WEBVTT

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Section twenty seven, the space time
continuum of the general theory of relativity is

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not a Euclidean continuum. In the
first part of this book we were able

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to make use of space time coordinates, which allowed of a simple and direct

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physical interpretation, and which, according
to section twenty six, can be regarded

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as four dimensional Cartesian coordinates. This
was possible on the basis of the law

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of the constancy of the velocity of
light, but according to section twenty one,

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the general theory of relativity cannot retain
this law. On the contrary,

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we arrived at the result that,
according to this latter theory, the velocity

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of light must always depend on the
coordinates. When a gravitational field is present.

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In connection with a specific illustration in
section twenty three, we found that

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the presence of a gravitational field in
valid. It's the definition of the coordinates

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and the time which led us to
our objective in the special theory of relativity.

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In view of the results of these
considerations, we are led to the

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conviction that, according to the general
principle of relativity, the space time continuum

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cannot be regarded as a Euclidean one, but that here we have the general

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case corresponding to the marble slab with
local variations of temperature, and with which

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we made acquaintance as an example of
a two dimensional continuum. Just as it

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was there impossible to construct a Cartesian
coordinate system from equal rods, so here

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it is impossible to build up a
system or reference body from rigid bodies and

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clocks, which shall be of such
a nature that measuring rods and clocks arranged

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rigidly with respect to one another shall
indicate position and time directly. Such was

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the essence of the difficulty with which
we were confronted in section twenty three,

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But the considerations of section twenty five
and twenty six show us the way to

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surmount this difficulty. We refer the
four dimensional space time continuum in an arbitrary

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manner to Gauss coordinates. We assign
to every point of the continuum or event

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four numbers x sub one, x
sub two, x sub three, and

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x sub four coordinates, which have
not the least direct physical significance, but

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only serve the purpose of numbering the
points of the continuum in a definite but

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arbitrary manner. This arrangement does not
even need to be of such a kind,

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that we must regard x sub one, x sub two, and x

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sub three as space coordinates, and
x sub four as a time coordinate.

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The reader may think that such a
description of the world would be quite inadequate.

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What does it mean to assign to
an event the particular coordinates x sub

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one, x sub two, x
sub three, and x sub four,

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if in themselves these coordinates have no
significance. More careful consideration shows, however,

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that this anxiety is unfounded. Let
us consider, for instance, a

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material point with any kind of motion. If this point had only a momentary

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existence without duration, then it would
be described in space time by a single

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system of values x sub one,
x sub two, x sub three,

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and x sub four. Thus,
its permanent existence must be characterized by an

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infinitely large number of such systems of
values, the coordinate values of which are

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so close together as to give continuity
corresponding to the material point. We thus

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have a unidimensional line in the four
dimensional continuum. In the same way,

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any such lines in our continuum correspond
to many points in motion. The only

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statements having regard to these points which
can claim a physical existence are, in

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reality, the statements about their encounters
in our mathematical treatment. Such an encounter

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is expressed in the fact that the
two lines which represent the motions of the

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points in question have a particular system
of coordinate values x sub one, x

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sub two, x sub three,
and x sub four in common. After

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mature consideration, the reader will doubtless
admit that, in reality, such encounters

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constitute the only actual evidence of a
time space nature with which we meet in

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physical statements. When we were describing
the motion of a material point relative to

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a body of reference, we stated
nothing more than the encounters of this point

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with particular points of the reference body. We can also determine the corresponding values

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of the time by the observation of
encounters of the body with clocks, in

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conjunction with the observation of the encounter
of the hands of clocks with particular points

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on the dials. It is just
the same in the case of space measurements

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by means of measuring rods. As
a little consideration will show the following statements

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hauled. Generally, every physical description
resolves itself into a number of statements,

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each of which refers to the space
time coincidence of two events A and B.

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In terms of Gaussian coordinates. Every
such statement is expressed by the agreement

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of therefore coordinates x sub one,
x sub two, x sub three,

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and x sub four. Thus,
in reality, the description of the time

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space continuum by means of Gaus coordinates
completely replaces the description with the aid of

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a body of reference, without suffering
from the defects of the latter mode of

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description. It is not tied down
to the Euclidean character of the continuum,

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which has to be represented section twenty
eight Exact formulation of the General principle of

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relativity. We are now in a
position to replace the provisional formulation of the

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general principle of relativity given in section
eighteen by an exact formulation the form they

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are used quote all bodies of reference
K, K, prime, et cetera

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are equivalent for the description of natural
phenom or formulation of the general laws of

