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necessary until we come to deal with the general theory of relativity,
treated in the second part of this book.



SPACE AND TIME IN CLASSICAL MECHANICS


The purpose of mechanics is to describe how bodies change their
position in space with "time." I should load my conscience with grave
sins against the sacred spirit of lucidity were I to formulate the
aims of mechanics in this way, without serious reflection and detailed
explanations. Let us proceed to disclose these sins.

It is not clear what is to be understood here by "position" and
"space." I stand at the window of a railway carriage which is
travelling uniformly, and drop a stone on the embankment, without
throwing it. Then, disregarding the influence of the air resistance, I
see the stone descend in a straight line. A pedestrian who observes
the misdeed from the footpath notices that the stone falls to earth in
a parabolic curve. I now ask: Do the "positions" traversed by the

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mechanics. But that a principle of such broad generality should hold
with such exactness in one domain of phenomena, and yet should be
invalid for another, is a priori not very probable.

We now proceed to the second argument, to which, moreover, we shall
return later. If the principle of relativity (in the restricted sense)
does not hold, then the Galileian co-ordinate systems K, K1, K2, etc.,
which are moving uniformly relative to each other, will not be
equivalent for the description of natural phenomena. In this case we
should be constrained to believe that natural laws are capable of
being formulated in a particularly simple manner, and of course only
on condition that, from amongst all possible Galileian co-ordinate
systems, we should have chosen one (K[0]) of a particular state of
motion as our body of reference. We should then be justified (because
of its merits for the description of natural phenomena) in calling
this system " absolutely at rest," and all other Galileian systems K "
in motion." If, for instance, our embankment were the system K[0] then
our railway carriage would be a system K, relative to which less
simple laws would hold than with respect to K[0]. This diminished
simplicity would be due to the fact that the carriage K would be in
motion (i.e."really")with respect to K[0]. In the general laws of
nature which have been formulated with reference to K, the magnitude
and direction of the velocity of the carriage would necessarily play a
part. We should expect, for instance, that the note emitted by an
organpipe placed with its axis parallel to the direction of travel
would be different from that emitted if the axis of the pipe were
placed perpendicular to this direction.

Now in virtue of its motion in an orbit round the sun, our earth is
comparable with a railway carriage travelling with a velocity of about
30 kilometres per second. If the principle of relativity were not
valid we should therefore expect that the direction of motion of the

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system K then corresponds to the embankment, and a co-ordinate system
K' to the train. An event, wherever it may have taken place, would be
fixed in space with respect to K by the three perpendiculars x, y, z
on the co-ordinate planes, and with regard to time by a time value t.
Relative to K1, the same event would be fixed in respect of space and
time by corresponding values x1, y1, z1, t1, which of course are not
identical with x, y, z, t. It has already been set forth in detail how
these magnitudes are to be regarded as results of physical
measurements.

Obviously our problem can be exactly formulated in the following
manner. What are the values x1, y1, z1, t1, of an event with respect
to K1, when the magnitudes x, y, z, t, of the same event with respect
to K are given ? The relations must be so chosen that the law of the
transmission of light in vacuo is satisfied for one and the same ray
of light (and of course for every ray) with respect to K and K1. For
the relative orientation in space of the co-ordinate systems indicated
in the diagram ([7]Fig. 2), this problem is solved by means of the
equations :

                         eq. 1: file eq01.gif

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man walking along the carriage, or of the moving point in the present
section. If we denote the velocity of the light relative to the tube
by W, then this is given by the equation (A) or (B), according as the
Galilei transformation or the Lorentz transformation corresponds to
the facts. Experiment * decides in favour of equation (B) derived
from the theory of relativity, and the agreement is, indeed, very
exact. According to recent and most excellent measurements by Zeeman,
the influence of the velocity of flow v on the propagation of light is
represented by formula (B) to within one per cent.

Nevertheless we must now draw attention to the fact that a theory of
this phenomenon was given by H. A. Lorentz long before the statement
of the theory of relativity. This theory was of a purely
electrodynamical nature, and was obtained by the use of particular
hypotheses as to the electromagnetic structure of matter. This
circumstance, however, does not in the least diminish the
conclusiveness of the experiment as a crucial test in favour of the
theory of relativity, for the electrodynamics of Maxwell-Lorentz, on
which the original theory was based, in no way opposes the theory of

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a question of detecting or of describing the motion involved, it is in
principle immaterial to what reference-body we refer the motion. As
already mentioned, this is self-evident, but it must not be confused
with the much more comprehensive statement called "the principle of
relativity," which we have taken as the basis of our investigations.

