Abstract
Presented on 18/XI 1925 in Halle at a meeting of the Gauverein Sachsen-Thüringen-Schlesien of the German Physical Society.
Full Text
NEW DATA FOR AND AGAINST THE THEORY OF RELATIVITY1
G. Joos.
Introduction. Around the theory of relativity a struggle has again flared up, but this time no longer on the barren ground of the question of the logical justification of the theory, but on a purely physical question: the experimental verification of the theory. The following coincidence is curious. In one of the issues of the Proceedings of the National Academy of Washington there was published an extremely interesting confirmation of the shift of spectral lines in a gravitational field toward the red end of the spectrum—a shift required by the general theory of relativity, but which until now had not yet received irreproachable confirmation. In the very next issue of these Proceedings there was published the result of a repetition of Michelson’s experiment, sharply contradicting the special theory of relativity. In addition, recently the phenomena of aberration have been subjected to discussion, in which apparent contradictions with the theory of relativity were discerned.
Proof of the Shift Toward the Red
Until now the searches for the shift of spectral lines predicted by Einstein had been limited to the study of the solar spectrum. However, the presumed influence of the gravitational field on the Sun is so small that it lies at the very limit of the accuracy of measurement. Moreover, other factors—for example, pressure—may cause analogous shifts of spectral lines and mask the expected effect. Recently K. F. Bottlinger, J. W. Weber, and A. S. Eddington have pointed to another luminary—the companion of Sirius—the conditions for studying whose spectrum are considerably more favorable for the search for the shift.
This star is one of the few about which we know sufficiently accurate data to determine, by elementary considerations, the order of magnitude of the gravitational field at its surface. We know,
that the period of revolution of this visual binary star is 49.3 years, and that the major axis of the companion’s orbit about the principal star is equal to 20 radii of the earth’s orbit. According to Kepler’s third law, it follows from these data that the total mass of the Sirius system exceeds the mass of the sun by 3.5 times. Moreover, it has been possible to measure not only the orbit of the companion relative to the principal star, but also the orbits of both stars relative to their center of gravity, which makes it possible to determine the ratio of the mass of the companion to the mass of the principal star; this ratio is \(2.5:1\). It follows that the mass of the companion is equal to the mass of the sun. To determine its radius, its brightness must be compared with that of the sun. The companion of Sirius, situated at a distance of 9 light-years from us, which corresponds to an annual parallax of \(0.37''\), is a star of magnitude 8.5. To compare its true brightness with the brightness of the sun, one must mentally transfer both these stars to the same standard distance, which is usually taken as corresponding to a parallax of \(0.1''\). The brightness of the companion of Sirius at this distance would be \((3.7)^2\) times less. Since, by definition, a decrease in the brightness of a star by \(2.512\) times corresponds to an increase in its magnitude by one unit, the absolute brightness of the companion of Sirius, i.e. its apparent magnitude at the standard distance, is equal to 11.34. The sun at the same distance from us would appear to us as a star of magnitude 4.9. Thus, the brightness of the companion of Sirius is \(2.5^{(11.3-4.9)} = 376\) times less than the brightness of the sun. If both these stars had the same temperature, and consequently the same brightness per unit surface, then from this one could conclude that the surface of the companion of Sirius is \(\frac{1}{376}\) of the surface of the sun. However, from the continuous spectrum and from the character of the line absorption spectrum, the temperature of the companion of Sirius is determined to be \(8000^\circ\), whereas the temperature of the sun is only \(5900^\circ\). Consequently, according to the Stefan–Boltzmann law, its surface brightness is
\[ \left(\frac{80}{59}\right)^4 \]
times greater than the surface brightness of the sun; therefore the just indicated value of its surface must be reduced by the same number of times, whence it follows that its radius amounts to 0.0285 solar radii, i.e. is equal to 79,800 km. Thus, the density of the companion of Sirius reaches the fantastic value of the order of 50,0001. Proceeding from several other considerations of stellar statis-
...of a [[unclear: word ending in “-stical”]] character, Seares1 calculated that the radius of the companion of Sirius should be 25,000 km if this star is assigned to spectral class F0, and 18,000 km if it is assumed to belong to class A5. With the aid of the values given for the quantities \(r\) and \(M\), the magnitude of the redshift according to Einstein’s formula
\[ \nu=\nu_0\left(1-\frac{\chi M}{c^2 r}\right) \]
is determined to be approximately 0.3 angstrom, as against 0.008 angstrom for the Sun.