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nature. Whatever may be their state
of motion cannot be maintained because the use

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of rigid reference bodies in the sense
of the method followed in the special theory

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of relativity is in general not possible
in space time description. The Gauss coordinate

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system has to take the place of
the body of reference. The following statement

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corresponds to the fundamental idea of the
general principle of relativity. All Gaussian coordinate

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systems are essentially equivalent for the formulation
of the general laws of nature. We

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can state this general principle of relativity
in still another form, which renders it

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yet more clearly intelligible than it is
when in the form of the natural extension

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of the special principle of relativity.
According to the special theory of relativity,

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the equation which express the general laws
of nature pass over into equations of the

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same form when, by making use
of the Lorentz transformation, we replace the

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space time variables x, y,
z, and t of a Galilean reference

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body k by the space time variables
x prime, y prime, z prime,

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and t prime of a new reference
body k prime. According to the

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general theory of relativity, on the
other hand, by application of arbitrary substitutions

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of the Gauss variables x sub one, x sub two, x sub three,

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and x sub four, the equations
must pass over into equations of the

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same form for every transformation. Not
only the Lorentz transformation corresponds to the transition

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of one Gaus coordinate system into another
if we desire to adhere to our old

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time three dimensional view of things,
then we can characterize the development which is

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being undergone by the fundamental idea of
the general theory of relativity as follows.

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The special theory of relativity has reference
to Galilean domains, ie to those in

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which no gravitational field exists. In
this connection, a Galilean reference body serves

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as body of reference, i e. A rigid body, the state of

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motion of which is so chosen that
the Galilean law of the uniform rectilinear motion

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of isolated material points holds relatively to
it. Certain considerations suggest that we should

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refer the same Galileyan domains to non
Galileyan reference bodies. Also, a gravitational

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field of a special kind is then
present with respect to these bodies see f

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Sections twenty and twenty three. In
gravitational fields there are no such things as

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rigid bodies with Euclidean properties. Thus
the fictitious rigid body of reference is of

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no avail in the general theory of
relativity. The motion of clocks is also

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influenced by gravitational fields, and in
such a way that a physical definition of

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time which is made directly with the
aid of clocks, has by no means

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the same degree of plausibility as in
the special theory of relativity. For this

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reason, non rigid reference bodies are
used, which are as a whole not

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only moving in any way whatsoever,
but which also suffer alterations in form AdLib

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during their motion. Clocks for which
the law of motion is of any kind,

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however irregular, serve for the definition
of time. We have to imagine

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each of these clocks fixed at a
point on the non rigid reference body.

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These clocks satisfy only the one condition
that the readings which are observed simultaneously on

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adjacent clocks in space differ from each
other by an indefinitely small amount. This

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non rigid reference body, which might
appropriately be termed a reference mollusc, is

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in the main equivalent to a Gaussian
four dimensional coordinate system chosen arbitrarily. That

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which gives the mollusc a certain comprehensibility
as compared with the Gause coordinate system is

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the really unjustified formal retention of the
separate existence of the space coordinates as opposed

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to the time coordinate. Every point
on the mollusc is treated as a space

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point and every material point which is
at rest relatively to it is at rest

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so long as the mollusc is considered
as reference body. The or principle of

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relativity requires that all these molarchs can
be used as reference bodies with equal right

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and equal success in the formulation of
the general laws of nature. The laws

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themselves must be quite independent of the
choice of mollusk. The great power possessed

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by the general principle of relativity lies
in the comprehensive limitation which is imposed on

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the laws of nature in consequence of
what we have seen above Section twenty nine

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the solution of the problem of gravitation
on the basis of the general principle of

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relativity. If the reader has followed
all our previous considerations, he will have

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no further difficulty in understanding the methods
leading to the solution of the problem of

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gravitation. We start off on a
consideration of a Galilean domain IE domain in

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which there is no gravitational field relative
to the Galileyan reference body K. The

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behavior of measuring rods and clocks with
reference to K is known from the special

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theory of relativity. Likewise, the
behavior of isolated material points the latter move

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uniformly and in straight lines. Now, let us refer this domain to a

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random Gaus coordinate system or to a
mollusk as reference body k prime. Then

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with respect to k prime, there
is a gravitational field G of a particular

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kind. We learn the behavior of
measuring rods and clocks, and also of

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freely moving material points with reference to
k prime simply by mathematical transformation. We

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interpret this behavior as the behavior of
measuring rods, clocks, and material points

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under the influence of the gravitational field
G. Hereupon we introduce a hypothesis that

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the influence of the gravitational field on
measuring rods, clocks, and freely moving

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material points continues to take place according
to the same laws, even in the

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case where the prevailing gravitational field is
not derivable from the Galilean special case simply

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by means of a transformation of coordinates. The next step is to investigate the

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space time behavior of the gravitational field
G, which was derived from the Galilean

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special case simply by transformation of the
coordinates. This behavior is formulated in a

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law which is always valid no matter
how the reference body or mollusc used in

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the description may be chosen. This
law is not yet the general law of

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the gravitational field, since the gravitational
field under consideration is of a special kind.

