uid. The contribution to the stress tensor is then evidently
p
in which  is the special symmetrical tensor. This term will
also be present in the case of a viscous 
uid. But in this case
there will also be pressure terms, which depend upon the space
derivatives of the u. We shall assume that this dependence is a
linear one. Since these terms must be symmetrical tensors, the
only ones which enter will be
ff@u
@x
+
@u
@x+ fi
@uff
@xff
(for
@uff
@xff
is a scalar). For physical reasons (no slipping) it
is assumed that for symmetrical dilatations in all directions,
i.e. when
@u1
@x1
=
@u2
@x2
=
@u3
@x3
;
@u1
@x2
; etc., = 0;
there are no frictional forces present, from which it follows that
fi = 􀀀
2
3
ff. If only
@u1
@x3
is different from zero, let p31 = 􀀀ff
@u1
@x3
,
by which ff is determined. We then obtain for the complete
stress tensor,
p = p􀀀ff@u
@x
+
@u
@x􀀀
2
3@u1
@x1
+
@u2
@x2
+
@u3
@x3: (18)
PRE-RELATIVITY PHYSICS 23
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.
@h3
@x2 􀀀
@h2
@x3
=
1
c
@e1
@t
+
1
c
i1;
@h1
@x3 􀀀
@h3
@x1
=
1
c
@e2
@t
+
1
c
i2;
@h2
@x1 􀀀
@h1
@x2
=
1
c
@e3
@t
+
1
c
i3;
@e1
@x1
+
@e2
@x2
+
@e3
@x3
= ;
9>
>>>>>=>>>>>>;
(19)
@e3
@x2 􀀀
@e2
@x3
= 􀀀
1
c
@h1
@t
;
@e1
@x3 􀀀
@e3
@x1
= 􀀀
1
c
@h2
@t
;
@e2
@x1 􀀀
@e1
@x2
= 􀀀
1
c
@h3
@t
;
@h1
@x1
+
@h2
@x2
+
@h3
@x3
= 0:
9>>>>>>=>
>>>>>;
(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
These considerations will make the reader familiar with tensor opera-
THE MEANING OF RELATIVITY 24
interpreted if h is regarded as a skew-symmetrical tensor of the
second rank. In this sense, we write h23, h31, h12, in place of
h1, h2, h3 respectively. Paying attention to the skew-symmetry
of h, the first three equations of (19) and (20) may be written
in the form
@h
@x
=
1
c
@e
@t
+
1
c
i; (19a)
@e
@x 􀀀
@e
@x
= +
1
c
@h
@t
: (20a)
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
@e
@x
= ; (19b)
@h
@x
+
@h
@x
+
@h
@x
= 0: (20b)
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 proved, if attention
is paid to the skew-symmetry of h). 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.
tions without the special diculties of the four-dimensional treatment; cor-
responding considerations in the theory of special relativity (Minkowski's
interpretation of the field) will then offer fewer diculties.
LECTURE II
THE THEORY OF SPECIAL RELATIVITY
The previous considerations concerning the configuration 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 coordinates,
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
25
THE MEANING OF RELATIVITY 26
other system K0 which moves uniformly and without rotation
relatively to K, is also an inertial system; the laws of nature are
in concordance 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 direction.
In order to be able to do this, we must first solve the following
problem. If we are given the Cartesian co-ordinates, x, and
the time, t, of an event relatively to one inertial system, K,
how can we calculate the co-ordinates, x0, and the time, t0, of
the same event relatively to an inertial system K0 which moves
with uniform translation relatively to K? In the pre-relativity
physics this problem was solved by making unconsciously two
hypotheses:|
1. The time is absolute; the time of an event, t0, relatively
to K0 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 in
uence 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 K0, 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 K0 which is in motion relatively to K.
If the axes of K and K0 are parallel to each other, a simple
calculation based on these two assumptions, gives the equations
SPECIAL RELATIVITY 27
of transformation
x0 = x 􀀀 a 􀀀 bt;
t0 = t 􀀀 b: ) (21)
This transformation is known as the \Galilean Transformation."
Differentiating twice by the time, we get
d2x0
dt2 =
d2x
dt2 :
Further, it follows that for two simultaneous events,
x0
(1) 􀀀 x0
(2) = x
(1) 􀀀 x
(2):
The invariance of the distance between the two points results
from squaring and adding. From this easily follows the covariance
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 electromagnetic
phenomena. The Maxwell-Lorentz electromagnetic 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 K0,
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 ther). But all experiments have shown that electromagnetic
and optical phenomena, relatively to the earth as the
THE MEANING OF RELATIVITY 28
body of reference, are not in
uenced 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 doubted.
On 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
(Fizeau), 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 therefore 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, U, 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
SPECIAL RELATIVITY 29
to it, and regulated according to the following scheme. A ray
of light is sent out from one of the clocks, Um, at the instant
when it indicates the time tm, and travels through a vacuum a
distance rmn, to the clock Un; at the instant when this ray meets
the clock Un the latter is set to indicate the time tn = tm+
rmn
c
.
The principle of the constancy of the velocity of light then states
that this adjustment of the clocks will 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 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 r^ole 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
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.
THE MEANING OF RELATIVITY 30
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, significance; 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, K0, 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 K0, in the way that
has been shown. Further, let a ray of light pass from one point P1
to another point P2 of K through a vacuum. If r is the measured
distance between the two points, then the propagation of light
must satisfy the equation
r = c delta  delta t:
If we square this equation, and express r2 by the differences
of the co-ordinates, delta x, in place of this equation we can write
X(delta x)2 􀀀 c2 delta t2 = 0: (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.
SPECIAL RELATIVITY 31
The same propagation of light may also be considered relatively
to K0, in which case also the principle of the constancy of
the velocity of light must be satisfied. Therefore, with respect
to K0, we have the equation
X(delta x0)2 􀀀 c2 delta t02 = 0: (22a)
Equations (22a) and (22) must be mutually consistent with
each other with respect to the transformation which transforms
from K to K0. 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 prerelativity
physics space and time were separate entities. Speci-
fications 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-simultaneous events happened at the same place had no objective
meaning (that is, independent of the space of reference).
But this relativity had no r^ole 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 x1, x2, x3, 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 immediately
recognized. It is neither the point in space, nor the instant in
THE MEANING OF RELATIVITY 32
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
x1, x2, x3, t as the four co-ordinates of an event in the fourdimensional
continuum. 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 r^ole of the time co-ordinate from that of the space
co-ordinates; for the term delta t2 has the opposite sign to the space
terms, delta x1
2, delta x2
2, delta x3
2.
SPECIAL RELATIVITY 33
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
delta x1
2 + delta x2
2 + delta x3
2 􀀀 delta l2 = 0 (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 l = ct, the imaginary
time co-ordinate
x4 = il = ict (p􀀀1 = i):
Then the equation defining the propagation of light, which must
be co-variant with respect to the Lorentz transformation, becomes
X(4)
delta x
2 = delta x1
2 + delta x2
2 + delta x3
2 + delta x4
2 = 0: (22c)
This condition is always satisfied if we satisfy the more general
condition that
s2 = delta x1
2 + delta x2
2 + delta x3
2 + delta x4
2 (23)
That this specialization lies in the nature of the case will be evident
later.
THE MEANING OF RELATIVITY 34
shall be an invariant with respect to the transformation. This
condition is satisfied only by linear transformations, that is,
transformations of the type
x0 = a + bffxff (24)
in which the summation over the ff is to be extended from ff = 1
to ff = 4. A glance at equations (23) and (24) shows that the
Lorentz transformation so defined is identical with the translational
and rotational transformations of the Euclidean geometry,
if we disregard the number of dimensions and the relations of reality.
We can also conclude that the coecients bff must satisfy
the conditions
bffbff =  = bffbff: (25)
Since the ratios of the x are real, it follows that all the a and
the bff are real, except a4, b41, b42, b43, b14, b24 and b34, 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 a, 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) furnish,
x01 = x1 cos  􀀀 x2 sin ;
x02 = x1 sin  + x2 cos ;
x03 = x3;
x04 = x4:
9>
>=>>;
(26)
SPECIAL RELATIVITY 35
This is a simple rotation in space of the (space) co-ordinate
system about x3-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,
x01 = x1 cos   􀀀 x4 sin  ;
x04 = x1 sin   + x4 cos  ;
x02 = x2;
x03 = x3:
9>
>=>>;
(26a)
On account of the relations of reality   must be taken as
imaginary. To interpret these equations physically, we introduce
the real light-time l and the velocity v of K0 relatively to K,
instead of the imaginary angle  . We have, first,
x01 = x1 cos   􀀀 il sin  ;
l0 = 􀀀ix1 sin   + l cos  :
Since for the origin of K0 i.e., for x1 = 0, we must have x1 = vl,
it follows from the first of these equations that
v = i tan  ; (27)
and also
sin   = 􀀀iv
p1 􀀀 v2
;
cos   =
1
p1 􀀀 v2
9>
>=>>;
(28)
THE MEANING OF RELATIVITY 36
so that we obtain
x01 =
x1 􀀀 vl
p1 􀀀 v2
;
l0 =
l 􀀀 vx1 p1 􀀀 v2
;
x02 = x2;
x03 = x3:
9>
>>>=>>>>;
(29)
These equations form the well-known special Lorentz transformation,
which in the general theory represents a rotation,
through an imaginary angle, of the four-dimensional system of
co-ordinates. If we introduce the ordinary time t, in place of the
light-time l, then in (29) we must replace l by ct and v by
v
c
.
We must now fill in a gap. From the principle of the constancy
of the velocity of light it follows that the equation
Xdelta x
2 = 0
has a significance which is independent of the choice of the inertial
system; but the invariance of the quantity Pdelta x
2 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 multiplied 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 measuring rod is equal
to R0, its radius R in motion, might be different from R0, 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
SPECIAL RELATIVITY 37
all directions in space must be equivalent to each other. R may
therefore 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 K0 the equation of its lateral
surface is
x02 + y02 = R0
2:
If we write the last two equations of (29) more generally
x02 = x2;
x03 = x3;
then the lateral surface of the cylinder referred to K satisfies the
equation
x2 + y2 =
R0
2
2 :
The factor  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, K00, which
moves relatively to K0 with velocity v in the direction of the
negative x-axis of K, we obtain, by applying (29) twice,
x001 = (v)(􀀀v)x1;
x002 = (v)(􀀀v)x2;
x003 = (v)(􀀀v)x3;
l00 = (v)(􀀀v)l:
Now, since (v) must be equal to (􀀀v), and since we assume
that we use the same measuring rods in all the systems, it follows
that the transformation of K00 to K must be the identical
THE MEANING OF RELATIVITY 38
transformation (since the possibility  = 􀀀1 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 = 0, the position of the points given by the integers
x01 = n, is with respect to K, given by x1 = np1 􀀀 v2; this follows
