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%%% Primary Title: The Experimental Confirmation of the General Theory of Relativity
%%% Primary Category Code: 04.20.-q
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\begin{document}
\subsection{The Experimental Confirmation of the General Theory 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}
From a systematic theoretical point of view, we may imagine the
process of evolution of an empirical science to be a continuous
process of induction. Theories are evolved and are expressed in short
compass as statements of a large number of individual observations in
the form of empirical laws, from which the general laws can be
ascertained by comparison. Regarded in this way, the development of a
science bears some resemblance to the compilation of a classified
catalogue. It is, as it were, a purely empirical enterprise.
But this point of view by no means embraces the whole of the actual
process; for it slurs over the important part played by intuition and
deductive thought in the development of an exact science. As soon as a
science has emerged from its initial stages, theoretical advances are
no longer achieved merely by a process of arrangement. Guided by
empirical data, the investigator rather develops a \htmladdnormallink{system}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} of thought
which, in general, is built up logically from a small number of
fundamental assumptions, the so-called axioms. We call such a system
of thought a {\it theory}. The theory finds the justification for its
existence in the fact that it correlates a large number of single
observations, and it is just here that the ``truth'' of the theory
lies.
Corresponding to the same complex of empirical data, there may be
several theories, which differ from one another to a considerable
extent. But as regards the deductions from the theories which are
capable of being tested, the agreement between the theories may be so
complete that it becomes difficult to find any deductions in which the
two theories differ from each other. As an example, a case of general
interest is available in the province of biology, in the Darwinian
theory of the development of species by selection in the struggle for
existence, and in the theory of development which is based on the
hypothesis of the hereditary \htmladdnormallink{transmission}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} of acquired characters.
We have another instance of far-reaching agreement between the
deductions from two theories in \htmladdnormallink{Newtonian mechanics}{http://planetphysics.us/encyclopedia/NewtonianMechanics.html} on the one hand,
and the \htmladdnormallink{general theory}{http://planetphysics.us/encyclopedia/GeneralTheory.html} of relativity on the other. This agreement goes
so far, that up to the preseat we have been able to find only a few
deductions from the general theory of relativity which are capable of
investigation, and to which the physics of pre-relativity days does
not also lead, and this despite the profound difference in the
fundamental assumptions of the two theories. In what follows, we shall
again consider these important deductions, and we shall also discuss
the empirical evidence appertaining to them which has hitherto been
obtained.
\subsection{Motion of the Perihelion of Mercury}
According to Newtonian mechanics and \htmladdnormallink{Newton's law of gravitation}{http://planetphysics.us/encyclopedia/PrincipleOfCorrespondingStates.html}, a
planet which is revolving round the sun would describe an ellipse
round the latter, or, more correctly, round the common centre of
gravity of the sun and the planet. In such a system, the sun, or the
common centre of gravity, lies in one of the foci of the orbital
ellipse in such a manner that, in the course of a planet-year, the
distance sun-planet grows from a minimum to a maximum, and then
decreases again to a minimum. If instead of \htmladdnormallink{Newton's law}{http://planetphysics.us/encyclopedia/Newtons3rdLaw.html} we insert a
somewhat different law of attraction into the calculation, we find
that, according to this new law, the \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} would still take place in
such a manner that the distance sun-planet exhibits periodic
variations; but in this case the angle described by the line joining
sun and planet during such a period (from perihelion--closest
proximity to the sun--to perihelion) would differ from $360^\circ$. The line
of the orbit would not then be a closed one but in the course of time
it would fill up an annular part of the orbital plane, viz. between
the circle of least and the circle of greatest distance of the planet
from the sun.
According also to the general theory of relativity, which differs of
course from the theory of Newton, a small variation from the
Newton-Kepler motion of a planet in its orbit should take place, and
in such away, that the angle described by the radius sun-planet
between one perhelion and the next should exceed that corresponding to
one complete revolution by an amount given by
$$+ \frac{24\pi^3a^2}{T^2e^2(I-e^2)}$$
\noindent (N.B. -- One complete revolution corresponds to the angle $2\pi$ in the
absolute angular measure customary in physics, and the above
expression giver the amount by which the radius sun-planet exceeds
this angle during the interval between one perihelion and the next.)
In this expression $a$ represents the major semi-axis of the ellipse, $e$
its eccentricity, $c$ the \htmladdnormallink{velocity}{http://planetphysics.us/encyclopedia/Velocity.html} of light, and $T$ the period of
revolution of the planet. Our result may also be stated as follows:
According to the general theory of relativity, the major axis of the
ellipse rotates round the sun in the same sense as the orbital motion
of the planet. Theory requires that this rotation should amount to 43
seconds of arc per century for the planet Mercury, but for the other
Planets of our solar system its \htmladdnormallink{magnitude}{http://planetphysics.us/encyclopedia/AbsoluteMagnitude.html} should be so small that it
would necessarily escape \htmladdnormallink{detection}{http://planetphysics.us/encyclopedia/MolecularOrbitals.html}.\footnotemark
In point of fact, astronomers have found that the theory of Newton
does not suffice to calculate the observed motion of Mercury with an
exactness corresponding to that of the delicacy of observation
attainable at the present time. After taking account of all the
disturbing influences exerted on Mercury by the remaining planets, it
was found (Leverrier: 1859; and Newcomb: 1895) that an unexplained
perihelial movement of the orbit of Mercury remained over, the amount
of which does not differ sensibly from the above mentioned +43 seconds
of arc per century. The uncertainty of the empirical result amounts to
a few seconds only.
