ALBERT EINSTEIN
Fundamental ideas and problems
of the theory of relativity
Lecture delivered to the Nordic Assembly of Naturalists at Gothenburg*
July 11, 1923
If we consider that part of the theory of relativity which may nowadays in
a sense be regarded as bona fide scientific knowledge, we note two aspects
which have a major bearing on this theory. The whole development of the
theory turns on the question of whether there are physically preferred states
of motion in Nature (physical relativity problem). Also, concepts and distinctions
are only admissible to the extent that observable facts can be assigned
to them without ambiguity (stipulation that concepts and distinctions
should have meaning). This postulate, pertaining to epistemology, proves to
be of fundamental importance.
These two aspects become clear when applied to a special case, e.g. to classical
mechanics. Firstly we see that at any point filled with matter there exists
a preferred state of motion, namely that of the substance at the point considered.
Our problem starts however with the question whether physically
preferred states of motion exist in reference to extensive regions. From the
viewpoint of classical mechanics the answer is in the affirmative; the physically
preferred states of motion from the viewpoint of mechanics are those of
the inertial frames.
This assertion, in common with the basis of the whole of mechanics as it
generally used to be described before the relativity theory, far from meets
the above "stipulation of meaning". Motion can only be conceived as the
relative motion of bodies. In mechanics, motion relative to the system of
coordinates is implied when merely motion is referred to. Nevertheless this
interpretation does not comply with the "stipulation of meaning" if the coordinate
system is considered as something purely imaginary. If we turn our
attention to experimental physics we see that there the coordinate system is
invariably represented by a "practically rigid" body. Furthermore it is assumed
that such rigid bodies can be positioned in rest relative to one another
* The Lecture was not delivered on the occasion of the Nobel Prize award, and did
not, therefore, concern the discovery of the photoelectric effect.
THEORY OF RELATIVITY
in common with the bodies of Euclidian geometry. Insofar as we may think
of the rigid measuring body as existing as an object which can be experienced,
the "system of coordinates" concept as well as the concept of the motion of
matter relative thereto can be accepted in the sense of the "stipulation of
meaning". At the same time Euclidian geometry, by this conception, has been
adapted to the requirements of the physics of the "stipulation of meaning".
The question whether Euclidian geometry is valid becomes physically significant;
its validity is assumed in classical physics and also later in the special
theory of relativity.
In classical mechanics the inertial frame and time are best defined together
by a suitable formulation of the law of inertia: It is possible to fix the time
and assign a state of motion to the system of coordinates (inertial frame) such
that, with reference to the latter, force-free material points undergo no acceleration;
furthermore it is assumed that this time can be measured without
disagreement by identical clocks (systems which run down periodically) in
any arbitrary state of motion. There are then an infinite number of inertial
frames which are in uniform translational motion relative to each other, and
hence there is also an infinite number of mutually equivalent, physically preferred
states of motion. Time is absolute, i.e.independent of the choice of
the particular inertial frame; it is defined by more characteristics than logically
necessary, although - as implied by mechanics - this should not lead
to contradictions with experience. Note in passing that the logical weakness
of this exposition from the point of view of the stipulation of meaning is
the lack of an experimental criterion for whether a material point is force-
free or not; therefore the concept of the inertial frame remains rather problematical.
This deficiency leads to the general theory of relativity. We shall
not consider it for the moment.
The concept of the rigid body (and that of the clock) has a key bearing
on the foregoing consideration of the fundamentals of mechanics, a bearing
which there is some justification for challenging. The rigid body is only approximately
achieved in Nature, not even with desired approximation; this
concept does not therefore strictly satisfy the "stipulation of meaning". It is
also logically unjustifiable to base all physical consideration on the rigid or
solid body and then finally reconstruct that body atomically by means of
elementary physical laws which in turn have been determined by means of
the rigid measuring body. I am mentioning these deficiencies of method
because in the same sense they are also a feature of the relativity theory in
the schematic exposition which I am advocating here. Certainly it would be
1921 A.EINSTEIN
logically more correct to begin with the whole of the laws and to apply the
"stipulation of meaning" to this whole first, i.e. to put the unambiguous relation
to the world of experience last instead of already fulfilling it in an imperfect
form for an artificially isolated part, namely the space-time metric.
We are not, however, sufficiently advanced in our knowledge of Nature’s
elementary laws to adopt this more perfect method without going out of our
depth. At the close of our considerations we shall see that in the most recent
studies there is an attempt, based on ideas by Levi-Civita, Weyl, and Eddington,
to implement that logically purer method.