from the first of equations (29) and expresses the Lorentz
contraction. A clock at rest at the origin x1 = 0 of K, whose
beats are characterized by l = n, will, when observed from K0,
have beats characterized by
l0 =
n
p1 􀀀 v2
;
this follows from the second of equations (29) and shows that
the clock goes slower than if it were at rest relatively to K0.
These two consequences, which hold, mutatis mutandis, for every
system of reference, form the physical content, free from
convention, of the Lorentz transformation.
Addition Theorem for Velocities. If we combine two special
Lorentz transformations with the relative velocities v1 and v2,
then the velocity of the single Lorentz transformation which
takes the place of the two separate ones is, according to (27),
given by
v12 = i tan( 1 +  2) = i
tan  1 + tan  2
1 􀀀 tan  1 tan  2
=
v1 + v2
1 + v1v2
: (30)
SPECIAL RELATIVITY 39
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 s2 (23).
Formally, it has the same r^ole in the four-dimensional space-time
continuum as the invariant delta x1
2+delta x2
2+delta x3
2 in the Euclidean
geometry and in the pre-relativity physics. The latter quantity
is not an invariant with respect to all the Lorentz transformations;
the quantity s2 of equation (23) assumes the r^ole of this
invariant. With respect to an arbitrary inertial system, s2 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 s2 differs, disregarding the number of dimensions,
from the corresponding invariant of the Euclidean geometry
in the following points. In the Euclidean geometry s2 is
necessarily positive; it vanishes only when the two points concerned
come together. On the other hand, from the vanishing
of
s2 =X(4)
delta x
2 = delta x1
2 + delta x2
2 + delta x3
2 􀀀 delta t2
it cannot be concluded that the two space-time points fall together;
the vanishing of this quantity s2, 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 fourdimensional
space of the x1, x2, x3, l, then all the \points" which
can be connected to P by means of a light signal lie upon the
cone s2 = 0 (compare Fig. 1, in which the dimension x3 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
THE MEANING OF RELATIVITY 40
x1
x2
l
Fig. 1.
of the cone will contain the \points" from which light signals
can be sent to P. The points P0 enclosed by the conical surface
furnish, with P, a negative s2; PP0, as well as P0P 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 l-axis may be drawn
That material velocities exceeding that of light are not possible, fol-
lows from the appearance of the radical p1 􀀀 v2 in the special Lorentz
SPECIAL RELATIVITY 41
in the direction of PP0 by suitably choosing the state of motion
of the inertial system. If P0 lies outside of the \light-cone" then
PP0 is of the nature of a space; in this case, by properly choosing
the inertial system, delta l can be made to vanish.
By the introduction of the imaginary time variable, x4 =
il, Minkowski has made the theory of invariants for the fourdimensional
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 threedimensional
space, therefore, only in the number of dimensions
and the relations of reality.
A physical entity which is specified by four quantities, A,
in an arbitrary inertial system of the x1, x2, x3, x4, is called
a 4-vector, with the components A, if the A correspond in
their relations of reality and the properties of transformation to
the delta x; it may be of the nature of a space or of a time. The
sixteen quantities A then form the components of a tensor of
the second rank, if they transform according to the scheme
A0 = bffbfiAfffi:
It follows from this that the A behave, with respect to
their properties of transformation and their properties of reality,
as the products of components, UV, of two 4-vectors,
(U) 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,
transformation (29).
THE MEANING OF RELATIVITY 42
contraction and differentiation for these tensors are wholly
analogous to the corresponding operations for tensors in threedimensional
space.
Before we apply the tensor theory to the four-dimensional
space-time continuum, we shall examine more particularly the
skew-symmetrical tensors. The tensor of the second rank has, in
general, 16 = 4delta 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 skewsymmetrical
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:
23 31 12 14 24 34
h23 h31 h12 􀀀 iex 􀀀 iey 􀀀 iez) (30a)
J1 J2 J3 J4
1
c
ix
1
c
iy
1
c
iz i9=;
(31)
with the convention that  shall be equal to 􀀀. Then
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 continuum.
SPECIAL RELATIVITY 43
Maxwell's equations may be combined into the forms
@
@x
= J; (32)
@
@x
+
@
@x
+
@
@x
= 0; (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  and the J have a tensor character, which we assume.
Consequently, the laws for transforming these quantities from
one to another allowable (inertial) system of co-ordinates are
uniquely determined. The progress in method which electrodynamics
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 of
@e
@l
,
we had, say,
@2e
@l2 , 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
THE MEANING OF RELATIVITY 44
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 transformation
furnishes, for the special case of the special Lorentz
transformation, the equations
e0x = ex h0x = hx;
e0y =
ey 􀀀 vhz p1 􀀀 v2
h0y =
hy + vez p1 􀀀 v2
;
e0z =
ez + vhy p1 􀀀 v2
h0z =
hz 􀀀 vey p1 􀀀 v2
:
9>
>>=>>>;
(34)
If there exists with respect to K only a magnetic field, h, but
no electric field, e, then with respect to K0 there exists an electric
field e0 as well, which would act upon an electric particle at rest
relatively to K0. 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 electricity,
k = e + [i; h]; (35)
in which i is the vector velocity of electricity, with the velocity
of light as the unit. If we introduce J and  according to
(30a) and (31), we obtain for the first component the expression
12J2 + 13J3 + 14J4:
SPECIAL RELATIVITY 45
Observing that 11 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 vector
K = J; (36)
and the fourth component is given by
K4 = 41J1 + 42J2 + 43J3 = i(exix + eyiy + eziz) = i: (37)
There is, therefore, a four-dimensional vector of force per unit
volume, whose first three components, K1, K2, K3, 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 p􀀀1.
A comparison of (36) and (35) shows that the theory of relativity
formally unites the ponderomotive force of the electric
field, e, 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 K. Let us
imagine a body upon which the electromagnetic field acts for
a time. In the symbolic figure (Fig. 2) Ox1 designates the x1-
axis, and is at the same time a substitute for the three space axes
Ox1, Ox2, Ox3; Ol designates the real time axis. In this diagram
a body of finite extent is represented, at a definite time l, 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 to the l-axis. Between the time sections, l = l1
and l = l2, but not extending to them, a portion of the strip is
shaded. This represents the portion of the space-time manifold
THE MEANING OF RELATIVITY 46
x1
l
l1
l
l2
O
A B
Fig. 2.
in which the electromagnetic 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,
delta Ix, delta Iy, delta Iz, and the change in energy, delta E, are then given
SPECIAL RELATIVITY 47
by the expressions
delta Ix = Z l2
l1
dl Z kx dx dy dz =
1
i Z K1 dx1 dx2 dx3 dx4;
delta Iy = Z l2
l1
dl Z ky dx dy dz =
1
i Z K2 dx1 dx2 dx3 dx4;
delta Iz = Z l2
l1
dl Z kz dx dy dz =
1
i Z K3 dx1 dx2 dx3 dx4;
delta E = Z l2
l1
dl Z  dx dy dz =
1
i Z 1
i
K4 dx1 dx2 dx3 dx4:
Since the four-dimensional element of volume is an invariant,
and (K1;K2;K3;K4) 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 l1 and l2, because
the portion of the region which is not shaded contributes nothing
to the integral. It follows, therefore, that delta Ix, delta Iy, delta Iz, idelta 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
Ix; Iy; Iz; iE
has itself the properties of a vector; these quantities are referred
to an instantaneous condition of the body (e.g. at the time l =
l1).
This 4-vector may also be expressed in terms of the mass m,
and the velocity of the body, considered as a material particle.
To form this expression, we note first, that
􀀀ds2 = d2 = 􀀀(dx1
2 +dx2
2 +dx3
2)􀀀dx4
2 = dl2(1􀀀q2) (38)
THE MEANING OF RELATIVITY 48
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 d 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 d = 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  the proper
time of the material particle. As opposed to dl, d 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
u =
dx
d
(39)
has, just as the dx, the character of a vector; we shall designate
(u) as the four-dimensional vector (in brief, 4-vector) of
velocity. Its components satisfy, by (38), the condition
Xu
2 = 􀀀1: (40)
We see that this 4-vector, whose components in the ordinary
notation are
qx p1 􀀀 q2
;
qy p1 􀀀 q2
;
qz p1 􀀀 q2
;
i
p1 􀀀 q2
(41)
is the only 4-vector which can be formed from the velocity components
of the material particle which are defined in three dimensions
by
qx =
dx
dl
; qy =
dy
dl
; qz =
dz
dl
:
SPECIAL RELATIVITY 49
We therefore see that m
dx
d  (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,
Ix =
mqx p1 􀀀 q2
;
Iy =
mqy p1 􀀀 q2
;
Iz =
mqz p1 􀀀 q2
;
E =
m
p1 􀀀 q2
:
9>
>>>>>=>>>>>>;
(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 = 0), we see that the energy, E0, of a body at rest is
equal to its mass. Had we chosen the second as our unit of time,
we would have obtained
E0 = mc2: (44)
Mass and energy are therefore essentially alike; they are only
different expressions for the same thing. The mass of a body
THE MEANING OF RELATIVITY 50
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 1, the velocity of light. If we develop E in powers
of q2, we obtain,
E = m +
m
2
q2 +
3
8
mq4 + : : : : (45)
The second term of this expansion corresponds to the kinetic
energy of the material particle in classical mechanics.
Equations of Motion of Material Particles. From (43) we
obtain, by differentiating by the time l, and using the principle
of momentum, in the notation of three-dimensional vectors,
K =
d
dl   mq
p1 􀀀 q2!: (46)
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 fi-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 electromagnetic field. The four-dimensional
formulation of these principles leads to an important conception,
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.
SPECIAL RELATIVITY 51
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 volume,
K = J;
using the field equations (32), we express J in terms of the
field intensities, , we obtain, after some transformations and
repeated application of the field equations (32) and (33), the
expression
K = 􀀀
@T
@x
; (47)
where we have written
T = 􀀀1
4fffi
2 + ffff: (48)
The physical meaning of equation (47) becomes evident if in
place of this equation we write, using a new notation,
kx = 􀀀
@pxx
@x 􀀀
@pxy
@y 􀀀
@pxz
@z 􀀀
@(ibx)
@(il)
;
ky = 􀀀
@pyx
@x 􀀀
@pyy
@y 􀀀
@pyz
@z 􀀀
@(iby)
@(il)
;
kz = 􀀀
@pzx
@x 􀀀
@pzy
@y 􀀀
@pzz
@z 􀀀
@(ibz)
@(il)
;
i = 􀀀
@(isx)
@x 􀀀
@(isy)
@y 􀀀
@(isz)
@z 􀀀
@(􀀀)
@(il)
9>
>>>>>=>>>>>>;
(47a)
To be summed for the indices ff and fi.
THE MEANING OF RELATIVITY 52
or, on eliminating the imaginary,
kx = 􀀀
@pxx
@x 􀀀
@pxy
@y 􀀀
@pxz
@z 􀀀
@bx
@l
;
ky = 􀀀
@pyx
@x 􀀀
@pyy
@y 􀀀
@pyz
@z 􀀀
@by
@l
;
kz = 􀀀
@pzx
@x 􀀀
@pzy
@y 􀀀
@pzz
@z 􀀀
@bz
@l
;
 = 􀀀
@sx
@x 􀀀
@sy
@y 􀀀
@sz
@z 􀀀
@
@l
:
9>
>>>>>=>>>>>>;
(47b)
When expressed in the latter form, we see that the first three
equations state the principle of momentum; pxx,. . . , pzx are the
Maxwell stresses in the electromagnetic field, and (bx; by; bz) is
the vector momentum per unit volume of the field. The last of
equations (47b) expresses the energy principle; s is the vector

