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%TCIDATA{Created=Wed Oct 16 16:28:19 1996}
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\markright{{\sf  History of Relativity}}
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\section{11.1\quad History of Relativity.}

\subsection{11.1.1}

\quad \thinspace \textbf{a. }Newton's equations are invariant under: 
\begin{eqnarray*}
-\text{space rotations}\quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad x_\alpha ^{\prime } &=&R_{\alpha \beta }x_\beta \quad \quad
(R_{\alpha \beta }R_{\alpha \gamma }=\delta _{\beta \gamma }) \\
-\text{space inversion (parity)\quad }\quad \quad \quad \quad \quad \quad
x_\alpha ^{\prime } &=&-x_\alpha \\
-\text{time inversion (time reversal)}\quad \,\,\quad \quad \quad
\,t^{\prime }\,\,\, &=&-t \\
-\text{space translations\quad }\quad \,\,\,\quad \quad \quad \quad \quad
\quad \quad x_\alpha ^{\prime } &=&x_\alpha +const. \\
-\text{time translations\quad }\quad \,\,\quad \quad \quad \quad \quad \quad
\quad \quad t^{\prime }\,\,\, &=&t+const. \\
-\text{Galilean transformations}\quad \quad \,\,\,\,\quad \quad \quad \quad
x_\alpha ^{\prime } &=&x_\alpha -v_\alpha t\quad \quad \quad t^{\prime }=t
\end{eqnarray*}
(Repeated indices are summed; all indices sum over 1,2,3). Note that the
vector $\mathbf{x}$ is odd under parity and even (i.e. unchanged) under time
reversal, but other vectors derived from $\mathbf{x}$ can behave differently
under $P$ and $T:$ the velocity $\mathbf{v}=d\mathbf{x}/dt$ is odd under
both $P$ and $T,$ the angular momentum $\mathbf{L}=m\mathbf{x\times v}$ is
odd under $T$ and, being an axial vector (or pseudovector), it is even under 
$P$.

\smallskip\ 

\textbf{b}. Maxwell's equations are invariant under the same
transformations, \textsl{except} time reversal and Galilean transformations.
Time reversal invariance is restored if one realizes that $t\rightarrow
t^{\prime }=-t$ must also imply $\mathbf{B\rightarrow B}^{\prime }=-\mathbf{%
B,}$ i.e. $\mathbf{B}$ is odd under $T$ (it is also even under $P,$ being an
axial vector). But Galilean frames are inequivalent.

\smallskip\ 

\textbf{c.} Classical physicists were not disturbed by this. They considered
Maxwell's equations to be akin to many other wave equations which have a
simple form when the source and the observer are at rest with respect to the
medium and become messy otherwise.

\smallskip\ 

\textbf{d.} A primary example of classical wave equations, that are formally
analogous to Maxwell's equations, are the elasticity equations for
transverse waves in an isotropic medium. Thus ether was introduced,
notwithstanding its rather bizarre properties as an elastic medium (lighter
than air yet stronger than steel).\textbf{\ }

\smallskip\ 

\textbf{e. }It appeared reasonable to assume that ether was at rest with
respect to the frame of reference of the ``fixed stars''. However, there
were puzzling results and contradictions.

\newpage

\subsection{11.1.2}

\quad \quad \textbf{a. }\textsl{The aberration of stars} was consistent with
the idea that the ether is fixed to the stars and is not dragged along by
the earth in its orbit. The classical explanation should be obvious to
anyone who has driven a car (better yet a bicycle) in the rain. The
aberration angle $\alpha \simeq \tan \alpha =v/c$ is about $10^{-4}$
radians. The stars describe little (apparent) circles due to the daily and
annual changes of $\mathbf{v.}$

\smallskip\ 

\textbf{b.} \textsl{Fizeau's experiment} on the other hand gave a strange
result. The speed of light in a medium of index of refraction $n$ moving
with velocity $v$ was found to be 
\[
\frac{c}{n}\pm v\left( 1-\frac{1}{n^{2}}\right) 
\]
The $\pm $ goes with the two signs of flow. One would expect $c/n\pm v$ if
the ether is dragged. It appears thus that only the fraction 
\[
\frac{n^{2}-1}{n^{2}}=\frac{\varepsilon -1}{\varepsilon }
\]
of the ``luminiferous medium'' is being dragged. One can make some sense out
of this by noting that this fraction is equal to the ratio of the
polarization current $\partial P\mathbf{/}\partial t$ to the total current $%
\partial D\mathbf{/}\partial t.$ Thus one can say that only the atomic
dipoles in the medium are moving. The ether could still be attached to the
stars (or to the earth, which gives the same result to order $v/c).$