It more clearly follows from the above what is implied by "preferred states
of motion". They are preferred as regards the laws of Nature. States of motion
are preferred when, relative to the formulation of the laws of Nature,
coordinate systems within them are distinguished in that with respect to them
those laws assume a form preferred by simplicity. According to classical mechanics
the states of motion of the inertial frames in this sense are physically
preferred. Classical mechanics permits a distinction to be made between (absolutely)
unaccelerated and accelerated motions; it also claims that velocities
have only a relative existence (dependent on the selection of the inertial
frame), while accelerations and rotations have an absolute existence (independent
of the selection of the inertial frame). This state of affairs can be
expressed thus: According to classical mechanics "velocity relativity" exists,
but not "acceleration relativity". After these preliminary considerations we
can pass to the actual topic of our contemplations, the relativity theory, by
characterizing its development so far in terms of principles.
The special theory of relativity is an adaptation of physical principles to
Maxwell-Lorentz electrodynamics. From earlier physics it takes the assumption
that Euclidian geometry is valid for the laws governing the position of
rigid bodies, the inertial frame and the law of inertia. The postulate of equivalence
of inertial frames for the formulation of the laws of Nature is assumed
to be valid for the whole of physics (special relativity principle). From Maxwell-
Lorentz electrodynamics it takes the postulate of invariance of the velocity
of light in a vacuum (light principle).
To harmonize the relativity principle with the light principle, the assumption
that an absolute time (agreeing for all inertial frames) exists, had to
be abandoned. Thus the hypothesis is abandoned that arbitrarily moved and
suitably set identical clocks function in such a way that the times shown by
two of them, which meet, agree. A specific time is assigned to each inertial
frame; the state of motion and the time of the inertial frame are defined, in
THEORY OF RELATIVITY
accordance with the stipulation of meaning, by the requirement that the
light principle should apply to it. The existence of the inertial frame thus
defined and the validity of the law of inertia with respect to it are assumed.
The time for each inertial frame is measured by identical clocks that are stationary
relative to the frame.
The laws of transformation for space coordinates and time for the transition
from one inertial frame to another, the Lorentz transformations as they
are termed, are unequivocally established by these definitions and the hypotheses
concealed in the assumption that they are free from contradiction. Their
immediate physical significance lies in the effect of the motion relative to the
used inertial frame on the form of rigid bodies (Lorentz contraction) and on
the rate of the clocks. According to the special relativity principle the laws of
Nature must be covariant relative to Lorentz transformations; the theory
thus provides a criterion for general laws of Nature. It leads in particular to
a modification of the Newtonian point motion law in which the velocity of
light in a vacuum is considered the limiting velocity, and it also leads to the
realization that energy and inertial mass are of like nature.
The special relativity theory resulted in appreciable advances. It reconciled
mechanics and electrodynamics. It reduced the number of logically independent
hypotheses regarding the latter. It enforced the need for a clarification
of the fundamental concepts in epistemological terms. It united the momentum
and energy principle, and demonstrated the like nature of mass and
energy. Yet it was not entirely satisfactory - quite apart from the quantum
problems, which all theory so far has been incapable of really solving. In
common with classical mechanics the special relativity theory favours certain
states of motion - namely those of the inertial frames - to all other states of
motion. This was actually more difficult to tolerate than the preference for
a single state of motion as in the case of the theory of light with a stationary
ether, for this imagined a real reason for the preference, i.e. the light ether.
A theory which from the outset prefers no state of motion should appear more
satisfactory. Moreover the previously mentioned vagueness in the definition
of the inertial frame or in the formulation of the law of inertia raises doubts
which obtain their decisive importance, owing to the empirical principle for
the equality of the inertial and heavy mass, in the light of the following consideration.
Let K be an inertial frame without a gravitational field, K’ a system of coordinates
accelerated uniformly relative to K. The behaviour of material
points relative to K’ is the the same as if K’ were an inertial frame in respect
1921 A.EINSTEIN
of which a homogeneous gravitational field exists. On the basis of the empirically
known properties of the gravitational field, the definition of the
inertial frame thus proves to be weak. The conclusion is obvious that any
arbitrarily moved frame of reference is equivalent to any other for the formulation
of the laws of Nature, that there are thus no physically preferred
states of motion at all in respect of regions of finite extension (general relativity
principle).
