Section twenty seven, the space time continuum of the general theory of relativity is
not a Euclidean continuum. In the first part of this book we were able
to make use of space time coordinates, which allowed of a simple and direct
physical interpretation, and which, according to section twenty six, can be regarded
as four dimensional Cartesian coordinates. This was possible on the basis of the law
of the constancy of the velocity of light, but according to section twenty one,
the general theory of relativity cannot retain this law. On the contrary,
we arrived at the result that, according to this latter theory, the velocity
of light must always depend on the coordinates. When a gravitational field is present.
In connection with a specific illustration in section twenty three, we found that
the presence of a gravitational field in valid. It's the definition of the coordinates
and the time which led us to our objective in the special theory of relativity.
In view of the results of these considerations, we are led to the
conviction that, according to the general principle of relativity, the space time continuum
cannot be regarded as a Euclidean one, but that here we have the general
case corresponding to the marble slab with local variations of temperature, and with which
we made acquaintance as an example of a two dimensional continuum. Just as it
was there impossible to construct a Cartesian coordinate system from equal rods, so here
it is impossible to build up a system or reference body from rigid bodies and
clocks, which shall be of such a nature that measuring rods and clocks arranged
rigidly with respect to one another shall indicate position and time directly. Such was
the essence of the difficulty with which we were confronted in section twenty three,
But the considerations of section twenty five and twenty six show us the way to
surmount this difficulty. We refer the four dimensional space time continuum in an arbitrary
manner to Gauss coordinates. We assign to every point of the continuum or event
four numbers x sub one, x sub two, x sub three, and
x sub four coordinates, which have not the least direct physical significance, but
only serve the purpose of numbering the points of the continuum in a definite but
arbitrary manner. This arrangement does not even need to be of such a kind,
that we must regard x sub one, x sub two, and x
sub three as space coordinates, and x sub four as a time coordinate.
The reader may think that such a description of the world would be quite inadequate.
What does it mean to assign to an event the particular coordinates x sub
one, x sub two, x sub three, and x sub four,
if in themselves these coordinates have no significance. More careful consideration shows, however,
that this anxiety is unfounded. Let us consider, for instance, a
material point with any kind of motion. If this point had only a momentary
existence without duration, then it would be described in space time by a single
system of values x sub one, x sub two, x sub three,
and x sub four. Thus, its permanent existence must be characterized by an
infinitely large number of such systems of values, the coordinate values of which are
so close together as to give continuity corresponding to the material point. We thus
have a unidimensional line in the four dimensional continuum. In the same way,
any such lines in our continuum correspond to many points in motion. The only
statements having regard to these points which can claim a physical existence are, in
reality, the statements about their encounters in our mathematical treatment. Such an encounter
is expressed in the fact that the two lines which represent the motions of the
points in question have a particular system of coordinate values x sub one, x
sub two, x sub three, and x sub four in common. After
mature consideration, the reader will doubtless admit that, in reality, such encounters
constitute the only actual evidence of a time space nature with which we meet in
physical statements. When we were describing the motion of a material point relative to
a body of reference, we stated nothing more than the encounters of this point
with particular points of the reference body. We can also determine the corresponding values
of the time by the observation of encounters of the body with clocks, in
conjunction with the observation of the encounter of the hands of clocks with particular points
on the dials. It is just the same in the case of space measurements
by means of measuring rods. As a little consideration will show the following statements
hauled. Generally, every physical description resolves itself into a number of statements,
each of which refers to the space time coincidence of two events A and B.
