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%%% Primary Title: A Few Inferences from the General Principle of Relativity
%%% Primary Category Code: 04.20.-q
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\begin{document}
\subsection{A Few Inferences from the General Principle of Relativity}
From \htmladdnormallink{Relativity: The Special and General Theory}{http://planetphysics.us/encyclopedia/SpecialTheoryOfRelativity.html} by \htmladdnormallink{Albert Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html}
The considerations of \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} 20 show that the general principle of
relativity puts us in a \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} to derive properties of \htmladdnormallink{The Gravitational Field}{http://planetphysics.us/encyclopedia/GravitationalField.html} in a purely theoretical manner. Let us suppose,
for instance, that we know the \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html} ``course'' for any natural
process whatsoever, as regards the manner in which it takes place in
the Galileian \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} relative to a Galileian body of reference $K$. By
means of purely theoretical \htmladdnormallink{operations}{http://planetphysics.us/encyclopedia/Cod.html} ({\it i.e.} simply by calculation) we
are then able to find how this known natural process appears, as seen
from a reference-body $K'$ which is accelerated relatively to $K$. But
since a gravitational \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} exists with respect to this new body of
reference $K$, our consideration also teaches us how the gravitational
field influences the process studied.
For example, we learn that a body which is in a state of uniform
rectilinear \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} with respect to $K$ (in accordance with the law of
Galilei) is executing an accelerated and in general curvilinear motion
with respect to the accelerated reference-body $K'$ (chest). This
\htmladdnormallink{acceleration}{http://planetphysics.us/encyclopedia/Acceleration.html} or curvature corresponds to the influence on the moving
body of the gravitational field prevailing relatively to $K$. It is
known that a gravitational field influences the movement of bodies in
this way, so that our consideration supplies us with nothing
essentially new.
However, we obtain a new result of fundamental importance when we
carry out the analogous consideration for a ray of light. With respect
to the Galileian reference-body $K$, such a ray of light is transmitted
rectilinearly with the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} $c$. It can easily be shown that the
path of the same ray of light is no longer a straight line when we
consider it with reference to the accelerated chest (reference-body
$K'$). From this we conclude, that, in general, rays of light are
propagated curvilinearly in gravitational fields. In two respects this
result is of great importance.
In the first place, it can be compared with the reality. Although a
detailed examination of the question shows that the curvature of light
rays required by the \htmladdnormallink{general theory}{http://planetphysics.us/encyclopedia/GeneralTheory.html} of relativity is only exceedingly
small for the gravitational fields at our disposal in practice, its
estimated \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} for light rays passing the sun at grazing
incidence is nevertheless 1.7 seconds of arc. This ought to manifest
itself in the following way. As seen from the earth, certain fixed
stars appear to be in the neighbourhood of the sun, and are thus
capable of observation during a total eclipse of the sun. At such
times, these stars ought to appear to be displaced outwards from the
sun by an amount indicated above, as compared with their apparent
position in the sky when the sun is situated at another part of the
heavens. The examination of the correctness or otherwise of this
deduction is a problem of the greatest importance, the early solution
of which is to be expected of astronomers.\footnotemark
In the second place our result shows that, according to the general
theory of relativity, the law of the constancy of the velocity of
light in vacuo, which constitutes one of the two fundamental
assumptions in the special theory of relativity and to which we have
already frequently referred, cannot claim any unlimited validity. A
curvature of rays of light can only take place when the velocity of
propagation of light varies with position. Now we might think that as
a consequence of this, the special theory of relativity and with it
the whole theory of relativity would be laid in the dust. But in
reality this is not the case. We can only conclude that the special
theory of relativity cannot claim an unlinlited domain of validity;
its results hold only so long as we are able to disregard the
influences of gravitational fields on the phenomena ({\it e.g.} of light).
Since it has often been contended by opponents of the theory of
relativity that the special theory of relativity is overthrown by the
general theory of relativity, it is perhaps advisable to make the
facts of the case clearer by means of an appropriate comparison.
Before the development of electrodynamics the laws of electrostatics
were looked upon as the laws of electricity. At the present time we
know that \htmladdnormallink{Electric Fields}{http://planetphysics.us/encyclopedia/ElectricField.html} can be derived correctly from electrostatic
considerations only for the case, which is never strictly realised, in
which the electrical \htmladdnormallink{masses}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} are quite at rest relatively to each
other, and to the co-ordinate \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}. Should we be justified in saying
that for this reason electrostatics is overthrown by the
field-equations of Maxwell in electrodynamics? Not in the least.
Electrostatics is contained in electrodynamics as a limiting case;
the laws of the latter lead directly to those of the former for the
case in which the fields are invariable with regard to time. No fairer
destiny could be allotted to any physical theory, than that it should
of itself point out the way to the introduction of a more
comprehensive theory, in which it lives on as a limiting case.
In the example of the \htmladdnormallink{transmission}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} of light just dealt with, we have
seen that the general theory of relativity enables us to derive
theoretically the influence of a gravitational field on the course of
natural processes, the Iaws of which are already known when a
gravitational field is absent. But the most attractive problem, to the
solution of which the general theory of relativity supplies the key,
concerns the investigation of the laws satisfied by the gravitational
field itself. Let us consider this for a moment.
We are acquainted with space-time domains which behave (approximately)
in a ``Galileian'' fashion under suitable choice of reference-body,
{\it i.e.} domains in which gravitational fields are absent. If we now refer
such a domain to a reference-body $K'$ possessing any kind of motion,
then relative to $K'$ there exists a gravitational field which is
variable with respect to space and time.\footnotemark\ The character of this
field will of course depend on the motion chosen for $K'$. According to
the general theory of relativity, the general law of the gravitational
field must be satisfied for all gravitational fields obtainable in
this way. Even though by no means all gravitationial fields can be
produced in this way, yet we may entertain the hope that the general
law of gravitation will be derivable from such gravitational fields of
a special kind. This hope has been realised in the most beautiful
manner. But between the clear vision of this goal and its actual
realisation it was necessary to surmount a serious difficulty, and as
this lies deep at the root of things, I dare not withhold it from the
reader. We require to extend our ideas of the space-time continuum
still farther.
\subsection{References}
This article is derived from the Einstein Reference Archive (marxists.org) 1999, 2002. \htmladdnormallink{Einstein Reference Archive}{http://www.marxists.org/reference/archive/einstein/index.htm} which is under the FDL copyright.
\footnotetext[1]{By means of the star photographs of two expeditions equipped by
a Joint Committee of the Royal and Royal Astronomical Societies, the
existence of the deflection of light demanded by theory was first
confirmed during the solar eclipse of 29th May, 1919. (Cf. Appendix
III.)}
\footnotetext{This follows from a generalisation of the discussion in
Section 20}
\end{document}
