FROM CURRENT LITERATURE

New Experimental Confirmations of Consequences of the General Theory of Relativity

S. N. Vashlov.

1. One of the most important consequences of the general theory of relativity is the dependence of local time on the gravitational potential at a given place. An atom emitting fine spectral lines on the surface of a star may be regarded as a most precise clock, on which the influence of gravitation must inevitably be reflected; the greater the gravitational potential at the given point, the more slowly the clocks will run, and the more the line radiation of the atoms will be shifted toward the red side of the spectrum. If we denote by \(\nu_0\) the unaltered frequency of the light oscillations, and by \(\Delta \nu\) the change of frequency required by the general theory of relativity, then

\[ \Delta \nu = - \frac{k}{c^2} \cdot \frac{M}{r}\,\nu_0 . \tag{1} \]

Here \(k\) is the gravitational constant, \(M\) the mass of the star, \(r\) its radius, and \(c\) the velocity of light. The magnitude of the “red shift” even for the surface of the sun is extremely small (a few hundredths of an Å), but it is nevertheless accessible to spectroscopic observation. Numerous attempts to detect the “red shift” on the sun have led, however, almost always to rather indefinite results. In the gaseous envelope of the sun there are several causes which, in their turn, may produce a considerable change in the positions of spectral lines (relative motion in the solar atmosphere, pressure, anomalous dispersion). It is difficult to isolate the pure relativistic effect. In this sense solar spectral observations can hardly give entirely irreproachable proof of the presence of the red shift, or of its absence. On the other hand, it is very essential for the theory to clarify this point; in Einstein’s words, the absence of the red shift is tantamount to the unacceptability of the general theory of relativity. For unquestionable proof of the red shift it is necessary to find a place with a gravitational potential considerably greater than on the sun. To look for a star with a mass many times greater than the sun’s is apparently hopeless. Masses of the order of \(10^{33}\)—\(10^{34}\) g, as Eddington has shown, must be limiting. But the limitation of mass does not exclude wide variations in the density, or radius, of stars; in the universe there are giant stars and dwarf stars of approximately the same mass. The dwarf stars are a particularly convenient object for detecting the relativistic shift. Especially favorable in this respect is the companion of Sirius, as Eddington has shown1. This companion belongs to the type of “white dwarfs,” being a star in an early stage of development (spectral type \(F0\) or \(A5\)). From the elements of the companion’s orbit its mass and velocity relative to Sirius are determined.

The parallax of Sirius is well known. Knowing, in addition, the apparent stellar “magnitude” of the companion, one can, on the basis of these data, determine the absolute dimensions of the companion. In this way Eddington calculated for the companion a radius of 19,600 km and a colossal mean density of 53,000 (mass \(1.6 \cdot 10^{33}\) g). From these data one may expect on the surface of this star a relatively enormous red shift of several tenths of an Å. Adams, at the Mount Wilson Observatory,^1) made the corresponding spectral observations. The work was very difficult, since the companion’s own radiation is masked to a considerable extent by the light of Sirius (whose brightness is, roughly, 10,000 times greater). On the other hand, the inevitable presence of the scattered light of Sirius in the spectrograms makes it possible at once to compare the shifted lines of the companion with the unshifted lines of Sirius. On the basis of a photometric measurement of the spectrograms and after applying corrections connected with the superposition of the spectrum of Sirius upon the spectrum of the companion, Adams obtained the following figures for the displacement of the spectral lines:

The displacement is expressed here in \(\dfrac{\mathrm{km}}{\mathrm{sec}}\), corresponding to the equivalent Doppler shift. The value obtained must also be corrected for the velocity of the companion relative to Sirius, \(1.7 \dfrac{\mathrm{km}}{\mathrm{sec}}\). The corrected value will be approximately \(+21 \dfrac{\mathrm{km}}{\mathrm{sec}}\), or \(+0.32\) Å. From this, conversely, one can calculate the companion’s radius, 18,000 km, and density, 64,000.

Thus the consequence of the theory of relativity concerning the red shift has been satisfactorily confirmed. If, conversely, one assumes in advance the validity of this consequence, then a new method is obtained for determining the dimensions and densities of luminaries (of course only in the case when the mass of the luminary is known).

2. The initial experimental basis of the special theory of relativity is the negative result of the Michelson experiment and of other similar experiments. Attempts are made in various ways to undermine this basis. Criticism of the “classical,” ether theory of the Michelson experiment continues, although up to now in all cases this criticism has proved to be the fruit of one misunderstanding or another. In the well-known experiments of Miller,^2) it is as though the presence of a certain “ether wind” is detected when the interferometer is transferred to a height. These experiments are not yet finished, details have not been published, and the results are equally surprising from both the relativistic and the ether point of view. Judgment about these experiments, and the application to them of the theory of relativity or the theory of the ether, should more cautiously be postponed until the reliability of the results is finally clarified. Very often attempts are made to explain the negative results of Michelson by saying that the ether, at least in the layers adjacent to the earth, is carried along by the earth in its motion, just as the atmosphere is carried along. If this is so, then not only the annual translational motion of the earth will not be accompanied by an “ether wind,” but also the daily rotation will not be reflected in any way in the interferometer. As early as 1913 Sagnac^3) performed an interference experiment in a rotating system,