In terms of Gaussian coordinates. Every such statement is expressed by the agreement
of therefore coordinates x sub one, x sub two, x sub three,
and x sub four. Thus, in reality, the description of the time
space continuum by means of Gaus coordinates completely replaces the description with the aid of
a body of reference, without suffering from the defects of the latter mode of
description. It is not tied down to the Euclidean character of the continuum,
which has to be represented section twenty eight Exact formulation of the General principle of
relativity. We are now in a position to replace the provisional formulation of the
general principle of relativity given in section eighteen by an exact formulation the form they
are used quote all bodies of reference K, K, prime, et cetera
are equivalent for the description of natural phenom or formulation of the general laws of
nature. Whatever may be their state of motion cannot be maintained because the use
of rigid reference bodies in the sense of the method followed in the special theory
of relativity is in general not possible in space time description. The Gauss coordinate
system has to take the place of the body of reference. The following statement
corresponds to the fundamental idea of the general principle of relativity. All Gaussian coordinate
systems are essentially equivalent for the formulation of the general laws of nature. We
can state this general principle of relativity in still another form, which renders it
yet more clearly intelligible than it is when in the form of the natural extension
of the special principle of relativity. According to the special theory of relativity,
the equation which express the general laws of nature pass over into equations of the
same form when, by making use of the Lorentz transformation, we replace the
space time variables x, y, z, and t of a Galilean reference
body k by the space time variables x prime, y prime, z prime,
and t prime of a new reference body k prime. According to the
general theory of relativity, on the other hand, by application of arbitrary substitutions
of the Gauss variables x sub one, x sub two, x sub three,
and x sub four, the equations must pass over into equations of the
same form for every transformation. Not only the Lorentz transformation corresponds to the transition
of one Gaus coordinate system into another if we desire to adhere to our old
time three dimensional view of things, then we can characterize the development which is
being undergone by the fundamental idea of the general theory of relativity as follows.
The special theory of relativity has reference to Galilean domains, ie to those in
which no gravitational field exists. In this connection, a Galilean reference body serves
as body of reference, i e. A rigid body, the state of
motion of which is so chosen that the Galilean law of the uniform rectilinear motion
of isolated material points holds relatively to it. Certain considerations suggest that we should
refer the same Galileyan domains to non Galileyan reference bodies. Also, a gravitational
field of a special kind is then present with respect to these bodies see f
Sections twenty and twenty three. In gravitational fields there are no such things as
rigid bodies with Euclidean properties. Thus the fictitious rigid body of reference is of
no avail in the general theory of relativity. The motion of clocks is also
influenced by gravitational fields, and in such a way that a physical definition of
time which is made directly with the aid of clocks, has by no means
the same degree of plausibility as in the special theory of relativity. For this
reason, non rigid reference bodies are used, which are as a whole not
only moving in any way whatsoever, but which also suffer alterations in form AdLib
during their motion. Clocks for which the law of motion is of any kind,
however irregular, serve for the definition of time. We have to imagine
each of these clocks fixed at a point on the non rigid reference body.
These clocks satisfy only the one condition that the readings which are observed simultaneously on
adjacent clocks in space differ from each other by an indefinitely small amount. This
non rigid reference body, which might appropriately be termed a reference mollusc, is
in the main equivalent to a Gaussian four dimensional coordinate system chosen arbitrarily. That
which gives the mollusc a certain comprehensibility as compared with the Gause coordinate system is
the really unjustified formal retention of the separate existence of the space coordinates as opposed
to the time coordinate. Every point on the mollusc is treated as a space
point and every material point which is at rest relatively to it is at rest
so long as the mollusc is considered as reference body. The or principle of
relativity requires that all these molarchs can be used as reference bodies with equal right
and equal success in the formulation of the general laws of nature. The laws
themselves must be quite independent of the choice of mollusk. The great power possessed
by the general principle of relativity lies in the comprehensive limitation which is imposed on
the laws of nature in consequence of what we have seen above Section twenty nine
the solution of the problem of gravitation on the basis of the general principle of
relativity. If the reader has followed all our previous considerations, he will have
no further difficulty in understanding the methods leading to the solution of the problem of
gravitation. We start off on a consideration of a Galilean domain IE domain in
which there is no gravitational field relative to the Galileyan reference body K. The
behavior of measuring rods and clocks with reference to K is known from the special
theory of relativity. Likewise, the behavior of isolated material points the latter move
uniformly and in straight lines. Now, let us refer this domain to a
random Gaus coordinate system or to a mollusk as reference body k prime. Then
with respect to k prime, there is a gravitational field G of a particular
kind. We learn the behavior of measuring rods and clocks, and also of
freely moving material points with reference to k prime simply by mathematical transformation. We
interpret this behavior as the behavior of measuring rods, clocks, and material points
under the influence of the gravitational field G. Hereupon we introduce a hypothesis that
the influence of the gravitational field on measuring rods, clocks, and freely moving
material points continues to take place according to the same laws, even in the
case where the prevailing gravitational field is not derivable from the Galilean special case simply
by means of a transformation of coordinates. The next step is to investigate the
space time behavior of the gravitational field G, which was derived from the Galilean
special case simply by transformation of the coordinates. This behavior is formulated in a
law which is always valid no matter how the reference body or mollusc used in
the description may be chosen. This law is not yet the general law of
the gravitational field, since the gravitational field under consideration is of a special kind.
