Translation:On the Interpretation of Sagnac's Experiment in the General Theory of Relativity

by Rudolf Ortvay

On the Interpretation of Sagnac's Experiment in the General Theory of Relativity^[1]

By Rudolf Ortvay.

In the well known experiment of Sagnac^[2], two light rays traversing in opposite directions the periphery of a rotating circle (more exactly a polygon), will be brought to interfere, and on that occasion a displacement of the interference fringes in respect to the stationary state was observed, by an amount that corresponds to a phase difference of ${\tfrac {2lv}{c^{2}}}$, where $l$ is the length of the periphery, $v=\omega r_{0}$ is the velocity of a point of the periphery, $c$ is the speed of light. The theory of the experiment for the resting system of earth was given by M. v. Laue, before the experiment was even conducted.^[3] Since the experiment proves that in the rotating system the propagation of light does not correspond to that of an uniform moving system, a discussion from the standpoint of general relativity is desired.

I.

The method that was used by H. Thirring in his paper^[4] on the effect of distant rotating masses, provides the means for this investigation, and we will treat the problem with the same approximation, as the centrifugal- and Coriolis forces were treated by Thirring.

We assume with Thirring, that a spherical shell of mass $M$ and radius $r$ is uniformly rotating around the origin of our coordinate system with the angular velocity $\omega$ around the $z$-axis. Let $r$ be large against the circle radius.

In infinity we take for the coefficients $g_{ik}$ of the line element, its values in the pseudo-euclidean case and apply the approximate solution of Einstein with retarded potentials.

When we neglect the terms that include $\omega ^{2}$, we obtain for the energy-momentum-tensor^[4]

$T_{ik}=\varrho \left({\frac {dx_{4}}{ds}}\right)^{2}\left\{{\begin{array}{ccccccc}0&&0&&0&&-r\omega \sin \vartheta \sin \varphi \\0&&0&&0&&r\omega \sin \vartheta \cos \varphi \\0&&0&&0&&0\\-r\omega \sin \vartheta \sin \varphi &&r\omega \sin \vartheta \cos \varphi &&0&&1\end{array}}\right.$

We denote by $x_{4}=ct$, $c=1=$ speed of light, $\varrho$ = mass density.

We obtain the covariant components of the line element,^[4] when we neglect the terms that include $\omega ^{2}$ as well,

$g_{ik}=\left\{{\begin{array}{ccccccc}-(1+\alpha )&&0&&0&&+{\frac {2}{3}}\alpha \omega y\\\\0&&-(1+\alpha )&&0&&-{\frac {2}{3}}\alpha \omega x\\\\0&&0&&0&&0\\\\+{\frac {2}{3}}\alpha \omega y&&-{\frac {2}{3}}\alpha \omega x&&0&&+(1+\alpha )\end{array}}\right.$

where $\alpha ={\frac {xM}{4\pi r}}$, $x$ = gravitational constant.

For the line element we obtain

$ds^{2}=-(1+\alpha )\left(dx^{2}+dy^{2}\right)+{\frac {4}{3}}\alpha \omega (xdy-ydx)dt+(1-\alpha )dt^{2}$

and we determine the speed of light from the equation

$ds^{2}=0$

It is

$xdy-ydx=r^{2}d\varphi =r^{2}{\frac {d\varphi }{dt}}\cdot dt=r\cdot c\cdot dt$

where $c$ is the speed of light in the direction of the circumference.

From equation $ds^{2}=0$ we obtain after substitution of that value and division by $dt^{2}$

$-(1+\alpha )\left\{\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}\right\}+{\frac {4}{3}}\alpha \omega rc+1-\alpha =0$

