Skip to main content

Full text of "The meaning of relativity"

See other formats



( MAY 2 1933 J 



Digitized by the Internet Archive 
in 2019 with funding from 
Princeton Theological Seminary Library 







VUY 9 10' 

‘ r\ I AJ 



BY / 






Copyright 1922 
Princeton University Press 
Published iq22 


Note.— The translation of these lectures into English 
was made by Edwin Plimpton Adams, Professor 
of Physics in Princeton University 



Space and Time in Pre-Relativity Physics 


The Theory of Special Relativity 


The General Theory of Relativity 


The General Theory of Relativity ( co ? itinued ) 






HE theory of relativity is intimately connected with 

-1 the theory of space and time. I shall therefore begin 
with a brief investigation of the origin of our ideas of space 
and time, although in doing so I know that I introduce a 
controversial subject. The object of all science, whether 
natural science or psychology, is to co-ordinate our experi¬ 
ences and to bring them into a logical system. How are 
our customary ideas of space and time related to the 
character of our experiences ? 

The experiences of an individual appear to us arranged 
in a series of events ; in this series the single events which 
we remember appear to be ordered according to the criterion 
of “ earlier ” and “ later,” which cannot be analysed further. 
There exists, therefore, for the individual, an I-time, or 
subjective time. This in itself is not measurable. I can, 
indeed, associate numbers with the events, in such a way 
that a greater number is associated with the later event 
than with an earlier one ; but the nature of this association 
may be quite arbitrary. This association I can define by 
means of a clock by comparing the order of events furnished 



by the clock with the order of the given series of events. 
We understand by a clock something which provides a 
series of events which can be counted, and which has other 
properties of which we shall speak later. 

By the aid of speech different individuals can, to a certain 
extent, compare their experiences. In this way it is shown 
that certain sense perceptions of different individuals 
correspond to each other, while for other sense perceptions 
no such correspondence can be established. We are ac¬ 
customed to regard as real those sense perceptions which 
are common to different individuals, and which therefore 
are, in a measure, impersonal. The natural sciences, and 
in particular, the most fundamental of them, physics, deal 
with such sense perceptions. The conception of physical 
bodies, in particular of rigid bodies, is a relatively constant 
complex of such sense perceptions. A clock is also a body, 
or a system, in the same sense, with the additional property 
that the series of events which it counts is formed of 
elements all of which can be regarded as equal. 

The only justification for our concepts and system of 
concepts is that they serve to represent the complex of 
our experiences; beyond this they have no legitimacy. I 
am convinced that the philosophers have had a harmful 
effect upon the progress of scientific thinking in removing 
certain fundamental concepts from the domain of empiric¬ 
ism, where they are under our control, to the intangible 
heights of the a priori. For even if it should appear that 
the universe of ideas cannot be deduced from experience 
by logical means, but is, in a sense, a creation of the human 
mind, without which no science is possible, nevertheless 



this universe of ideas is just as little independent of the 
nature of our experiences as clothes are of the form of 
the human body. This is particularly true of our con¬ 
cepts of time and space, which physicists have been 
obliged by the facts to bring down from the Olympus of 
the a priori in order to adjust them and put them in a 
serviceable condition. 

We now come to our concepts and judgments of space. 
It is essential here also to pay strict attention to the 
relation of experience to our concepts. It seems to me 
that Poincare clearly recognized the truth in the account 
he gave in his book, “ La Science et l’Hypothese.” 
Among all the changes which we can perceive in a rigid 
body those are marked by their simplicity which can be 
made reversibly by an arbitrary motion of the body; 
Poincare calls these, changes in position. By means of 
simple changes in position we can bring two bodies into 
contact. The theorems of congruence, fundamental in 
geometry, have to do with the laws that govern such 
changes in position. For the concept of space the follow¬ 
ing seems essential. We can form new bodies by bringing 
bodies B, C, ... up to body A ; we say that we continue 
body A. We can continue body A in such a way that 
it comes into contact with any other body, X. The 
ensemble of all continuations of body A we can designate 
as the “space of the body A.” Then it is true that all 
bodies are in the “space of the (arbitrarily chosen) body 
AA In this sense we cannot speak of space in the 
abstract, but only of the “space belonging to a body AA 
The earth’s crust plays such a dominant role in our daily 


life in judging the relative positions of bodies that it has 
led to an abstract conception of space which certainly 
cannot be defended. In order to free ourselves from this 
fatal error we shall speak only of “bodies of reference,” 
or “ space of reference.” It was only through the theory 
of general relativity that refinement of these concepts 
became necessary, as we shall see later. 

I shall not go into detail concerning those properties 
of the space of reference which lead to our conceiving 
points as elements of space, and space as a continuum. 
Nor shall I attempt to analyse further the properties of 
space which justify the conception of continuous series 
of points, or lines. If these concepts are assumed, together 
with their relation to the solid bodies of experience, then 
it is easy to say what we mean by the three-dimensionality 
of space; to each point three numbers, x v x 2 , x 3 (co¬ 
ordinates), may be associated, in such a way that this 
association is uniquely reciprocal, and that x v x v and x 2 
vary continuously when the point describes a continuous 
series of points (a line). 

It is assumed in pre-relativity physics that the laws of 
the orientation of ideal rigid bodies are consistent with 
Euclidean geometry. What this means may be expressed 
as follows: Two points marked on a rigid body form 
an interval. Such an interval can be oriented at rest, 
relatively to our space of reference, in a multiplicity of 
ways. If, now, the points of this space can be referred 
to co-ordinates^, x v x v in such a way that the differences 
of the co-ordinates, Ax v Aq, Ar 3 , of the two ends of the 
interval furnish the same sum of squares, 

s 2 = A*i 2 + Ax 2 2 + kx 3 . . (i) 



for every orientation of the interval, then the space of 
reference is called Euclidean, and the co-ordinates 
Cartesian.* It is sufficient, indeed, to make this assump¬ 
tion in the limit for an infinitely small interval. Involved 
in this assumption there are some which are rather less 
special, to which we must call attention on account of 
their fundamental significance. In the first place, it is 
assumed that one can move an ideal rigid body in an 
arbitrary manner. In the second place, it is assumed 
that the behaviour of ideal rigid bodies towards orienta¬ 
tion is independent of the material of the bodies and their 
changes of position, in the sense that if two intervals can 
once be brought into coincidence, they can always and 
everywhere be brought into coincidence. Both of these 
assumptions, which are of fundamental importance for 
geometry and especially for physical measurements, 
naturally arise from experience ; in the theory of general 
relativity their validity needs to be assumed only for 
bodies and spaces of reference which are infinitely small 
compared to astronomical dimensions. 

The quantity s we call the length of the interval. In 
order that this may be uniquely determined it is necessary 
to fix arbitrarily the length of a definite interval; for 
example, we can put it equal to I (unit of length). Then 
the lengths of all other intervals may be determined. If 
we make the x v linearly dependent upon a parameter X, 

x v = d v + X^„, 

* This relation must hold for an arbitrary choice of the origin and of the 
direction (ratios Ax l : Ax 2 : Ax 3 ) of the interval. 


we obtain a line which has all the properties of the straight 
lines of the Euclidean geometry. In particular, it easily 
follows that by laying off n times the interval n upon a 
straight line, an interval of length n's is obtained. A 
length, therefore, means the result of a measurement 
carried out along a straight line by means of a unit 
measuring rod. It has a significance which is as inde¬ 
pendent of the system of co-ordinates as that of a straight 
line, as will appear in the sequel. 

We come now to a train of thought which plays an 
analogous role in the theories of special and general 
relativity. We ask the question : besides the Cartesian 
co-ordinates which we have used are there other equivalent 
co-ordinates ? An interval has a physical meaning which 
is independent of the choice of co-ordinates; and so has 
the spherical surface which we obtain as the locus of the 
end points of all equal intervals that we lay off from an 
arbitrary point of our space of reference. If x v as well as 
x v {y from I to 3) are Cartesian co-ordinates of our space 
of reference, then the spherical surface will be expressed 
in our two systems of co-ordinates by the equations 

A^ 2 = const. . . (2) 

= const. . . . (2a) 

How must the x v be expressed in terms of thex v in order 
that equations (2) and (2a) may be equivalent to each 
other ? Regarding the x v expressed as functions of the 
x vy we can write, by Taylor’s theorem, for small values of 
the Ax u , 



AY. = J 




+ 5 . 


c)V v 

Ax a i\xp . 

If we substitute (2a) in this equation and compare with 
(i), we see that the x v must be linear functions of the x v . 
If we therefore put 

x v — a v + b va x a . . . (3) 


or Ar'„ = ^<Ar a . • • • (3 a ) 

then the equivalence of equations (2) and (2a) is expressed 
in the form 

= \^Jix 2 (X independent of Ax v ) . (2b) 

It therefore follows that X must be a constant. If we put 
X = 1, (2b) and (3a) furnish the conditions 

j3 ^aj8 • • • (4) 

in which S ai3 = 1, cr 8 af} = o, according as a = /3 or 
a / 3 . The conditions (4) are called the conditions of ortho¬ 
gonality, and the transformations (3), (4), linear orthogonal 

transformations. If we stipulate that s ' 1 = ^Ax 2 shall be 

equal to the square of the length in every system of 
co-ordinates, and if we always measure with the same unit 
scale, then X must be equal to 1. Therefore the linear 
orthogonal transformations are the only ones by means of 
which we can pass from one Cartesian system of co¬ 
ordinates in our space of reference to another. We see 


that in applying such transformations the equations of 
a straight line become equations of a straight line. 
Reversing equations (3a) by multiplying both sides by b vfi 
and summing for all the vs, we obtain 

v = ^bvofoa. ~ / $ a p/\X a = l\Xp . (5) 

va a 

The same coefficients, b, also determine the inverse 
substitution of Ax v . Geometrically, b va is the cosine of the 
angle between the x v axis and the ;r a axis. 

To sum up, we can say that in the Euclidean geometry 
there are (in a given space of reference) preferred systems 
of co-ordinates, the Cartesian systems, which transform 
into each other by linear orthogonal transformations. 
The distance s between two points of our space of 
reference, measured by a measuring rod, is expressed in 
such co-ordinates in a particularly simple manner. The 
whole of geometry may be founded upon this conception 
of distance. In the present treatment, geometry is 
related to actual things (rigid bodies), and its theorems 
are statements concerning the behaviour of these things, 
which may prove to be true or false. 

One is ordinarily accustomed to study geometry 
divorced from any relation between its concepts and 
experience. There are advantages in isolating that 
which is purely logical and independent of what is, in 
principle, incomplete empiricism. This is satisfactory to 
the pure mathematician. He is satisfied if he can deduce 
his theorems from axioms correctly, that is, without 
errors of logic. The question as to whether Euclidean 



geometry is true or not does not concern him. But for 
our purpose it is necessary to associate the fundamental 
concepts of geometry with natural objects ; without such 
an association geometry is worthless for the physicist. 
The physicist is concerned with the question as to 
whether the theorems of geometry are true or not. That 
Euclidean geometry, from this point of view, affirms 
something more than the mere deductions derived 
logically from definitions may be seen from the following 
simple consideration. 

Between n points of space there are 


; between these and the 3 n co-ordinates we have the 

S ^v“ — X \ (t-)) 2 *1" C*- 2 (M) '*2(»')) + • • • 

n(n - 1) . 

From these - equations the 3^ co-ordinates 

may be eliminated, and from this elimination at least 
n(n - 1) 

-- 3 n equations in the s will result.* Since 

the are measurable quantities, and by definition are 
independent of each other, these relations between the 
s^ v are not necessary a priori. 

From the foregoing it is evident that the equations of 
transformation (3), (4) have a fundamental significance in 
Euclidean geometry, in that they govern the transforma¬ 
tion from one Cartesian system of co-ordinates to another. 
The Cartesian systems of co-ordinates are characterized 

n(n - 1 ) 

In reality there are 

- 3« + 6 equations. 


by the property that in them the measurable distance 
between two points, s , is expressed by the equation 

If K( Xv ) and K\ Xv) are two Cartesian systems of co¬ 
ordinates, then 

^>Auy 2 = ^Aa'V 2 . 

The right-hand side is identically equal to the left-hand 

side on account of the equations of the linear orthogonal 
transformation, and the right-hand side differs from the 
left-hand side only in that the x v are replaced by the x v . 

This is expressed by the statement that is an 

invariant with respect to linear orthogonal transforma¬ 
tions. It is evident that in the Euclidean geometry only 
such, and all such, quantities have an objective signifi¬ 
cance, independent of the particular choice of the Cartesian 
co-ordinates, as can be expressed by an invariant with 
respect to linear orthogonal transformations. This is the 
reason that the theory of invariants, which has to do with 
the laws that govern the form of invariants, is so important 
for analytical geometry. 

As a second example of a geometrical invariant, con¬ 
sider a volume. This is expressed by 

V — j j jdz 1 dx 2 dx 3 . 

By means of Jacobi’s theorem we may write 

dx\dx\dx 3 


ap'i, x'. 2 , x 3 ) 
x. 2 , x 3 ) 

dx Y dx 2 dx 3 



where the integrand in the last integral is the functional 
determinant of the x v with respect to the x v , and this by 
(3) is equal to the determinant | b^ v | of the coefficients 
of substitution, b V0L . If we form the determinant of the 
S, Aa from equation (4), we obtain, by means of the theorem 
of multiplication of determinants, 


If we limit ourselves to those transformations which have 
the determinant + I,* and only these arise from con¬ 
tinuous variations of the systems of co-ordinates, then V 
is an invariant. 

Invariants, however, are not the only forms by means 
of which we can give expression to the independence of 
the particular choice of the Cartesian co-ordinates. Vectors 
and tensors are other forms of expression. Let us express 
the fact that the point with the current co-ordinates x v lies 
upon a straight line. We have 

x v - A v = \B V (v from 1 to 3). 

Without limiting the generality we can put 

]>A. 2 = 1. 

If we multiply the equations by b^ v (compare (3a) and 
(5)) and sum for all the p’s, we get 

x p — A p = \B p 

* There are thus two kinds of Cartesian systems which are designated 
as “right-handed” and “left-handed” systems. The difference between 
these is familiar to every physicist and engineer. It is interesting to note 
that these two kinds of systems cannot be defined geometrically, but only 
the contrast between them. 


where we have written 

^0 = y bp v B v ; Ap = '^bp v A v . 

V V 

These are the equations of straight lines with respect 
to a second Cartesian system of co-ordinates K'. They 
have the same form as the equations with respect to the 
original system of co-ordinates. It is therefore evident 
that straight lines have a significance which is independent 
of the system of co-ordinates. Formally, this depends 
upon the fact that the quantities (x v - A v ) - \B V are 
transformed as the components of an interval, The 

ensemble of three quantities, defined for every system of 
Cartesian co-ordinates, and which transform as the com¬ 
ponents of an interval, is called a vector. If the three 
components of a vector vanish for one system of Cartesian 
co-ordinates, they vanish for all systems, because the equa¬ 
tions of transformation are homogeneous. We can thus 
get the meaning of the concept of a vector without referring 
to a geometrical representation. This behaviour of the 
equations of a straight line can be expressed by saying 
that the equation of a straight line is co-variant with respect 
to linear orthogonal transformations. 

We shall now show briefly that there are geometrical 
entities which lead to the concept of tensors. Let P 0 be 
the centre of a surface of the second degree, P any point 
on the surface, and the projections of the interval P 0 P 
upon the co-ordinate axes. Then the equation of the 
surface is 



In this, and in analogous cases, we shall omit the sign of 
summation, and understand that the summation is to be 
carried out for those indices that appear twice. We thus 
write the equation of the surface 

The quantities a^ v determine the surface completely, for 
a given position of the centre, with respect to the chosen 
system of Cartesian co-ordinates. From the known law 
of transformation for the (3a) for linear orthogonal 
transformations, we easily find the law of transformation 
for the a^ v * : 

^ err ' 

This transformation is homogeneous and of the first degree 
in the a^ v . On account of this transformation, the a^ v 
are called components of a tensor of the second rank (the 
latter on account of the double index). If all the com¬ 
ponents, of a tensor with respect to any system of 
Cartesian co-ordinates vanish, they vanish with respect to 
every other Cartesian system. The form and the position 
of the surface of the second degree is described by this 
tensor (a). 

Analytic tensors of higher rank (number of indices) 
may be defined. It is possible and advantageous to 
regard vectors as tensors of rank 1, and invariants (scalars) 
as tensors of rank o. In this respect, the problem of the 
theory of invariants may be so formulated : according to 
what laws may new tensors be formed from given tensors ? 

* The equation aVrlV^'r = 1 may, by (5), be replaced by &’ errb fxa-bpT^o-^T 
= i, from which the result stated immediately follows. 