In order to find out the general law of field of gravitation, we
still require to obtain a generalization of the law as found above. This can
be obtained without caprice, however, by taking into consideration the following demands.
A. The required generalization must likewise satisfy the general postulate of relativity. B.
If there is any matter in the domain under consideration, only its inertial
mass, and thus, according to section fifteen only its energy is of importance
for its effect in exciting a field. C. Gravitational field and matter together
must satisfy the law of the conservation of energy and of impulse. Finally,
the general principle of relativity permits us to determine the influence of the gravitational fear
field on the course of all those processes which take place according to known laws
when a gravitational field is absent, ie which have already been fitted into the
frame of the special theory of relativity. In this connection, we proceed in
principle according to the method which has already been explained for measuring rods, clocks,
and freely moving material points. The theory of gravitation, derived in this
way from the general postulate of relativity excels not only in its beauty, nor
in removing the defect attaching to classical mechanics, which was brought to light in
section twenty one, nor in interpreting the empirical law of the equality of inertial
and gravitational mass. But it has also already explained a result of observation in
astronomy, against which classical mechanics is powerless. If we confine the application of
the theory to the case where the gravitational fields can be regarded as being weak,
and in which all masses move with respect to the coordinate system with velocities
which are small compared with the velocity of light, we then obtain as a
first approximation the Newtonian theory. Thus, the latter theory is obtained here without
any particular assumption, whereas Newton had to introduce the hypothesis that the force of
attraction between mutually attracting material points is inversely proportional to the square of the distance
between them. If we increase the accuracy of the calculation. Deviations from the
theory of Newton make their appearance practically, all of which must nevertheless escape the
test of observation owing to their smallness. We must draw attention here to one
of these deviations. According to Newton's theory, a planet moves around the Sun
in an ellipse, which would permanently maintain its position with respect to the fixed
stars if we could disregard the motion of the fixed stars themselves and the action
of the other planets under consideration. Thus, if we correct the observed motion
of the planets for these two influences, and if Newton's theory be strictly correct,
we ought to obtain for the orbit of the planet and ellipse which is
fixed with reference to the fixed stars. This deduction, which can be tested
with great accuracy, has been confirmed for all the planets save one, with
the precision that is capable of being obtained by the delicacy of observation attainable at
the present time. The sole exception is Mercury, the planet which lies nearest
the Sun. Since the time of la Verier, it has been known that
the eclipse corresponding to the orbit of Mercury after it has been corected for the
influences mentioned above, is not stationary with respect to the fixed stars, but
that it rotates exceedingly slowly in the plane of the orbit and in the sense
of the orbital motion. The value obtained for this rotary movement of the orbital
ellipse was forty three seconds of arc per century, an amount ensured to be
correct to within a few seconds of arc. This effect can be explained by
means of classical mechanics only on the assumption of hypotheses which have little probability and
which were devised solely for this purpose. On the basis of the general theory
of relativity, it is found that the ellipse of every planet round the Sun
must necessarily rotate in the manner indicated above. That for all the planets,
with the exception of Mercury, this rotation is too small to be detected with
the delicacy of observation possible at the present time, but that in the case
of Mercury it must amount to forty three seconds of arc per century, a
result which is strictly an agreement with observation. Apart from this one, it
has hitherto been possible to make only two deductions from the theory which admit of
being tested by observation, to wit, the curvature of light rays by the
gravitational field of the Sun first observed by Eddington and others in nineteen nineteen,
and a displacement of the spectral lines of light reaching us from large stars as
compared with the corresponding lines for light produced in an analogous manner terrestrially i e.
By the same kind of atom, established by Adams in nineteen twenty four.
These two deductions from the theory have both been confirmed end of Section twenty nine.