ow of energy, and  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 electrodynamics,
pxx = 􀀀 hxhx + 1
2 (hx
2 + hy
2 + hz
2)
􀀀 exex + 1
2 (ex
2 + ey
2 + ez
2);
pxy = 􀀀 hxhy pxz = 􀀀 hxhz
􀀀 exey; 􀀀 exez;
...
bx = sx = eyhz 􀀀 ezhy;
by = sy = ezhx 􀀀 exhz;
bz = sz = exhy 􀀀 eyhx;
 = +1
2 (ex
2 + ey
2 + ez
2 + hx
2 + hy
2 + hz
2):
9>
>>>>>>>>=>>>>>>>>>;
(48a)
SPECIAL RELATIVITY 53
We conclude from (48) that the energy tensor of the electromagnetic
field is symmetrical; with this is connected the fact
that the momentum per unit volume and the 
ow 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),
@T
@x
= 0: (47c)
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.
THE MEANING OF RELATIVITY 54
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, 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 principle.
Let us consider a spatially bounded, closed system, which,
four-dimensionally, we may represent as a strip, outside of which
the T vanish. Integrate equation (47c) over a space section.
Since the integrals of
@T1
@x1
,
@T2
@x2
and
@T3
@x3
vanish because
the T vanish at the limits of integration, we obtain
@
@l Z T4 dx1 dx2 dx3= 0: (49)
Inside the parentheses are the expressions for the momentum of
the whole system, multiplied by i, 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 considerations.
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 therefore obliged to make use of an
SPECIAL RELATIVITY 55
x1
l
Fig. 3.
inexact description of matter, which corresponds to that of classical
mechanics. The density , of a material substance and the
hydrodynamical pressures are the fundamental concepts upon
which such a description is based.
Let 0 be the density of matter at a place, estimated with
reference to a system of co-ordinates moving with the matter.
Then 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
THE MEANING OF RELATIVITY 56
velocity components, u and 0. We secure the tensor character
of T by putting
T = 0uu; (50)
in which the u, in the three-dimensional representation, are
given by (41). In fact, it follows from (50) that for q = 0, T44 =
􀀀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, K)
acts upon the matter, by the principles of momentum and energy
the equation
K =
@T
@x
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
x1, x2, x3, we obtain
Z K1 dx1 dx2 dx3 = Z @T14
@x4
dx1 dx2 dx3
= 􀀀i
d
dl Z 0
dx1
d
dx4
d
dx1 dx2 dx3:
Now R dx1 dx2 dx3 dx4 is an invariant, as is, therefore, also R 0 dx1 dx2 dx3 dx4. 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
SPECIAL RELATIVITY 57
of the thread for which 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 dV0 respectively, then we have
Z 0 dV dl = Z 0 dV0 d
and therefore also
Z 0 dV = Z 0 dV0
d
dl
= Z dmi
d
dx4
:
If we substitute the right-hand side for the left-hand side in
the former integral, and put
dx1
d
outside the sign of integration,
we obtain,
Kx =
d
dl m
dx1
d  =
d
dl   mqx p1 􀀀 q2!:
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 
uid in which the pressure is determined
by a scalar p. Since the tangential stresses pxy, etc., vanish in
this case, the contribution to the energy tensor must be of the
form p. We must therefore put
T = uu + p: (51)
THE MEANING OF RELATIVITY 58
At rest, the density of the matter, or the energy per unit volume,
is in this case, not  but  􀀀 p. For
􀀀T44 = 􀀀
dx4
d
dx4
d 􀀀 p44 =  􀀀 p:
In the absence of any force, we have
@T
@x
= u
@u
@x
+ u
@(u)
@x
+
@p
@x
= 0:
If we multiply this equation by u =
dx
d  and sum for the
's we obtain, using (40),
􀀀
@(u)
@x
+
dp
d
= 0; (52)
where we have put
@p
@x
dx
d
=
dp
d
. This is the equation of
continuity, which differs from that of classical mechanics by
the term
dp
d
, which, practically, is vanishingly small. Observing
(52), the conservation principles take the form