We shall consider these laws now, in order to be able to 
apply them later. We shall deal first only with the 
properties of tensors with respect to the transformation 
from one Cartesian system to another in the same space 
of reference, by means of linear orthogonal transforma¬ 
tions. As the laws are wholly independent of the number 
of dimensions, we shall leave this number, n, indefinite at 

Definition. If a figure is defined with respect to every 
system of Cartesian co-ordinates in a space of reference of 
n dimensions by the n a numbers A^ p . . . (a = number 
of indices), then these numbers are the components of a 
tensor of rank a if the transformation law is 

i u.'v'p' • * • ^\u.'p.^v'v^p’p ■ * • ^ju .vp * • * • (7) 

Remark. From this definition it follows that 

jj.vp • • * ^ fid y D p ... . . (^) 

is an invariant, provided that ( B ), (Q, (Z?) . . . are 
vectors. Conversely, the tensor character of ( A ) may be 
inferred, if it is known that the expression (8) leads to an 
invariant for an arbitrary choice of the vectors ( B ), (C), 

Addition and Subtraction. By addition and subtraction 
of the corresponding components of tensors of the same 
rank, a tensor of equal rank results : 


± B 

— P-Vp 

The proof follows from the definition of a tensor given 

Multiplication. From a tensor of rank a and a tensor 



of rank /3 we may obtain a tensor of rank a + (3 by 
multiplying all the components of the first tensor by all 
the components of the second tensor : 

• • • afi • • • • • • -^a/3y • • • (^ O) 

Contraction. A tensor of rank a - 2 may be obtained 
from one of rank a by putting two definite indices equal 
to each other and then summing for this single index : 

T — a (- 

p • • • • • • v 

Y A 

y ■'■Vmp 

• • •) • (ii) 

The proof is 

A' — h h h A 

-* 1 fxp.p • * • t/ /ua t/ p.j3 c/ py • • • •‘- 1 afiy ’ • 

^afi^py • •• A 

= ... A 


a ay 

In addition to these elementary rules of operation 
there is also the formation of tensors by differentiation 
(“ erweiterung ”): 

( 12 ) 




New tensors, in respect to linear orthogonal transforma¬ 
tions, may be formed from tensors according to these rules 
of operation. 

Symmetrical Properties of Tensors. Tensors are called 
symmetrical or skew-symmetrical in respect to two of 
their indices, ^ and v , if both the components which result 
from interchanging the indices and v are equal to each 
other or equal with opposite signs. 

Condition for symmetry: A pvp = A pvp . 

Condition for skew-symmetry: A p „ p = - A,, pp . 

Theorem. The character of symmetry or skew-symmetry 
exists independently of the choice of co-ordinates, and in 


this lies its importance. The proof follows from the 
equation defining tensors. 

Special Tensors. 

I. The quantities 8 pcr (4) are tensor components (funda¬ 
mental tensor). 

Proof. If in the right-hand side of the equation of 
transformation A\ v = b^ a b vfi A a ^ we substitute for A afi the 
quantities 8 afi (which are equal to I or o according as 
a = ft or a / 3 ), we get 

/]' _ h h _ £ 

The justification for the last sign of equality becomes 
evident if one applies (4) to the inverse substitution (5). 

II. There is a tensor ( 8 ^ vp . . .) skew-symmetrical with 
respect to all pairs of indices, whose rank is equal to the 
number of dimensions, n, and whose components are 
equal to + I or - 1 according as [xvp ... is an even 
or odd permutation of 123 . . . 

The proof follows with the aid of the theorem proved 
above \ b pa \ = 1. 

These few simple theorems form the apparatus from 
the theory of invariants for building the equations of pre¬ 
relativity physics and the theory of special relativity. 

We have seen that in pre-relativity physics, in order to 
specify relations in space, a body of reference, or a space 
of reference, is required, and, in addition, a Cartesian 
system of co-ordinates. We can fuse both these concepts 
into a single one by thinking of a Cartesian system of 
co-ordinates as a cubical frame-work formed of rods each 
of unit length. The co-ordinates of the lattice points of 



this frame are integral numbers. It follows from the 
fundamental relation 

s 2 = Arp + Ar 2 2 + Ar 3 2 

that the members of such a space-lattice are all of unit 
length. To specify relations in time, we require in 
addition a standard clock placed at the origin of our 
Cartesian system of co-ordinates or frame of reference. 
If an event takes place anywhere we can assign to it three 
co-ordinates, x vi and a time t, as soon as we have specified 
the time of the clock at the origin which is simultaneous 
with the event. We therefore give an objective signifi¬ 
cance to the statement of the simultaneity of distant 
events, while previously we have been concerned only 
with the simultaneity of two experiences of an individual. 
The time so specified is at all events independent of the 
position of the system of co-ordinates in our space of 
reference, and is therefore an invariant with respect to 
the transformation (3). 

It is postulated that the system of equations expressing 
the laws of pre-relativity physics is co-variant with respect 
to the transformation (3), as are the relations of Euclidean 
geometry. The isotropy and homogeneity of space is 
expressed in this way.* We shall now consider some of 

* The laws of physics could be expressed, even in case there were a 
unique direction in space, in such a way as to be co-variant with respect to 
the transformation (3); but such an expression would in this case be un¬ 
suitable. If there were a unique direction in space it would simplify the 
description of natural phenomena to orient the system of co-ordinates in a 
definite way in this direction. But if, on the other hand, there is no unique 
direction in space it is not logical to formulate the laws of nature in such 
a way as to conceal the equivalence of systems of co-ordinates that are 



the more important equations of physics from this point 
of view. 

The equations of motion of a material particle are 

**. y 

m ~d¥. “ 


(dx v ) is a vector ; dt , and therefore also an invariant; 

thus (^r) is a vector ; in the same way it may be shown 

/ dsx \ 

that is a vector. In general, the operation of dif¬ 

ferentiation with respect to time does not alter the tensor 
character. Since in is an invariant (tensor of rank o), 

f d 2 x v \ 

\ l ~df ) lS a vec i ;or ’ or t ensor of rank I (by the theorem 

of the multiplication of tensors). If the force (A v ) has 
a vector character, the same holds for the difference 

( d^x \ 

m ~d¥ ~ X v' ^ ese equations of motion are therefore 

valid in every other system of Cartesian co-ordinates in 
the space of reference. In the case where the forces are 
conservative we can easily recognize the vector character 
of (X v ). For a potential energy, <F, exists, which depends 
only upon the mutual distances of the particles, and is 
therefore an invariant. The vector character of the force, 

X v = - ^7, is then a consequence of our general theorem 
about the derivative of a tensor of rank o. 

oriented differently. We shall meet with this point of view again in the 
theories of special and general relativity. 



Multiplying by the velocity, a tensor of rank i, we 
obtain the tensor equation 

d l x 


r - X, 



= o. 

dt 2 " v ) dt 

By contraction and multiplication by the scalar dt we 
obtain the equation of kinetic energy 

, 2 \ 

mq ‘ 

= X v dx v . 

If denotes the difference of the co-ordinates of 
the material particle and a point fixed in space, then 
the % v have the character of vectors. We evidently 


d 2 x v d 2 ^ v 

dt 2 ~ ~dd' SO ^ a t e 9 ua ^ 10ns °f m °ti° n of the 
particle may be written 

Multiplying this equation by f we obtain a tensor 

(>*w - 

Contracting the tensor on the left and taking the time 
average we obtain the virial theorem, which we shall 
not consider further. By interchanging the indices and 
subsequent subtraction, we obtain, after a simple trans¬ 
formation, the theorem of moments, 

It is evident in this way that the moment of a vector 


is not a vector but a tensor. On account of their skew- 
symmetrical character there are not nine, but only three 
independent equations of this system. The possibility of 
replacing skew-symmetrical tensors of the second rank in 
space of three dimensions by vectors depends upon the 
formation of the vector 

A — - A 8 

[i. CTT^CTTfl. 

If we multiply the skew-symmetrical tensor of rank 2 
by the special skew-symmetrical tensor 8 introduced 
above, and contract twice, a vector results whose compon¬ 
ents are numerically equal to those of the tensor. These 
are the so-called axial vectors which transform differ¬ 
ently, from a right-handed system to a left-handed system, 
from the There is a gain in picturesqueness in 

regarding a skew-symmetrical tensor of rank 2 as a vector 
in space of three dimensions, but it does not represent 
the exact nature of the corresponding quantity so well as 
considering it a tensor. 

We consider next the equations of motion of a con¬ 
tinuous medium. Let p be the density, u v the velocity 
components considered as functions of the co-ordinates and 
the time, X v the volume forces per unit of mass, and p va 
the stresses upon a surface perpendicular to the c-axis 
in the direction of increasing x v . Then the equations of 
motion are, by Newton’s law, 

~^Pvcr -yjr 

PHi = " 55 “ + P x * 

in which is the acceleration of the particle which at 


time t has the co-ordinates x* If we express this 
acceleration by partial differential coefficients, we obtain, 
after dividing by p , 

"du v 

1 st 




+ X v 

(i 6) 

We must show that this equation holds independently 
of the special choice of the Cartesian system of co-ordinates. 

lsu v 'bu v . 

(«„) is a vector, and therefore -r— is also a vector, r— is 

a tensor of rank 2, ^~^u T is a tensor of rank 3. The second 


term on the left results from contraction in the indices 
cr, r. The vector character of the second term on the right 
is obvious. In order that the first term on the right may 
also be a vector it is necessary for p v(J to be a tensor. 


Then by differentiation and contraction r—^ results, and 


is therefore a vector, as it also is after multiplication by 

the reciprocal scalar - • That p v(T is a tensor, and therefore 
transforms according to the equation 

P = ^na^vfipafl ) 

is proved in mechanics by integrating this equation over 
an infinitely small tetrahedron. It is also proved there 
by application of the theorem of moments to an infinitely 
small parallelopipedon, that p v(J = p av) and hence that the 
tensor of the stress is a symmetrical tensor. From what 
has been said it follows that, with the aid of the rules 


given above, the equation is co-variant with respect to 
orthogonal transformations in space (rotational trans¬ 
formations) ; and the rules according to which the 
quantities in the equation must be transformed in order 
that the equation may be co-variant also become evident. 

The co-variance of the equation of continuity, 

tp Xp u v) 

3 7 + 

requires, from the foregoing, no particular discussion. 

We shall also test for co-variance the equations which 
express the dependence of the stress components upon 
the properties of the matter, and set up these equations 
for the case of a compressible viscous fluid with the aid 
of the conditions of co-variance. If we neglect the vis¬ 
cosity, the pressure, />, will be a scalar, and will depend 
only upon the density and the temperature of the fluid. 
The contribution to the stress tensor is then evidently 


in which is the special symmetrical tensor. This term 
will also be present in the case of a viscous fluid. But in 
this case there will also be pressure terms, which depend 
upon the space derivatives of the u v . We shall assume 
that this dependence is a linear one. Since these terms 
must be symmetrical tensors, the only ones which enter 
will be 

(for r * is a scalar). For physical reasons (no slipping) 



it is assumed that for symmetrical dilatations in all 
directions, i.e. when 

bu 2 bu 3 bUj 
'bx 1 ~ bx 2 ~ bx 2 ’ bx 2 

, etc., = o, 

bx 2 bx 3 ’ bx 2 
there are no frictional forces present, from which it 

2 1 bu, 

follows that /3 = - -a. If only ^7 is different from 

bu 1 

zero, let p 3l = - 77- —, by which a is determined. We 

then obtain for the complete stress tensor, 

rY^u . ^y\ 2{bU, bu 2 bu 3 \ * “1 , 

(* s ) 

The heuristic value of the theory of invariants, which 
arises from the isotropy of space (equivalence of all 
directions), becomes evident from this example. 

We consider, finally, Maxwell’s equations in the form 
which are the foundation of the electron theory of Lorentz. 

u 3 


be x 



dx 2 

^x 3 





be 2 














1 + 


0 _ 


bx 1 

bx. 2 




^3 _ 

'be 2 

I bh x 


bx 2 



be 1 


I bh 2 


bx l 




• ( 20 ) 


i is a vector, because the current density is defined as 
the density of electricity multiplied by the vector velocity 
of the electricity. According to the first three equations 
it is evident that e is also to be regarded as a vector. 
Then h cannot be regarded as a vector.* The equations 
may, however, easily be interpreted if h is regarded as a 
symmetrical tensor of the second rank. In this sense, we 
write /z 23 , k 31) ^ 12j in place of h x , k 2i h z respectively. Pay¬ 
ing attention to the skew-symmetry of k^, the first three 
equations of (19) and (20) may be written in the form 

_ 1 + L { 

bx v C bt C 11 

'K. - ^ = + I 

bx v bx^ C bt 



In contrast to e, h appears as a quantity which has the 
same type of symmetry as an angular velocity. The 
divergence equations then take the form 

= P . . . (i 9 b) 

+ 'bhyp bh ?iX _ o 

bx p bx^ bx v 

The last equation is a skew-symmetrical tensor equation 
of the third rank (the skew-symmetry of the left-hand 
side with respect to every pair of indices may easily be 

* These considerations will make the reader familiar with tensor opera¬ 
tions without the special difficulties of the four-dimensional treatment; 
corresponding considerations in the theory of special relativity (Minkowski’s* 
interpretation of the field) will then offer fewer difficulties, 




proved, if attention is paid to the skew-symmetry cf k^). 
This notation is more natural than the usual one, because, 
in contrast to the latter, it is applicable to Cartesian left- 
handed systems as well as to right-handed systems without 
change of sign. 



T HE previous considerations concerning the configura¬ 
tion of rigid bodies have been founded, irrespective 
of the assumption as to the validity of the Euclidean 
geometry, upon the hypothesis that all directions in space, 
or all configurations of Cartesian systems of co-ordinates, 
are physically equivalent. We may express this as the 
“ principle of relativity with respect to direction,” and it 
has been shown how equations (laws of nature) may be 
found, in accord with this principle, by the aid of the 
calculus of tensors. We now inquire whether there is a 
relativity with respect to the state of motion of the space 
of reference; in other words, whether there are spaces of 
reference in motion relatively to each other which are 
physically equivalent. From the standpoint of mechanics 
it appears that equivalent spaces of reference do exist. 
For experiments upon the earth tell us nothing of the 
fact that we are moving about the sun with a velocity of 
approximately 30 kilometres a second. On the other 
hand, this physical equivalence does not seem to hold for 
spaces of reference in arbitrary motion; for mechanical 
effects do not seem to be subject to the same laws in a 
jolting railway train as in one moving with uniform 




velocity; the rotation of the earth must be considered in 
writing down the equations of motion relatively to the 
earth. It appears, therefore, as if there were Cartesian 
systems of co-ordinates, the so-called inertial systems, with 
reference to which the laws of mechanics (more generally 
the laws of physics) are expressed in the simplest form. 
We may infer the validity of the following theorem : If 
K is an inertial system, then every other system K' which 
moves uniformly and without rotation relatively to K , is 
also an inertial system; the laws of nature are in con¬ 
cordance for all inertial systems. This statement we shall 
call the “ principle of special relativity.” We shall draw 
certain conclusions from this principle of “ relativity of 
translation ” just as we have already done for relativity of 

In order to be able to do this, we must first solve the 
following problem. If we are given the Cartesian co¬ 
ordinates,^, and the time /, of an event relatively to one 
inertial system, K , how can we calculate the co-ordinates, 
x v , and the time, of the same event relatively to an 
inertial system K' which moves with uniform trans¬ 
lation relatively to K ? In the pre-relativity physics 
this problem was solved by making unconsciously two 
hypotheses :— 

i. The time is absolute; the time of an event, t\ 
relatively to K' is the same as the time relatively to K. 
If instantaneous signals could be sent to a distance, and 
if one knew that the state of motion of a clock had no 
influence on its rate, then this assumption would be 
physically established. For then clocks, similar to one 


another, and regulated alike, could be distributed over 
the systems K and K\ at rest relatively to them, and 
their indications would be independent of the state of 
motion of the systems ; the time of an event would then 
be given by the clock in its immediate neighbourhood. 

2. Length is absolute ; if an interval, at rest relatively 
to K, has a length s, then it has the same length s, 
relatively to a system K' which is in motion relatively 
to K. 

If the axes of K and K' are parallel to each other, a 
simple calculation based on these two assumptions, gives 
the equations of transformation 

x v = x v - a v - b v t 
t' = t - b 

This transformation is known as the “ Galilean Trans¬ 
formation.” Differentiating twice by the time, we get 

d 2 x v d 2 x v 

~dF = ~dF m 

Further, it follows that for two simultaneous events, 

J a) _ 


( 2 ) = ^ ( 1 ) _ 

( 2 ) 

The invariance of the distance between the two points 
results from squaring and adding. From this easily 
follows the co-variance of Newton’s equations of motion 
with respect to the Galilean transformation (21). Hence 
it follows that classical mechanics is in accord with the 
principle of special relativity if the two hypotheses 
respecting scales and clocks are made. 

But this attempt to found relativity of translation upon 
the Galilean transformation fails when applied to electron 



magnetic phenomena. The Maxwell-Lorentz electro¬ 
magnetic equations are not co-variant with respect to the 
Galilean transformation. In particular, we note, by (21), 
that a ray of light which referred to K has a velocity c, 
has a different velocity referred to K\ depending upon 
its direction. The space of reference of K is therefore 
distinguished, with respect to its physical properties, from 
all spaces of reference which are in motion relatively to it 
(quiescent sether). But all experiments have shown that 
electro-magnetic and optical phenomena, relatively to the 
earth as the body of reference, are not influenced by the 
translational velocity of the earth. The most important 
of these experiments are those of Michelson and Morley, 
which I shall assume are known. The validity of the 
principle of special relativity can therefore hardly be 

O11 the other hand, the Maxwell-Lorentz equations 
have proved their validity in the treatment of optical 
problems in moving bodies. No other theory has 
satisfactorily explained the facts of aberration, the 
propagation of light in moving bodies (Flzeau), and 
phenomena observed in double stars (De Sitter). The 
consequence of the Maxwell-Lorentz equations that in a 
vacuum light is propagated with the velocity c, at least 
with respect to a definite inertial system K, must there¬ 
fore be regarded as proved. According to the principle 
of special relativity, we must also assume the truth of 
this principle for every other inertial system. 