The implementation of this concept necessitates an even more profound
modification of the geometric-kinematical principles than the special relativity
theory. The Lorentz contraction, which is derived from the latter,
leads to the conclusion that with regard to a system K’ arbitrarily moved relative
to a (gravity field free) inertial frame K, the laws of Euclidian geometry
governing the position of rigid (at rest relative to K’) bodies do not apply.
Consequently the Cartesian system of coordinates also loses its significance
in terms of the stipulation of meaning. Analogous reasoning applies to time;
with reference to K’ the time can no longer meaningfully be defined by the
indication on identical clocks at rest relative to K’, nor by the law governing
the propagation of light. Generalizing, we arrive at the conclusion that gravitational
field and metric are only different manifestations of the same physical
field.
We arrive at the formal description of this field by the following consideration.
For each infinitesimal point-environment in an arbitrary gravitational
field a local frame of coordinates can be given for such a state of motion
that relative to this local frame no gravitational field exists (local inertial
frame). In terms of this inertial frame we may regard the results of the special
relativity theory as correct to a first approximation for this infinitesimally
small region. There are an infinite number of such local inertial frames at
any space-time point; they are associated by Lorentz transformations. These
latter are characterised in that they leave invariant the "distance" ds of two
infinitely adjacent point events - defined by the equation:
which distance can be measured by means of scales and clocks. For, x, y, z, t
represent coordinates and time measured with reference to a local inertial
frame.
To describe space-time regions of finite extent arbitrary point coordinates
in four dimensions are required which serve no other purpose than to pro-
Y
THEORY OF RELATIVITY
vide an unambiguous designation of the space-time points by four numbers
each, x1, x2, x3 and x4, which takes account of the continuity of this four-
dimensional manifold (Gaussian coordinates). The mathematical expression
of the general relativity principle is then, that the systems of equations expressing
the general laws of Nature are equal for all such systems of coordinates.
Since the coordinate differentials of the local inertial frame are expressed
linearly by the differentials dxof a Gaussian system of coordinates, when
v
the latter is used, for the distance ds of two events an expression of the form
= =
is obtained. The guv which arc continuous functions of xv, determine the
metric in the four-dimensional manifold where ds is defined as an (absolute)
parameter measurable by means of rigid scales and clocks. These same parameters
guv however also describe with reference to the Gaussian system of
coordinates the gravitational field which we have previously found to be
identical with the physical cause of the metric. The case as to the validity of
the special relativity theory for finite regions is characterised in that when
the system of coordinates is suitably chosen, the values of gfor finite regions
uv
are independent of x.
v
In accordance with the general theory of relativity the law of point motion
in the pure gravitational field is expressed by the equation for the geodetic
line. Actually the geodetic line is the simplest mathematically which
in the special case of constant gbecomes rectilinear. Here therefore we
uv
are confronted with the transfer of Galileo’s law of inertia to the general
theory of relativity.
In mathematical terms the search for the field equations amounts to ascertaining
the simplest generally covariant differential equations to which the
gravitational potentials gcan be subjected. By definition these equations
uv
should not contain higher derivatives of gwith respect to xthan the sec
uv v
ond, and these only linearly, which condition reveals these equations to be a
logical transfer of the Poisson field equation of the Newtonian theory of gravity
to the general theory of relativity.
The considerations mentioned led to the theory of gravity which yields
the Newtonian theory as a first approximation and furthermore it yields the
motion of the perihelion of Mercury, the deflection of light by the sun, and
the red shift of spectral lines in agreement with experience.*
* As regards the red shift, the agreement with experience is not yet completely assured,
however.
1921 A.EINSTEIN
To complete the basis of the general theory of relativity, the electromagnetic
field must still be introduced into it which, according to our present
conviction, is also the material from which we must build up the elementary
structures of matter. The Maxwellian field equations can readily
be adopted into the general theory of relativity. This is a completely unambiguous
adoption provided it is assumed that the equations contain no
differential quotients of ghigher than the first, and that in the customary
uv
Maxwellian form they apply in the local inertial frame. It is also easily possible
to supplement the gravitational field equations by electromagnetic
terms in a manner specified by the Maxwellian equations so that they contain
the gravitational effect of the electromagnetic field.
These field equations have not provided a theory of matter. To incorporate
the field generating effect of ponderable masses in the theory, matter
had therefore (as in classical physics) to be introduced into the theory in an
approximate, phenomenological representation.