du
d
+ u
dp
d
+
@p
@x
= 0: (53)
The equations for the first three indices evidently correspond 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.
LECTURE III
THE GENERAL THEORY OF RELATIVITY
All 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 in
uenced 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. Accord-
59
THE MEANING OF RELATIVITY 60
ing 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
field (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 Eotvos), 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 extension
of the principle of relativity, follows from the following consideration.
A little re
ection will show that the theorem of the
THE GENERAL THEORY 61
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) delta  (Acceleration) = (Intensity of the
gravitational field) delta  (Gravitational mass):
It is only when there is numerical equality between the inert
and gravitational mass that the acceleration is independent of
the nature of the body. Let now K be an inertial system. Masses
which are suciently far from each other and from other bodies
are then, with respect to K, free from acceleration. We shall
also refer these masses to a system of co-ordinates K0, uniformly
accelerated with respect to K. Relatively to K0 all the masses
have equal and parallel accelerations; with respect to K0 they
behave just as if a gravitational field were present and K0 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 K0 is \at rest" and a gravitational
field is present we may consider as equivalent to the conception
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 K0,
we call the \principle of equivalence;" 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
THE MEANING OF RELATIVITY 62
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 gravitation (with respect to K0). 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 diculties 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 experiment 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 suciently far from other bodies; we
know that it is suciently far from other bodies only by the fact
that it moves without acceleration. 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 perturbations 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 reT
HE GENERAL THEORY 63
gions as a special case of known properties.
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 acceleration and rotation. If,
further, we are going to do away completely with the dicult
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 con
ict with that physical interpretation
of space and time to which we were led by the special
theory of relativity. For let K0 be a system of co-ordinates whose
z0-axis coincides with the z-axis of K, and which rotates about
the latter axis with constant angular velocity. Are the configurations
of rigid bodies, at rest relatively to K0, in accordance with
the laws of Euclidean geometry? Since K0 is not an inertial system,
we do not know directly the laws of configuration of rigid
bodies with respect to K0, 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 K0. Imagine
a circle drawn about the origin in the x0-y0 plane of K0, 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 K0. If U is the number of these rods
along the periphery, D the number along the diameter, then, if
K0 does not rotate relatively to K, we shall have
U
D
= :
THE MEANING OF RELATIVITY 64
But if K0 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 contraction (along their lengths!). It therefore
follows that
U
D
> :
It therefore follows that the laws of configuration of rigid
bodies with respect to K0 do not agree with the laws of configuration
of rigid bodies that are in accordance with Euclidean geometry.
If, further, we place two similar clocks (rotating with K0),
one upon the periphery, and the other at the centre of the circle,
then, judged from K, the clock on the periphery will go
slower than the clock at the centre. The same thing must take
place, judged from K0, if we define time with respect to K0 in a
not wholly unnatural way, that is, in such a way that the laws
with respect to K0 depend explicitly upon the time. Space and
time, therefore, cannot be defined with respect to K0 as they
were in the special theory of relativity with respect to inertial
systems. But, according to the principle of equivalence, K0 is
also to be considered 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 in
uences 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
These considerations assume that the behaviour of rods and clocks
depends only upon velocities, and not upon accelerations, or, at least, that
the in
uence of acceleration does not counteract that of velocity.
THE GENERAL THEORY 65
of a gravitational field the geometry is not Euclidean.
The case that we have been considering is analogous to that
which is presented in the two-dimensional treatment of surfaces.
It is impossible in the latter case also, to introduce coordinates
on a surface (e.g. the surface of an ellipsoid) which
have a simple metrical significance, while on a plane the Cartesian
co-ordinates, x1, x2, signify directly lengths measured by a
unit measuring rod. Gauss overcame this diculty, 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 coordinates,
x1, x2, x3, x4, which shall number uniquely the spacetime
points, so that neighbouring events are associated with
neighbouring 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
THE MEANING OF RELATIVITY 66
distance may be expressed by the formula ds2 = dx1
2 + dx2
2.
We may base upon this quantity the concepts of the straight
line as the geodesic (Rds = 0), 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, X1, X2, upon
such a small portion of the surface, and the distance between
two points, measured by a measuring rod, is given by
ds2 = dX1
2 + dX2
2:
If we introduce arbitrary curvilinear co-ordinates, x1, x2, on the
surface, then dX1, dX2, may be expressed linearly in terms of
dx1, dx2. Then everywhere upon the surface we have
ds2 = g11 dx1
2 + 2g12 dx1 dx2 + g22 dx2
2;
where g11, g12, g22 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 ds2 exactly as plane geometry
is based upon the corresponding expression.
There are analogous relations in the four-dimensional spacetime
continuum of physics. In the immediate neighbourhood of
an observer, falling freely in a gravitational 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, X1, X2, X3, and the time
THE GENERAL THEORY 67
co-ordinate X4) 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,
dX1
2 + dX2
2 + dX3
2 􀀀 dX4
2;
or its negative,
ds2 = 􀀀dX1
2 􀀀 dX2
2 􀀀 dX3
2 + dX4
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
THE MEANING OF RELATIVITY 68
expressed in arbitrary co-ordinates. If one observes that the
local dX may be expressed linearly in terms of the co-ordinate
differentials dx, ds2 may be expressed in the form
ds2 = g dx dx: (55)
The functions g describe, with respect to the arbitrarily
chosen system of co-ordinates, the metrical relations 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 concepts and operations of
this calculus of tensors.
We designate four quantities, which are defined as functions
of the x with respect to every system of co-ordinates, as comT
HE GENERAL THEORY 69
ponents, A, of a contra-variant vector, if they transform in a
change of co-ordinates as the co-ordinate differentials dx. We
therefore have
A0 =
@x0
@x
A: (56)
Besides these contra-variant vectors, there are also co-variant
vectors. If B are the components of a co-variant vector, these
vectors are transformed according to the rule
B0 =
@x
@x0
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,
 = BA (summed over the ).
Accordingly,
B0A0 =
@xff
@x0
@x0
@xfi
BffAfi = BffAff:
In particular, the derivatives
@
@xff
of a scalar , are components
of a co-variant vector, which, with the co-ordinate differentials,
form the scalar
@
@xff
dxff; we see from this example how natural
is the definition of the co-variant vectors.
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 designated by the position
THE MEANING OF RELATIVITY 70
of the index. For example, A
 denotes a tensor of the second
rank, which is co-variant with respect to the index , and contravariant
with respect to the index . The tensor character indicates
that the equation of transformation is
A0
 =
@xff
@x0
@x0
@xfi
Afi
ff: (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 example,
A
 + B
 = C
 : (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,
A
B = C
 : (60)
The proof follows directly from the rule of transformation.
Tensors may be formed by contraction with respect to two
indices of different character, for example,
A
 = B : (61)
The tensor character of A
 determines the tensor character
of B . Proof|
A0
 =
@xff
@x0
@x0
@xfi
@xs
@x0
@xt
@x0
Afi
ffst =
@xs
@x0
@xt
@x0
Aff
ffst:
THE GENERAL THEORY 71
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 invariance
of ds2 for an arbitrary choice of the dx, in connexion with
the condition of symmetry consistent with (55), that the g
are components of a symmetrical co-variant tensor (Fundamental
Tensor). Let us form the determinant, g, of the g, and
also the minors, divided by g, corresponding 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
gffgfi = fi
ff = (1 if ff = fi,
0 if ff 6= fi.
(62)
If we form the infinitely small quantities (co-variant vectors)
d = gff dxff; (63)
multiply by gfi and sum over the , we obtain, by the use
of (62),
dxfi = gfi d: (64)
Since the ratios of the d are arbitrary, and the dxfi as well as
the dx are components of vectors, it follows that the g are the
components of a contra-variant tensor (contra-variant fundamental
tensor). The tensor character of fi
ff (mixed fundamental
If we multiply (64) by @x0ff
@xfi
, sum over the fi, and replace the d by
THE MEANING OF RELATIVITY 72
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,
A = gffAff;
A = gffAff;
T
 = gT:
Volume Invariants. The volume element
Z dx1 dx2 dx3 dx4 = dx
is not an invariant. For by Jacobi's theorem,
dx0 =fifififi
dx0
dxfifififi dx: (65)
But we can complement dx so that it becomes an invariant. If
we form the determinant of the quantities
g0 =
@xff
@x0
@xfi
@x0
gfffi;
a transformation to the accented system, we obtain
dx0ff = @x0
@x
@x0ff
@xfi
gfi d0:
The statement made above follows from this, since, by (64), we must also
have dx0ff = gff0 d0ff, and both equations must hold for every choice of
the d0.
THE GENERAL THEORY 73
we obtain, by a double application of the theorem of multiplication
of determinants,
g0 = jg0j =fifififi
@x
@x0fifififi
2
delta  jgj =fifififi
@x0
@xfifififi
􀀀2
g: (66)
We therefore get the invariant,
pg0 dx0 = pg dx:
Formation of Tensors 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 coecients of its transformation,
@x0
@x
, are independent of position only if the transformation
is a linear one. For then the vector components, A +
@A
@xff
dxff,
at 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
@A
@xff
. But if the
@x0
@x
are
variable this is no longer true.
That there are, nevertheless, in the general case, invariant
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. Let P1 and P2
THE MEANING OF RELATIVITY 74
be two infinitesimally near points of the continuum. For the in-
finitesimal region surrounding the point P1, there is, according
to our way of considering the matter, a co-ordinate system of
the X (with imaginary X4-co-ordinate) for which the continuum
is Euclidean. Let A
(1) be the co-ordinates of the vector at
the point P1. Imagine a vector drawn at the point P2, using the
local system of the X, with the same co-ordinates (parallel vector
through P2), then this parallel vector is uniquely determined
by the vector at P1 and the displacement. We designate this operation,
whose uniqueness will appear in the sequel, the parallel
displacement of the vector A from P1 to the infinitesimally near
point P2. If we form the vector difference of the vector (A) at
the point P2 and the vector obtained by parallel displacement
from P1 to P2, we get a vector which may be regarded as the
differential of the vector (A) for the given displacement (dx).
This vector displacement can naturally also be considered
with respect to the co-ordinate system of the x. If A are the
co-ordinates of the vector at P1, A + A the co-ordinates of
the vector displaced to P2 along the interval (dx), then the A
do not vanish in this case. We know of these quantities, which
do not have a vector character, that they must depend linearly
and homogeneously upon the dx and the A. We therefore put
A = 􀀀􀀀
fffiAff dxfi: (67)
In addition, we can state that the 􀀀
fffi must be symmetrical
with respect to the indices ff and fi. For we can assume from
a representation by the aid of a Euclidean system of local coordinates
that the same parallelogram will be described by the
displacement of an element d(1)x along a second element d(2)x
THE GENERAL THEORY 75
as by a displacement of d(2)x along d(1)x. We must therefore
have
d(2)x + (d(1)x 􀀀 􀀀
fffi d(1)xff d(2)xfi)
= d(1)x + (d(2)x 􀀀 􀀀
fffi d(2)xff d(1)xfi):
The statement made above follows from this, after interchanging
the indices of summation, ff and fi, on the right-hand side.
Since the quantities g determine all the metrical properties
of the continuum, they must also determine the 􀀀
fffi. If we
consider the invariant of the vector A, that is, the square of its
magnitude,
gAA;
which is an invariant, this cannot change in a parallel displacement.
We therefore have
0 = (gAA) =
@g
@xff
AA dxff + gAA + gAA
or, by (67), @g
@xff 􀀀 gfi􀀀fi
 ff 􀀀 gfi􀀀fi
ffAA dxff = 0:
Owing to the symmetry of the expression in the brackets
with respect to the indices  and , this equation can be valid
for an arbitrary choice of the vectors (A) and dx only when
the expression in the brackets vanishes for all combinations of
the indices. By a cyclic interchange of the indices , , ff, we
obtain thus altogether three equations, from which we obtain,
on taking into account the symmetrical property of the 􀀀ff
,