Before we draw any conclusions from these two 
principles we must first review the physical significance 


of the concepts “time” and “velocity.” It follows from 
what has gone before, that co-ordinates with respect to 
an inertial system are physically defined by means of 
measurements and constructions with the aid of rigid 
bodies. In order to measure time, we have supposed a 
clock, Uy present somewhere, at rest relatively to K. But 
we cannot fix the time, by means of this clock, of an event 
whose distance from the clock is not negligible ; for there 
are no “ instantaneous signals ” that we can use in order 
to compare the time of the event with that of the clock. 
In order to complete the definition of time we may 
employ the principle of the constancy of the velocity of 
light in a vacuum. Let us suppose that we place similar 
clocks at points of the system K , at rest relatively to it, 
and regulated according to the following scheme. A ray 
of light is sent out from one of the clocks, U m , at the 
instant when it indicates the time t m) and travels through 
a vacuum a distance r mn} to the clock U n ; at the instant 
when this ray meets the clock U n the latter is set to 

indicate the time t n = t m 4- — .* The principle of the 


constancy of the velocity of light then states that this 
adjustment of the clocks wall not lead to contradictions. 
With clocks so adjusted, we can assign the time to events 
which take place near any one of them. It is essential to 

* Strictly speaking, it would be more correct to define simultaneity first, 
somewhat as follows: two events taking place at the points A and B of 
the system K are simultaneous if they appear at the same instant when 
observed from the middle point, M, of the interval AB. Time is then 
defined as the ensemble of the indications of similar clocks, at rest 
relatively to K, which register the same simultaneously. 


note that this definition of time relates only to the inertial 
system K , since we have used a system of clocks at rest 
relatively to K. The assumption which was made in the 
pre-relativity physics of the absolute character of time 
^i.e. the independence of time of the choice of the inertial 
system) does not follow at all from this definition. 

The theory of relativity is often criticized for giving, 
without justification, a central theoretical role to the 
propagation of light, in that it founds the concept of time 
upon the law of propagation of light. The situation, 
however, is somewhat as follows. In order to give 
physical significance to the concept of time, processes of 
some kind are required which enable relations to be 
established between different places. It is immaterial 
what kind of processes one chooses for such a definition 
of time. It is advantageous, however, for the theory, 
to choose only those processes concerning which we know 
something certain. This holds for the propagation of 
light in vacuo in a higher degree than for any other process 
which could be considered, thanks to the investigations 
of Maxwell and H. A. Lorentz. 

From all of these considerations, space and time data 
have a physically real, and not a mere fictitious, signifi¬ 
cance ; in particular this holds for all the relations in 
which co-ordinates and time enter, e.g. the relations 
(21). There is, therefore, sense in asking whether those 
equations are true or not, as well as in asking what the 
true equations of transformation are by which we pass 
from one inertial system K to another, K\ moving 
relatively to it. It may be shown that this is uniquely 


settled by means of the principle of the constancy of the 
velocity of light and the principle of special relativity. 

To this end we think of space and time physically 
defined with respect to two inertial systems, K and K\ in 
the way that has been shown. Further, let a ray of light 
pass from one point P 1 to another point P 2 of K through 
a vacuum. If r is the measured distance between the two 
points, then the propagation of light must satisfy the 

r = c . At 

If we square this equation, and express r 2 by the 
differences of the co-ordinates, Ax v , in place of this equation 
we can write 

(A;r v ) 2 - c 2 A t 2 = o . . (22) 

This equation formulates the principle of the constancy 
of the velocity of light relatively to K. It must hold 
whatever may be the motion of the source which emits 
the ray of light. 

The same propagation of light may also be considered 
relatively to K\ in which case also the principle of the 
constancy of the velocity of light must be satisfied. 
Therefore, with respect to K', we have the equation 

^>(AT V ) 2 - c 2 A/ 2 = o . (22a) 

Equations (22a) and (22) must be mutually consistent 
with each other with respect to the transformation which 
transforms from K to K\ A transformation which effects 
this we shall call a “Lorentz transformation.” 

Before considering these transformations in detail we 



shall make a few general remarks about space and time. 
In the pre-relativity physics space and time were separ¬ 
ate entities. Specifications of time were independent of 
the choice of the space of reference. The Newtonian 
mechanics was relative with respect to the space of 
reference, so that, e.g. the statement that two non-simul- 
taneous events happened at the same place had no objective 
meaning (that is, independent of the space of reference). 
But this relativity had no role in building up the theory. 
One spoke of points of space, as of instants of time, as if 
they were absolute realities. It was not observed that 
the true element of the space-time specification was the 
event, specified by the four numbers x l} x 2 , x z , t. The 
conception of something happening was always that of a 
four-dimensional continuum ; but the recognition of this 
was obscured by the absolute character of the pre-relativity 
time. Upon giving up the hypothesis of the absolute 
character of time, particularly that of simultaneity, the 
four-dimensionality of the time-space concept was im¬ 
mediately recognized. It is neither the point in space, 
nor the instant in time, at which something happens that 
has physical reality, but only the event itself. There is 
no absolute (independent of the space of reference) relation 
in space, and no absolute relation in time between two 
events, but there is an absolute (independent of the space 
of reference) relation in space and time, as will appear in 
the sequel. The circumstance that there is no objective 
rational division of the four-dimensional continuum into 
a three-dimensional space and a one-dimensional time 
continuum indicates that the laws of nature will assume 


a form which is logically most satisfactory when expressed 
as laws in the four-dimensional space-time continuum. 
Upon this depends the great advance in method which 
the theory of relativity owes to Minkowski. Considered 
from this standpoint, we must regard x v x 2 , x 3) t as the 
four co-ordinates of an event in the four-dimensional con¬ 
tinuum. We have far less success in picturing to ourselves 
relations in this four-dimensional continuum than in the 
three-dimensional Euclidean continuum ; but it must be 
emphasized that even in the Euclidean three-dimensional 
geometry its concepts and relations are only of an abstract 
nature in our minds, and are not at all identical with the 
images we form visually and through our sense of touch. 
The non-divisibility of the four-dimensional continuum 
of events does not at all, however, involve the equivalence 
of the space co-ordinates with the time co-ordinate. On 
the contrary, we must remember that the time co-ordinate 
is defined physically wholly differently from the space 
co-ordinates. The relations (22) and (22a) which when 
equated define the Lorentz transformation show, further, 
a difference in the role of the time co-ordinate from that 
of the space co-ordinates ; for the term At 2 has the opposite 
sign to the space terms, Ax 2 , Ax 2 2 , Ax 3 2 . 

Before we analyse further the conditions which define 
the Lorentz transformation, we shall introduce the light¬ 
time, l = ct , in place of the time, t, in order that the 
constant c shall not enter explicitly into the formulas to 
be developed later. Then the Lorentz transformation is 
defined in such a way that, first, it makes the equation 

Ax 2 + Ax 2 + Ax 3 - Al 2 = o . (22b) 



a co-variant equation, that is, an equation which is satisfied 
with respect to every inertial system if it is satisfied in 
the inertial system to which we refer the two given events 
(emission and reception of the ray of light). Finally, 
with Minkowski, we introduce in place of the real time 
co-ordinate / = ct, the imaginary time co-ordinate 

= il = ict - I = z). 

Then the equation defining the propagation of light, 
which must be co-variant with respect to the Lorentz 
transformation, becomes 

}Ax 2 = A;tq 2 + A^ 2 2 + A;r 3 2 + A^ 4 2 = o (22c) 

This condition is always satisfied * if we satisfy the more 
general condition that 

s 2 = A^! 2 + A^ 2 2 + A^ 3 2 + A^ 4 2 . (23) 

shall be an invariant with respect to the transformation. 
This condition is satisfied only by linear transformations, 
that is, transformations of the type 

■ ■ ■ (24) 

in which the summation over the a is to be extended 
from a = I to a = 4. A glance at equations (23) and 
(24) shows that the Lorentz transformation so defined is 
identical with the translational and rotational transforma¬ 
tions of the Euclidean geometry, if we disregard the 
number of dimensions and the relations of reality. We 

* That this specialization lies in the nature of the case will be evident 


can also conclude that the coefficients b Ma must satisfy the 

^\iafyva iv ^ av 

. ( 25 ) 

Since the ratios of the x v are real, it follows that all the 
a ,a and the b Ma are real, except b iV b± 2 , b± 3 , b w b u , and 
£ 34 , which are purely imaginary. 

Special Lorentz Transformation. We obtain the 
simplest transformations of the type of (24) and (25) if 
only two of the co-ordinates are to be transformed, and if 
all the which determine the new origin, vanish. We 
obtain then for the indices 1 and 2, on account of the 
three independent conditions which the relations (25) 

x\ — x Y cos cf) - x 2 sin <£ 
x\ = x x sin <f> + x 2 cos <jf> 

x 3 = x 3 

x \ = 


This is a simple rotation in space of the (space) 
co-ordinate system about ^ 3 -axis. We see that the 
rotational transformation in space (without the time 
transformation) which we studied before is contained in 
the Lorentz transformation as a special case. For the 
indices 1 and 4 we obtain, in an analogous manner, 

x\ = x 1 cos \fr - x± sin yfr 
x\ — x x sin y/r + x 4 cos yjr 

X o =a X n 

X 3 = *3 



On account of the relations of reality yjr must be taken 
as imaginary. To interpret these equations physically, 
we introduce the real light-time l and the velocity v of 



K' relatively to K , instead of the imaginary angle yjr. We 
have, first, 

x\ = x \ cos ^ _ sin yjr 
l — - MTj sin yjr + / cos 

Since for the origin of K\ i.e., for x x = o, we must have 
Aq == it follows from the first of these equations that 

and also 

so that we obtain 

v = i tan -v/r 

sin y\r = 

- iv 

s/l - V 1 

COS yjr = 7= 

F 2 / 

-T, = 

/' = 

x x - vl ^ 

Jl - z / 2 

/ - ZUq 

v/T “ v ‘ 2 

X 2 


= ^ 




These equations form the well-known special Lorentz 
transformation, which in the general theory represents a 
rotation, through an imaginary angle, of the four-dimen¬ 
sional system of co-ordinates. If we introduce the ordinary 
time t, in place of the light-time /, then in (29) we must 


replace l by ct and v by -• 

We must now fill in a gap. From the principle of the 
constancy of the velocity of light it follows that the 

A.V 2 = o 


has a significance which is independent of the choice of 
the inertial system ; but the invariance of the quantity 

does not at all follow from this. This quantity 

might be transformed with a factor. This depends upon 
the fact that the right-hand side of (29) might be multi¬ 
plied by a factor independent of v. But the principle 
of relativity does not permit this factor to be different from 
1, as we shall now show. Let us assume that we have 
a rigid circular cylinder moving in the direction of its 
axis. If its radius, measured at rest with a unit measur¬ 
ing rod is equal to R 0i its radius R in motion, might be 
different from R 0 , since the theory of relativity does not 
make the assumption that the shape of bodies with respect 
to a space of reference is independent of their motion 
relatively to this space of reference. But all directions 
in space must be equivalent to each other. R may there¬ 
fore depend upon the magnitude q of the velocity, but 
not upon its direction; R must therefore be an even 
function of q. If the cylinder is at rest relatively to K' 
the equation of its lateral surface is 

;r ' 2 + / 2 = R 0 2 . 

If we write the last two equations of (29) more generally 

then the lateral surface of the cylinder referred to K 
satisfies the equation 

R 2 



The factor X therefore measures the lateral contraction of 
the cylinder, and can thus, from the above, be only an 
even function of v. 

If we introduce a third system of co-ordinates, K", 
which moves relatively to K' with velocity v in the direc¬ 
tion of the negative ^r-axis of K, we obtain, by apply¬ 
ing (29) twice, 

x\ — X(z;)X( - v)x ± 

• • • • 

/" = X(v)X( - v)l. 

Now, since \(v) must be equal to X( - v ), and since we 
assume that we use the same measuring rods in all the 
systems, it follows that the transformation of K" to K 
must be the identical transformation (since the possibility 
X = — I does not need to be considered). It is essential 
for these considerations to assume that the behaviour of 
the measuring rods does not depend upon the history of 
their previous motion. 

Moving Measuring Rods and Clocks. At the definite K- 
time, l=o, the position of the points given by the integers 
x\ = n, is with respect to K, given by x x = n yj 1 - v 2, ; 
this follows from the first of equations (29) and expresses 
the Lorentz contraction. A clock at rest at the origin 
x Y = o of K , whose beats are characterized by / = n, will, 
when observed from K', have beats characterized by 


1 = V r=7 ^ ; 

this follows from the second of equations (29) and shows 


that the clock goes slower than if it were at rest relatively 
to K'. These two consequences, which hold, mutatis 
mutandis , for every system of reference, form the physical 
content, free from convention, of the Lorentz transforma¬ 

Addition Theorem for Velocities. If we combine two 
special Lorentz transformations with the relative velocities 
v 1 and v 2 , then the velocity of the single Lorentz trans¬ 
formation which takes the place of the two separate ones 
is, according to (27), given by 

v u = 1 

, , , N . tan 'vk + tan aK 

tan (* x + _ t ; n ^ tan \ = 

V, + V, 

2 1 + ^2 ’ 

( 30 ) 

General Statements about the Lorentz Transformation 
and its Theory of Invariants. The whole theory of 
invariants of the special theory of relativity depends upon 
the invariant a 2 (23). Formally, it has the same role in 
the four-dimensional space-time continuum as the in¬ 
variant A-vp + Arq 2 + A^ 3 2 in the Euclidean geometry 
and in the pre-relativity physics. The latter quantity is 
not an invariant with respect to all the Lorentz transfor¬ 
mations; the quantity a 2 of equation (23) assumes the 
role of this invariant. With respect to an arbitrary 
inertial system, a 2 may be determined by measurements ; 
with a given unit of measure it is a completely determinate 
quantity, associated with an arbitrary pair of events. 

The invariant a 2 differs, disregarding the number of 
dimensions, from the corresponding invariant of the 
Euclidean geometry in the following points. In the 
Euclidean geometry a 2 is necessarily positive ; it vanishes 



only when the two points concerned come together. On 
the other hand, from the vanishing of 

s 2 = ^Aav = A-t'f + Aal 2 + Aa' 3 2 - A t 2 


it cannot be concluded that the two space-time points 
fall together; the vanishing of this quantity s 2 , is the 
invariant condition that the two space-time points can be 
connected by a light signal in vacuo. If P is a point 


(event) represented in the four-dimensional space of the 
x v x 2i x 3) /, then all the “ points ” which can be connected 
to P by means of a light signal lie upon the cone s 2 — o 
(compare Fig. I, in which the dimension x 3 is suppressed). 
The “ upper ” half of the cone may contain the “ points ” 
to which light signals can be sent from P; then the 
“ lower ” half of the cone will contain the “ points ” from 
which light signals can be sent to P. The points P' 
enclosed by the conical surface furnish, with P, a negative 
s 2 ; PP', as well as P'P is then, according to Minkowski, 
of the nature of a time. Such intervals represent elements 
of possible paths of motion, the velocity being less than 
that of light.* In this case the /-axis may be drawn in 
the direction of PP' by suitably choosing the state of 
motion of the inertial system. If P' lies outside of the 
“light-cone” then PP' is of the nature of a space; in 
this case, by properly choosing the inertial system, A/ 
can be made to vanish. 

By the introduction of the imaginary time variable, 
x± = z 7 , Minkowski has made the theory of invariants for 
the four-dimensional continuum of physical phenomena 
fully analogous to the theory of invariants for the three- 
dimensional continuum of Euclidean space. The theory 
of four-dimensional tensors of special relativity differs from 
the theory of tensors in three-dimensional space, therefore, 
only in the number of dimensions and the relations of 

* That material velocities exceeding that of light are not possible, 
follows from the appearance of the radical i - v 2 in the special Lorentz 
transformation (29). 



A physical entity which is specified by four quantities, 
A v , in an arbitrary inertial system of the x ly x 2 , x 3 , x±, is 
called a 4-vector, with the components A v , if the A v 
correspond in their relations of reality and the properties 
of transformation to the Ax v ; it may be of the nature of 
a space or of a time. The sixteen quantities, A^ v then 
form the components of a tensor of the second rank, if 
they transform according to the scheme 

A fi V * 

It follows from this that the A^ v behave, with respect to 
their properties of transformation and their properties 
of reality, as the products of components, U^V v , of two 
4-vectors, (£ 7 ) and ( V). All the components are real 
except those which contain the index 4 once, those being 
purely imaginary. Tensors of the third and higher ranks 
may be defined in an analogous way. The operations 
of addition, subtraction, multiplication, contraction and 
differentiation for these tensors are wholly analogous to 
the corresponding operations for tensors in three-dimen¬ 
sional space. 

Before we apply the tensor theory to the four-dimen¬ 
sional space-time continuum, we shall examine more 
particularly the skew-symmetrical tensors. The tensor 
of the second rank has, in general, 16 = 4.4 components. 
In the case of skew-symmetry the components with two 
equal indices vanish, and the components with unequal 
indices are equal and opposite in pairs. There exist, 
therefore, only six independent components, as is the 
case in the electromagnetic field. In fact, it will be shown 


when we consider Maxwell’s equations that these may 
be looked upon as tensor equations, provided we regard 
the electromagnetic field as a skew-symmetrical tensor. 
Further, it is clear that the skew-symmetrical tensor of 
the third rank (skew-symmetrical in all pairs of indices) 
has only four independent components, since there are 
only four combinations of three different indices. 