^1) W. S. Adams. Proc. of Nat. Washington, 11, 382, 1925.
^2) See Uspekhi Fizicheskikh Nauk, V, issue 3, p. 177, 1925.
^3) G. Sagnac. Journ. de Phys. Mars 1914.

when the light source, mirrors, and “observer” (photographic plate) rotated on a common platform; the experiment gave a positive result, i.e., a shift of the interference fringes of the magnitude that follows from the hypothesis of the stationary ether. At that time Sagnac regarded this result as a refutation of the theory of relativity, but it was soon clarified that there is no contradiction here with the theory of relativity. The special principle of relativity deals only with uniform rectilinear motion. Rotational motion is subject to the conduct of the general theory. The presence of acceleration is equivalent to the appearance of gravitational forces (in the generalized sense) that change the magnitude of the velocity of light. The application of the general theory of relativity to the case of uniform rotation of a solid body leads to the conclusion that the velocities of light here will be

\[ c' = c \pm r\omega \tag{2} \]

where \(c\) is the velocity of light in the case of uniform rectilinear motion, \(r\) is the distance of a point on the periphery from the axis of rotation, and \(\omega\) is the angular velocity1. The result in this case is the same as in the theory of the stationary ether. For uniform rectilinear motion, the principle of relativity and Lorentz’s ether theory equally establish the absence of first-order effects (relative to \(\frac{v}{c}\)); the distinguishing feature of the principle of relativity is the elimination of second-order effects.

In order that first-order effects may manifest themselves, the light ray must, in one way or another, be turned, “accelerated.”

For reconciling the positive results of Sagnac’s experiment with the negative results of Michelson’s experiment, it was necessary to suppose that Sagnac’s small apparatus does not carry along, in its rotation relative to the earth, the surrounding ether (Lenard). However, this position too lost its ground after the new Michelson–Gale experiment2, completed in 1925. This time the matter concerns an experiment with positive results.

Fig. 1.

The purpose of the experiment is to detect the diurnal rotation of the earth by optical means. The general theory of relativity and the hypothesis of the stationary ether require the presence of a first-order effect (2); the theory of the entrained ether requires the absence of the effect, as in Michelson’s first experiment. The experiment was arranged in the following manner (Fig. 1).

From west to east on the earth, water pipes \(CE\) and \(AF\), 30 cm in diameter, were laid. The lengths of \(CE\) and \(AF\) are 610 m. The pipes \(AC\), \(BD\), \(FE\), running from north to south, have a length of 340 m. With a single common pump, operated by a 50-horsepower motor, it was possible within 3 hours to pump the air out of the pipes to approximately 1 cm of pressure. A beam of light from an arc was split at the slightly silvered mirror \(A\) and, by means of the mirrors \(C, D, E, B, F\), could traverse the contours \(ABCDA\) and \(AFECA\) counterclockwise and clockwise. The adjustment of the mirrors was carried out with the help of rods protruding outward and provided with adjusting screws. When the air was pumped out of the pipes, the interference pattern was obtained quite distinctly.

The arms \(AF\) and \(CE\) run along different latitudes; accordingly, \(r \cdot \omega\) in formula (2) is different for these sections of the path of the light beam. The rays going counterclockwise and clockwise in the contour \(AFECA\) will meet at point \(A\) with a phase difference

\[ \Delta=\frac{4l(\psi_1-\psi_2)R}{c\cdot\lambda}\sin\psi \tag{3} \]

where \(l\) is the length of the arm \(AF\); \(\psi_1, \psi_2\) are the latitudes at which \(AF\) and \(CE\) are situated; \(\psi\) is the mean latitude of the place; \(R\) is the radius of the Earth; \(c\) is the speed of light; \(\lambda\) is the mean wavelength of the light source. To determine the displacement of the interference fringe, the auxiliary contour \(ABDCA\) is used, in which the arms \(AB\) and \(CD\) are very small and, consequently, \(\Delta\) is negligible. The light is passed once through this small contour, and a second time through the large one; the resulting displacement of the interference fringes then makes it possible to determine the phase difference (3). Expression (3), calculated on the basis of the hypothesis of an immobile ether or of the general theory of relativity in the Michelson–Gale apparatus, was equal to

\[ 0.236 \pm 0.002 \]

The observed value (the average of 13 groups of readings, with 20 measurements in each) was

\[ 0.230 \pm 0.005. \]

The numbers are given in fractions of the width of the interference fringe.

Thus a consequence of the general theory of relativity, or of the hypothesis of an immobile ether, has been confirmed with an enormous degree of accuracy.

For supporters of the ether theory, reconciling the negative results of the old and the positive results of Michelson’s new experiment is possible only on the assumption of a thin layer of ether carried along by translational motion and not carried along by the rotation of the Earth.

1. A simple derivation of formula (2) is given by C. Runge, Naturwissenschaften, 20, 440, 1925. ↩↩

2. A. A. Michelson and Henry G. Gale assisted by Fred Pearson, Astroph. Journ., 61, 137, 1925. ↩