ff = gfffi􀀀fi
; (68)
THE MEANING OF RELATIVITY 76
in which, following Christoffel, the abbreviation has been used,

ff = 1
2 @gff
@x
+
@gff
@x 􀀀
@g
@xff : (69)
If we multiply (68) by gff and sum over the ff, we obtain
􀀀ff
 = 1
2gff @gff
@x
+
@gff
@x 􀀀
@g
@xff = 
 	; (70)
in which 
 	 is the Christoffel symbol of the second kind.
Thus the quantities 􀀀 are deduced from the g. Equations
(67) and (70) are the foundation for the following discussion.
Co-variant Differentiation of Tensors. If (A + A) is
the vector resulting from an infinitesimal parallel displacement
from P1 to P2, and (A + dA) the vector A at the point P2,
then the difference of these two,
dA 􀀀 A = @A
@x
+ 􀀀
 ffAffdx;
is also a vector. Since this is the case for an arbitrary choice of
the dx, it follows that
A
;  =
@A
@x
+ 􀀀
 ffAff (71)
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),
􀀀
 = 1
2gff @gff
@x
=
1
pg
@pg
@x
: (72)
THE GENERAL THEORY 77
If we put, further,
Apg = A; (73)
a quantity designated by Weyl as the contra-variant tensor density
 of the first rank, it follows that,
A =
@ A
@x
(74)
is a scalar density.
We get the law of parallel displacement for the co-variant
vector B by stipulating that the parallel displacement shall be
effected in such a way that the scalar
 = AB
remains unchanged, and that therefore
A B + B A
vanishes for every value assigned to (A). We therefore get
B = 􀀀ff
Aff dx: (75)
From this we arrive at the co-variant derivative of the covariant
vector by the same process as that which led to (71),
B;  =
@B
@x 􀀀 􀀀ff
Bff: (76)
This expression is justified, in that Apg dx = A dx has a tensor
character. Every tensor, when multiplied by pg, changes into a tensor
density. We employ capital Gothic letters for tensor densities.
THE MEANING OF RELATIVITY 78
By interchanging the indices  and fi, and subtracting, we get
the skew-symmetrical tensor,
 =
@B
@x 􀀀
@B
@x
: (77)
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 ) be a co-variant tensor of the
second rank. Then AEF is a scalar, if E and F are vectors.
This expression must not be changed by the -displacement;
expressing this by a formula, we get, using (67), A , whence
we get the desired co-variant derivative,
A;  =
@A
@x 􀀀 􀀀ff
 Aff 􀀀 􀀀ff
 Aff: (78)
In order that the general law of co-variant differentiation of
tensors may be clearly seen, we shall write down two co-variant
derivatives deduced in an analogous way:
A
;  =
@A

@x 􀀀 􀀀ff
A
ff + 􀀀
ffAff
 ; (79)
A
;  =
@A
@x
+ 􀀀
ffAff + 􀀀
ffAff: (80)
The general law of formation now becomes evident. From these
formul we shall deduce some others which are of interest for
the physical applications of the theory.
In case A is skew-symmetrical, we obtain the tensor
A =
@A
@x
+
@A
@x
+
@A
@x
; (81)
THE GENERAL THEORY 79
which is skew-symmetrical in all pairs of indices, by cyclic interchange
and addition.
If, in (78), we replace A by the fundamental tensor, g ,
then the right-hand side vanishes identically; an analogous statement
holds for (80) with respect to g ; 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 is skew-symmetrical, we obtain from (80), by
contraction with respect to  and ,
A =
@ A
@x
: (82)
In the general case, from (79) and (80), by contraction with
respect to  and , we obtain the equations,
A =
@ Aff

@xff 􀀀 􀀀ff
fi Afi
ff; (83)
A =
@ Aff
@xff
+ 􀀀
fffi Afffi : (84)
The Riemann Tensor. If we have given a curve extending
from the point P to the point G of the continuum, then a vector
A, given at P, may, by a parallel displacement, be moved
along the curve to G. If the continuum is Euclidean (more generally,
if by a suitable choice of co-ordinates the g 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 displacement.
In this case, therefore, a vector suffers a change, delta A
(in its direction, not its magnitude), when it is carried from a
THE MEANING OF RELATIVITY 80
P G
Fig. 4.
point P of a closed curve, along the curve, and back to P. We
shall now calculate this vector change:
delta A = I 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),
delta A = 􀀀I 􀀀
fffiAff dxfi:
In this, 􀀀
fffi is the value of this quantity at the variable
point G of the path of integration. If we put
 = (x)G 􀀀 (x)P
THE GENERAL THEORY 81
and denote the value of 􀀀
fffi at P by 􀀀
fffi, then we have, with
sucient accuracy,
􀀀
fffi = 􀀀
fffi +
@􀀀
fffi
@x
:
Let, further, Aff be the value obtained from Aff by a parallel
displacement along the curve from P to G. It may now easily
be proved by means of (67) that A 􀀀 A is infinitely small of
the first order, while, for a curve of infinitely small dimensions
of the first order, delta A is infinitely small of the second order.
Therefore there is an error of only the second order if we put
Aff = Aff 􀀀 􀀀ff
 A  :
If we introduce these values of 􀀀
fffi and Aff into the integral,
we obtain, neglecting all quantities of a higher order of small
quantities than the second,
delta A = 􀀀@􀀀
fi
@xff 􀀀 􀀀
fi􀀀
ffA I ff dfi: (85)
The quantity removed from under the sign of integration refers
to the point P. Subtracting 1
2d(fffi) from the integrand, we
obtain
1
2 I (ff dfi 􀀀 fi dff):
This skew-symmetrical tensor of the second rank, ffffi, 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 ff and fi, we could
THE MEANING OF RELATIVITY 82
conclude its tensor character from (85). We can accomplish this
by interchanging the summation indices ff and fi in (85) and
adding the resulting equation to (85). We obtain
2delta A = 􀀀R
fffiAffffi; (86)
in which
R
fffi = 􀀀
@􀀀
ff
@xfi
+
@􀀀
fi
@xff
+ 􀀀
ff􀀀
fi 􀀀 􀀀
fi􀀀
ff: (87)
The tensor character of R
fffi 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 sucient
condition (disregarding the reality of the chosen co-ordinates)
that the continuum is Euclidean.
By contraction of the Riemann tensor with respect to the
indices , fi, we obtain the symmetrical tensor of the second
rank,
R = 􀀀
@􀀀ff