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In order to find out the general
law of field of gravitation, we

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still require to obtain a generalization of
the law as found above. This can

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be obtained without caprice, however,
by taking into consideration the following demands.

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A. The required generalization must likewise
satisfy the general postulate of relativity. B.

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If there is any matter in the
domain under consideration, only its inertial

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mass, and thus, according to
section fifteen only its energy is of importance

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for its effect in exciting a field. C. Gravitational field and matter together

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must satisfy the law of the conservation
of energy and of impulse. Finally,

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the general principle of relativity permits us
to determine the influence of the gravitational fear

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field on the course of all those
processes which take place according to known laws

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when a gravitational field is absent,
ie which have already been fitted into the

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frame of the special theory of relativity. In this connection, we proceed in

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principle according to the method which has
already been explained for measuring rods, clocks,

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and freely moving material points. The
theory of gravitation, derived in this

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way from the general postulate of relativity
excels not only in its beauty, nor

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in removing the defect attaching to classical
mechanics, which was brought to light in

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section twenty one, nor in interpreting
the empirical law of the equality of inertial

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and gravitational mass. But it has
also already explained a result of observation in

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astronomy, against which classical mechanics is
powerless. If we confine the application of

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the theory to the case where the
gravitational fields can be regarded as being weak,

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and in which all masses move with
respect to the coordinate system with velocities

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which are small compared with the velocity
of light, we then obtain as a

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first approximation the Newtonian theory. Thus, the latter theory is obtained here without

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any particular assumption, whereas Newton had
to introduce the hypothesis that the force of

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attraction between mutually attracting material points is
inversely proportional to the square of the distance

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between them. If we increase the
accuracy of the calculation. Deviations from the

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theory of Newton make their appearance practically, all of which must nevertheless escape the

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test of observation owing to their smallness. We must draw attention here to one

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of these deviations. According to Newton's
theory, a planet moves around the Sun

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in an ellipse, which would permanently
maintain its position with respect to the fixed

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stars if we could disregard the motion
of the fixed stars themselves and the action

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of the other planets under consideration.
Thus, if we correct the observed motion

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of the planets for these two influences, and if Newton's theory be strictly correct,

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we ought to obtain for the orbit
of the planet and ellipse which is

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fixed with reference to the fixed stars. This deduction, which can be tested

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with great accuracy, has been confirmed
for all the planets save one, with

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the precision that is capable of being
obtained by the delicacy of observation attainable at

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the present time. The sole exception
is Mercury, the planet which lies nearest

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the Sun. Since the time of
la Verier, it has been known that

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the eclipse corresponding to the orbit of
Mercury after it has been corected for the

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influences mentioned above, is not stationary
with respect to the fixed stars, but

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that it rotates exceedingly slowly in the
plane of the orbit and in the sense

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of the orbital motion. The value
obtained for this rotary movement of the orbital

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ellipse was forty three seconds of arc
per century, an amount ensured to be

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correct to within a few seconds of
arc. This effect can be explained by

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means of classical mechanics only on the
assumption of hypotheses which have little probability and

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which were devised solely for this purpose. On the basis of the general theory

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of relativity, it is found that
the ellipse of every planet round the Sun

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must necessarily rotate in the manner indicated
above. That for all the planets,

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with the exception of Mercury, this
rotation is too small to be detected with

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the delicacy of observation possible at the
present time, but that in the case

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of Mercury it must amount to forty
three seconds of arc per century, a

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result which is strictly an agreement with
observation. Apart from this one, it

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has hitherto been possible to make only
two deductions from the theory which admit of

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being tested by observation, to wit, the curvature of light rays by the

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gravitational field of the Sun first observed
by Eddington and others in nineteen nineteen,

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and a displacement of the spectral lines
of light reaching us from large stars as

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compared with the corresponding lines for light
produced in an analogous manner terrestrially i e.

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By the same kind of atom,
established by Adams in nineteen twenty four.

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These two deductions from the theory have
both been confirmed end of Section twenty nine.