can expand (1 k z sin 2 0)~ 1/2 in a 
rapidly convergent series and a mere integration of the initial 
terms will give a very good approximation to A<p. The multiplier 
of the integral is 



2 = x2M - = 2[1 - 2a(3 - e)]' 1 / 2 = 2[1 + a(3 - e)] 
and using J*' /2 sin 2 6 d8 = x/4 we find 



but F = 4ae to a first approximation so 

A?> = 7r{l + a(3 -e)}{! + ae] = 

Hence in a complete revolution the perihelion advances by an 
amount equal to 

m z _ 3m 

6<X 6 -rr -- 



- e 2 ) r 2 (l - e 2 ) 
of a complete revolution, T being the period in our units. If we 



118 VECTOR ANALYSIS AND RELATIVITY 

wish to use the period in seconds and measure A in kilometers 
then the unit of time in the formula given is the time it takes 
light to travel 1 kilometer = 1/3. 10 5 seconds; hence if T is the 
period in seconds the fractional advance of the perihelion per 



revolution is , ftin<M/ , - 57 On substituting the values of 
9.10 1 (1 (,) 

A, T, and e for Mercury's path this works out to be an advance 
of 43" per century. For the other planets e is much smaller than 
for Mercury and the amount of advance of perihelion is much 
smaller; save in the case of Mars the predicted advance is too 
small to be detected by observation. 

4. THE PATH OF A LIGHT RAY IN THE GRAVITATIONAL FIELD 
OF A SINGLE ATTRACTING CENTER 

These paths satisfy the equation (ds) 2 = or ds = 0; they 
are geodesies since, ds being the non-negative root of the expres- 
sion for (ds) 2 , no curve can have a negative length. The method 
of the preceding paragraph does not, however, immediately 
apply since the arc length along a light ray, being a constant, 
cannot be used as an independent variable or parameter in terms 
of which the coordinates x may be expressed. Further in the 
discussion of Oh. 6, 1, it was assumed that the integral 



could be expanded in a Taylor series in powers of a so that the 
existence of the derivative (d//da) a= , was presupposed. It is 
apparent, however, on differentiation of 

ds = ^g tm dx (l) dx (m) (I, m umbral) 

a 

that if ds = when a = 0, (ds) becomes meaningless when 

da 

a on account of the zero factor (cfo)._o which occurs in the 
denominator. These difficulties are overcome in the following 
manner. If we investigate those curves (non-minimal) for which 



PROBELMS IN RELATIVITY 119 



the first variation of the integral I = J*(ds)~* is zero we are led 
to exactly the same differential equations as those of Ch. 6, 1, 
which express the fact that the first variation of / = fds is zero. 
Accordingly we now derive the equations of the minimal geodesies 
from the fact that the first variation of I = S(ds) z is zero, ds 
being zero along the curves. The coordinates x are supposed 
expressed in terms of any convenient parameter v and differentia- 
tions with respect to this parameter are denoted by primes. 
The Euler-Lagrangian equations are (cf. Ch. 6, 1) 

dF __ d ( dF \ _ 

I ~i I \T if ' ' ' t *) 



F = (fo) = g lm x^'x^' (I, m umbral) 

= 2 (g r ix (l)t ) (I, m umbral) 

= 2 




or 

_(!)" | f] m .1^.(/)'/-(m)' A (1 tnt ifm\\f.-i}\ 

ffrix ~T~ [i Tiit f T\X c \i) m uiiiorai,/ 

Multiplying by g 1 " and using r as an umbral symbol we obtain 

z<*>" + {lm, p}x^'x^' = (p = 1, 2, 3, 4) 

which are exactly the equations of Ch. 6, 1. The first integral 
of these equations which has already been mentioned may be 
very briefly obtained as follows. Since F = gi m x (l) 'x (m) ' is 

* Attention should, however, be called to the fact that this integral is not, 
properly speaking, a line integral at all; its value depends not only on the 
curve over which it is extended but on the particular parametric mode of 
representation chosen for this curve. In order that the value of the integral 
should not be dependent on the parametric representation the integrand should 
be positively homogeneous of degree unity in the derivatives z'. 



120 VECTOR ANALYSIS AND RELATIVITY 

homogeneous of degree 2 in the x' we have, by Eider's theorem 
on homogeneous functions, 



a result immediately verifiable directly (r umbral). On multiply- 
ing the equations 



by x (r)/ and using r as an umbral symbol we obtain 

(r) , dF d f (r) , dF \, dF (r) /, 

x {r > I x (r) - I H 1 x (T > = 

oc uD V dx / doc 



or 

^_2^=0 

r/r '/r 

showing that F is constant along the geodesies. The constant 
is now zero instead of unity as it was in the case of the non- 
minimal geodesies. 

Before proceeding to calculate the deflection of the light rays 
it will be well to prove an often quoted property of them. In a 
statical gravitational field the time coordinate z (4) does not enter 
into F = (d*) 2 . Hence 

d f dF \ dF 

I 1=0 or 



\ 
. 1 
V 



or - . = const. 