The principle we have made use of not only maintains that we may
equally well choose the carriage or the embankment as our
reference-body for the description of any event (for this, too, is
self-evident). Our principle rather asserts what follows : If we
formulate the general laws of nature as they are obtained from
experience, by making use of

(a) the embankment as reference-body,
(b) the railway carriage as reference-body,

then these general laws of nature (e.g. the laws of mechanics or the
law of the propagation of light in vacuo) have exactly the same form
in both cases. This can also be expressed as follows : For the
physical description of natural processes, neither of the reference
bodies K, K1 is unique (lit. " specially marked out ") as compared
with the other. Unlike the first, this latter statement need not of
necessity hold a priori; it is not contained in the conceptions of "
motion" and " reference-body " and derivable from them; only
experience can decide as to its correctness or incorrectness.

Up to the present, however, we have by no means maintained the
equivalence of all bodies of reference K in connection with the
formulation of natural laws. Our course was more on the following
Iines. In the first place, we started out from the assumption that
there exists a reference-body K, whose condition of motion is such
that the Galileian law holds with respect to it : A particle left to
itself and sufficiently far removed from all other particles moves
uniformly in a straight line. With reference to K (Galileian
reference-body) the laws of nature were to be as simple as possible.
But in addition to K, all bodies of reference K1 should be given
preference in this sense, and they should be exactly equivalent to K
for the formulation of natural laws, provided that they are in a state
of uniform rectilinear and non-rotary motion with respect to K ; all
these bodies of reference are to be regarded as Galileian
reference-bodies. The validity of the principle of relativity was
assumed only for these reference-bodies, but not for others (e.g.
those possessing motion of a different kind). In this sense we speak
of the special principle of relativity, or special theory of
relativity.

In contrast to this we wish to understand by the "general principle of
relativity" the following statement : All bodies of reference K, K1,
etc., are equivalent for the description of natural phenomena
(formulation of the general laws of nature), whatever may be their
state of motion. But before proceeding farther, it ought to be pointed
out that this formulation must be replaced later by a more abstract
one, for reasons which will become evident at a later stage.

Since the introduction of the special principle of relativity has been
justified, every intellect which strives after generalisation must
feel the temptation to venture the step towards the general principle
of relativity. But a simple and apparently quite reliable
consideration seems to suggest that, for the present at any rate,
there is little hope of success in such an attempt; Let us imagine
ourselves transferred to our old friend the railway carriage, which is
travelling at a uniform rate. As long as it is moving unifromly, the

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general principle of relatvity. But in what follows we shall soon see
that this conclusion cannot be maintained.



THE GRAVITATIONAL FIELD


"If we pick up a stone and then let it go, why does it fall to the
ground ?" The usual answer to this question is: "Because it is
attracted by the earth." Modern physics formulates the answer rather
differently for the following reason. As a result of the more careful
study of electromagnetic phenomena, we have come to regard action at a
distance as a process impossible without the intervention of some
intermediary medium. If, for instance, a magnet attracts a piece of
iron, we cannot be content to regard this as meaning that the magnet
acts directly on the iron through the intermediate empty space, but we
are constrained to imagine -- after the manner of Faraday -- that the
magnet always calls into being something physically real in the space
around it, that something being what we call a "magnetic field." In
its turn this magnetic field operates on the piece of iron, so that

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rest with respect to the original reference-body K. But the observer
on the disc may regard his disc as a reference-body which is " at rest
" ; on the basis of the general principle of relativity he is
justified in doing this. The force acting on himself, and in fact on
all other bodies which are at rest relative to the disc, he regards as
the effect of a gravitational field. Nevertheless, the
space-distribution of this gravitational field is of a kind that would
not be possible on Newton's theory of gravitation.* But since the
observer believes in the general theory of relativity, this does not
disturb him; he is quite in the right when he believes that a general
law of gravitation can be formulated- a law which not only explains
the motion of the stars correctly, but also the field of force
experienced by himself.

The observer performs experiments on his circular disc with clocks and
measuring-rods. In doing so, it is his intention to arrive at exact
definitions for the signification of time- and space-data with
reference to the circular disc K1, these definitions being based on
his observations. What will be his experience in this enterprise ?