\smallskip\ 

\textbf{c. }\textsl{The negative result of} the \textsl{Trouton-Noble }%
experiment, on the other hand, seemed to show that the ether moves with the
earth. The experiment is very simple: one looks for a torque on a torsion
pendulum carrying charges $\pm q$. The interpretation, however, brings in
the Lorentz force $(\mathbf{v/}c\mathbf{)\times B,}$ in addition to
Maxwell's equations. If $\mathbf{v}$ is to be reckoned with respect to the
ether, as it seems reasonable, the conclusion is as stated. 

\newpage

\subsection{11.1.3}

\quad \quad \textbf{a. }\textsl{The negative result of} the \textsl{%
Michelson-Morley} experiment is in more direct contradiction to the
aberration of stars, within an ether theory. Only simple concepts of wave
propagation (not even Maxwell's equations in their full glory) enter in the
interpretation of this experiment. One looks for a difference in time of
travel for light propagating equal distances in two perpendicular directions
when the apparatus is (supposedly) moving with respect to the ether. (In
practice it is not possible to make $PS_1=PS_2$ exactly, so the experiment
looked for a change when the apparatus was rotated $180^o$ around the dotted
axis, interchanging $S_1$ and $S_2$). No effect was found due to the orbital
motion of the earth. The experiment looks for an effect of order $v^2/c^2,$
one order smaller than Fizeau's; thus it is difficult.

\smallskip\ 

\textbf{b. }\textsl{The Fitzgerald contraction} hypothesis, that the length
of a body moving with velocity $\mathbf{v}$ through the ether is shortened
by a factor $\sqrt{1-v^{2}/c^{2}}$ in the direction of motion (because it
pushes against the ether?!) could reconcile the Michelson-Morley result with
the aberration of starlight.

\smallskip\ 

\textbf{c. }\textsl{Lorentz} showed that the Fitzgerald contraction could be
understood if one assumes that all interatomic forces are electromagnetic in
origin. He and Poincar\'{e} found that Maxwell's equations are invariant
under a group of transformations which reduce to the Galilean group for
small $v/c.$ The transformed fields in a moving body are such that the body
shrinks in the direction of motion.

\smallskip\ 

\textbf{d. }\textsl{Einstein} clarified the meaning of Lorentz's and
Poincar\'{e}'s work. It is uncertain whether he was aware of Poincar\'{e}'s
theories and of the Michelson-Morley result; he was mostly influenced by
Lorentz and Fizeau.

\newpage 

\subsection{11.1.4}

\quad \quad \textbf{a. }Einstein based his reasonong on two postulates are
extremely simple, yet lead to a radically new viewpoint. They are the
postulate of relativity and the postulate of the constancy of the speed of
light. The first had been strongly advocated by Poincar\'{e}; the latter had
probably occurred to many people, but nobody realized that a unique and
consistent theory in full accord with the experiment could be derived from
these postulates alone.

\smallskip\ 

\textbf{b. }\textsl{Postulate of Relativity:}

\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad The
laws of nature and the results of all experiments performed in a given frame
of reference are independent of the translational motion of the system as a
whole.

\smallskip\ 

\textbf{c. }\textsl{Constancy of the speed of light:}

\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad
\quad \quad \quad \quad The speed of light is independent of the motion of
its source. (It is also independent of the motion of the observers, but this
is not an additional postulate once relativity is assumed).

\smallskip\ 

\textbf{d. }Einstein's relativity theory discarded the ether, kept Maxwell's
equations unchanged and introduced a modification of Newton's equations to
make them conform with the (Lorentz-Poincar\'{e}) transformation properties
of the Maxwell equations.