And that exhausts the direct consequences of the relativity principle. I shall
turn to those problems which are related to the development which I have
traced. Already Newton recognized that the law of inertia is unsatisfactory
in a context so far unmentioned in this exposition, namely that it gives no
real cause for the special physical position of the states of motion of the inertial
frames relative to all other states of motion. It makes the observable
material bodies responsible for the gravitational behaviour of a material
point, yet indicates no material cause for the inertial behaviour of the material
point but devises the cause for it (absolute space or inertial ether). This
is not logically inadmissible although it is unsatisfactory. For this reason
E. Mach demanded a modification of the law of inertia in the sense that the
inertia should be interpreted as an acceleration resistance of the bodies against
one another and not against "space". This interpretation governs the expectation
that accelerated bodies have concordant accelerating action in the same
sense on other bodies (acceleration induction).
This interpretation is even more plausible according to general relativity
which eliminates the distinction between inertial and gravitational effects.
It amounts to stipulating that, apart from the arbitrariness governed by the
free choice of coordinates, the guv -field shall be completely determined by
the matter. Mach’s stipulation is favoured in general relativity by the circumstance
that acceleration induction in accordance with the gravitational field
equations really exists, although of such slight intensity that direct detection
by mechanical experiments is out of the question.
THEORY OF RELATIVITY
Mach’s stipulation can be accounted for in the general theory of relativity
by regarding the world in spatial terms as finite and self-contained. This hypothesis
also makes it possible to assume the mean density of matter in the
world as finite, whereas in a spatially infinite( quasi-Euclidian) world it should
disappear. It cannot, however, be concealed that to satisfy Mach’s postulate
in the manner referred to a term with no experimental basis whatsoever
must be introduced into the field equations, which term logically is in no
way determined by the other terms in the equations. For this reason this
solution of the "cosmological problem" will not be completely satisfactory
for the time being.
A second problem which at present is the subject of lively interest is the
identity between the gravitational field and the electromagnetic field. The
mind striving after unification of the theory cannot be satisfied that two
fields should exist which, by their nature, are quite independent. A mathematically
unified field theory is sought in which the gravitational field and
the electromagnetic field are interpreted only as different components or
manifestations of the same uniform field, the field equations where possible
no longer consisting of logically mutually independent summands.
The gravitational theory, considered in terms of mathematical formalism,
i.e.Riemannian geometry, should be generalized so that it includes the laws
of the electromagnetic field. Unfortunately we are unable here to base ourselves
on empirical facts as when deriving the gravitational theory (equality
of the inertial and heavy mass), but we are restricted to the criterion of mathematical
simplicity which is not free from arbitrariness. The attempt which
at present appears the most successful is that, based on the ideas of Levi-
Civita, Weyl and Eddington, to replace Riemannian metric geometry by
the more general theory of affine correlation.
The characteristic assumption of Riemannian geometry is the attribution
to two infinitely adjacent points of a "distance" ds, the square of which is a
homogeneous second order function of the coordinate differentials. It follows
from this that (apart from certain conditions of reality) Euclidian geometry
is valid in any infinitely small region. Hence to every line element
(or vector) at a point P is assigned a parallel and equal line element (or vector)
through any given infinitesimally adjacent point P’ (affine correlation).
Riemannian metric determines an affine correlation. Conversely, however,
when an affine correlation( law of infinitesimal parallel displacement) is mathematically
given, generally no Riemannian metric determination exists from
which it can be derived.
1921 A.EINSTEIN
The most important concept of Riemannian geometry, "space curvature",
on which the gravitational equations are also based, is based exclusively on
the "affine correlation". If one is given in a continuum, without first proceeding
from a metric, it constitutes a generalization of Riemannian geometry
but which still retains the most important derived parameters. By
seeking the simplest differential equations which can be obeyed by an affine
correlation there is reason to hope that a generalization of the gravitation
equations will be found which includes the laws of the electromagnetic field.
This hope has in fact been fulfilled although I do not know whether the formal
connection so derived can really be regarded as an enrichment of physics
as long as it does not yield any new physical connections. In particular a field
theory can, to my mind, only be satisfactory when it permits the elementary
electrical bodies to be represented as solutions free from singularities.
Moreover it should not be forgotten that a theory relating to the elementary
electrical structures is inseparable from the quantum theory problems.
So far also relativity theory has proved ineffectual in relation to this most
profound physical problem of the present time. Should the form of the general
equations some day, by the solution of the quantum problem, undergo
a change however profound, even if there is a complete change in the parameters
by means of which we represent the elementary process, the relativity
principle will not be relinquished and the laws previously derived therefrom
will at least retain their significance as limiting laws.