If, now, in the discussion of Ch. 6, 1, instead of keeping both 
ends of the " varied curve " C(a) fixed, we had allowed the ends 
to vary also, the part of 81 which came outside the sign of integra- 
tion when we integrated by parts would not vanish automatically. 
Since the first variation is to vanish when the end points are 
fixed as well as when they vary the part under the sign of integra- 
tion vanishes as before yielding the Eulerian equations but in 



PROBLEMS IN RELATIVITY 121 

addition we have the end condition 

2 " = (r umbral) 



f 
d*' 

If now all the coordinates but z (4) are kept fixed 



ait 

and we find since - . is constant over the extremal curve 
' 



dF | =o or 5/iW = 



and as 

dF 



we have 

= o 



which is known as the Fermat or Huyghens' Principle of Least 
Time. It is an immediate consequence of the absence of x w 
from (<fo) 2 ; there is a similar theorem for the symmetrical 
attracting center: 

= o 



but this has no special utility. The Fermat Principle states that, 
given two fixed points in space (by fixed is meant that the three 
space coordinates for an observer attached to the gravitating 
center are constant), a light signal passes from one to the other 
in such a way that the first variation of the time interval is a 
minimum. 
With the same notation as that employed in 3 we find 

g*v' = h; git' - C 

where h and C are constants and we find exactly as before that a 
proper choice of our initial conditions for 6 enables us to 



122 VECTOR ANALYSIS AND RELATIVITY 

write 6 = r/2. The only difference is that (Z)') is replaced by 

(1 - 2m/r)- 1 (r') 8 + rV) 2 - (1 - 2m/r)(O 2 = 
whence on writing r = l/u and using rV' = A we find 

"A 8 

In order to get an idea of the order of magnitude of the constants 
C, h of integration we make a first trial-approximation. The 
largest value that u can have is IfR where R, the radius of the 
sun, = 697,000, the units being kilometers. Hence we neglect, 
for the moment, the u 3 term in comparison with the others and 
find at once 

u = sin (<p <po) 
h 

where ^o is a constant of integration. Hence Cfh is the largest 
value of u and is therefore a small quantity of the order 1/10 6 . 
Denoting this small quantity by a (a is the positive square root 
of CVA 2 ) we have 



The discriminant (cf. Ch. 7, 3) of the cubic on the right is 
4o 2 (l 27m 2 a 2 ), a positive quantity, so that the three roots are 
real. When a = they reduce to l/2m, 0, 0, so that trying in 
turn ka and (l/2m) -H ka we find the first approximation to the 
three roots u 9 = a, v* = a, Ui = l/2m where we have ar- 
ranged the roots so that u% < Wj < u\. For a second approxi- 
mation, we try in turn a + fco 2 , a + kc?, (l/2m) -f kc? for 
W|, ut, Ui respectively and find 

w 8 = a + wio 2 ; ut = a + mo?; u\ = (l/2m) 2mot 2 . 



We now, as before, determine a linear transformation which 
sends u u 3 into z = 0; u = u* into z = 1. It is u = a -f- bz 



PROBLEMS IN RELATIVITY 123 

where = 2/3, b = u z w 3 and then the third root uu\ 
goes into z = 1/fc 2 where 



Now the cubic 2mu 3 w 2 -f- a 2 cannot be negative in our problem 
nor can u itself. At remote distances from the sun u - so 
that initially u = and it increases to u = u z at which point u 

has a maximum value, since ( -y- ) = there. Then M- begins to 

\d<pj _ 

decrease and the radical V2wm 3 if + 2 in the expression for 
d(f> 

, du 



a 2 

also changes sign so that d<p keeps its sign. The angle < between 
a point at a remote distance and the perihelion of the light ray 

is given by the integral I ,. The excess of 

Jo V2mw 3 - w 2 + a 2 

twice this over TT is the deflection D experienced by the ray. 
Hence 

du 



=2 r 

Jo 



9 I 9 

u H- or 

which on writing u = a-\- bz becomes 

dz 



">(H 



On making the final substitution 2 = sin 2 6 this becomes 

d0 



nj 

D -f 7T = 



_ 

J_!9_ Vl - fc 2 sin 2 

V , M, 

Now 

= 4ma -\- higher powers in a 



-- f- a 
2m 



124 VECTOR ANALYSIS AND RELATIVITY 

so that, m being 1.47, k 2 is a small quantity of the same order 
as a. Hence (1 A^ sin 2 0)~ 1/2 can be expanded in a rapidly 
convergent series and an integration of the initial terms of this 
series gives a high approximation to D + TT. On substituting 
the values of k and 6 the multiplier of the integral becomes 



= 4(1 -f 2roa)- 1 / 2 = 4(1 - ma) 

whilst the lower limit of the integral is 

. _, /I ma ._,!,, i v 
sin 1 ^ = sm l -^ (1 -- ma) 

Here it is necessary to use the second approximation since u^ is 
to be divided by w 3 M? itself a small quantity of the first order. 
On expanding 



by Taylor's theorem we have for the lower limit (ir/4) \ma 
so that 

D + T = 4(1 - m 



In the term multiplied by 



it is sufficient to take the rough approximation r/4 to the lower 
limit and we have 



gvng 

D = 4ma 

a, in this expression, is the maximum value v* of u (to a first 



PROBLEMS IN RELATIVITY 125 

approximation), i.e., is the reciprocal of the radius of the sun. 
An idea as to the closeness of this approximation is obtained 
by using the second approximation 

= U2 = a + mo? 
R 

The positive root a of this quadratic is 

so that writing a = l/R is equivalent to neglecting m/R in 
comparison with unity or to a neglect of 1 part in 5.10 5 . On 
substituting m = 1.47, R = 697,000 in the expression D = 4m/ R 
and converting this radian measure into seconds of arc we find 
the value 1.73" predicted by Einstein for a light ray which just 
grazes the sun.* 

*For a fuller discussion of the problems dealt with in this chapter reference 
is made to two papers by the author in the Phil. Mag. of dates Jan. (1922) 
and March (1922) respectively. For an alternative treatment of the subject 
matter of 2 the reader should consult the paper Concomitants of Quadratic 
Differential Forms by A. R. Forsyth in the Proc. Roy. Soc. Edin. May (1922) . 



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