\smallskip\ 

\textbf{e.} Many other theories were proposed. The most successful was
Ritz's emission theory, which also discarded the ether, but kept Newton's
equations unchanged and modified Maxwell's. It was preferred by many on
philosophical grounds, because it kept an absolute meaning for time and
avoided the strange ``contradictions'' and ``dilations'' of relativity \`{a}
la Einstein.

\newpage 

\subsection{11.1.5}

\quad \quad \textbf{a. }The\textbf{\ }\textsl{Kennedy-Thorndike} experiment
showed that the Fitzgerald-Lorentz contraction was not enough. The classical
analysis of the Michelson-Morley experiment shows that, with $\beta =v/c,$
the optical path difference for the apparatus sketched in 3a is 
\[
\Delta =c(t_1-t_2)=\frac 2{\sqrt{1-\beta ^2}}\left( \frac{l_1}{\sqrt{1-\beta
^2}}-l_2\right) . 
\]
With the F-L contraction, $l_1$ must be replaced by $l_1\sqrt{1-\beta ^2}$
(note the direction of $\mathbf{v}$ in the sketch), so that $\Delta $
vanishes if $l_1=l_2$ and is anyhow independent of the direction of $\mathbf{%
v.}$ However, there is still a dependence on $v^2,$ which changes daily and
annually as the earth rotates and revolves around the sun. Kennedy and
Thorndike found no dependence, which is consistent with special relativity,
but also with the emission theory of Ritz.

\smallskip\ 

\textbf{b.} In fact the emission theory gives the same results as special
relativity for all experiments with light quoted so far. The principle of
the theory is that the velocity of light is $c/n$ with respect to the
source, regardless of the state of motion of the medium, The velocity
becomes $\left( c/n\right) \pm v$ for an observer moving with velocity $\mp v
$ relative to the source, again regardless of the state of motion of the
medium. It disagrees with special relativity to second order in $v/c$ for
sources moving with respect to the observer. Michelson-Morley done with
starlight was regarded as a crucial test in favor of relativity.

\smallskip\ 

\textbf{c.} The analysis of light-propagation phenomena is in fact very
complex because of the almost unavoidable presence of a material medium. The
simple theory for a stationary medium of constant $\varepsilon $ shows that
the incident light of speed $c$ is replaced by the refracted light of speed $%
c/n$. But what happens for a moving medium?

\newpage 

\subsection{11.1.6}

\quad \quad \textbf{a. }If the interatomic spacing is much shorter than a
wavelength, so that a continuum description can be applied, it is easy to
show that any disturbance arriving at the surface of the medium can be
Fourier analyzed into components with a given frequency $\omega $ and that
for each such component the wave number in the medium is $c\omega /n$, or
the phase velocity is $c/n,$ regardless of anything else. Even in this case,
one must be careful to draw conclusions about the speed of propagation of a
finite pulse. It is clear that the change in speed cannot be ascertained
until the phase shift due to the medium is comparable to $\pi $ (say). Since 
$\Delta \varphi =\omega (n-1)x/c,$ this implies a distance 
\[
x\simeq \frac{\pi c}{\omega \left| n-1\right| }=\frac{\lambda }{2\left|
n-1\right| }\,.
\]
For glass, this is about $10^{-4}$ cm.

\smallskip\ 

\textbf{b.} The identification of $\varepsilon $ with $1+4\pi Nf_0/k^2,$
where $f_0$ is the forward scattering amplitude (Sect. 9.14) allows us to
extend the above result to a dilute medium, with $\lambda $ less than the
interparticle spacing $d,$ provided that 
\[
\frac \lambda {2\left| n-1\right| }>d. 
\]

\smallskip\ 

\textbf{c.} As discussed by Jackson, the emission theory can be finally
ruled out by recent experiments using $\gamma $-rays emitted in high vacuum
by decaying pions moving at $0.99975c.$ Of course, there are many other
convincing confirmations of relativity that do not involve directly a
measurement of the speed of light from a moving source.

\smallskip\ 

\textbf{d.} Jackson discusses also a modern test for an ether drift based on
the search of frequency shifts for two M\"{o}ssbauer sources on a rotating
turntable. The effect should be of order $\Omega Rv/c^{2},$ where $v$ is the
plane of the turntable. No effect was found.

\end{document}
