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%%% Primary Title: Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference
%%% Primary Category Code: 04.20.-q
%%% Filename: BehaviourOfClocksAndMeasuringRodsOnARotatingBodyOfReference.tex
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\begin{document}
\subsection{Behaviour of Clocks and Measuring-Rods on a Rotating Body of Reference}
From \htmladdnormallink{Relativity: The Special and General Theory}{http://planetphysics.us/encyclopedia/SpecialTheoryOfRelativity.html} by \htmladdnormallink{Albert Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html}
Hitherto I have purposely refrained from speaking about the physical
interpretation of space- and time-data in the case of the \htmladdnormallink{general theory}{http://planetphysics.us/encyclopedia/GeneralTheory.html} of relativity. As a consequence, I am guilty of a certain
slovenliness of treatment, which, as we know from the special theory
of relativity, is far from being unimportant and pardonable. It is now
high time that we remedy this defect; but I would mention at the
outset, that this matter lays no small claims on the patience and on
the \htmladdnormallink{power}{http://planetphysics.us/encyclopedia/Power.html} of abstraction of the reader.
We start off again from quite special cases, which we have frequently
used before. Let us consider a space time \htmladdnormallink{domain}{http://planetphysics.us/encyclopedia/Bijective.html} in which no
gravitational \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} exists relative to a reference-body $K$ whose state
of \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} has been suitably chosen. $K$ is then a Galileian
reference-body as regards the domain considered, and the results of
the special theory of relativity hold relative to $K$. Let us supposse
the same domain referred to a second body of reference $K'$, which is
rotating uniformly with respect to $K$. In order to fix our ideas, we
shall imagine $K'$ to be in the form of a plane circular disc, which
rotates uniformly in its own plane about its centre. An observer who
is sitting eccentrically on the disc $K'$ is sensible of a \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html} which
acts outwards in a radial direction, and which would be interpreted as
an effect of inertia (centrifugal force) by an observer who was at
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.\footnotemark\ 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 $K'$, 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
the centre of the circular disc, and the other on the edge of the
disc, so that they are at rest relative to it. We now ask ourselves
whether both clocks go at the same rate from the standpoint of the
non-rotating Galileian reference-body $K$. As judged from this body, the
clock at the centre of the disc has no \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html}, whereas the clock at
the edge of the disc is in motion relative to $K$ in consequence of the
rotation. According to a result obtained in \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} 12, it follows
that the latter clock goes at a rate permanently slower than that of
the clock at the centre of the circular disc, {\it i.e.} as observed from $K$.
It is obvious that the same effect would be noted by an observer whom
we will imagine sitting alongside his clock at the centre of the
circular disc. Thus on our circular disc, or, to make the case more
general, in every gravitational field, a clock will go more quickly or
less quickly, according to the \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} in which the clock is situated
(at rest). For this reason it is not possible to obtain a reasonable
definition of time with the aid of clocks which are arranged at rest
with respect to the body of reference. A similar difficulty presents
itself when we attempt to apply our earlier definition of simultaneity
in such a case, but I do not wish to go any farther into this
question.
Moreover, at this stage the definition of the space co-ordinates also
presents insurmountable difficulties. If the observer applies his
standard measuring-rod (a rod which is short as compared with the
radius of the disc) tangentially to the edge of the disc, then, as
judged from the Galileian \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html}, the length of this rod will be less
than I, since, according to Section 12, moving bodies suffer a
shortening in the direction of the motion. On the other hand, the
measaring-rod will not experience a shortening in length, as judged
from $K$, if it is applied to the disc in the direction of the radius.
If, then, the observer first measures the circumference of the disc
with his measuring-rod and then the diameter of the disc, on dividing
the one by the other, he will not obtain as quotient the familiar
number $\pi$ = 3.14 . . ., but a larger number,\footnotemark\ whereas of course,
for a disc which is at rest with respect to $K$, this \htmladdnormallink{operation}{http://planetphysics.us/encyclopedia/Cod.html} would
yield $\pi$ exactly. This proves that the \htmladdnormallink{propositions}{http://planetphysics.us/encyclopedia/Predicate.html} of Euclidean
geometry cannot hold exactly on the rotating disc, nor in general in a
gravitational field, at least if we attribute the length I to the rod
in all positions and in every orientation. Hence the idea of a
straight line also loses its meaning. We are therefore not in a
position to define exactly the co-ordinates $x, y, z$ relative to the
disc by means of the method used in discussing the special theory, and
as long as the co-ordinates and times of events have not been
defined, we cannot assign an exact meaning to the natural laws in
which these occur.
Thus all our previous conclusions based on \htmladdnormallink{general relativity}{http://planetphysics.us/encyclopedia/SR.html} would
appear to be called in question. In reality we must make a subtle
detour in order to be able to apply the postulate of general
relativity exactly. I shall prepare the reader for this in the
following paragraphs.
\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]{The field disappears at the centre of the disc and increases
proportionally to the distance from the centre as we proceed outwards.}
\footnotetext[2]{Throughout this consideration we have to use the Galileian
(non-rotating) system $K$ as reference-body, since we may only assume
the validity of the results of the special theory of relativity
relative to $K$ (relative to $K'$ a gravitational field prevails).}
\end{document}