Fundamental ideas and problems
of the theory of relativity
Lecture delivered to the Nordic Assembly of Naturalists at Gothenburg*
July 11, 1923
If we consider that part of the theory of relativity which may nowadays in
a sense be regarded as bona fide scientific knowledge, we note two aspects
which have a major bearing on this theory. The whole development of the
theory turns on the question of whether there are physically preferred states
of motion in Nature (physical relativity problem). Also, concepts and distinctions
are only admissible to the extent that observable facts can be assigned
to them without ambiguity (stipulation that concepts and distinctions
should have meaning). This postulate, pertaining to epistemology, proves to
be of fundamental importance.
These two aspects become clear when applied to a special case, e.g. to classical
mechanics. Firstly we see that at any point filled with matter there exists
a preferred state of motion, namely that of the substance at the point considered.
Our problem starts however with the question whether physically
preferred states of motion exist in reference to extensive regions. From the
viewpoint of classical mechanics the answer is in the affirmative; the physically
preferred states of motion from the viewpoint of mechanics are those of
the inertial frames.
This assertion, in common with the basis of the whole of mechanics as it
generally used to be described before the relativity theory, far from meets
the above "stipulation of meaning". Motion can only be conceived as the
relative motion of bodies. In mechanics, motion relative to the system of
coordinates is implied when merely motion is referred to. Nevertheless this
interpretation does not comply with the "stipulation of meaning" if the coordinate
system is considered as something purely imaginary. If we turn our
attention to experimental physics we see that there the coordinate system is
invariably represented by a "practically rigid" body. Furthermore it is assumed
that such rigid bodies can be positioned in rest relative to one another
* The Lecture was not delivered on the occasion of the Nobel Prize award, and did
not, therefore, concern the discovery of the photoelectric effect.
THEORY OF RELATIVITY
in common with the bodies of Euclidian geometry. Insofar as we may think
of the rigid measuring body as existing as an object which can be experienced,
the "system of coordinates" concept as well as the concept of the motion of
matter relative thereto can be accepted in the sense of the "stipulation of
meaning". At the same time Euclidian geometry, by this conception, has been
adapted to the requirements of the physics of the "stipulation of meaning".
The question whether Euclidian geometry is valid becomes physically significant;
its validity is assumed in classical physics and also later in the special
theory of relativity.
In classical mechanics the inertial frame and time are best defined together
by a suitable formulation of the law of inertia: It is possible to fix the time
and assign a state of motion to the system of coordinates (inertial frame) such
that, with reference to the latter, force-free material points undergo no acceleration;
furthermore it is assumed that this time can be measured without
disagreement by identical clocks (systems which run down periodically) in
any arbitrary state of motion. There are then an infinite number of inertial
frames which are in uniform translational motion relative to each other, and
hence there is also an infinite number of mutually equivalent, physically preferred
states of motion. Time is absolute, i.e.independent of the choice of
the particular inertial frame; it is defined by more characteristics than logically
necessary, although - as implied by mechanics - this should not lead
to contradictions with experience. Note in passing that the logical weakness
of this exposition from the point of view of the stipulation of meaning is
the lack of an experimental criterion for whether a material point is force-
free or not; therefore the concept of the inertial frame remains rather problematical.
This deficiency leads to the general theory of relativity. We shall
not consider it for the moment.
The concept of the rigid body (and that of the clock) has a key bearing
on the foregoing consideration of the fundamentals of mechanics, a bearing
which there is some justification for challenging. The rigid body is only approximately
achieved in Nature, not even with desired approximation; this
concept does not therefore strictly satisfy the "stipulation of meaning". It is
also logically unjustifiable to base all physical consideration on the rigid or
solid body and then finally reconstruct that body atomically by means of
elementary physical laws which in turn have been determined by means of
the rigid measuring body. I am mentioning these deficiencies of method
because in the same sense they are also a feature of the relativity theory in
the schematic exposition which I am advocating here. Certainly it would be
1921 A.EINSTEIN
logically more correct to begin with the whole of the laws and to apply the
"stipulation of meaning" to this whole first, i.e. to put the unambiguous relation
to the world of experience last instead of already fulfilling it in an imperfect
form for an artificially isolated part, namely the space-time metric.
We are not, however, sufficiently advanced in our knowledge of Nature’s
elementary laws to adopt this more perfect method without going out of our
depth. At the close of our considerations we shall see that in the most recent
studies there is an attempt, based on ideas by Levi-Civita, Weyl, and Eddington,
to implement that logically purer method.