We now turn to Maxwell’s equations (19a), (19b), (20a)> 
(20b), and introduce the notation : * 




h 3 i 


hi 2 


- ie* 

024 03il 

- tey - ie zj 

■ ( 30 a) 







i . 

• (30 

'c l * 

- i y 


- \ z 

c z 

ip \ 


with the convention 


</> M „ shall 


equal to 


Then Maxwell’s equations may be combined into the 


3 ^ iii' 3<Lcr 3 rf- 1 u )L 


( 32 ) 

( 33 ) 

as one can easily verify by substituting from (30a) and 
(31). Equations (32) and (33) have a tensor character, 
and are therefore co-variant with respect to Lorentz 
transformations, if the <£ MJ , and the J, x have a tensor 
character, which we assume. Consequently, the laws for 

* In order to avoid confusion from now on we shall use the three- 
dimensional space indices, x, y, z instead of 1, 2, 3, and we shall reserve 
the numeral indices 1, 2, 3, 4 for the four-dimensional space-time con¬ 



transforming these quantities from one to another allow¬ 
able (inertial) system of co-ordinates are uniquely 
determined. The progress in method which electro¬ 
dynamics owes to the theory of special relativity lies 
principally in this, that the number of independent 
hypotheses is diminished. If we consider, for example, 
equations (19a) only from the standpoint of relativity of 
direction, as we have done above, we see that they have 
three logically independent terms. The way in which 
the electric intensity enters these equations appears to 
be wholly independent of the way in which the magnetic 
intensity enters them ; it would not be surprising if instead 

c) 2 e 

of -^jr, we had, say, or if this term were absent. On 

the other hand, only two independent terms appear in 
equation (32). The electromagnetic field appears as a 
formal unit; the way in which the electric field enters 
this equation is determined by the way in which the 
magnetic field enters it. Besides the electromagnetic 
field, only the electric current density appears as an 
independent entity. This advance in method arises from 
the fact that the electric and magnetic fields draw their 
separate existences from the relativity of motion. A 
field which appears to be purely an electric field, judged 
from one system, has also magnetic field components 
when judged from another inertial system. When applied 
to an electromagnetic field, the general law of transforma¬ 
tion furnishes, for the special case of the special Lorentz 
transformation, the equations 



h'* = h* -j 

Cy — V\\ z 

, , K + 

y \J I - V 2 

> x/i -^4 

e* + vh y 

, , - ve y 

*/ I - V 2 

* v/l -vJ 

If there exists with respect to K only a magnetic field, 
h, but no electric field, e, then with respect to K' there 
exists an electric field e' as well, which would act upon 
an electric particle at rest relatively to K'. An observer 
at rest relatively to K would designate this force as the 
Biot-Savart force, or the Lorentz electromotive force. It 
therefore appears as if this electromotive force had become 
fused with the electric field intensity into a single entity. 

In order to view this relation formally, let us consider 
the expression for the force acting upon unit volume of 

k = pe + [i, h] . . . (35) 

in which i is the vector velocity of electricity, with the 
velocity of light as the unit. If we introduce and 
according to (30a) and (31), we obtain for the first 
component the expression 

$12 J2 + ^ 13^3 + 

Observing that <f) n vanishes on account of the skew- 
symmetry of the tensor (<£), the components of k are given 
by the first three components of the four-dimensional 

= Jv ( 3 ^) 

and the fourth component is given by 

+ fy&J2 fy&J3 ~ "f" = ^ • ( 37 ) 



There is, therefore, a four-dimensional vector of force per 
unit volume, whose first three components, k v k. 2) k 3 , are 
the ponderomotive force components per unit volume, and 
whose fourth component is the rate of working of the field 

per unit volume, multiplied by ^ - I. 

A comparison of (36) and (35) shows that the theory 
of relativity formally unites the ponderomotive force of 
the electric field, pe, and the Biot-Savart or Lorentz 
force [i, h]. 


Mass and Energy . An important conclusion can be 
drawn from the existence and significance of the 4-vector 
Let us imagine a body upon which the electro¬ 
magnetic field acts for a time. In the symbolic figure 
(Fig. 2) Ox l designates the ^-axis, and is at the same 
time a substitute for the three space axes Ox v Ox 2 , Ox s ; 
01 designates the real time axis. In this diagram a body 
of finite extent is represented, at a definite time /, by the 
interval AB ; the whole space-time existence of the body 
is represented by a strip whose boundary is everywhere 
inclined less than 45 0 to the /-axis. Between the time 
sections, l = and / = / 2 , but not extending to them, 
a portion of the strip is shaded. This represents the 
portion of the space-time manifold in which the electro¬ 
magnetic field acts upon the body, or upon the electric 
charges contained in it, the action upon them being 
transmitted to the body. We shall now consider the 
changes which take place in the momentum and energy 
of the body as a result of this action. 

We shall assume that the principles of momentum 
and energy are valid for the body. The change in 
momentum, A I X) A l y , A A, and the change in energy, A E, 
are then given by the expressions 


M, - UW X dxdydz — J r ^K 1 dx 1 dx 2 dx 3 dx 4 

A E = 


~K\dxyix 2 dx z dx i 



Since the four-dimensional element of volume is an 
invariant, and (K v K 2 , K 3 , W 4 ) forms a 4-vector, the four- 
dimensional integral extended over the shaded portion 
transforms as a 4-vector, as does also the integral between 
the limits / x and / 2 , because the portion of the region which 
is not shaded contributes nothing to the integral. It 
follows, therefore, that A/*, A/ v , A I z , i^E form a 4-vector. 
Since the quantities themselves transform in the same 
way as their increments, it follows that the aggregate of 
the four quantities 

A, 4 4 i E 

has itself the properties of a vector; these quantities are 
referred to an instantaneous condition of the body (e.g. at 
the time l = 4. 

This 4-vector may also be expressed in terms of the 
mass ;//, and the velocity of the body, considered as a 
material particle. To form this expression, we note first, 

- ds°- = dr 2 = - (dx 2 + dxd + dx 2 ) - dx 2 = dl 2 {\ - q 2 ') (38; 

is an invariant which refers to an infinitely short portion 
of the four-dimensional line which represents the motion 
of the material particle. The physical significance of the 
invariant dr may easily be given. If the time axis is 
chosen in such a way that it has the direction of the line 
differential which we are considering, or, in other words, 
if we reduce the material particle to rest, we shall then 
have dr = dl ; this will therefore be measured by the 
light-seconds clock which is at the same place, and at 
rest relatively to the material particle. We therefore call 


r the proper time of the material particle. As opposed 
to dl , dr is therefore an invariant, and is practically 
equivalent to dl for motions whose velocity is small 
compared to that of light. Hence we see that 

• • • ( 39 ) 


has, just as the dx vi the character of a vector ; we shall 
designate (u v ) as the four-dimensional vector (in brief, 
4-vector) of velocity. Its components satisfy, by (38), 
the condition 

= -1. . . . (40) 

We see that this 4-vector, whose components in the 
ordinary notation are 

Qx Q y Q s i 

T^l 1 ’ Vi - 7 7^1- 7^7 


is the only 4-vector which can be formed from the velocity 
components of the material particle which are defined in 
three dimensions by 

_ dx _ dy _ dz 

q * ~ Jl' ^ tl' q ’~ ic 

We therefore see that 

( 42 ) 

must be that 4-vector which is to be equated to the 
4-vector of momentum and energy whose existence we 
have proved above. By equating the components, we 
obtain, in three-dimensional notation, 



L = 

x / 1 " f 

E = 


n/I - ? 2 J 

( 43 ) 

We recognize, in fact, that these components of 
momentum agree with those of classical mechanics for 
velocities which are small compared to that of light. For 
large velocities the momentum increases more rapidly 
than linearly with the velocity, so as to become infinite 
on approaching the velocity of light. 

If we apply the last of equations (43) to a material 
particle at rest (q = o), we see that the energy, if 0 , of a 
body at rest is equal to its mass. Had we chosen the 
second as our unit of time, we would have obtained 

E 0 = me 1 . . . (44) 

Mass and energy are therefore essentially alike ; they are 
only different expressions for the same thing. The mass 
of a body is not a constant; it varies with changes in its 
energy.* We see from the last of equations (43) that E 
becomes infinite when q approaches I, the velocity of 
light. If we develop E in powers of q 2 , we obtain, 

r* in o 3 4 / v 

E = m + —q* + | m q* + . . . . (45) 

2 o 

* The emission of energy in radioactive processes is evidently connected 
with the fact that the atomic weights are not integers. Attempts have 
been made to draw conclusions from this concerning the structure and 
stability of the atomic nuclei. 


The second term of this expansion corresponds to the 
kinetic energy of the material particle in classical 

Equations of Motion of Material Particles. From (43) 
we obtain, by differentiating by the time /, and using 
the principle of momentum, in the notation of three- 
dimensional vectors, 

This equation, which was previously employed by 
H. A. Lorentz for the motion of electrons, has been 
proved to be true, with great accuracy, by experiments 
with /5-rays. 

Energy Tensor of the Electromagnetic Field. Before the 
development of the theory of relativity it was known 
that the principles of energy and momentum could 
be expressed in a differential form for the electro¬ 
magnetic field. The four-dimensional formulation of 
these principles leads to an important conception, that of 
the energy tensor, which is important for the further 
development of the theory of relativity. 

If in the expression for the 4-vector of force per unit 

using the field equations (32), we express in terms of 
the field intensities, we obtain, after some trans¬ 
formations and repeated application of the field equations 
(32) and (33), the expression 




where we have written * 

( 48 ) 

The physical meaning of equation (47) becomes evident 
if in place of this equation we write, using a new 

iX = 

'tip XX: 














• • 

2 ( 1 %) 

2 (>s.) 

X - v ) 






or, on eliminating the imaginary, 


x = 

tip XX 

tip xy 

tip XX 






• • • • 

tiS x 


tis x 






When expressed in the latter form, we see that the 
first three equations state the principle of momentum ; 
Pxx • • • pxx are the Maxwell stresses in the electro¬ 
magnetic field, and (b x , b yi b z ) is the vector momentum 
per unit volume of the field. The last of equations (47b) 
expresses the energy principle; s is the vector flow of 
energy, and 77 the energy per unit volume of the field. 
In fact, we get from (48) by introducing the well-known 
expressions for the components of the field intensity from 

* To be summed for the indices a and / 3 , 


pxx — — + -J(h * 2 + h/ + hy) 

- e x e y + ie, 2 + e/ + e, 2 ) 


Pxy ^hj/ P xz 

— Q x Gy 

- h x h z 

(48 a) 

b x s x ©jh s 62 hy 

We conclude from (48) that the energy tensor of the 
electromagnetic field is symmetrical; with this is con¬ 
nected the fact that the momentum per unit volume and 
the flow of energy are equal to each other (relation 
between energy and inertia). 

We therefore conclude from these considerations that 
the energy per unit volume has the character of a tensor. 
This has been proved directly only for an electromagnetic 
field, although we may claim universal validity for it. 
Maxwell’s equations determine the electromagnetic field 
when the distribution of electric charges and currents is 
known. But we do not know the laws which govern 
the currents and charges. We do know, indeed, that 
electricity consists of elementary particles (electrons, 
positive nuclei), but from a theoretical point of view we 
cannot comprehend this. We do not know the energy 
factors which determine the distribution of electricity in 
particles of definite size and charge, and all attempts to 
complete the theory in this direction have failed. If then 
we can build upon Maxwell’s equations in general, the 



energy tensor of the electromagnetic field is known only 
outside the charged particles.* In these regions, outside 
of charged particles, the only regions in which we can 
believe that we have the complete expression for the 
energy tensor, we have, by (47), 



= O 


General Expressions for the Conservation Principles. We 
can hardly avoid making the assumption that in all other 
cases, also, the space distribution of energy is given by a 
symmetrical tensor, T )X , and that this complete energy 
tensor everywhere satisfies the relation (47c). At any 
rate we shall see that by means of this assumption we 
obtain the correct expression for the integral energy 

Let us consider a spatially bounded, closed system, 
which, four-dimensionally, we may represent as a strip, 
outside of which the T^ v vanish. Integrate equation 
(47c) over a space section. Since the integrals of 



r—, -r—^- and -r— 1 — vanish because the T uv vanish at the 
ox l ’ ^x» Lr 3 ^ 

limits of integration, we obtain 


57I I T^dxydx^ 


( 49 ) 

Inside the parentheses are the expressions for the 

* It has been attempted to remedy this lack of knowledge by considering 
the charged particles as proper singularities. But in my opinion this means 
giving up a real understanding of the structure of matter. It seems to me 
much better to give in to our present inability rather than to be satisfied 
by a solution that is only apparent. 


momentum of the whole system, multiplied by z, together 
with the negative energy of the system, so that (49) 
expresses the conservation principles in their integral 
form. That this gives the right conception of energy and 


the conservation principles will be seen from the following 

Phenomenological Representation of the 
Energy Tensor of Matter. 

Hydrodynamical Equations. We know that matter is 
built up of electrically charged particles, but we do not 



know the laws which govern the constitution of these 
particles. In treating mechanical problems, we are there¬ 
fore obliged to make use of an inexact description of 
matter, which corresponds to that of classical mechanics. 
The density <7, of a material substance and the hydro- 
dynamical pressures are the fundamental concepts upon 
which such a description is based. 

Let cr 0 be the density of matter at a place, estimated 
with reference to a system of co-ordinates moving with 
the matter. Then cr 0 , the density at rest, is an invariant. 
If we think of the matter in arbitrary motion and neglect 
the pressures (particles of dust in vacuo , neglecting the 
size of the particles and the temperature), then the energy 
tensor will depend only upon the velocity components, 
u v and cr 0 . We secure the tensor character of T^ v by 

T, V &Q M^jUy . . . ( 50 ) 

in which the u in the three-dimensional representation, 
are given by (41). In fact, it follows from (50) that for 
q — o, 7 " 44 = - <7 0 (equal to the negative energy per unit 
volume), as it should, according to the theorem of the 
equivalence of mass and energy, and according to the 
physical interpretation of the energy tensor given above. 
If an external force (four-dimensional vector, WJ acts 
upon the matter, by the principles of momentum and 
energy the equation 


must hold. We shall now show that this equation leads 
to the same law of motion of a material particle as that 
already obtained. Let us imagine the matter to be of 
infinitely small extent in space, that is, a four-dimensional 
thread ; then by integration over the whole thread with 
respect to the space co-ordinates a q, x 2 , ;r 3 , we obtain 

^K‘ 1 dx 1 dx 2 dx 2 

^dL*dx, dx.dxo = 

J Ltq 1 ^ 3 



dx, dx., , j 
dr dr 123 

Now j dx Y dxtflx % dx± is an invariant, as is, therefore, also 


j (T^dx^dx^dx % dx^. We shall calculate this integral, first 

with respect to the inertial system which we have chosen, 
and second, with respect to a system relatively to which 
the matter has the velocity zero. The integration is to 
be extended over a filament of the thread for which cr 0 
may be regarded as constant over the whole section. If 
the space volumes of the filament referred to the two 
systems are dV and dV 0 respectively, then we have 

L 0 dV<il = 


and therefore also 

Jcr 0 dV = = jd/n i 

. dr 


If we substitute the right-hand side for the left-hand 

, . dx 

side in the former integral, and put ~ outside the sign 




of integration, we obtain, 

is _ d (,J x i\ _ d ( m \ 

K * ~ diK^) ~ dkjnr?) 

We see, therefore, that the generalized conception of the 
energy tensor is in agreement with our former result. 

The Eulerian Equations for Perfect Fluids. In order 
to get nearer to the behaviour of real matter we must add 
to the energy tensor a term which corresponds to the 
pressures. The simplest case is that of a perfect fluid in 
which the pressure is determined by a scalar p. Since 
the tangential stresses p xy , etc., vanish in this case, the 
contribution to the energy tensor must be of the form 
p 8 vll . We must therefore put 

T ^ — <?upu v + pS^ . . (5 0 

At rest, the density of the matter, or the energy per unit 
volume, is in this case, not a but a - p. For 


dx 4 dx i 
dr dr 

a - p. 

In the absence of any force, we have 




du u d(auh) 
<ru vZ —- + u )L + 



dX v 


= o. 

If we multiply this equation by u a 
the fs we obtain, using (40), 

and sum for 


where we have put This is the equation of 

'bx fL dr 


continuity, which differs from that of classical mechanics 

by the term which, practically, is vanishingly small. 

Observing (52), the conservation principles take the form 

+ «, 


L dr 


The equations for the first three indices evidently corre¬ 
spond to the Eulerian equations. That the equations 
(52) and (53) correspond, to a first approximation, to the 
hydrodynamical equations of classical mechanics, is a 
further confirmation of the generalized energy principle. 
The density of matter and of energy has the character of 
a symmetrical tensor. 



A LL of the previous considerations have been based 
upon the assumption that all inertial systems are 
equivalent for the description of physical phenomena, but 
that they are preferred, for the formulation of the laws 
of nature, to spaces of reference in a different state of 
motion. We can think of no cause for this preference 
for definite states of motion to all others, according to 
our previous considerations, either in the perceptible 
bodies or in the concept of motion ; on the contrary, it 
must be regarded as an independent property of the 
space-time continuum. The principle of inertia, in 
particular, seems to compel us to ascribe physically 
objective properties to the space-time continuum. Just 
as it was necessary from the Newtonian standpoint to 
make both the statements, tempus est absolutum , spatium 
est absolutum , so from the standpoint of the special theory 
of relativity we must say, continuum spatii et temporis est 
absolutum. In this latter statement absolutum means not 
only “physically real,” but also “independent in its 
physical properties, having a physical effect, but not itself 
influenced by physical conditions.” 