@xff
+ 􀀀ff
fi􀀀fi
 ff +
@􀀀ff
ff
@x 􀀀 􀀀ff
􀀀fi
fffi: (88)
The last two terms vanish if the system of co-ordinates is so
chosen that g = constant. From R we can form the scalar,
R = gR: (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
 dx
ds = 􀀀􀀀
fffi
dxff
ds
dxfi:
THE GENERAL THEORY 83
The left-hand side is to be replaced by
d2x
ds2 , so that we have
d2x
ds2 + 􀀀
fffi
dxff
ds
dxfi
ds
= 0: (90)
We get the same line if we find the line which gives a stationary
value to the integral
Z ds or Z pg dx dx
between two points (geodetic line).
The direction vector at a neighbouring point of the curve results, by
a parallel displacement along the line element (dxfi), from the direction
vector of each point considered.
LECTURE IV
THE GENERAL THEORY OF RELATIVITY
(continued)
Weare 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
completeness, but single results and possibilities will
be developed 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,
d2x
ds2 + 􀀀
fffi
dxff
ds
dxfi
ds
= 0: (90)
In fact, this equation reduces to that of a straight line if all the
components, 􀀀
fffi, of the gravitational field vanish.
How are these equations connected with Newton's equations
of motion? According to the special theory of relativity, the g
as well as the g, have the values, with respect to an inertial
84
THE GENERAL THEORY 85
system (with real time co-ordinate and suitable choice of the
sign of ds2),
􀀀1 0 0 0
0 􀀀1 0 0
0 0 􀀀1 0
0 0 0 1
9>
>=>>;
: (91)
The equations of motion then become
d2x
ds2 = 0:
We shall call this the \first approximation" to the g-field. In
considering approximations it is often useful, as in the special
theory of relativity, to use an imaginary x4-co-ordinate, as then
the g, to the first approximation, assume the values
􀀀1 0 0 0
0 􀀀1 0 0
0 0 􀀀1 0
0 0 0 􀀀1
9>
>=>>;
: (91a)
These values may be collected in the relation
g = 􀀀:
To the second approximation we must then put
g = 􀀀 + 
; (92)
where the 
 are to be regarded as small of the first order.
THE MEANING OF RELATIVITY 86
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
ds2 = 􀀀dx
2 = dl2(1 􀀀 q2); (93)
􀀀
fffi = 􀀀 fffi
 = 􀀀fffi
 =
1
2 @
fffi
@x 􀀀
@
ff
@xfi 􀀀
@
fi
@xff : (94)
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 same as the time differential,
dl. Further,
dx1
ds
,
dx2
ds
,
dx3
ds
will vanish compared to
dx4
ds
.
We shall assume, in addition, that the gravitational field varies
so little with the time that the derivatives of the 
 by x4 may
be neglected. Then the equation of motion (for  = 1; 2; 3)
reduces to
d2x
dl2 =
@
@x 
44
2 : (90a)
This equation is identical with Newton's equation of motion for
a material particle in a gravitational field, if we identify 
44
2  with the potential of the gravitational field; whether or not this
is allowable, naturally depends upon the field equations of gravitation,
that is, it depends 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 􀀀
fffi actually do play the r^ole of
the intensity of the gravitational field. These quantities do not
have a tensor character.
Equations (90) express the in
uence of inertia and gravitation
upon the material particle. The unity of inertia and graviT
HE GENERAL THEORY 87
tation 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 expression
for inertia, and the second as the expression for the
gravitational force.
We must next attempt to find the laws of the gravitational
field. For this purpose, Poisson's equation,
delta  = 4K
of the Newtonian theory must serve as a model. This equation
has its foundation in the idea that the gravitational field
arises from the density  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 sucient knowledge 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,
THE MEANING OF RELATIVITY 88
whose structure we do not know provisionally, and which includes
in itself the energy density of the electromagnetic field
and of ponderable 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) denotes the co-variant energy tensor
of matter, T
 the corresponding mixed tensor density, then,
in accordance with (83), we must require that
0 =
@ Tff

@xff 􀀀 􀀀ff
fi Tfi
ff (95)
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 gravitational 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 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. We have to find this differenT
HE GENERAL THEORY 89
tial tensor. It is completely determined by the following three
conditions:|
1. It may contain no differential coecients of the g higher
than the second.
2. It must be linear and homogeneous in these second differential
coecients.
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
R + agR;
in which R and R are defined by (88) and (89) respectively.
Further, it may be proved that the third condition requires a
to have the value 􀀀1
2 . For the law of the gravitational field we
therefore get the equation
R 􀀀 1
2gR = 􀀀T: (96)
Equation (95) is a consequence of this equation.  denotes 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 =
@ Tff

@xff 􀀀 􀀀ff
fi Tfi
ff; (97)
THE MEANING OF RELATIVITY 90
in which
Tff
 = T gffp􀀀g:
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 x4-co-ordinate imaginary,
at the given point,
g = g = 􀀀 = (􀀀1 if  = ,
0 if  6= ,
and for which the first derivatives of the g and the g vanish.
We shall verify the vanishing of the divergence of the left-hand
side at this point. At this point the components 􀀀ff
fi vanish, so
that we have to prove the vanishing only of
@
@x p􀀀g g(R 􀀀 1
2gR):
Introducing (88) and (70) into this expression, we see that the
only terms that remain are those in which third derivatives of
the g 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 coordinates
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
THE GENERAL THEORY 91
lead to the Newtonian theory as a first approximation. For this
purpose we must introduce 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 relativity, we take the
fourth co-ordinate imaginary, this means that we must put
g = 􀀀 + 
; (98)
in which the 
 are so small compared to 1 that we can neglect
the higher powers of the 
 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 in
uence of neighbouring masses upon physical phenomena.
Before carrying through this approximation we shall transform
(96). We multiply (96) by g, summed over the  and ;
observing the relation which follows from the definition of
the g,
gg = 4;
we obtain the equation
R = gT = T:
If we put this value of R in (96) we obtain
R = 􀀀(T 􀀀 1
2gT) = 􀀀T : (96a)
When the approximation which has been mentioned is carried
out, we obtain for the left-hand side,
􀀀1
2 @2

@xff
2 +
@2
ffff
@x @x 􀀀
@2
ff
@x @xff 􀀀
@2
ff
@x @xff
THE MEANING OF RELATIVITY 92
or
􀀀1
2
@2

@xff
2 + 1
2
@
@x @
0ff
@xff + 1
2
@
@x @
0ff
@xff ;
in which has been put

0 = 
 􀀀 1
2
: (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 differ infinitely little from the constant values
􀀀. 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 con
ict with the conditions for the order of magnitude of
the 
. We shall now assume that the system of co-ordinates
is so chosen that the four relations|
0 =
@
0
@x
=
@

@x 􀀀 1
2
@

@x
(100)
are satisfied. Then (96a) takes the form
@2

@xff
2 = 2T : (96b)
These equations may be solved by the method, familiar in
electrodynamics, of retarded potentials; we get, in an easily
understood notation,

 = 􀀀

2 Z T (x0; y0; z0; t 􀀀 r)
r
dV0: (101)
THE GENERAL THEORY 93
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 phenomenologically, 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 1, 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 sucient for our
purpose if we put
T = 
dx
ds
dx
ds
;
ds2 = g dx dx:
9=;
(102)
In this,  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
by 􀀀, so that we put
ds2 = 􀀀Xdx
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
THE MEANING OF RELATIVITY 94
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 1, with our choice
of the unit of time. We therefore get an approximation which is
sucient for nearly all practical purposes if in (101) we replace
the retarded potential by the ordinary (non-retarded) potential,
and if, for the masses which generate the field, we put
dx1
ds
=
dx2
ds
=
dx3
ds
= 0;
dx4
ds
=
p􀀀1 dl
dl
= p􀀀1: (103)
Then we get for T and T the values
0 0 0 0
0 0 0 0
0 0 0 0
0 0 0 􀀀
9>>=>>;
: (104)
For T we get the value , and, finally, for T  the values,