Since

$\left({\frac {dx}{dt}}\right)^{2}+\left({\frac {dy}{dt}}\right)^{2}=e^{2}$

we further obtain

$c^{2}-{\frac {4}{3}}\cdot {\frac {\alpha }{1+\alpha }}\omega rc-{\frac {1-\alpha }{1+\alpha }}=0$

and from that for $c$

$c=+{\frac {2}{3}}{\frac {\alpha }{1+\alpha }}\omega r\pm {\sqrt {{\frac {4}{9}}\left({\frac {\alpha }{1+\alpha }}\right)^{2}+{\frac {1-\alpha }{1+\alpha }}}}$

and by restriction to terms of first order in $\alpha$

$c_{1}=1-\alpha +{\frac {2}{3}}\alpha \omega r=\left(1-{\frac {xM}{4\pi \cdot r}}\right)+{\frac {x}{6\pi }}\cdot {\frac {M}{r}}\cdot v$ $c_{2}=-(1-\alpha )+{\frac {2}{3}}\alpha \omega r=-\left(1-{\frac {x}{4\pi }}\cdot {\frac {M}{r}}\right)+{\frac {x}{6\pi }}\cdot {\frac {M}{r}}\cdot v$

Here,

$c_{0}=1-{\frac {xM}{4\pi r}}$

means the speed of light; when the shell is at rest, it differs only by a small extant from unity, only as much as it is determined by the potential of the shell. The second term determines that the velocity in both direction is not equal.

The speed of light is not $c=c_{0}\pm v$ as in Sagnac's experiment, because to the second term the dragging coefficient

${\frac {x}{6\pi }}\cdot {\frac {M}{r}}$

is added, which is very small. Another result was not to be expected, since the distant masses that determine the normal values of $g_{ik}$, don't share the rotation.

II.

In the first section I've tried to interpret Sagnac's experiment by direct calculation of the effect of the distant rotating masses, however, the line element could only be calculated as long as the velocity of the distant masses is small.

However, Sagnac's experiment can be interpreted exactly by general relativity, when we, like in the treatment of the centrifugal forces^[5], relate the line element to a coordinate system that rotates against the system of fixed stars.

If the line element is in a system that is stationary against the fixed stars by using spatial cylindric coordinates^[6] (where the unit time is chosen so that the speed of light is unity)

$ds^{2}=dz^{2}+dr^{2}+r^{2}d\varphi '^{2}-dt'^{2}$

then we can transform it to a system that rotates around the $z$-axis by a constant angular velocity $\omega$, by the substitution

$\varphi '=\varphi +\omega t,\ t'=t$

thus

$ds^{2}=dz^{2}+dr^{2}+r^{2}d\varphi ^{2}-2r^{2}\omega d\varphi dt-dt^{2}\left(1-\omega ^{2}r^{2}\right)$

We determine the speed of light from equation $ds^{2}=0$. By considering, that for a light ray propagating in a plane perpendicular to the rotation axis, $dz=0$, $dr=0$, $d{\frac {d\varphi }{dt}}=c=$ speed of light, $\omega \cdot r=v=$ the velocity of a point that is at rest at distance $r$ to the origin, we obtain (after division by $dt^{2}$) a quadratic equation for the determination of the speed of light

$c^{2}=2vc-\left(1-v^{2}\right)=0$

hence

$c=v\pm 1$

in agreement with Sagnac's experiment.

If we would have assumed a spatially finite world according to Einstein or de Sitter, then we would have to start with a line element, that (in the proximity of the origin) differs from the assumed expression only by magnitudes of second order, and the result would be the same except the magnitudes of the same order.

Summary

1. It was shown that the gravitational forces emanating from the rotating distant masses, are dragging the light in the direction of motion.

2. In a coordinate system that is rotating against the system of fixed stars, light is moving in agreement with the experiment of Sagnac.

Budapest, February 10, 1922

(Received February 28, 1922)

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1. Partially according to a paper of the author, submitted to the Kgl. Ung. Akademie der Wissensch. in the meeting of February 13, 1922
2. Sagnac, Paris, C.R. 157, 708 a. 1410, 1913
3. M. v. Laue, Münch. Ber. 1911. p. 404; Das Relativitätsprinzip, 3. Aufl. 1919; Ann. d. Phys. 62, 448, 1920
4. H. Thirring, this journal 19, 33, 1918; 22, 29, 1921
5. H. Thirring, this journal 19, 33, 1918 a. 22, 29, 1921; A. Kopff, this journal 22, 24. a. 179, 1921.
6. H. Weyl, Raum, Zeit, Materie IV. Aufl. S. 203