As long as the principle of inertia is regarded as the 



keystone of physics, this standpoint is certainly the only 
one which is justified. But there are two serious criticisms 
of the ordinary conception. In the first place, it is contrary 
to the mode of thinking in science to conceive of a thing 
(the space-time continuum) which acts itself, but which 
cannot be acted upon. This is the reason why E. Mach 
was led to make the attempt to eliminate space as an 
active cause in the system of mechanics. According to 
him, a material particle does not move in unaccelerated 
motion relatively to space, but relatively to the centre of 
all the other masses in the universe; in this way the 
series of causes of mechanical phenomena was closed, in 
contrast to the mechanics of Newton and Galileo. In 
order to develop this idea within the limits of the modern 
theory of action through a medium, the properties of 
the space-time continuum which determine inertia must 
be regarded as field properties of space, analogous to 
the electromagnetic field. The concepts of classical 
mechanics afford no way of expressing this. For this 
reason Mach’s attempt at a solution failed for the time 
being. We shall come back to this point of view later. 
In the second place, classical mechanics indicates a 
limitation which directly demands an extension of the 
principle of relativity to spaces of reference which are not 
in uniform motion relatively to each other. The ratio of 
the masses of two bodies is defined in mechanics in two 
ways which differ from each other fundamentally; in the 
first place, as the reciprocal ratio of the accelerations 
which the same motional force imparts to them (inert 
mass), and in the second place, as the ratio of the forces 



which act upon them in the same gravitational held 
(gravitational mass). The equality of these two masses, 
so differently defined, is a fact which is confirmed by 
experiments of very high accuracy (experiments of Edtvos), 
and classical mechanics offers no explanation for this 
equality. It is, however, clear that science is fully justified 
in assigning such a numerical equality only after this 
numerical equality is reduced to an equality of the real 
nature of the two concepts. 

That this object may actually be attained by an exten¬ 
sion of the principle of relativity, follows from the follow¬ 
ing consideration. A little reflection will show that the 
theorem of the equality of the inert and the gravitational 
mass is equivalent to the theorem that the acceleration 
imparted to a body by a gravitational field is independent 
of the nature of the body. For Newton’s equation of 
motion in a gravitational field, written out in full, is 

(Inert mass). (Acceleration) = (Intensity of the 

gravitational field) . (Gravitational mass). 

It is only when there is numerical equality between the 
inert and gravitational mass that the acceleration is in¬ 
dependent of the nature of the body. Let now K be an 
inertial system. Masses which are sufficiently far from 
each other and from other bodies are then, with respect 
to Y, free from acceleration. We shall also refer these 
masses to a system of co-ordinates K\ uniformly acceler¬ 
ated with respect to K. Relatively to K' all the masses 
have equal and parallel accelerations ; with respect to K' 
they behave just as if a gravitational field were present and 


K' were unaccelerated. Overlooking for the present the 
question as to the “ cause ” of such a gravitational field, 
which will occupy us later, there is nothing to prevent 
our conceiving this gravitational field as real, that is, the 
conception that K' is “ at rest ” and a gravitational field 
is present we may consider as equivalent to the concep¬ 
tion that only K is an “ allowable ” system of co-ordinates 
and no gravitational field is present. The assumption of 
the complete physical equivalence of the systems of co¬ 
ordinates, K and K\ we call the “ principle of equival¬ 
ence;” this principle is evidently intimately connected 
with the theorem of the equality between the inert and 
the gravitational mass, and signifies an extension of the 
principle of relativity to co-ordinate systems which are 
in non-uniform motion relatively to each other. In fact, 
through this conception we arrive at the unity of the 
nature of inertia and gravitation. For according to our 
way of looking at it, the same masses may appear to be 
either under the action of inertia alone (with respect to 
K) or under the combined action of inertia and gravita¬ 
tion (with respect to K). The possibility of explaining 
the numerical equality of inertia and gravitation by the 
unity of their nature gives to the general theory of 
relativity, according to my conviction, such a superiority 
over the conceptions of classical mechanics, that all the 
difficulties encountered in development must be considered 
as small in comparison. 

What justifies us in dispensing with the preference 
for inertial systems over all other co-ordinate systems, a 
preference that seems so securely established by experi- 


ment based upon the principle of inertia ? The weakness 
of the principle of inertia lies in this, that it involves an 
argument in a circle : a mass moves without acceleration 
if it is sufficiently far from other bodies; we know that 
it is sufficiently far from other bodies only by the fact 
that it moves without acceleratioa Are there, in general, 
any inertial systems for very extended portions of the 
space-time continuum, or, indeed, for the whole universe? 
We may look upon the principle of inertia as established, 
to a high degree of approximation, for the space of our 
planetary system, provided that we neglect the perturba¬ 
tions due to the sun and planets. Stated more exactly, 
there are finite regions, where, with respect to a suitably 
chosen space of reference, material particles move freely 
without acceleration, and in which the laws of the special 
theory of relativity, which have been developed above, 
hold with remarkable accuracy. Such regions we shall 
call “Galilean regions.” We shall proceed from the 
consideration of such regions as a special case of known 

The principle of equivalence demands that in dealing 
with Galilean regions we may equally well make use of 
non-inertial systems, that is, such co-ordinate systems as, 
relatively to inertial systems, are not free from accelera¬ 
tion and rotation. If, further, we are going to do away 
completely with the difficult question as to the objective 
reason for the preference of certain systems of co-ordinates, 
then we must allow the use of arbitrarily moving systems 
of co-ordinates. As soon as we make this attempt seriously 



we come into conflict with that physical interpretation of 
space and time to which we were led by the special theory 
of relativity. For let K' be a system of co-ordinates whose 
/-axis coincides with the -S'-axis of K , and which rotates 
about the latter axis with constant angular velocity. Are 
the configurations of rigid bodies, at rest relatively to K\ 
in accordance with the laws of Euclidean geometry? 
Since K' is not an inertial system, we do not know 
directly the laws of configuration of rigid bodies with 
respect to K', nor the laws of nature, in general. But 
we do know these laws with respect to the inertial system 
K : and we can therefore estimate them with respect to K'. 
Imagine a circle drawn about the origin in the x'y plane 
of K\ and a diameter of this circle. Imagine, further, that 
we have given a large number of rigid rods, all equal to 
each other. We suppose these laid in series along the 
periphery and the diameter of the circle, at rest relatively 
to K'. If U is the number of these rods along the peri¬ 
phery, D the number along the diameter, then, if K does 
not rotate relatively to K, we shall have 


d ~ 77 • 

But if K rotates we get a different result. Suppose 
that at a definite time t } of K we determine the ends of 
all the rods. With respect to K all the rods upon the 
periphery experience the Lorentz contraction, but the 
rods upon the diameter do not experience this contrac- 



tion (along their lengths !).* It therefore follows that 


D >lr ■ 

It therefore follows that the laws of configuration of 
rigid bodies with respect to K' do not agree with the 
laws of configuration of rigid bodies that are in accord¬ 
ance with Euclidean geometry. If, further, we place two 
similar clocks (rotating withTT), one upon the periphery, 
and the other at the centre of the circle, then, judged 
from K y the clock on the periphery will go slower than 
the clock at the centre. The same thing must take place, 
judged from K\ if we define time with respect to K' in 
a not wholly unnatural way, that is, in such a way that 
the laws with respect to K' depend explicitly upon the 
time. Space and time, therefore, cannot be defined 
with respect to K' as they were in the special theory of 
relativity with respect to inertial systems. But, accord¬ 
ing to the principle of equivalence, K' is also to be con¬ 
sidered as a system at rest, with respect to which there 
is a gravitational field (field of centrifugal force, and 
force of Coriolis). We therefore arrive at the result: 
the gravitational field influences and even determines the 
metrical laws of the space-time continuum. If the laws 
of configuration of ideal rigid bodies are to be expressed 
geometrically, then in the presence of a gravitational 
field the geometry is not Euclidean. 

* These considerations assume that the behaviour of rods and clocks 
depends only upon velocities, and not upon accelerations, or, at least, that 
the influence of acceleration does not counteract that of velocity. 


The case that we have been considering is analogous 
to that which is presented in the two-dimensional treat¬ 
ment of surfaces. It is impossible in the latter case 
also, to introduce co-ordinates on a surface (e.g. the 
surface of an ellipsoid) which have a simple metrical 
significance, while on a plane the Cartesian co-ordinates, 
x v x 2 , signify directly lengths measured by a unit 
measuring rod. Gauss overcame this difficulty, in his 
theory of surfaces, by introducing curvilinear co-ordinates 
which, apart from satisfying conditions of continuity, 
were wholly arbitrary, and afterwards these co-ordinates 
were related to the metrical properties of the surface. 
In an analogous way we shall introduce in the general 
theory of relativity arbitrary co-ordinates, x v x 2 , x v x^ 
which shall number uniquely the space-time points, so 
that neighbouring events are associated with neighbour¬ 
ing values of the co-ordinates ; otherwise, the choice of 
co-ordinates is arbitrary. We shall be true to the 
principle of relativity in its broadest sense if we give 
such a form to the laws that they are valid in every 
such four-dimensional system of co-ordinates, that is, if 
the equations expressing the laws are co-variant with 
respect to arbitrary transformations. 

The most important point of contact between Gauss’s 
theory of surfaces and the general theory of relativity 
lies in the metrical properties upon which the concepts 
of both theories, in the main, are based. In the case 
of the theory of surfaces, Gauss’s argument is as follows. 
Plane geometry may be based upon the concept of the 
distance ds, between two indefinitely near points. The 



concept of this distance is physically significant because 
the distance can be measured directly by means of a 
rigid measuring rod. By a suitable choice of Cartesian 
co-ordinates this distance may be expressed by the 
formula ds 2 = dx 2 + dx 2 2 . We may base upon this 
quantity the concepts of the straight line as the geodesic 
(h\ds = o), the interval, the circle, and the angle, upon 
which the Euclidean plane geometry is built. A 
geometry may be developed upon another continuously 
curved surface, if we observe that an infinitesimally 
small portion of the surface may be regarded as plane, 
to within relatively infinitesimal quantities. There are 
Cartesian co-ordinates, X lt X% t upon such a small 
portion of the surface, and the distance between two 
points, measured by a measuring rod, is given by 

ds 1 = dX , 2 + dX*. 

If we introduce arbitrary curvilinear co-ordinates, x Y , x 2 , 
on the surface, then dX lt dX 2 , may be expressed linearly 
in terms of dx lt dx 2 . Then everywhere upon the sur¬ 
face we have 

ds 2 = g n dx^ + 2g u dx 1 dx 2 + g^dxg 

where g n , g 12 , g 22 are determined by the nature of the 
surface and the choice of co-ordinates ; if these quantities 
are known, then it is also known how networks of rigid 
rods may be laid upon the surface. In other words, the 
geometry of surfaces may be based upon this expression 
for ds 2 exactly as plane geometry is based upon the 
corresponding expression. 

There are analogous relations in the four-dimensional 


space-time continuum of physics. In the immediate 
neighbourhood of an observer, falling freely in a gravi¬ 
tational field, there exists no gravitational field. We 
can therefore always regard an infinitesimally small 
region of the space-time continuum as Galilean. For 
such an infinitely small region there will be an inertial 
system (with the space co-ordinates, X lt X 2 , AG, and the 
time co-ordinate A” 4 ) relatively to which we are to regard 
the laws of the special theory of relativity as valid. The 
quantity which is directly measurable by our unit 
measuring rods and clocks, 

dx 2 + dX A + dX 3 2 - dX 2 
or its negative, 

ds 1 = - dX 2 - dX 2 - dX 2 + dX 2 . (54) 

is therefore a uniquely determinate invariant for two 
neighbouring events (points in the four-dimensional 
continuum), provided that we use measuring rods that 
are equal to each other when brought together and 
superimposed, and clocks whose rates are the same 
when they are brought together. In this the physical 
assumption is essential that the relative lengths of two 
measuring rods and the relative rates of two clocks are 
independent, in principle, of their previous history. But 
this assumption is certainly warranted by experience; 
if it did not hold there could be no sharp spectral lines ; 
for the single atoms of the same element certainly do 
not have the same history, and it would be absurd to 
suppose any relative difference in the structure of the 



single atoms due to their previous history if the mass 
and frequencies of the single atoms of the same element 
were always the same. 

Space-time regions of finite extent are, in general, 
not Galilean, so that a gravitational field cannot be done 
away with by any choice of co-ordinates in a finite 
region. There is, therefore, no choice of co-ordinates 
for which the metrical relations of the special theory of 
relativity hold in a finite region. But the invariant ds 
always exists for two neighbouring points (events) of 
the continuum. This invariant ds may be expressed in 
arbitrary co-ordinates. If one observes that the local 
dX v may be expressed linearly in terms of the co¬ 
ordinate differentials dx„ ds 2 may be expressed in the 

ds 1 = g^dx/lXy . . • (55) 

The functions g^ v describe, with respect to the arbit¬ 
rarily chosen system of co-ordinates, the metrical rela¬ 
tions of the space-time continuum and also the 
gravitational field. As in the special theory of relativity, 
we have to discriminate between time-like and space¬ 
like line elements in the four-dimensional continuum ; 
owing to the change of sign introduced, time-like 
line elements have a real, space-like line elements an 
imaginary ds. The time-like ds can be measured directly 
by a suitably chosen clock. 

According to what has been said, it is evident that 
the formulation of the general theory of relativity 
assumes a generalization of the theory of invariants and 
the theory of tensors; the question is raised as to the 


form of the equations which are co-variant with respect 
to arbitrary point transformations. The generalized 
calculus of tensors was developed by mathematicians 
long before the theory of relativity. Riemann first 
extended Gauss’s train of thought to continua of any 
number of dimensions; with prophetic vision he saw 
the physical meaning of this generalization of Euclid’s 
geometry. Then followed the development of the theory 
in the form of the calculus of tensors, particularly by 
Ricci and Levi-Civita. This is the place for a brief 
presentation of the most important mathematical con¬ 
cepts and operations of this calculus of tensors. 

We designate four quantities, which are defined as 
functions of the x v with respect to every system of co¬ 
ordinates, as components, A u y of a contra-variant vector, 
if they transform in a change of co-ordinates as the co¬ 
ordinate differentials dx v . We therefore have 


A*' = z-^A\ 


Besides these contra-variant vectors, there are also co¬ 
variant vectors. If B v are the components of a co-variant 
vector, these vectors are transformed according to the 

B\ = 

( 57 ) 

The definition of a co-variant vector is chosen in such a 
way that a co-variant vector and a contra-variant vector 
together form a scalar according to the scheme, 

<ft = B V A V (summed over the v). 




B\A*' = ^ 


BAf* = B n A\ 


In particular, the derivatives of a scalar 6 , are com- 

ponents of a co-variant vector, which, with the co-ordinate 

differentials, form the scalar ; we see from this 


example how natural is the definition of the co-variant 

There are here, also, tensors of any rank, which may 
have co-variant or contra-variant character with respect 
to each index ; as with vectors, the character is desig¬ 
nated by the position of the index. For example, A / 
denotes a tensor of the second rank, which is co-variant 
with respect to the index /i, and contra-variant with re¬ 
spect to the index v. The tensor character indicates 
that the equation of transformation is 

^ a ' 

( 58 ) 

Tensors may be formed by the addition and subtraction 
of tensors of equal rank and like character, as in the 
theory of invariants of orthogonal linear substitutions, for 

a; + b;= q. . . . (59) 

The proof of the tensor character of C* depends upon (58). 

Tensors may be formed by multiplication, keeping the 
character of the indices, just as in the theory of invariants 
of linear orthogonal transformations, for example, 

. (60) 

r v 



The proof follows directly from the rule of transforma¬ 

Tensors may be formed by contraction with respect to 
two indices of different character, for example, 

A% t = B aT . . . . (61) 

The tensor character of A£ ar determines the tensor 
character of B aT . Proof— 

_ 3*. 3£V 

Lr s ~dX t 
~dx' a dx' T 


ast • 

The properties of symmetry and skew-symmetry of a 
tensor with respect to two indices of like character have 
the same significance as in the theory of invariants. 

With this, everything essential has been said with 
regard to the algebraic properties of tensors. 

The Fundamental Tensor. It follows from the invari¬ 
ance of ds 2 for an arbitrary choice of the dx v , in connexion 
with the condition of symmetry consistent with (55), that 
the g^ v are components of a symmetrical co-variant tensor 
(Fundamental Tensor). Let us form the determinant, 
g, of the g^ vi and also the minors, divided by g, cor¬ 
responding to the single g^. These minors, divided by 
g } will be denoted by g and their co-variant character 
is not yet known. Then we have 


<rtf = = 1 U r a 7 ^ 

a O if a ft 


If we form the infinitely small quantities (co-variant 

— g^oF^o. • • • (^ 3 ) 


multiply by g' x & and sum over the //,, we obtain, by the 
use of (62), 

dxi = g^d^. . . ■ (64) 

Since the ratios of the d^ are arbitrary, and the dx$ as 
well as the dx^ are components of vectors, it follows that 
the g* v are the components of a contra-variant tensor * 
(contra-variant fundamental tensor). The tensor character 
of Sf (mixed fundamental tensor) accordingly follows, 
by (62). By means of the fundamental tensor, instead 
of tensors with co-variant index character, we can 
introduce tensors with contra-variant index character, 
and conversely. For example, 

= g»*A a 

A = 

T a = . 

fJ- 5 fj-v 

Volume Invariants. The volume element 

S dx Y dx. L dx z dx A = dx 

is not an invariant. For by Jacobi’s theorem, 

dx = 






* If we multiply (64) by , sum over the £, and replace the dj-n by a 
transformation to the accented system, we obtain 

dx' a 

dx'q- 'dx'a 
dx^ dxp 


<T • 

The statement made above follows from this, since, by (64), we must also 
have dx'a = g^'d}-'a , and both equations must hold for every choice of the 



But we can complement dx so that it becomes an in¬ 
variant. If we form the determinant of the quantities 

, ^x a Dx,, 

~ ix'^ 7>x' g ^ 

we obtain, by a double application of the theorem of 
multiplication of determinants, 




<b MV 

We therefore get the invariant, 

Jgdx = Jgdx. 