2
0 0 0
0

2
0 0
0 0

2
0
0 0 0 􀀀

2
9>
>>>=>>>>;
: (104a)
We thus get, from (101),

11 = 
22 = 
33 = 􀀀

4 Z  dV0
r
;

44 = +

4 Z  dV0
r
9>
>=>>;
(101a)
THE GENERAL THEORY 95
while all the other 
 vanish. The least of these equations,
in connexion with equation (90a), contains Newton's theory of
gravitation. If we replace l by ct we get
d2x
dt2 =
c2
8
@
@x Z  dV0
r : (90b)
We see that the Newtonian gravitation constant K, is connected
with the constant  that enters into our field equations by the
relation
K =
c2
8
: (105)
From the known numerical value of K, it therefore follows that
 =
8K
c2 =
8 delta  6:67 delta  10􀀀8
9 delta  1020 = 1:86 delta  10􀀀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 g44, to a first approximation, enters
the equations 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 falling, and without rotation). We
can make the same statement for local systems of co-ordinates
THE MEANING OF RELATIVITY 96
which, relatively to these, have small accelerations, 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 events,
ds2 = 􀀀dX1
2 􀀀 dX2
2 􀀀 dX3
2 + dT 2 = 􀀀dS2 + 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 ds2, on the other hand, is
known in terms of the co-ordinates x employed in finite regions,
in the form
ds2 = 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 ds2 we get two relations. If, by (101a), we put
ds2 = 􀀀1 +

4 Z  dV0
r (dx1
2 + dx2
2 + dx3
2)
+ 1 􀀀

4 Z  dV0
r dl2;
we obtain, to a suciently close approximation,
pdX1
2 + dX2
2 + dX3
2
= 1 +

8 Z  dV0
r pdx1
2 + dx2
2 + dx3
2;
dT = 1 􀀀

8 Z  dV0
r dl:
9>
>>=>>>;
(106)
THE GENERAL THEORY 97
The unit measuring rod has therefore the length,
1 􀀀

8 Z  dV0
r
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,
delta x1, delta x2, delta x3, corresponding to the ends of a unit measuring
rod, oriented in any way, shall always satisfy the relation
delta x1
2 + delta x2
2 + delta x3
2 = 1. 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 = 1)
corresponds to the \time"
1 +

8 Z  dV0
r
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 therefore 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 delta  10􀀀6 of their wave-lengths.
At first, this important consequence of the theory appeared to
con
ict with experiment; but results obtained during the past
year seem to make the existence of this effect more probable, and
THE MEANING OF RELATIVITY 98
it can hardly be doubted that this consequence of the theory will
be confirmed within the next year.
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
ds2 = 0:
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
1+

4 Z  dV0
r (dx1
2+dx2
2+dx3
2) = 1􀀀

4 Z  dV0
r dl2:
The velocity of light L, is therefore expressed in our co-ordinates
by pdx1
2 + dx2
2 + dx3
2
dl
= 1 􀀀

4 Z  dV0
r
: (107)
We can therefore draw the conclusion from this, that a ray of
light passing near a large mass is de
ected. If we imagine the
sun, of mass M concentrated at the origin of our system of coordinates,
then a ray of light, travelling parallel to the x3-axis,
in the x1-x3 plane, at a distance delta  from the origin, will be
de
ected, in all, by an amount
ff = Z +1
􀀀1
1
L
@L
@x1
dx3
THE GENERAL THEORY 99
towards the sun. On performing the integration we get
ff =
M
2delta 
: (108)
The existence of this de
ection, which amounts to 1:700 for
delta  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 in
uenced 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
sucient for a comparison of theory and observation. It is necessary
to go back to the general field equations (96). To solve
this problem I made use of the method of successive 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.
THE MEANING OF RELATIVITY 100
In the case of a statical field, ds2 must have the form
ds2 = 􀀀d2 + f2 dx4
2;
d2 =X1{3

fffi dxff dxfi;9=;
(109)
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 
 to be of the form,

fffi = fffi + xffxfi; (110)
f2,  and  are functions of r = px1
2 + x2
2 + x3
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
x04 = x4;
x0ff = F(r)xff;
we can always insure that one of these three functions shall be
an assigned function of r0. In place of (110) we can therefore
put, without limiting the generality,

fffi = fffi + xffxfi: (110a)
In this way the g are expressed in terms of the two quantities
 and f. These are to be determined as functions of r,
by introducing them into equation (96), after first calculating
THE GENERAL THEORY 101
the 􀀀
 from (109) and (110a). We have
􀀀
fffi = 1
2
x
r delta 
0xffxfi + 2r fffi
1 + r2 (for ff; fi;  = 1; 2; 3);
􀀀4
44 = 􀀀ff
4fi = 􀀀4
fffi = 0 (for ff; fi = 1; 2; 3);
􀀀4
4ff = 1
2f􀀀2 @f2
@xff
; 􀀀ff
44 = 􀀀1
2gfffi @f2
@xfi
:
9>
>>=>>>;
(110b)
With the help of these results, the field equations furnish
Schwarzschild's solution:
ds2 = 1 􀀀
A
r dl2 􀀀264
dr2
1 􀀀
A
r
+ r2(sin2  d2 + d2)375 ; (109a)
in which we have put
x4 = l;
x1 = r sin  sin ;
x2 = r sin  cos ;
x3 = r cos ;
A =
M
4
:
9>>>>=>
>>>;
(109b)
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 vanish. If the motion
of the planet takes place in the x1-x2 plane then we must replace
(109a) by
ds2 = 1 􀀀
A
r dl2 􀀀
dr2
1 􀀀
A
r
􀀀 r2 d2: (109c)
THE MEANING OF RELATIVITY 102
The calculation of the planetary motion depends upon equation
(90). From the first of equations (110b) and (90) we get,
for the indices 1; 2; 3,
d
ds xff
dxfi
ds 􀀀 xfi
dxff
ds = 0;
or, if we integrate, and express the result in polar co-ordinates,
r2 d
ds
= constant: (111)
From (90), for  = 4, we get
0 =
d2l
ds2 +
1
f2
df2
dxff
dxff
ds
dl
ds
=
d2l
ds2 +
1
f2
df2
ds
dl
ds
:
From this, after multiplication by f2 and integration, we have
f2 dl
ds
= constant: (112)
In (109c), (111) and (112) we have three equations between
the four variables s, r, l 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
243a2
(1 􀀀 e2)c2T2 ; (113)
THE GENERAL THEORY 103
where
a = the semi-major axis of the planetary orbit in centimetres.
e = the numerical eccentricity.
c = 3 delta  10+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 diculty 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  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,
 =
@
@x 􀀀
@
@x
: (114)
The second of Maxwell's systems of equations is then defined by
the tensor equation, resulting from this,
@
@x
+
@
@x
+
@
@x
= 0; (114a)
and the first of Maxwell's systems of equations is defined by the
tensor-density relation
@ F
@x
= J; (115)
THE MEANING OF RELATIVITY 104
in which
F = p􀀀g gg ;
J = p􀀀g 
dx
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 J = 0, 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 relativity 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
THE GENERAL THEORY 105
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 suciently 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 would
be constant at infinity, as they are in the special theory of relativity,
while within finite regions the g would differ from these
constant values by small amounts only, with a suitable choice
of co-ordinates, as a result of the in
uence of the masses in fi-
nite regions. The physical properties of space would not then be
wholly independent, that is, unin
uenced 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 Riklm
THE MEANING OF RELATIVITY 106
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, according 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 ponderable
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 acceleration.
3. A rotating hollow body must generate inside of itself
a \Coriolis field," which de
ects 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).
THE GENERAL THEORY 107
First, we consider 
44 as small of the first order. The square
of the velocity of masses moving under the in
uence 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 1
2 .
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 = pg44 dl = 1 􀀀

44
2 dl:
From (90) we obtain, at first,
d
dl 1 +

44
2 dx
dl = 􀀀􀀀
fffi
dxff
dl
dxfi
dl 1 +

44
2 : (116)
From (101) we get, to the approximation sought for,
􀀀
11 = 􀀀
22 = 􀀀
33 = 
44 =

4 Z  dV0
r
;

4ff = 􀀀
i
2 Z 
dxff
ds
dV0
r
;

fffi = 0;
9>
>>>=>>>>;
(117)
in which, in (117), ff and fi denote the space indices only.
THE MEANING OF RELATIVITY 108
On the right-hand side of (116) we can replace 1 +