\subsection{Deflection of Light by a Gravitational Field}
In \htmladdnormallink{section}{http://planetphysics.us/encyclopedia/IsomorphicObjectsUnderAnIsomorphism.html} 22 it has been already mentioned that according to the
general theory of relativity, a ray of light will experience a
curvature of its path when passing through a gravitational \htmladdnormallink{field}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html}, this
curvature being similar to that experienced by the path of a body
which is projected through a gravitational field. As a result of this
theory, we should expect that a ray of light which is passing close to
a heavenly body would be deviated towards the latter. For a ray of
light which passes the sun at a distance of $\Delta$ sun-radii from its
centre, the angle of deflection (a) should amount to
$$a = \frac{1.7 \mbox{seconds of arc}}{\Delta}$$
It may be added that, according to the theory, half of Figure 05 this
deflection is produced by the Newtonian field of attraction of the
sun, and the other half by the geometrical modification (``curvature")
of space caused by the sun.
\begin{figure}[hbtp]
\centering
\caption{}
\label{fig:5}
\begin{picture}(110,250)(0,30)
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\put(22,135){S}
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\put(35,90){\vector(1,4){3}}
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This result admits of an experimental test by means of the
photographic registration of stars during a total eclipse of the sun.
The only reason why we must wait for a total eclipse is because at
every other time the atmosphere is so strongly illuminated by the
light from the sun that the stars situated near the sun's disc are
invisible. The predicted effect can be seen clearly from the
accompanying diagram. If the sun (S) were not present, a star which is
practically infinitely distant would be seen in the direction $D_1$, as
observed front the earth. But as a consequence of the deflection of
light from the star by the sun, the star will be seen in the direction
$D_2$, {\it i.e.} at a somewhat greater distance from the centre of the sun
than corresponds to its real position.
In practice, the question is tested in the following way. The stars in
the neighborhood of the sun are photographed during a solar eclipse.
In addition, a second photograph of the same stars is taken when the
sun is situated at another position in the sky, {\it i.e.} a few months
earlier or later. As compared whh the standard photograph, the
positions of the stars on the eclipse-photograph ought to appear
displaced radially outwards (away from the centre of the sun) by an
amount corresponding to the angle a.
We are indebted to the [British] Royal Society and to the Royal
Astronomical Society for the investigation of this important
deduction. Undaunted by the [first world] war and by difficulties of
both a material and a psychological nature aroused by the war, these
societies equipped two expeditions---to Sobral (Brazil), and to the
island of Principe (West Africa)---and sent several of Britain's most
celebrated astronomers (Eddington, Cottingham, Crommelin, Davidson),
in order to obtain photographs of the solar eclipse of 29th May, 1919.
The relative discrepancies to be expected between the stellar
photographs obtained during the eclipse and the comparison photographs
amounted to a few hundredths of a millimetre only. Thus great accuracy
was necessary in making the adjustments required for the taking of the
photographs, and in their subsequent measurement.