Formation of Te?isors by Differentiation. Although 
the algebraic operations of tensor formation have proved 
to be as simple as in the special case of invariance with 
respect to linear orthogonal transformations, nevertheless 
in the general case, the invariant differential operations 
are, unfortunately, considerably more complicated. The 
reason for this is as follows. If A* is a contra-variant 

vector, the coefficients of its transformation, are in- 

Dx v 

dependent of position only if the transformation is a linear 


one. For then the vector components, A* + —— dx a , at 

oX a 

a neighbouring point transform in the same way as the 
A*, from which follows the vector character of the vector 

differentials, and the tensor character of 



Dx v 

are variable this is no longer true. 

But if the 



That there are, nevertheless, in the general case, in¬ 
variant differential operations for tensors, is recognized 
most satisfactorily in the following way, introduced by 
Levi-Civita and Weyl. Let (A*) be a contra-variant vector 
whose components are given with respect to the co¬ 
ordinate system of the x v . Let P 1 and P 2 be two in¬ 
finitesimally near points of the continuum. For the 
infinitesimal region surrounding the point P v there is, 
according to our way of considering the matter, a co¬ 
ordinate system of the X v (with imaginary ^-co¬ 
ordinates) for which the continuum is Euclidean. Let 
A f x) be the co-ordinates of the vector at the point P v 
Imagine a vector drawn at the point P v using the local 
system of the X v , with the same co-ordinates (parallel 
vector through P^) } then this parallel vector is uniquely 
determined by the vector at P 1 and the displacement. 
We designate this operation, whose uniqueness will appear 
in the sequel, the parallel displacement of the vector An 
from P 1 to the infinitesimally near point P 2 If we form 
the vector difference of the vector (A*) at the point P 2 
and the vector obtained by parallel displacement from P x 
to P 2 , we get a vector which may be regarded as the 
differential of the vector ( A for the given displacement 

This vector displacement can naturally also be con¬ 
sidered with respect to the co-ordinate system of the x v . 
If A v are the co-ordinates of the vector at P lf A v + &A V 
the co-ordinates of the vector displaced to P 2 along the 
interval (dx v ), then the SA U do not vanish in this case. 
We know of these quantities, which do not have a vector 


character, that they must depend linearly and homo¬ 
geneously upon the dx v and the A v . We therefore put 

SA V = - T^A'dxp . . (67) 

In addition, we can state that the T v a p must be sym¬ 
metrical with respect to the indices a and { 3 . For we 
can assume from a representation by the aid of a Euclid¬ 
ean system of local co-ordinates that the same parallelo¬ 
gram will be described by the displacement of an element 
d [ 1 ) x v along a second element d^x v as by a displacement 
of d^x v along d^x v . We must therefore have 

d^\x v + (d [X) x v - T^K^Xp) 

= d { 1 ) x v + (d [ ~ ) x v - V^xj^xp). 

The statement made above follows from this, after inter¬ 
changing the indices of summation, a and / 3 , on the 
right-hand side. 

Since the quantities g^ v determine all the metrical 
properties of the continuum, they must also determine 
the T^. If we consider the invariant of the vector A v , 
that is, the square of its magnitude, 


which is an invariant, this cannot change in a parallel 
displacement. We therefore have 

o = S(g^A»A') = jgA*A"dx a + g^A^SA* + g^ASA* 
or, by (67), 

- g^ r i ~ g^K)^A”dx a = o. 


Owing to the symmetry of the expression in the 
brackets with respect to the indices and v , this equation 
can be valid for an arbitrary choice of the vectors ( A a ) 
and dx v only when the expression in the brackets vanishes 
for all combinations of the indices. By a cyclic inter¬ 
change of the indices fi, v , a, we obtain thus altogether 
three equations, from which we obtain, on taking into 
account the symmetrical property of the 

• • • ( 68 ) 

in which, following Christoffel, the abbreviation has been 

If we multiply (68) by g acr and sum over the a, we 


in which {'7} is the Christoffel symbol of the second 
kind. Thus the quantities T are deduced from the g^ v . 
Equations (67) and (70) are the foundation for the 
following discussion. 

Co-variant Differentiation of Tensors. If (A 11 + SAf is 
the vector resulting from an infinitesimal parallel displace¬ 
ment from P 1 to P 2, and ( A “ + dA the vector A* at the 
point P 2l then the difference of these two, 

dA* - 8A* = 

+ Y^A^dX" 


is also a vector. Since this is the case for an arbitrary 
choice of the dx vi it follows that 


' dA fX 


is a tensor, which we designate as the co-variant derivative 
of the tensor of the first rank (vector). Contracting this 
tensor, we obtain the divergence of the contra-variant 
tensor A\ In this we must observe that according to 
( 70 ), 

If we put, further, 

() gr 

_ JL rr<ra <7> (Ta 


JL h/A 


A* Jg = B* 



a quantity designated by Weyl as the contra-variant tensor 
density * of the first rank, it follows that, 


is a scalar density. 

We get the law of parallel displacement for the 
co-variant vector Z? by stipulating that the parallel 
displacement shall be effected in such a way that the 

cf) = A^B^ 

remains unchanged, and that therefore 

Ar-ZBp + 

*This expression is justified, in that Av-Jgdx = 21 ^dx has a tensor 
character. Every tensor, when multiplied by Jg, changes into a tensor 
density. We employ capital Gothic letters for tensor densities. 



vanishes for every value assigned to (A"-). We therefore 

BB P = ri„AJx„. . . . (75) 

From this we arrive at the co-variant derivative of the 
co-variant vector by the same process as that which led 
to (71), 

b p , <T - ¥?* - r %b v . . (76) 

ox 9 

By interchanging the indices ^ and a, and subtracting, 
we get the skew-symmetrical tensor, 





For the co-variant differentiation of tensors of the 
second and higher ranks we may use the process by 
which (75) was deduced. Let, for example, ( A ar ) be a 
co-variant tensor of the second rank. Then A^E^F 7 is 
a scalar, if E and F are vectors. This expression must 
not be changed by the 8-displacement; expressing this 
by a formula, we get, using (67), SA aT , whence we get the 
desired co-variant derivative, 




or; p 

te n 

_ r ,a A — P a A 

A (Tp**- 1 aT A Tp 


• (78) 

In order that the general law of co-variant differ¬ 
entiation of tensors may be clearly seen, we shall write 
down two co-variant derivatives deduced in an analogous 

Al. a 

... aA; 

a, p 

te p 


'dA (TT 

. p 

- r a A T + T T A a 

A ap xx a * A ap x± (T 

+ + F' p A™. 





The general law of formation now becomes evident. 
From these formulae we shall deduce some others which 
are of interest for the physical applications of the theory. 
In case A ar is skew-symmetrical, we obtain the tensor 



+ 3 ^ + 3 A fT 
~bx p ~bx, ix T 


which is skew-symmetrical in all pairs of indices, by cyclic 
interchange and addition. 

If, in (78), we replace A ar by the fundamental tensor* 
g aT , then the right-hand side vanishes identically ; an 
analogous statement holds for (80) with respect to g aT ; 
that is, the co-variant derivatives of the fundamental 
tensor vanish. That this must be so we see directly in 
the local system of co-ordinates. 

In case A aT is skew-symmetrical, we obtain from (80), 
by contraction with respect to t and p, 


In the general case, from (79) and (80), by contraction 
with respect to t and p, we obtain the equations, 

= Tiffri. . . ( 83 ) 

dX a 

a- = ^ . . (84) 

The Riemann Tensor . If we have given a curve ex¬ 
tending from the point P to the point G of the continuum, 
then a vector A*, given at P, may, by a parallel displace¬ 
ment, be moved along the curve to G. If the continuum 



is Euclidean (more generally, if by a suitable choice of 
co-ordinates the^ v are constants) then the vector obtained 
at G as a result of this displacement does not depend 
upon the choice of the curve joining P and G. But 
otherwise, the result depends upon the path of the dis¬ 
placement. In this case, therefore, a vector suffers a 
change, A A* (in its direction, not its magnitude), when it 
is carried from a point P of a closed curve, along the 


curve, and back to P. We shall now calculate this vector 

A A* = 

As in Stokes’ theorem for the line integral of a vector 
around a closed curve, this problem may be reduced to 
the integration around a closed curve with infinitely small 
linear dimensions; we shall limit ourselves to this case. 


We have, first, by (67), 

A A* = 


T%A a d*p 



In this, Tjjg is the value of this quantity at the variable 
point G of the path of integration. If vve put 

Z !L ~ ( x v)g ~ ( x v)p 

and denote the value of Y^p at P by T^, then we have, 
with sufficient accuracy, 

- 7nT va 

"PM _ pM 1 UJ - 
L< * “ ^ + ■ 

Let, further, A a be the value obtained from A a by a 
parallel displacement along the curve from P to G. It 
may now easily be proved by means of (67) that A M - A* 
is infinitely small of the first order, while, for a curve of 
infinitely small dimensions of the first order, A A* is 
infinitely small of the second order. Therefore there is 
an error of only the second order if we put 

A a = ~A* - f l T A~ a F- 

If we introduce these values of Y^p and A a into the 
integral, we obtain, neglecting all quantities of a higher 
order of small quantities than the second, 

a a* = - gy - . (85) 


The quantity removed from under the sign of integration 



refers to the point P. Subtracting from the 

integrand, we obtain 


This skew-symmetrical tensor of the second rank, f aPi 
characterizes the surface element bounded by the curve 
in magnitude and position. If the expression in the 
brackets in (85) were skew-symmetrical with respect to 
the indices a and ft, we could conclude its tensor char¬ 
acter from (85). We can accomplish this by interchanging 
the summation indices a and ft in (85) and adding the 
resulting equation to (85). We obtain 

2AA* = - R\* Tmfi A*f+ . . (86) 

in which 

r a > 

+ W, - r&rk (87) 

The tensor character of follows from (86); this is 
the Riemann curvature tensor of the fourth rank, whose 
properties of symmetry we do not need to go into. Its 
vanishing is a sufficient condition (disregarding the reality 
of the chosen co-ordinates) that the continuum is 

By contraction of the Riemann tensor with respect to 
the indices fi, ft, we obtain the symmetrical tensor of the 
second rank, 



+ r^r? 




pa p/3 
A 1 aj3‘ 

( 88 ) 

The last two terms vanish if the system of co-ordinates 


is so chosen that^ = constant. From R^ v we can form 
the scalar, 

R = • • • (89) 

Straightest ( Geodetic ) Lines. A line may be constructed 
in such a way that its successive elements arise from each 
other by parallel displacements. This is the natural 
generalization of the straight line of the Euclidean 
geometry. For such a line, we have 

The left-hand side is to be replaced by 

ds 1 ’ 

so that we 



dx a dxp 
ds ds 



We get the same line if we find the line which gives a 
stationary value to the integral 

[ds or L/. 


between two points (geodetic line). 

* The direction vector at a neighbouring point of the curve results, by a 
parallel displacement along the line element (<^^), from the direction vector 
of each point considered. 



( Continued) 

W E are now in possession of the mathematical 
apparatus which is necessary to formulate the 
laws of the general theory of relativity. No attempt 
will be made in this presentation at systematic complete¬ 
ness, but single results and possibilities will be devel¬ 
oped progressively from what is known and from the 
results obtained. Such a presentation is most suited 
to the present provisional state of our knowledge. 

A material particle upon which no force acts moves, 
according to the principle of inertia, uniformly in a 
straight line. In the four-dimensional continuum of the 
special theory of relativity (with real time co-ordinate) 
this is a real straight line. The natural, that is, the 
simplest, generalization of the straight line which is 
plausible in the system of concepts of Riemann’s general 
theory of invariants is that of the straightest, or geodetic, 
line. We shall accordingly have to assume, in the sense 
of the principle of equivalence, that the motion of a 
material particle, under the action only of inertia and 
gravitation, is described by the equation, 

ds 2 

dx a dx { 3 
+ l ^ds ds 




In fact, this equation reduces to that of a straight line 
if all the components, of the gravitational field 


How are these equations connected with Newton’s 
equations of motion? According to the special theory 
of relativity, the g^ v as well as the g^ v , have the values, 
with respect to an inertial system (with real time co¬ 
ordinate and suitable choice of the sign of ds 2 ), 

- i o o 

o-i o 

o o - i 






• (91) 

The equations of motion then become 

ds 2 

= o. 

We shall call this the “ first approximation ” to the g !XV - 
field. In considering approximations it is often useful, 
as in the special theory of relativity, to use an imaginary 
^-co-ordinate, as then the g fJLV , to the first approxima¬ 
tion, assume the values 


These values may be collected in the relation 

cr — — $ 

& fj-v '-'txv' 

To the second approximation we must then put 

S>y,v — “b ’ 




where the y^ v are to be regarded as small of the first 

Both terms of our equation of motion are then small 
of the first order. If we neglect terms which, relatively 
to these, are small of the first order, we have to put 

1 /dy aj 8 

2 \ 'bx tL 

We shall now introduce an approximation of a second 
kind. Let the velocity of the material particles be very 
small compared to that of light. Then ds will be the 

dx x dx^ dx 3 

same as the time differential, dl. Further, 

will vanish compared to We shall assume, in addi¬ 

tion, that the gravitational field varies so little with the 
time that the derivatives of the y^ v by x i may be 
neglected. Then the equation of motion (for fi= I, 2, 3) 
reduces to 

d 2 x, 


dl 1 

Lr^\ 2 / 


This equation is identical with Newton’s equation of 
motion for a material particle in a gravitational field, if 

we identify with the potential of the gravitational 

field ; whether or not this is allowable, naturally depends 
upon the field equations of gravitation, that is, it de¬ 
pends upon whether or not this quantity satisfies, to a 
first approximation, the same laws of the field as the 


gravitational potential in Newton’s theory. A glance 
at (90) and (90a) shows that the Tjh actually do play 
the role of the intensity of the gravitational field. 
These quantities do not have a tensor character. 

Equations (90) express the influence of inertia and 
gravitation upon the material particle. The unity of 
inertia and gravitation is formally expressed by the fact 
that the whole left-hand side of (90) has the character 
of a tensor (with respect to any transformation of co¬ 
ordinates), but the two terms taken separately do not 
have tensor character, so that, in analogy with Newton’s 
equations, the first term would be regarded as the ex¬ 
pression for inertia, and the second as the expression 
for the gravitational force. 

We must next attempt to find the laws of the gravita¬ 
tional field. For this purpose, Poisson’s equation, 

A<£ = \irKp 

of the Newtonian theory must serve as a model. This 
equation has its foundation in the idea that the gravi¬ 
tational field arises from the density p of ponderable 
matter. It must also be so in the general theory of 
relativity. But our investigations of the special theory 
of relativity have shown that in place of the scalar 
density of matter we have the tensor of energy per unit 
volume. In the latter is included not only the tensor 
of the energy of ponderable matter, but also that of the 
electromagnetic energy. We have seen, indeed, that 
in a more complete analysis the energy tensor can be 
regarded only as a provisional means of representing 



matter. In reality, matter consists of electrically charged 
particles, and is to be regarded itself as a part, in fact, 
the principal part, of the electromagnetic field. It is 
only the circumstance that we have not sufficient know¬ 
ledge of the electromagnetic field of concentrated charges 
that compels us, provisionally, to leave undetermined 
in presenting the theory, the true form of this tensor. 
From this point of view our problem now is to introduce 
a tensor, T^, of the second rank, whose structure we do 
not know provisionally, and which includes in itself the 
energy density of the electromagnetic field and of ponder¬ 
able matter; we shall denote this in the following as 
the “ energy tensor of matter.” 

According to our previous results, the principles of 
momentum and energy are expressed by the statement 
that the divergence of this tensor vanishes (47c). In 
the general theory of relativity, we shall have to assume 
as valid the corresponding general co-variant equation. 
If (T^ v ) denotes the co-variant energy tensor of matter, 
XKJ. the corresponding mixed tensor density, then, in 
accordance with (83), we must require that 

o = 


be satisfied. It must be remembered that besides the 
energy density of the matter there must also be given 
an energy density of the gravitational field, so that there 
can be no talk of principles of conservation of energy 
and momentum for matter alone. This is expressed 
mathematically by the presence of the second term in 


(95), which makes it impossible to conclude the existence 
of an integral equation of the form of (49). The gravi¬ 
tational field transfers energy and momentum to the 
“matter,” in that it exerts forces upon it and gives it 
energy; this is expressed by the second term in (95). 