It more clearly follows from the above what is implied by "preferred states
of motion". They are preferred as regards the laws of Nature. States of motion
are preferred when, relative to the formulation of the laws of Nature,
coordinate systems within them are distinguished in that with respect to them
those laws assume a form preferred by simplicity. According to classical mechanics
the states of motion of the inertial frames in this sense are physically
preferred. Classical mechanics permits a distinction to be made between (absolutely)
unaccelerated and accelerated motions; it also claims that velocities
have only a relative existence (dependent on the selection of the inertial
frame), while accelerations and rotations have an absolute existence (independent
of the selection of the inertial frame). This state of affairs can be
expressed thus: According to classical mechanics "velocity relativity" exists,
but not "acceleration relativity". After these preliminary considerations we
can pass to the actual topic of our contemplations, the relativity theory, by
characterizing its development so far in terms of principles.
The special theory of relativity is an adaptation of physical principles to
Maxwell-Lorentz electrodynamics. From earlier physics it takes the assumption
that Euclidian geometry is valid for the laws governing the position of
rigid bodies, the inertial frame and the law of inertia. The postulate of equivalence
of inertial frames for the formulation of the laws of Nature is assumed
to be valid for the whole of physics (special relativity principle). From Maxwell-
Lorentz electrodynamics it takes the postulate of invariance of the velocity
of light in a vacuum (light principle).
To harmonize the relativity principle with the light principle, the assumption
that an absolute time (agreeing for all inertial frames) exists, had to
be abandoned. Thus the hypothesis is abandoned that arbitrarily moved and
suitably set identical clocks function in such a way that the times shown by
two of them, which meet, agree. A specific time is assigned to each inertial
frame; the state of motion and the time of the inertial frame are defined, in
THEORY OF RELATIVITY
accordance with the stipulation of meaning, by the requirement that the
light principle should apply to it. The existence of the inertial frame thus
defined and the validity of the law of inertia with respect to it are assumed.
The time for each inertial frame is measured by identical clocks that are stationary
relative to the frame.
The laws of transformation for space coordinates and time for the transition
from one inertial frame to another, the Lorentz transformations as they
are termed, are unequivocally established by these definitions and the hypotheses
concealed in the assumption that they are free from contradiction. Their
immediate physical significance lies in the effect of the motion relative to the
used inertial frame on the form of rigid bodies (Lorentz contraction) and on
the rate of the clocks. According to the special relativity principle the laws of
Nature must be covariant relative to Lorentz transformations; the theory
thus provides a criterion for general laws of Nature. It leads in particular to
a modification of the Newtonian point motion law in which the velocity of
light in a vacuum is considered the limiting velocity, and it also leads to the
realization that energy and inertial mass are of like nature.
The special relativity theory resulted in appreciable advances. It reconciled
mechanics and electrodynamics. It reduced the number of logically independent
hypotheses regarding the latter. It enforced the need for a clarification
of the fundamental concepts in epistemological terms. It united the momentum
and energy principle, and demonstrated the like nature of mass and
energy. Yet it was not entirely satisfactory - quite apart from the quantum
problems, which all theory so far has been incapable of really solving. In
common with classical mechanics the special relativity theory favours certain
states of motion - namely those of the inertial frames - to all other states of
motion. This was actually more difficult to tolerate than the preference for
a single state of motion as in the case of the theory of light with a stationary
ether, for this imagined a real reason for the preference, i.e. the light ether.
A theory which from the outset prefers no state of motion should appear more
satisfactory. Moreover the previously mentioned vagueness in the definition
of the inertial frame or in the formulation of the law of inertia raises doubts
which obtain their decisive importance, owing to the empirical principle for
the equality of the inertial and heavy mass, in the light of the following consideration.
Let K be an inertial frame without a gravitational field, K’ a system of coordinates
accelerated uniformly relative to K. The behaviour of material
points relative to K’ is the the same as if K’ were an inertial frame in respect
1921 A.EINSTEIN
of which a homogeneous gravitational field exists. On the basis of the empirically
known properties of the gravitational field, the definition of the
inertial frame thus proves to be weak. The conclusion is obvious that any
arbitrarily moved frame of reference is equivalent to any other for the formulation
of the laws of Nature, that there are thus no physically preferred
states of motion at all in respect of regions of finite extension (general relativity
principle).