The results of the measurements confirmed the theory in a thoroughly
satisfactory manner. The rectangular components of the observed and of
the calculated deviations of the stars (in seconds of arc) are set
forth in the following table of results:
$$
\begin{array}{r|rr|rr}
\mbox{Number of the Star} & \mbox{First} & \mbox{Co-ordinate~~} & \mbox{Second} & \mbox{Co-ordinate~~} \\
\hline
& \mbox{Observed} & \mbox{Calculated} & \mbox{Observed} & \mbox{Calculated} \\
11 & -0'19 & -0'22 & +0'16 & +0'02 \\
5 & +0'29 & +0'31 & -0'46 & -0'43 \\
4 & +0'11 & +0'10 & +0'83 & +0'73 \\
3 & +0'22 & +0'12 & +1'00 & +0'87 \\
6 & +0'10 & +0'04 & +0'57 & +0'40 \\
10 & -0'08 & +0'09 & +0'35 & +0'32 \\
2 & +'095 & +0'85 & -0'27 & -0'09
\end{array}
$$
\subsection{Displacement of Spectral Lines Towards the Red}
In Section 23 it has been shown that in a system $K^1$ which is in
rotation with regard to a Galileian system $K$, clocks of identical
construction, and which are considered at rest with respect to the
rotating reference-body, go at rates which are dependent on the
positions of the clocks. We shall now examine this dependence
quantitatively. A clock, which is situated at a distance $r$ from the
centre of the disc, has a velocity relative to $K$ which is given by
$$V = \omega r$$
\noindent where $\omega$ represents the angular velocity of rotation of the disc $K^1$
with respect to $K$. If $v_0$, represents the number of ticks of the
clock per unit time (``rate'' of the clock) relative to $K$ when the
clock is at rest, then the ``rate'' of the clock ($v$) when it is moving
relative to $K$ with a velocity $V$, but at rest with respect to the disc,
will, in accordance with Section 12, be given by
$$v = v_2\sqrt{I-\frac{v^2}{c^2}}$$
\noindent or with sufficient accuracy by
$$v = v_0 \left( I-\frac{1}{2} \frac{v^2}{c^2} \right)$$
\noindent This expression may also be stated in the following form:
$$v = v_0 \left( I-\frac{1}{c^2} \frac{\omega^2r^2}{2} \right)$$
If we represent the difference of potential of the centrifugal \htmladdnormallink{force}{http://planetphysics.us/encyclopedia/Thrust.html} between the \htmladdnormallink{position}{http://planetphysics.us/encyclopedia/Position.html} of the clock and the centre of the disc by $\phi$,
{\it i.e.} the \htmladdnormallink{work}{http://planetphysics.us/encyclopedia/Work.html}, considered negatively, which must be performed on the
unit of \htmladdnormallink{mass}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html} against the centrifugal force in order to transport it
from the position of the clock on the rotating disc to the centre of
the disc, then we have
$$\phi = \frac{\omega^2r^2}{2}$$
\noindent From this it follows that
$$v = v_0 \left( I + \frac{\phi}{c^2} \right)$$
In the first place, we see from this expression that two clocks of
identical construction will go at different rates when situated at
different distances from the centre of the disc. This result is aiso
valid from the standpoint of an observer who is rotating with the
disc.
Now, as judged from the disc, the latter is in a gravititional field
of potential $\phi$, hence the result we have obtained will hold quite
generally for gravitational fields. Furthermore, we can regard an atom
which is emitting spectral lines as a clock, so that the following
statement will hold:
{\it An atom absorbs or emits light of a frequency which is dependent on
the potential of \htmladdnormallink{The Gravitational Field}{http://planetphysics.us/encyclopedia/GravitationalField.html} in which it is situated.}
The frequency of an atom situated on the surface of a heavenly body
will be somewhat less than the frequency of an atom of the same
element which is situated in free space (or on the surface of a
smaller celestial body).
Now $\phi = - K (M/r)$, where $K$ is Newton's constant of gravitation, and $M$
is the mass of the heavenly body. Thus a displacement towards the red
ought to take place for spectral lines produced at the surface of
stars as compared with the spectral lines of the same element produced
at the surface of the earth, the amount of this displacement being
$$\frac{v_0-v}{v_0} = \frac{K}{c^2} \frac{M}{r}$$
For the sun, the displacement towards the red predicted by theory
amounts to about two millionths of the wave-length. A trustworthy
calculation is not possible in the case of the stars, because in
general neither the mass $M$ nor the radius $r$ are known.
It is an open question whether or not this effect exists, and at the
present time (1920) astronomers are working with great zeal towards
the solution. Owing to the smallness of the effect in the case of the
sun, it is difficult to form an opinion as to its existence. Whereas
Grebe and Bachem (Bonn), as a result of their own measurements and
those of Evershed and Schwarzschild on the cyanogen bands, have placed
the existence of the effect almost beyond doubt, while other
investigators, particularly St. John, have been led to the opposite
opinion in consequence of their measurements.
Mean displacements of lines towards the less refrangible end of the
\htmladdnormallink{spectrum}{http://planetphysics.us/encyclopedia/CohomologyTheoryOnCWComplexes.html} are certainly revealed by statistical investigations of the
fixed stars; but up to the present the examination of the available
data does not allow of any definite decision being arrived at, as to
whether or not these displacements are to be referred in reality to
the effect of gravitation. The results of observation have been
collected together, and discussed in detail from the standpoint of the
question which has been engaging our attention here, in a paper by E.
Freundlich entitled ``Zur Pr\"ufung der allgemeinen
Relativit\"ats-Theorie" ({\it Die Naturwissenschaften}, 1919, No. 35,
p. 520: Julius Springer, Berlin).
At all events, a definite decision will be reached during the next few
years. If the displacement of spectral lines towards the red by the
gravitational potential does not exist, then the general theory of
relativity will be untenable. On the other hand, if the cause of the
displacement of spectral lines be definitely traced to the
gravitational potential, then the study of this displacement will
furnish us with important information as to the mass of the heavenly
bodies. \footnotemark
\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]{Especially since the next planet Venus has an orbit that is
almost an exact circle, which makes it more difficult to locate the
perihelion with precision.}
\footnotetext[2]{The displacentent of spectral lines towards the red end of the
spectrum was definitely established by Adams in 1924, by observations
on the dense companion of Sirius, for which the effect is about thirty
times greater than for the Sun. R.W.L. -- translator}
\end{document}