If there is an analogue of Poisson’s equation in the 
general theory of relativity, then this equation must be 
a tensor equation for the tensor g^ v of the gravitational 
potential; the energy tensor of matter must appear on 
the right-hand side of this equation. On the left-hand 
side of the equation there must be a differential tensor 
in the g^ v . We have to find this differential tensor. 
It is completely determined by the following three 

1. It may contain no differential coefficients of the^ 
higher than the second. 

2. It must be linear and homogeneous in these second 
differential coefficients. 

3. Its divergence must vanish identically. 

The first two of these conditions are naturally taken 
from Poisson’s equation. Since it may be proved 
mathematically that all such differential tensors can be 
formed algebraically (i.e. without differentiation) from 
Riemann’s tensor, our tensor must be of the form 

K v + 

in which R^ v and R are defined by (88) and (89) respec¬ 
tively. Further, it may be proved that the third condi¬ 
tion requires a to have the value - For the law 


of the gravitational field we therefore get the equa¬ 

Equation (95) is a consequence of this equation, tc de¬ 
notes a constant, which is connected with the Newtonian 
gravitation constant. 

In the following I shall indicate the features of the 
theory which are interesting from the point of view of 
physics, using as little as possible of the rather involved 
mathematical method. It must first be shown that the 
divergence of the left-hand side actually vanishes. The 
energy principle for matter may be expressed, by (83), 

0 - £ - 

in which 

Z", = - 



The analogous operation, applied to the left-hand side 
of (96), will lead to an identity. 

In the region surrounding each world-point there are 
systems of co-ordinates for which, choosing the ^^-co¬ 
ordinate imaginary, at the given point, 

- g* = _ = o if ^ ={= V) 

and for which the first derivatives of the g^ v and the 
g* v vanish. We shall verify the vanishing of the diverg¬ 
ence of the left-hand side at this point. At this point 
the components T^ a vanish, so that we have to prove 
the vanishing only of 


Introducing (88) and (70) into this expression, we see 
that the only terms that remain are those in which third 
derivatives of the g^ v enter. Since the g are to be 
replaced by - we obtain, finally, only a few terms 
which may easily be seen to cancel each other. Since 
the quantity that we have formed has a tensor character, 
its vanishing is proved for every other system of co-ordin¬ 
ates also, and naturally for every other four-dimensional 
point. The energy principle of matter (97) is thus a 
mathematical consequence of the field equations (96). 

In order to learn whether the equations (96) are 
consistent with experience, we must, above all else, find 
out whether they lead to the Newtonian theory as a 
first approximation. For this purpose we must intro¬ 
duce various approximations into these equations. We 
already know that Euclidean geometry and the law of the 
constancy of the velocity of light are valid, to a certain 
approximation, in regions of a great extent, as in the 
planetary system. If, as in the special theory of rela¬ 
tivity, we take the fourth co-ordinate imaginary, this 
means that we must put 

~ ~~ y^v • • • ( 9 ^) 

in which the y^ v are so small compared to 1 that we 
can neglect the higher powers of the y^ and their 
derivatives. If we do this, we learn nothing about the 
structure of the gravitational field, or of metrical space of 
cosmical dimensions, but we do learn about the influence 
of neighbouring masses upon physical phenomena. 

Before carrying through this approximation we shall 



transform (96). We multiply (96) by g* v , summed over 
the fi and v ; observing the relation which follows from 
the definition of the g^, 

= 4 

we obtain the equation 

R = /cg llv T flv = kT. 

If we put this value of R in (96) we obtain 

= - k{T^ - = - tcTl,. . (96a) 

When the approximation which has been mentioned is 
carried out, we obtain for the left-hand side, 

+ 'dx y ^x v dx y Zx a ^xg)xj 

l 3 ^- , 3 ■ 3 va\ 

7 ix* 5 .r a / 2 J.r a / 

in which has been put 

y h-v ~ y^v ~ \y • • ( 99 ) 

We must now note that equation (96) is valid for any 
system of co-ordinates. We have already specialized the 
system of co-ordinates in that we have chosen it so that 
within the region considered the g^ v differ infinitely little 
from the constant values - 8 ^. But this condition 
remains satisfied in any infinitesimal change of co¬ 
ordinates, so that there are still four conditions to which 
the may be subjected, provided these conditions do 
not conflict with the conditions for the order of magnitude 


of the y^. We shall now assume that the system of co¬ 
ordinates is so chosen that the four relations— 

_ l-i-v ~^y iav i ^ycrcr 

~ ~dX v “ 1x v 2 IXp. 
are satisfied. Then (96a) takes the form 

= 2 * t % . . . (96b) 

These equations may be solved by the method, familiar 
in electrodynamics, of retarded potentials; we get, in an 
easily understood notation, 

y ixv 

f 9V f ~ r ) JV 
27 rJ r " ( 


In order to see in what sense this theory contains the 
Newtonian theory, we must consider in greater detail 
the energy tensor of matter. Considered phenomeno¬ 
logically, this energy tensor is composed of that of the 
electromagnetic field and of matter in the narrower sense. 
If we consider the different parts of this energy tensor 
with respect to their order of magnitude, it follows 
from the results of the special theory of relativity that 
the contribution of the electromagnetic field practically 
vanishes in comparison to that of ponderable matter. In 
our system of units, the energy of one gram of matter is 
equal to I, compared to which the energy of the electric 
fields may be ignored, and also the energy of deformation 
of matter, and even the chemical energy. We get an 
approximation that is fully sufficient for our purpose if 


we put 

dx„ dx v 1 

^ ■*[ 

ds 2 = g^dxjx J 

In this, <j is the density at rest, that is, the density of the 
ponderable matter, in the ordinary sense, measured with 
the aid of a unit measuring rod, and referred to a Galilean 
system of co-ordinates moving with the matter. 

We observe, further, that in the co-ordinates we have 
chosen, we shall make only a relatively small error if we 
replace the g^ v by - 8 ^, so that we put 

ds 2 = - ^dx 2 . . . (102a) 

The previous developments are valid however rapidly 
the masses which generate the field may move relatively 
to our chosen system of quasi-Galilean co-ordinates. But 
in astronomy we have to do with masses whose velocities, 
relatively to the co-ordinate system employed, are always 
small compared to the velocity of light, that is, small 
compared to i, with our choice of the unit of time. 
We therefore get an approximation which is sufficient 
for nearly all practical purposes if in (ioi) we replace 
the retarded potential by the ordinary (non-retarded) 
potential, and if, for the masses which generate the field, 
we put 

_ dx.j, dx 3 dx± ./ — \dl - 

ds ds ~ ds ~ |ds ~ = ^ ~ I ’ ( I0 3 a ) 



Then we get for and T nv the values 





o o 

o o 

o o 

o o 








For T we get the value cr, and, finally, for T*„ the 









o o 

We thus get, from (101), 
7ll = 722 = 733 = 





O - 




o I 
2 J 

744 = + 


'odV Q \ 

47 T. 



m <rdV 0 

477 . 

r J 



while all the other y^ v vanish. The least of these equa¬ 
tions, in connexion with equation (90a), contains New¬ 
ton’s theory of gravitation. If we replace / by ct we 


drx kc 2 7 ) ff odV 0 [ 

~df ~ J 

We see that the Newtonian gravitation constant W, is 
connected with the constant tc that enters into our field 
equations by the relation 

K = 

fCC 1 





From the known numerical value of K , it therefore 
follows that 

k = 

877 K 877.6-67 . 10 



9 . 10 


1 -86 . 1 o~ 27 . (105a) 

From (101) we see that even in the first approximation 
the structure of the gravitational field differs fundamentally 
from that which is consistent with the Newtonian theory ; 
this difference lies in the fact that the gravitational 
potential has the character of a tensor and not a scalar. 
This was not recognized in the past because only the 
component g 44 , to a first approximation, enters the equa¬ 
tions of motion of material particles. 

In order now to be able to judge the behaviour of 
measuring rods and clocks from our results, we must 
observe the following. According to the principle of 
equivalence, the metrical relations of the Euclidean 
geometry are valid relatively to a Cartesian system of 
reference of infinitely small dimensions, and in a suitable 
state of motion (freely failing, and without rotation). 
We can make the same statement for local systems of 
co-ordinates which, relatively to these, have small ac¬ 
celerations, and therefore for such systems of co-ordinates 
as are at rest relatively to the one we have selected. For 
such a local system, we have, for two neighbouring point 

ds 2 = - dX 2 - dX 2 - dX 2 + dT 2 = - dS 2 + dT 2 

where dS is measured directly by a measuring rod and 
dT by a clock at rest relatively to the system : these are 


the naturally measured lengths and times. Since ds\ on 
the other hand, is known in terms of the co-ordinates x v 
employed in finite regions, in the form 

ds- = g^dx^dx. 

we have the possibility of getting the relation between 
naturally measured lengths and times, on the one hand, 
and the corresponding differences of co-ordinates, on the 
other hand. As the division into space and time is in 
agreement with respect to the two systems of co-ordinates, 
so when we equate the two expressions for ds 2 we get 
two relations. If, by (ioia), we put 

we obtain, to a sufficiently close approximation, 


The unit measuring rod has therefore the length, 

in respect to the system of co-ordinates we have selected. 
The particular system of co-ordinates we have selected 



insures that this length shall depend only upon 
the place, and not upon the direction. If we had 
chosen a different system of co-ordinates this would not 
be so. But however we may choose a system of co¬ 
ordinates, the laws of configuration of rigid rods do not 
agree with those of Euclidean geometry ; in other words, 
we cannot choose any system of co-ordinates so that the 
co-ordinate differences, Ax l} Ax 2 , Ax s , corresponding to the 
ends of a unit measuring rod, oriented in any way, shall 
always satisfy the relation Ax} + Ax} + Ax-} = i. In 
this sense space is not Euclidean, but “ curved.” It 
follows from the second of the relations above that the 
interval between two beats of the unit clock ( dT = i) 
corresponds to the “ time ” 

in the unit used in our system of co-ordinates. The rate 
of a clock is accordingly slower the greater is the mass of 
the ponderable matter in its neighbourhood. We there¬ 
fore conclude that spectral lines which are produced on 
the sun’s surface will be displaced towards the red, 
compared to the corresponding lines produced on the 
earth, by about 2. io~° of their wave-lengths. At first, 
this important consequence of the theory appeared to 
conflict with experiment; but results obtained during the 
past year seem to make the existence of this effect more 
probable, and it can hardly be doubted that this con¬ 
sequence of the theory will be confirmed within the next 


Another important consequence of the theory, which 
can be tested experimentally, has to do with the path of 
rays of light. In the general theory of relativity also 
the velocity of light is everywhere the same, relatively to 
a local inertial system. This velocity is unity in our 
natural measure of time. The law of the propagation of 
light in general co-ordinates is therefore, according to the 
general theory of relativity, characterized, by the equation 

ds 2 = o. 

To within the approximation which we are using, and in 
the system of co-ordinates which we have selected, the 
velocity of light is characterized, according to (106), by 
the equation 


I + 

q* dx<£ q- dxd 

The velocity of light A, is therefore expressed in our 
co-ordinates by 

v / dx 2 q- dx 2 q- dx 2 

k [crdVr, , v 

— —-*• ( I0 7) 

47tJ r 

We can therefore draw the conclusion from this, that a 
ray of light passing near a large mass is deflected. If 
we imagine the sun, of mass M , concentrated at the 
origin of our system of co-ordinates, then a ray of light, 
travelling parallel to the ^ 3 -axis, in the x 1 - x s plane, 
at a distance A from the origin, will be deflected, in all, 
by an amount 



+ » 

f 1 , 

a = — —ax^ 

JL dx x 

towards the sun. 

On performing the integration we get 





The existence of this deflection, which amounts to 
i 7" for A equal to the radius of the sun, was confirmed, 
with remarkable accuracy, by the English Solar Eclipse 
. Expedition in 1919, and most careful preparations have 
been made to get more exact observational data at the 
solar eclipse in 1922. It should be noted that this 
result, also, of the theory is not influenced by our 
arbitrary choice of a system of co-ordinates. 

This is the place to speak of the third consequence of 
the theory which can be tested by observation, namely, 
that which concerns the motion of the perihelion 
of the planet Mercury. The secular changes in the 
planetary orbits are known with such accuracy that the 
approximation we have been using is no longer sufficient 
for a comparison of theory and observation. It is neces¬ 
sary to go back to the general field equations (96). To 
solve this problem I made use of the method of succes¬ 
sive approximations. Since then, however, the problem 
of the central symmetrical statical gravitational field has 
been completely solved by Schwarzschild and others; 
the derivation given by H. Weyl in his book, “ Raum- 
Zeit-Materie,” is particularly elegant. The calculation 
can be simplified somewhat if we do not go back directly 


to the equation (96), but base it upon a principle of 
variation that is equivalent to this equation. I shall 
indicate the procedure only in so far as is necessary for 
understanding the method. 

In the case of a statical field, ds 2 must have the form 

1 ds 2 = - dc r 2 + f 2 dx± 

da- = ^Yapdxjxp 

where the summation on the right-hand side of the last 
equation is to be extended over the space variables only, 
The central symmetry of the field requires the y^ v to be 
of the form, 

Ya£ = /^a/3 + '^X 0 X ) 3 * * ( 1 1 °) 

f 2 , n and \ are functions of r — ^/x 2 + x£ + x z 2 only. 
One of these three functions can be chosen arbitrarily, 
because our system of co-ordinates is, a priori , completely 
arbitrary ; for by a substitution 

^4 = *4 

V« = F(r)x a 

we can always insure that one of these three functions 
shall be an assigned function of r. In place of (i io) we 
can therefore put, without limiting the generality, 

7a£ = ^ a 8 + . . (uoa) 

In this way the g^ v are expressed in terms of the two 
quantities \ and f. These are to be determined as func¬ 
tions of r, by introducing them into equation (96), after 



first calculating the from (107) and (108a). We 

-L afi 

r 4 

1 44 

r 4 

1 4a 

+ 2\r8 a p 

— u 

r I+ ^r-^(fora,A^ = 1 , 2 , 3 ) 
r“ 3 = = o (for a, /3 = I, 2, 3) 

- 2 y 2 

— if 

= - */ 

- 2 ¥! 


With the help of these results, the field equations 
furnish Schwarzschild’s solution : 

ds 2 = 

“ dr 2 



4- r 2 (sin 2 0dcf) 2 + d0 2 ) 

in which we have put 


x A = l 


x Y = r sin 6 sin (p 
x. 2 = r sin 0 cos </> 
x z = r cos 0 

A = 




M denotes the sun’s mass, centrally symmetrically 
placed about the origin of co-ordinates ; the solution (109) 
is valid only outside of this mass, where all the T^ v vanish. 
If the motion of the planet takes place in the x 1 - x. 2 
plane then we must replace (109) by 

/ A \ dv 2 

ds 2 — [1 - -yjdl 2 - -^ - r 2 d(p 2 . (109b) 

1 - — 


The calculation of the planetary motion depends upon 
equation (90). From the first of equations (108b) and 
(90) we get, for the indices 1, 2, 3, 

d ( dxp dx \ 
ds\ X *ds ~ x *ds) ~° 

or, if we integrate, and express the result in polar co¬ 


r = constant. 


From (90), for jj, = 4, we get 

dr l I df 2 dx a dr l I df~ 
0 ds 2 + f' 1 dx a ds ~ ds 2 + f 2 ds' 

From this, after multiplication by/ 2 and integration, we 


= constant. 


In (109b), (ill) and (112) we have three equations 
between the four variables j, r, / and </>, from which the 
motion of the planet may be calculated in the same way 
as in classical mechanics. The most important result we 
get from this is a secular rotation of the elliptic orbit of 
the planet in the same sense as the revolution of the 
planet, amounting in radians per revolution to 



a = the semi-major axis of the planetary orbit in 

e = the numerical eccentricity. 

c = 3 . io+ 10 , the velocity of light in vacuo . 

T = the period of revolution in seconds. 

This expression furnishes the explanation of the motion 
of the perihelion of the planet Mercury, which has been 
known for a hundred years (since Leverrier), and for 
which theoretical astronomy has hitherto been unable 
satisfactorily to account. 

There is no difficulty in expressing Maxwell’s theory 
of the electromagnetic field in terms of the general theory 
of relativity; this is done by application of the tensor 
formation (81), (82) and (77). Let (p^ be a tensor of the 
first rank, to be denoted as an electromagnetic 4-potential; 
then an electromagnetic field tensor may be defined by 
the relations, 




The second of Maxwell’s systems of equations is then 
defined by the tensor equation, resulting from this, 



and the first of Maxwell’s systems of equations is defined 
by the tensor-density relation 


in which 

fl*” = V - gg^g 1 ^ 

J s SP ds■ 


If we introduce the energy tensor of the electromagnetic 
field into the right-hand side of (96), we obtain (115), 
for the special case 3^ = o, as a consequence of (96) by 
taking the divergence. This inclusion of the theory of 
electricity in the scheme of the general theory of relativity 
has been considered arbitrary and unsatisfactory by 
many theoreticians. Nor can we in this way conceive of 
the equilibrium of the electricity which constitutes the 
elementary electrically charged particles. A theory in 
which the gravitational field and the electromagnetic field 
enter as an essential entity would be much preferable. 
H. Weyl, and recently Th. Kaluza, have discovered some 
ingenious theorems along this direction; but concerning 
them, I am convinced that they do not bring us nearer to 
the true solution of the fundamental problem. I shall 
not go into this further, but shall give a brief discussion 
of the so-called cosmological problem, for without this, 
the considerations regarding the general theory of rela¬ 
tivity would, in a certain sense, remain unsatisfactory. 