The implementation of this concept necessitates an even more profound
modification of the geometric-kinematical principles than the special relativity
theory. The Lorentz contraction, which is derived from the latter,
leads to the conclusion that with regard to a system K’ arbitrarily moved relative
to a (gravity field free) inertial frame K, the laws of Euclidian geometry
governing the position of rigid (at rest relative to K’) bodies do not apply.
Consequently the Cartesian system of coordinates also loses its significance
in terms of the stipulation of meaning. Analogous reasoning applies to time;
with reference to K’ the time can no longer meaningfully be defined by the
indication on identical clocks at rest relative to K’, nor by the law governing
the propagation of light. Generalizing, we arrive at the conclusion that gravitational
field and metric are only different manifestations of the same physical
field.
We arrive at the formal description of this field by the following consideration.
For each infinitesimal point-environment in an arbitrary gravitational
field a local frame of coordinates can be given for such a state of motion
that relative to this local frame no gravitational field exists (local inertial
frame). In terms of this inertial frame we may regard the results of the special
relativity theory as correct to a first approximation for this infinitesimally
small region. There are an infinite number of such local inertial frames at
any space-time point; they are associated by Lorentz transformations. These
latter are characterised in that they leave invariant the "distance" ds of two
infinitely adjacent point events - defined by the equation:
which distance can be measured by means of scales and clocks. For, x, y, z, t
represent coordinates and time measured with reference to a local inertial
frame.
To describe space-time regions of finite extent arbitrary point coordinates
in four dimensions are required which serve no other purpose than to pro-
Y
THEORY OF RELATIVITY
vide an unambiguous designation of the space-time points by four numbers
each, x1, x2, x3 and x4, which takes account of the continuity of this four-
dimensional manifold (Gaussian coordinates). The mathematical expression
of the general relativity principle is then, that the systems of equations expressing
the general laws of Nature are equal for all such systems of coordinates.
Since the coordinate differentials of the local inertial frame are expressed
linearly by the differentials dxof a Gaussian system of coordinates, when
v
the latter is used, for the distance ds of two events an expression of the form
= =
is obtained. The guv which arc continuous functions of xv, determine the
metric in the four-dimensional manifold where ds is defined as an (absolute)
parameter measurable by means of rigid scales and clocks. These same parameters
guv however also describe with reference to the Gaussian system of
coordinates the gravitational field which we have previously found to be
identical with the physical cause of the metric. The case as to the validity of
the special relativity theory for finite regions is characterised in that when
the system of coordinates is suitably chosen, the values of gfor finite regions
uv
are independent of x.
v
In accordance with the general theory of relativity the law of point motion
in the pure gravitational field is expressed by the equation for the geodetic
line. Actually the geodetic line is the simplest mathematically which
in the special case of constant gbecomes rectilinear. Here therefore we
uv
are confronted with the transfer of Galileo’s law of inertia to the general
theory of relativity.
In mathematical terms the search for the field equations amounts to ascertaining
the simplest generally covariant differential equations to which the
gravitational potentials gcan be subjected. By definition these equations
uv
should not contain higher derivatives of gwith respect to xthan the sec
uv v
ond, and these only linearly, which condition reveals these equations to be a
logical transfer of the Poisson field equation of the Newtonian theory of gravity
to the general theory of relativity.
The considerations mentioned led to the theory of gravity which yields
the Newtonian theory as a first approximation and furthermore it yields the
motion of the perihelion of Mercury, the deflection of light by the sun, and
the red shift of spectral lines in agreement with experience.*
* As regards the red shift, the agreement with experience is not yet completely assured,
however.
1921 A.EINSTEIN
To complete the basis of the general theory of relativity, the electromagnetic
field must still be introduced into it which, according to our present
conviction, is also the material from which we must build up the elementary
structures of matter. The Maxwellian field equations can readily
be adopted into the general theory of relativity. This is a completely unambiguous
adoption provided it is assumed that the equations contain no
differential quotients of ghigher than the first, and that in the customary
uv
Maxwellian form they apply in the local inertial frame. It is also easily possible
to supplement the gravitational field equations by electromagnetic
terms in a manner specified by the Maxwellian equations so that they contain
the gravitational effect of the electromagnetic field.
These field equations have not provided a theory of matter. To incorporate
the field generating effect of ponderable masses in the theory, matter
had therefore (as in classical physics) to be introduced into the theory in an
approximate, phenomenological representation.
