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A priori it is by no means certain that this last measurement will
supply us with the same result as the first. Thus the length of the
train as measured from the embankment may be different from that
obtained by measuring in the train itself. This circumstance leads us
to a second objection which must be raised against the apparently
obvious consideration of Section 6. Namely, if the man in the
carriage covers the distance w in a unit of time -- measured from the
train, -- then this distance -- as measured from the embankment -- is
not necessarily also equal to w.


  Notes

*) e.g. the middle of the first and of the hundredth carriage.



THE LORENTZ TRANSFORMATION


The results of the last three sections show that the apparent
incompatibility of the law of propagation of light with the principle
of relativity (Section 7) has been derived by means of a
consideration which borrowed two unjustifiable hypotheses from
classical mechanics; these are as follows:

(1) The time-interval (time) between two events is independent of the
condition of motion of the body of reference.

(2) The space-interval (distance) between two points of a rigid body
is independent of the condition of motion of the body of reference.

If we drop these hypotheses, then the dilemma of Section 7
disappears, because the theorem of the addition of velocities derived
in Section 6 becomes invalid. The possibility presents itself that
the law of the propagation of light in vacuo may be compatible with
the principle of relativity, and the question arises: How have we to
modify the considerations of Section 6 in order to remove the
apparent disagreement between these two fundamental results of
experience? This question leads to a general one. In the discussion of
Section 6 we have to do with places and times relative both to the
train and to the embankment. How are we to find the place and time of
an event in relation to the train, when we know the place and time of
the event with respect to the railway embankment ? Is there a
thinkable answer to this question of such a nature that the law of
transmission of light in vacuo does not contradict the principle of
relativity ? In other words : Can we conceive of a relation between
place and time of the individual events relative to both
reference-bodies, such that every ray of light possesses the velocity
of transmission c relative to the embankment and relative to the train
? This question leads to a quite definite positive answer, and to a
perfectly definite transformation law for the space-time magnitudes of
an event when changing over from one body of reference to another.

Before we deal with this, we shall introduce the following incidental
consideration. Up to the present we have only considered events taking
place along the embankment, which had mathematically to assume the
function of a straight line. In the manner indicated in Section 2
we can imagine this reference-body supplemented laterally and in a
vertical direction by means of a framework of rods, so that an event
which takes place anywhere can be localised with reference to this
framework. Fig. 2 Similarly, we can imagine the train travelling with
the velocity v to be continued across the whole of space, so that
every event, no matter how far off it may be, could also be localised
with respect to the second framework. Without committing any
fundamental error, we can disregard the fact that in reality these
frameworks would continually interfere with each other, owing to the
impenetrability of solid bodies. In every such framework we imagine
three surfaces perpendicular to each other marked out, and designated
as " co-ordinate planes " (" co-ordinate system "). A co-ordinate
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

                                y1 = y
                                z1 = z

                         eq. 2: file eq02.gif

This system of equations is known as the " Lorentz transformation." *

If in place of the law of transmission of light we had taken as our
basis the tacit assumptions of the older mechanics as to the absolute
character of times and lengths, then instead of the above we should
have obtained the following equations:

                             x1 = x - vt
                                y1 = y
                                z1 = z
                                t1 = t

This system of equations is often termed the " Galilei
transformation." The Galilei transformation can be obtained from the
Lorentz transformation by substituting an infinitely large value for
the velocity of light c in the latter transformation.

Aided by the following illustration, we can readily see that, in
accordance with the Lorentz transformation, the law of the
transmission of light in vacuo is satisfied both for the
reference-body K and for the reference-body K1. A light-signal is sent
along the positive x-axis, and this light-stimulus advances in
accordance with the equation

                               x = ct,

i.e. with the velocity c. According to the equations of the Lorentz
transformation, this simple relation between x and t involves a
relation between x1 and t1. In point of fact, if we substitute for x
the value ct in the first and fourth equations of the Lorentz
transformation, we obtain:

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in consequence of what we have seen above.



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 Galileian domain, i.e. a domain
in which there is no gravitational field relative to the Galileian
reference-body K. The behaviour of measuring-rods and clocks with
reference to K is known from the special theory of relativity,
likewise the behaviour of "isolated" material points; the latter move
uniformly and in straight lines.

Now let us refer this domain to a random Gauss coordinate system or to
a "mollusc" as reference-body K1. Then with respect to K1 there is a
gravitational field G (of a particular kind). We learn the behaviour
of measuring-rods and clocks and also of freely-moving material points
with reference to K1 simply by mathematical transformation. We
interpret this behaviour as the behaviour of measuring-rods, docks and
material points tinder the influence of the gravitational field G.
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
postulate of relativity.

(b) If there is any matter in the domain under consideration, only its
inertial mass, and thus according to Section 15 only its energy is
of importance for its etfect 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 field on the course of all those
processes which take place according to known laws when a
gravitational field is absent i.e. 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 21; 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 round 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 an 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 Leverrier, it has
been known that the ellipse corresponding to the orbit of Mercury,
after it has been corrected 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 43 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 purponse.

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 43 seconds of arc per century, a
result which is strictly in 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