Our previous considerations, based upon the field 
equations (96), had for a foundation the conception that 
space on the whole is Galilean-Euclidean, and that this 
character is disturbed only by masses embedded in it. 
This conception was certainly justified as long as we were 
dealing with spaces of the order of magnitude of those 



that astronomy has to do with. But whether portions of 
the universe, however large they may be, are quasi- 
Euclidean, is a wholly different question. We can make 
this clear by using an example from the theory of surfaces 
which we have employed many times. If a portion of a 
surface is observed by the eye to be practically plane, it 
does not at all follow that the whole surface has the form 
of a plane ; the surface might just as well be a sphere, for 
example, of sufficiently large radius. The question as to 
whether the universe as a whole is non-Euclidean was 
much discussed from the geometrical point of view before 
the development of the theory of relativity. But with the 
theory of relativity, this problem has entered upon a 
new stage, for according to this theory the geometrical 
properties of bodies are not independent, but depend 
upon the distribution of masses. 

If the universe were quasi-Euclidean, then Mach was 
wholly wrong in his thought that inertia, as well as 
gravitation, depends upon a kind of mutual action between 
bodies. For in this case, with a suitably selected system 
of co-ordinates, the g^ v would be constant at infinity, as 
they are in the special theory of relativity, while within 
finite regions the g ixv would differ from these constant 
values by small amounts only, with a suitable choice of 
co-ordinates, as a result of the influence of the masses in 
finite regions. The physical properties of space would 
not then be wholly independent, that is, uninfluenced by 
matter, but in the main they would be, and only in 
small measure, conditioned by matter. Such a dualistic 
conception is even in itself not satisfactory; there are, 


however, some important physical arguments against it, 
which we shall consider. 

The hypothesis that the universe is infinite and 
Euclidean at infinity, is, from the relativistic point of 
view, a complicated hypothesis. In the language of the 
general theory of relativity it demands that the Riemann 
tensor of the fourth rank R^i mi shall vanish at infinity, 
which furnishes twenty independent conditions, while only 
ten curvature components R } enter into the laws of the 
gravitational field. It is certainly unsatisfactory to 
postulate such a far-reaching limitation without any 
physical basis for it. 

But in the second place, the theory of relativity makes 
it appear probable that Mach was on the right road in 
his thought that inertia depends upon a mutual action of 
matter. For we shall show in the following that, accord¬ 
ing to our equations, inert masses do act upon each other 
in the sense of the relativity of inertia, even if only very 
feebly. What is to be expected along the line of Mach’s 
thought ? 

1. The inertia of a body must increase when ponder¬ 

able masses are piled up in its neighbourhood. 

2 . A body must experience an accelerating force when 

neighbouring masses are accelerated, and, in fact, 
the force must be in the same direction as the 

3. A rotating hollow body must generate inside of 

itself a “ Coriolis field,” which deflects moving 
bodies in the sense of the rotation, and a radial 
centrifugal field as well. 



We shall now show that these three effects, which are 
to be expected in accordance with Mach’s ideas, are 
actually present according to our theory, although their 
magnitude is so small that confirmation of them by 
laboratory experiments is not to be thought of. For this 
purpose we shall go back to the equations of motion of 
a material particle (90), and carry the approximations 
somewhat further than was done in equation (90a). 

First, we consider y 4l as small of the first order. The 
square of the velocity of masses moving under the influence 
of the gravitational force is of the same order, according 
to the energy equation. It is therefore logical to regard 
the velocities of the material particles we are considering, 
as well as the velocities of the masses which generate the 
field, as small, of the order -J. We shall now carry out the 
approximation in the equations that arise from the field 
equations (101) and the equations of motion (90) so far 
as to consider terms, in the second member of (90), that 
are linear in those velocities. Further, we shall not put 
ds and dl equal to each other, but, corresponding to the 
higher approximation, we shall put 

ds = JFJi = 0 ~ 

From (90) we obtain, at first, 


744 \^ 

2 ) dl J 

1 a/3 

dx„ dx 


a dxj y 44 \ , 

+ fM Il6 > 

From (101) we get, to the approximation sought for, 



722 = 

7 3 3 = 744 

K | o 


i x 

C dx 0 

7 4 a = - 


G ds 

Y«0 = 0 




(i 17) 

in which, in (117), a and /3 denote the space indices only. 
On the right-hand side of (116) we can replace 


1 +2'by 1 and - I 7 by [“/]. It is easy to see, in 

addition, that to this degree of approximation we must 

M = 

ail _ 1 P>V* 

[; 4 ] - 
[f] - ° 

4 a 



p- 1 

in which a, /3 and fi denote space indices. We therefore 
obtain from (116), in the usual vector notation, 

d _ ^B 

.1 + <J>] = grad a- + + [rot B, v] 

k |WF 0 

H = 

8 ttJ r 

K r «%dv. 

\. (118) 



The equations of motion, (i 18), show now, in fact, that 


1. The inert mass is proportional to I + a, and 

therefore increases when ponderable masses 
approach the test body. 

2. There is an inductive action of accelerated masses, 

of the same sign, upon the test body. This is 

, m 

the term 

3. A material particle, moving perpendicularly to the 

axis of rotation inside a rotating hollow body, 
is deflected in the sense of the rotation (Coriolis 
field). The centrifugal action, mentioned above, 
inside a rotating hollow body, also follows from 
the theory, as has been shown by Thirring.* 

Although all of these effects are inaccessible to experi¬ 
ment, because k is so small, nevertheless they certainly 
exist according to the general theory of relativity. We 
must see in them a strong support for Mach’s ideas as to 
the relativity of all inertial actions. If we think these 
ideas consistently through to the end we must expect the 
whole inertia, that is, the whole ^-field, to be determined 
by the matter of the universe, and not mainly by the 
boundary conditions at infinity. 

For a satisfactory conception of the ^,,-field of cosmical 
dimensions, the fact seems to be of significance that the 
relative velocity of the stars is small compared to the 
velocity of light. It follows from this that, with a suit- 

* That the centrifugal action must be inseparably connected with the 
existence of the Coriolis field may be recognized, even without calculation, 
in the special case of a co-ordinate system rotating uniformly relatively to 
an inertial system ; our general co-variant equations naturally must apply 
to such a case. 



able choice of co-ordinates, g u is nearly constant in the 
universe, at least, in that part of the universe in which 
there is matter. The assumption appears natural, more¬ 
over, that there are stars in all parts of the universe, so 
that we may well assume that the inconstancy of g^ 
depends only upon the circumstance that matter is not 
distributed continuously, but is concentrated in single 
celestial bodies and systems of bodies. If we are willing 
to ignore these more local non-uniformities of the density 
of matter and of the ^-field, in order to learn something 
of the geometrical properties of the universe as a whole, 
it appears natural to substitute for the actual distribution 
of masses a continuous distribution, and furthermore to 
assign to this distribution a uniform density a. In this 
imagined universe all points with space directions will 
be geometrically equivalent; with respect to its space 
extension it will have a constant curvature, and will be 
cylindrical with respect to its ^ 4 -co-ordinate. The pos¬ 
sibility seems to be particularly satisfying that the universe 
is spatially bounded and thus, in accordance with our 
assumption of the constancy of a, is of constant curvature, 
being either spherical or elliptical; for then the boundary 
conditions at infinity which are so inconvenient from the 
standpoint of the general theory of relativity, may be 
replaced by the much more natural conditions for a closed 

According to what has been said, we are to put 

ds 1 = dx£ - 7 [) . v dxgix v . . (i 19) 

in which the indices fi and v run from 1 to 3 only. The 



7 M „ will be such functions of x x , x 2 , x z as correspond 
to a three-dimensional continuum of constant positive 
curvature. We must now investigate whether such an 
assumption can satisfy the field equations of gravitation. 

In order to be able to investigate this, we must first 
find what differential conditions the three-dimensional 
manifold of constant curvature satisfies. A spherical 
manifold of three dimensions, embedded in a Euclidean 
continuum of four dimensions,* is given by the equations 

x \ + x 2 + x./ + x 2 = a 2 
dx^ + dx o 2 + dxd + dx 2 = ds 1 . 

By eliminating x i} we get 
ds 1 — dx± + dx 2 + dx 2 2 + 

(x 1 dx l + x 2 dx 2 + x-^dxj 2 
d 2 - X 2 - x 2 - x 2 

As far as terms of the third and higher degrees in the 
x v , we can put, in the neighbourhood of the origin of 

ds 1 = (s„, + X -^)dx v dx v . 

Inside the brackets are the g^ v of the manifold in the 
neighbourhood of the origin. Since the first derivatives 
of the g^ v , and therefore also the Y vanish at the 
origin, the calculation of the R^ v for this manifold, by 
(88), is very simple at the origin. We have 

2 <N _ 2 

5 - jgpr 

* The aid of a fourth space dimension has naturally no significance 
except that of a mathematical artifice. 



Since the relation is universally co-variant, 

and since all points of the manifold are geometrically 
equivalent, this relation holds for every system of co¬ 
ordinates, and everywhere in the manifold. In order to 
avoid confusion with the four-dimensional continuum, 
we shall, in the following, designate quantities that refer 
to the three-dimensional continuum by Greek letters, 
and put 


P,uv = — -tfnv • • • (i 20) 

We now proceed to apply the field equations (96) to 
our special case. From (119) we get for the four-dimen¬ 
sional manifold, 

R^ v = P M „ for the indices 1 to 3 

^14 = ^24 = ^34 = ^44 = O 

( 121 ) 

For the right-hand side of (96) we have to consider 
the energy tensor for matter distributed like a cloud of 
dust. According to what has gone before we must 
therefore put 

T> xv = 

dx, L dx v 
a—- •— 
ds ds 

specialized for the case of rest. But in addition, we 
shall add a pressure term that may be physically estab¬ 
lished as follows. Matter consists of electrically charged 
particles. On the basis of Maxwell’s theory these 
cannot be conceived of as electromagnetic fields free 
from singularities. In order to be consistent with the 



facts, it is necessary to introduce energy terms, not con¬ 
tained in Maxwell’s theory, so that the single electric 
particles may hold together in spite of the mutual re¬ 
pulsions between their elements, charged with electricity 
of one sign. For the sake of consistency with this fact, 
Poincare has assumed a pressure to exist inside these 
particles which balances the electrostatic repulsion. It 
cannot, however, be asserted that this pressure vanishes 
outside the particles. We shall be consistent with this 
circumstance if, in our phenomenological presentation, 
we add a pressure term. This must not, however, be 
confused with a hydrodynamical pressure, as it serves 
only for the energetic presentation of the dynamical 
relations inside matter. In this sense we put 

^ ax a ax b , \ 

T H.V ~ ~ g^p. • 

In our special case we have, therefore, to put 

T^ v = y^p (for and v from 1 to 3) 

! = - y^y^p + cr - p = a - 4/. 

Observing that the field equation (96) may be written 
in the form 

R - kg^T) 

we get from (96) the equations, 


From this follows 

If the universe is quasi-Euclidean, and its radius of 
curvature therefore infinite, then a would vanish. But 
it is improbable that the mean density of matter in the 
universe is actually zero; this is our third argument 
against the assumption that the universe is quasi- 
Euclidean. Nor does it seem possible that our hypo¬ 
thetical pressure can vanish ; the physical nature of this 
pressure can be appreciated only after we have a better 
theoretical knowledge of the electromagnetic field. 
According to the second of equations (123) the radius, 
a, of the universe is determined in terms of the total 
mass, M, of matter, by the equation 

47r 2 


The complete dependence of the geometrical upon the 
physical properties becomes clearly apparent by means 
of this equation. 

Thus we may present the following arguments against 
the conception of a space-infinite, and for the conception 
of a space-bounded, universe :— 

I. From the standpoint of the theory of relativity, 
the condition for a closed surface is very much simpler 
than the corresponding boundary condition at infinity 
of the quasi-Euclidean structure of the universe. 


2. The idea that Mach expressed, that inertia depends 
upon the mutual action of bodies, is contained, to a 
first approximation, in the equations of the theory of 
relativity; it follows from these equations that inertia 
depends, at least in part, upon mutual actions between 
masses. As it is an unsatisfactory assumption to make 
that inertia depends in part upon mutual actions, and 
in part upon an independent property of space, Mach’s 
idea gains in probability. But this idea of Mach’s 
corresponds only to a finite universe, bounded in space, 
and not to a quasi-Euclidean, infinite universe. From 
the standpoint of epistemology it is more satisfying to 
have the mechanical properties of space completely de¬ 
termined by matter, and this is the case only in a space- 
bounded universe. 

3. An infinite universe is possible only if the mean 
density of matter in the universe vanishes. Although 
such an assumption is logically possible, it is less prob¬ 
able than the assumption that there is a finite mean 
density of matter in the universe. 



Accelerated masses—inductive ac¬ 
tion of, 113. 

Addition and subtraction of tensors, 
* 4 - 

— theorem of velocities, 40. 


Biot-Savart force, 46. 


Centrifugal force, 67. 

Clocks—moving, 39. 

Compressible viscous fluid, 22. 
Concept of space, 3. 

-time, 30. 

Conditions of orthogonality, 7. 
Congruence—theorems of, 3. 
Conservation principles, 55. 
Continuum—four-dimensional, 33. 
Contraction of tensors, 15. 
Contra-variant vectors, 72, 

-tensors, 75. 

Co-ordinates—preferred systems of, 

8 . 

Co-variance of equation of contin¬ 
uity, 22. 

Co-variant, 12 et seq. 

-vector, 72. 

Criticism of principle of inertia, 65. 
Criticisms of theory of relativity, 31. 
Curvilinear co-ordinates, t8. 


Differentiation of tensors, 76, 79. 
Displacement of spectral lines, 101. 

8 * 


Energy and mass, 48, 51. 

— tensor oi' electromagnetic field, 

52 . 

— — of matter, 56. 

Equation of continuity—co-variance 
of, 22. 

Equations of motion of materia 
particle, 52. 

Equivalence of mass and energy, 51. 
Equivalent spaces of reference, 26. 
Euclidean geometry, 4. 


Finiteness of universe, no. 

Fizeau, 29. 

Four-dimensional continuum, 33. 
Four-vector, 43. 

Fundamental tensor, 74. 


Galilean regions, 65. 

— transformation, 28. 

Gauss, 68. 

Geodetic lines, 86. 

Geometry, Euclidean, 4. 
Gravitational mass, 63. 

Gravitation constant, 98. 


Homogeneity of space, 17. 
Hydrodynamical equations, 56. 
Hypotheses of pre-relativity physics, 
77 . 



Inductive action of accelerated 
masses, 113. 

Inert and gravitational mass—equal¬ 
ity of, 63. 

Invariant, 10 et seq. 

Isotropy of space, 17. 


Kaluza, 108. 


Levi-Civita, 77. 

Light-cone, 42. 

Light ray—path of, 102. 

Light-time, 34. 

Linear orthogonal transformation, 7. 
Lorentz electromotive force, 46. 

— transformation, 32. 


Mach, 62, log, no, m, 113, 119. 
Mass and Energy, 48, 51. 

— equality of gravitational and 

inert, 63. 

— gravitational, 63. 

Maxwell’s equations, 23. 

Mercury—perihelion of, 103, 107. 
Michelson and Morley, 29. 
Minkowski, 34, 

Motion of particle—equations of, 52. 
Moving measuring rods and clocks, 
39 - 

Multiplication ol tensors, 14. 


Newtonian gravitation constant, 98. 

Operations on tensors, 14 et seq. 
Orthogonality—conditions of, 7. 
Orthogonal transformations—linear, 


Path of light ray, 102. 

Perihelion of Mercury, 103, 107. 
Poisson’s equation, 90. 

Preferred systems of co-ordinates, 8. 
Pre-relativity physics—hypotheses 
of, 27. 

Principle of equivalence, 64. 

-inertia—criticism of, 65. 

Principles of conservation, 55. 


Radius of Universe, 118. 

Rank of tensor, 14. 

Ray of light—path of, 102. 

Reference—space of, 4. 

Riemann, 72. 

— tensor, 82, 85, no. 

Rods (measuring) and clocks in mo¬ 
tion, 39. 

Rotation, 66. 


Simultaneity, 17, 30. 

Sitter, 2g. 

Skew-symmetrical tensor, 15. 

Solar Eclipse expedition (1919), 103. 
Space—concept of, 3. 

— homogeneity of, 17. 

— Isotropy of, 17. 

Spaces of reference, 4 ; equivalence 
of, 26. 

Special Lorentz transformation, 36. 
Spectral lines—displacement of, 101. 
Straightest lines, 86. 

Stress tensor, 22. 

Symmetrical tensor, 15. 

Systems of co-ordinates—preferred, 

8 . 


Tensor, 12 et seq, 72 et seq. 

— Addition and subtraction of, 14. 

— Contraction of, 15. 

— Fundamental, 74. 

— Multiplication of, 14. 



Tensor, operations, 14 et seq. 

— Rank of, 14. 

— Symmetrical and Skew-symmet¬ 

rical, 15. 

Tensors—formation by differenti¬ 
ation, 76. 

Theorem for addition of velocities, 

Theorems of congruence, 3. 

Theory of relativity, criticisms of, 31. 
Thirring, 113. 

Time-concept, 30. 

Time-space concept, 33. 
Transformation—Galilean, 28. 

— Linear orthogonal, 7. 


Universe—finiteness of, no. 

— radius of, 118. 


Vector—co-variant, 72. 

— contra-variant, 72. 

Velocities—addition theorem of, 40. 
Viscous compressible fluid, 22. 


Weyl, 77, 103, 108.