And that exhausts the direct consequences of the relativity principle. I shall
turn to those problems which are related to the development which I have
traced. Already Newton recognized that the law of inertia is unsatisfactory
in a context so far unmentioned in this exposition, namely that it gives no
real cause for the special physical position of the states of motion of the inertial
frames relative to all other states of motion. It makes the observable
material bodies responsible for the gravitational behaviour of a material
point, yet indicates no material cause for the inertial behaviour of the material
point but devises the cause for it (absolute space or inertial ether). This
is not logically inadmissible although it is unsatisfactory. For this reason
E. Mach demanded a modification of the law of inertia in the sense that the
inertia should be interpreted as an acceleration resistance of the bodies against
one another and not against "space". This interpretation governs the expectation
that accelerated bodies have concordant accelerating action in the same
sense on other bodies (acceleration induction).
This interpretation is even more plausible according to general relativity
which eliminates the distinction between inertial and gravitational effects.
It amounts to stipulating that, apart from the arbitrariness governed by the
free choice of coordinates, the guv -field shall be completely determined by
the matter. Mach’s stipulation is favoured in general relativity by the circumstance
that acceleration induction in accordance with the gravitational field
equations really exists, although of such slight intensity that direct detection
by mechanical experiments is out of the question.
THEORY OF RELATIVITY
Mach’s stipulation can be accounted for in the general theory of relativity
by regarding the world in spatial terms as finite and self-contained. This hypothesis
also makes it possible to assume the mean density of matter in the
world as finite, whereas in a spatially infinite( quasi-Euclidian) world it should
disappear. It cannot, however, be concealed that to satisfy Mach’s postulate
in the manner referred to a term with no experimental basis whatsoever
must be introduced into the field equations, which term logically is in no
way determined by the other terms in the equations. For this reason this
solution of the "cosmological problem" will not be completely satisfactory
for the time being.
A second problem which at present is the subject of lively interest is the
identity between the gravitational field and the electromagnetic field. The
mind striving after unification of the theory cannot be satisfied that two
fields should exist which, by their nature, are quite independent. A mathematically
unified field theory is sought in which the gravitational field and
the electromagnetic field are interpreted only as different components or
manifestations of the same uniform field, the field equations where possible
no longer consisting of logically mutually independent summands.
The gravitational theory, considered in terms of mathematical formalism,
i.e.Riemannian geometry, should be generalized so that it includes the laws
of the electromagnetic field. Unfortunately we are unable here to base ourselves
on empirical facts as when deriving the gravitational theory (equality
of the inertial and heavy mass), but we are restricted to the criterion of mathematical
simplicity which is not free from arbitrariness. The attempt which
at present appears the most successful is that, based on the ideas of Levi-
Civita, Weyl and Eddington, to replace Riemannian metric geometry by
the more general theory of affine correlation.
The characteristic assumption of Riemannian geometry is the attribution
to two infinitely adjacent points of a "distance" ds, the square of which is a
homogeneous second order function of the coordinate differentials. It follows
from this that (apart from certain conditions of reality) Euclidian geometry
is valid in any infinitely small region. Hence to every line element
(or vector) at a point P is assigned a parallel and equal line element (or vector)
through any given infinitesimally adjacent point P’ (affine correlation).
Riemannian metric determines an affine correlation. Conversely, however,
when an affine correlation( law of infinitesimal parallel displacement) is mathematically
given, generally no Riemannian metric determination exists from
which it can be derived.
1921 A.EINSTEIN
The most important concept of Riemannian geometry, "space curvature",
on which the gravitational equations are also based, is based exclusively on
the "affine correlation". If one is given in a continuum, without first proceeding
from a metric, it constitutes a generalization of Riemannian geometry
but which still retains the most important derived parameters. By
seeking the simplest differential equations which can be obeyed by an affine
correlation there is reason to hope that a generalization of the gravitation
equations will be found which includes the laws of the electromagnetic field.
This hope has in fact been fulfilled although I do not know whether the formal
connection so derived can really be regarded as an enrichment of physics
as long as it does not yield any new physical connections. In particular a field
theory can, to my mind, only be satisfactory when it permits the elementary
electrical bodies to be represented as solutions free from singularities.
Moreover it should not be forgotten that a theory relating to the elementary
electrical structures is inseparable from the quantum theory problems.
So far also relativity theory has proved ineffectual in relation to this most
profound physical problem of the present time. Should the form of the general
equations some day, by the solution of the quantum problem, undergo
a change however profound, even if there is a complete change in the parameters
by means of which we represent the elementary process, the relativity
principle will not be relinquished and the laws previously derived therefrom
will at least retain their significance as limiting laws.