To start with, he places one of two identically constructed clocks at

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So far, these considerations hold for a continuum of two dimensions.
But the Gaussian method can be applied also to a continuum of three,
four or more dimensions. If, for instance, a continuum of four
dimensions be supposed available, we may represent it in the following
way. With every point of the continuum, we associate arbitrarily four
numbers, x[1], x[2], x[3], x[4], which are known as " co-ordinates."
Adjacent points correspond to adjacent values of the coordinates. If a
distance ds is associated with the adjacent points P and P1, this
distance being measurable and well defined from a physical point of
view, then the following formula holds:

ds2 = g[11]dx[1]^2 + 2g[12]dx[1]dx[2] . . . . g[44]dx[4]^2,

where the magnitudes g[11], etc., have values which vary with the
position in the continuum. Only when the continuum is a Euclidean one
is it possible to associate the co-ordinates x[1] . . x[4]. with the
points of the continuum so that we have simply

ds2 = dx[1]^2 + dx[2]^2 + dx[3]^2 + dx[4]^2.

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"distance," small parts of the continuum under consideration behave
more nearly like a Euclidean system, the smaller the part of the
continuum under our notice.



THE SPACE-TIME CONTINUUM OF THE SPEICAL THEORY OF RELATIVITY CONSIDERED AS A
EUCLIDEAN CONTINUUM


We are now in a position to formulate more exactly the idea of
Minkowski, which was only vaguely indicated in Section 17. In
accordance with the special theory of relativity, certain co-ordinate
systems are given preference for the description of the
four-dimensional, space-time continuum. We called these " Galileian
co-ordinate systems." For these systems, the four co-ordinates x, y,
z, t, which determine an event or -- in other words, a point of the
four-dimensional continuum -- are defined physically in a simple
manner, as set forth in detail in the first part of this book. For the
transition from one Galileian system to another, which is moving
uniformly with reference to the first, the equations of the Lorentz

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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.



EXACT FORMULATION OF THE GENERAL PRINCIPLE OF RELATIVITY


We are now in a position to replace the pro. visional formulation of
the general principle of relativity given in Section 18 by an exact
formulation. The form there used, "All bodies of reference K, K1,
etc., are equivalent for the description of natural phenomena
(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 co-ordinate 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
co-ordinate 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
equations 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, t, of a
(Galileian) reference-body K by the space-time variables x1, y1, z1,

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appropriately be termed a "reference-mollusc", is in the main
equivalent to a Gaussian four-dimensional co-ordinate system chosen
arbitrarily. That which gives the "mollusc" a certain
comprehensibility as compared with the Gauss co-ordinate system is the
(really unjustified) formal retention of the separate existence of the
space co-ordinates as opposed to the time co-ordinate. Every point on
the mollusc is treated as a space-point, and every material point
which is at rest relatively to it as at rest, so long as the mollusc
is considered as reference-body. The general principle of relativity
requires that all these molluscs 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 mollusc.

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.



THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL

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Hereupon we introduce a hypothesis: that the influence of the
gravitational field on measuringrods, 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 Galfleian special care, simply by means of a
transformation of co-ordinates.

The next step is to investigate the space-time behaviour of the
gravitational field G, which was derived from the Galileian special
case simply by transformation of the coordinates. This behaviour is
formulated in a law, which is always valid, no matter how the
reference-body (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 generalisation of the law as found above. This can
be obtained without caprice, however, by taking into consideration the
following demands:

(a) The required generalisation must likewise satisfy the general

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transmitted according to the equation

                                x = ct

or

                 x - ct = 0     .     .     .    (1).

Since the same light-signal has to be transmitted relative to K1 with
the velocity c, the propagation relative to the system K1 will be
represented by the analogous formula

                x' - ct' = O     .     .     .    (2)

Those space-time points (events) which satisfy (x) must also satisfy
(2). Obviously this will be the case when the relation

          (x' - ct') = l (x - ct)     .     .     .    (3).

is fulfilled in general, where l indicates a constant ; for, according
to (3), the disappearance of (x - ct) involves the disappearance of

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                        eq. 40: file eq40.gif

or, if we square this equation, according to the equation

          x2 + y2 + z2 = c^2t2 = 0    .     .     .    (10).

It is required by the law of propagation of light, in conjunction with
the postulate of relativity, that the transmission of the signal in
question should take place -- as judged from K1 -- in accordance with
the corresponding formula

                               r' = ct'

or,

       x'2 + y'2 + z'2 - c^2t'2 = 0    .     .     .    (10a).

In order that equation (10a) may be a consequence of equation (10), we
must have