44
2
by 1
and 􀀀􀀀
fffi by fffi
 . It is easy to see, in addition, that to this
degree of approximation we must put
44
 = 􀀀1
2
@
44
@x
+
@
4
@x4
;
ff4
 = 1
2 @
4
@xff 􀀀
@
4ff
@x ;
fffi
 = 0;
in which ff, fi and  denote space indices. We therefore obtain
from (116), in the usual vector notation,
d
dl (1 + )v= grad  +
@ A
@l
+ [rot A; v];
 =

8 Z  dV0
r
;
A =

2 Z 
dxff
dl
dV0
r
:
9>>>>>=>
>>>>;
(118)
The equations of motion, (118), show now, in fact, that
1. The inert mass is proportional to 1 + , 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 the
term
@ A
@l
.
THE GENERAL THEORY 109
3. A material particle, moving perpendicularly to the axis
of rotation inside a rotating hollow body, is de
ected 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 experiment,
because  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 g-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 g-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 suitable choice of co-ordinates,
g44 is nearly constant in the universe, at least, in that part of
the universe in which there is matter. The assumption appears
natural, moreover, that there are stars in all parts of the universe,
so that we may well assume that the inconstancy of g44 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
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.
THE MEANING OF RELATIVITY 110
non-uniformities of the density of matter and of the g-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 . 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 x4-co-ordinate. The possibility seems to be particularly
satisfying that the universe is spatially bounded and thus, in
accordance with our assumption of the constancy of , 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 surface.
According to what has been said, we are to put
ds2 = dx4
2 􀀀 
 dx dx; (119)
in which the indices  and  run from 1 to 3 only. The 

will be such functions of x1, x2, x3 as correspond to a threedimensional
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 dimenT
HE GENERAL THEORY 111
sions, is given by the equations
x1
2 + x2
2 + x3
2 + x4
2 = a2;
dx1
2 + dx2
2 + dx3
2 + dx4
2 = ds2:
By eliminating x4, we get
ds2 = dx1
2 + dx2
2 + dx3
2 +
(x1 dx1 + x2 dx2 + x3 dx3)2
a2 􀀀 x1
2 􀀀 x2
2 􀀀 x3
2 :
As far as terms of the third and higher degrees in the x, we
can put, in the neighbourhood of the origin of co-ordinates,
ds2 =  +
xx
a2 dx dx:
Inside the brackets are the g of the manifold in the neighbourhood
of the origin. Since the first derivatives of the g,
and therefore also the 􀀀
, vanish at the origin, the calculation
of the R for this manifold, by (88), is very simple at the origin.
We have
R = 􀀀
2
a2  =
2
a2 g:
Since the relation R =
2
a2 g 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 aid of a fourth space dimension has naturally no significance ex-
cept that of a mathematical artifice.
THE MEANING OF RELATIVITY 112
the four-dimensional continuum, we shall, in the following, designate
quantities that refer to the three-dimensional continuum
by Greek letters, and put
P = 􀀀
2
a2 
: (120)
We now proceed to apply the field equations (96) to our special
case. From (119) we get for the four-dimensional manifold,
R = P for the indices 1 to 3;
R14 = R24 = R34 = R44 = 0: ) (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 = 
dx
ds
dx
ds
specialized for the case of rest. But in addition, we shall add
a pressure term that may be physically established 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
contained in Maxwell's theory, so that the single electric particles
may hold together in spite of the mutual repulsions 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
THE GENERAL THEORY 113
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
T = ggfi
dxff
ds
dxfi
ds 􀀀 gp: (122)
In our special case we have, therefore, to put
T = 
p (for  and  from 1 to 3);
T44 =  􀀀 p;
T = 􀀀

p +  􀀀 p =  􀀀 4p:
Observing that the field equation (96) may be written in the
form
R = 􀀀(T 􀀀 1
2gT);
we get from (96) the equations,
+
2
a2 
 =  
2 􀀀 p
;
0 = 􀀀 
2
+ p:
From this follows
p = 􀀀

2
;
a = r 2

:
9>
=>
;
(123)
If the universe is quasi-Euclidean, and its radius of curvature
therefore infinite, then  would vanish. But it is improbable that
THE MEANING OF RELATIVITY 114
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 hypothetical
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
a =
M
42 : (124)
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 spacebounded,
universe:|
1. 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 episT
HE GENERAL THEORY 115
temology it is more satisfying to have the mechanical properties
of space completely determined 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 probable than the assumption that
there is a finite mean density of matter in the universe.
INDEX
A
Accelerated masses, inductive
action of, 108
Addition and subtraction of
tensors, 14
| theorem of velocities, 38
B
Biot-Savart force, 44
C
Centrifugal force, 64
Clocks, moving, 38
Compressible viscous 
uid, 22
Concept of space, 3
| time, 28
Conditions of orthogonality, 7
Congruence, theorems of, 3
Conservation principles, 54
Continuum, four-dimensional,
31
Contraction of tensors, 14
Contra-variant vectors, 69
| tensors, 71
Co-ordinates, preferred systems
of, 8
Co-variance of equation of
continuity, 21
Co-variant, 12 et seq.
| vector, 68
Criticism of principle of inertia,
62
Criticisms of theory of
relativity, 29
Curvilinear co-ordinates, 65
D
Differentiation of tensors, 73, 76
Displacement of spectral lines,
97
E
Energy and mass, 45, 49
| tensor of electromagnetic
field, 50
| | of matter, 54
Equation of continuity,
co-variance of, 21
Equations of motion of material
particle, 50
Equivalence of mass and
energy, 49
Equivalent spaces of reference,
25
Euclidean geometry, 4
F
Finiteness of universe, 105
Fizeau, 28
Four-dimensional continuum, 31
Four-vector, 41
116
INDEX 117
Fundamental tensor, 71
G
Galilean regions, 62
| transformation, 27
Gauss, 65
Geodetic lines, 82
Geometry, Euclidean, 4
Gravitation constant, 95
Gravitational mass, 60
H
Homogeneity of space, 17
Hydrodynamical equations, 54
Hypotheses of pre-relativity
physics, 73
I
Inductive action of accelerated
masses, 108
Inert and gravitational mass,
equality of, 60
Invariant, 9 et seq.
Isotropy of space, 17
K
Kaluza, 104
L
Levi-Civita, 73
Light-cone, 41
Light ray, path of, 98
Light-time, 33
Linear orthogonal
transformation, 7
Lorentz electromotive force, 44
| transformation, 31
M
Mach, 59, 105, 106, 109, 114
Mass and Energy, 45, 49
| equality of gravitational and
inert, 60
| gravitational, 60
Maxwell's equations, 23
Mercury, perihelion of, 99, 103
Michelson and Morley, 28
Minkowski, 32
Motion of particle, equations of,
50
Moving measuring rods and
clocks, 38
Multiplication of tensors, 14
N
Newtonian gravitation
constant, 95
O
Operations on tensors, 13 et
seq.
Orthogonal transformations,
linear, 7
Orthogonality, conditions of, 7
THE MEANING OF RELATIVITY 118
P
Path of light ray, 98
Perihelion of Mercury, 99, 103
Poisson's equation, 87
Preferred systems of
co-ordinates, 8
Pre-relativity physics,
hypotheses of, 26
Principle of equivalence, 61
| inertia, criticism of, 62
Principles of conservation, 54
R
Radius of Universe, 113
Rank of tensor, 13
Ray of light, path of, 98
Reference, space of, 3
Riemann, 68
| tensor, 79, 82, 105
Rods (measuring) and clocks in
motion, 38
Rotation, 63
S
Simultaneity, 17, 29
Sitter, 28
Skew-symmetrical tensor, 15
Solar Eclipse expedition (1919),
99
Space, Concept of, 2
| Homogeneity of, 17
| Isotropy of, 17
Spaces of reference, 3
| equivalence of, 25
Special Lorentz transformation,
34
Spectral lines, displacement of,
97
Straightest lines, 82
Stress tensor, 22
Symmetrical tensor, 15
Systems of co-ordinates,
preferred, 8
T
Tensor, 12 et seq., 68 et seq.
| Addition and subtraction of,
14
| Contraction of, 14
| Fundamental, 71
| Multiplication of, 14
| operations, 13 et seq.
| Rank of, 13
| Symmetrical and
Skew-symmetrical, 15
Tensors, formation by
differentiation, 73
Theorem for addition of
velocities, 38
Theorems of congruence, 3
Theory of relativity, criticisms
of, 29
Thirring, 109
Time-concept, 28
INDEX 119
Time-space concept, 31
Transformation, Galilean, 27
| Linear orthogonal, 7
U
Universe, Finiteness of, 105
| Radius of, 113
V
Vector, co-variant, 69
| contra-variant, 69
Velocities, addition theorem of,
38
Viscous compressible 
uid, 22
W
Weyl, 73, 99, 104
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