Despite so considerable a magnitude of the shift, even under the most favorable position of the stars it is nevertheless very difficult to obtain flawless photographs of the spectrum of the companion of Sirius, since the spectrum of the main star is superposed on its spectrum, the brightness of the former exceeding that of the companion by 10 magnitudes. As a result, only four of the photographs taken by W. Adams2 at Mount Wilson proved suitable for measurements. In these photographs the spectrum of the companion was taken in the middle, while above and below it the spectrum of the main star was faintly taken, which made it possible to compare the wavelengths of the companion and of the main star. The density of the main star is so small that the influence of gravitation on it may be neglected in comparison with the influence of gravitation on the main star. The intensity of the spectrum of the main star superposed on the spectrum of the companion was still very small at the H\(\beta\) line; however, at H\(\gamma\) it was already of the same order of magnitude as the intensity of the companion’s spectrum. These relations are due to the higher temperature of the main star. Since the spectrum of Sirius was also superposed on the middle part of the photographs, then, when the intensities of the superposed spectra are equal, the measurements of the middle part of the photographs correspond to the arithmetic mean of the wavelengths of the companion and Sirius, and the true difference of the wavelength on the companion from the wavelength on Sirius became twice as large as the measured one. The corresponding corrections were introduced into the measurements at H\(\gamma\); these corrections are fully justified.
If the redshift is expressed in the equivalent velocity along the line of sight that produces an equal Doppler shift, then the magnitude of the shift of the H\(\beta\) line is on average \(26 \ \frac{\mathrm{km}}{\mathrm{sec}}\) (extreme values 31 and 17), while that of the H\(\gamma\) line, after the correction is introduced, is \(21 \ \frac{\mathrm{km}}{\mathrm{sec}}\).
In addition, some other lines were measured, including Mg\(^+\) 4481, which gave on average 22 km. The mean of all measurements is 23 km. From this one must subtract the actual velocity of the companion rel-
relative to Sirius, having reached, at the moment of observation, 1.7 km in the direction of the line of sight. Thus the shift toward the red proves to be equal to \(21 \frac{km}{sec}\), which corresponds to a change in wavelength of 0.32 angstrom. One might try to ascribe this shift to the influence of pressure; although, as is directly apparent from comparison with conditions at the surface of the earth, a considerable value of the gravitational potential is associated with a small pressure. Adams’s statement that the lines are very diffuse points to an influence of pressure. The relation between the broadening and the displacement of lines caused by pressure has been studied most carefully for the mercury line 2536 Å by Füchtbauer, G. Ioos, and O. Dinkelacker¹). The results of this investigation, at least as regards order of magnitude, can probably be applied to all spectral lines of non-hydrogen-like atoms. According to this investigation, the observed magnitude of the shift, 0.3 angstrom, may be caused by a hydrogen pressure of approximately 75 atmospheres. At the same time the line width would reach 2 angstroms, which agrees with Adams’s indications. Thus the shift of the lines not belonging to hydrogen is not conclusive. The matter is different, however, with the hydrogen lines. In contrast to other lines, whose displacement in an electric field is insignificant and proportional to the square of the field strength, the Stark effect in hydrogen lines is proportional to the field strength. In view of this it is to be expected that the broadening of hydrogen lines under the influence of pressure will be considerably greater, for this broadening is caused by intermolecular electric fields. And indeed, according to A. Rossi’s measurements²), although rather crude, the broadening of the Balmer lines (in emission) is so large that at 75 atm. their width would have to amount to several thousand angstroms. Thus the influence of pressure must be regarded as excluded. Since other causes capable of producing a shift toward the red are unknown to us, it appears legitimate to conclude that in the spectrum of the companion of Sirius we are in fact dealing with the sought-for influence of gravitation³).
¹) Chr. Füchtbauer, G. Ioos und O. Dinkelacker. Ann. d. Phys. 71, 204, 1923.
²) A. Rossi. Astrophys. Journ. 34, 299, 1911. Compare the theory of Holtsmark, J. Holtsmark. Ann. d. Phys. 58, 623, 1919, and Hülburt’s measurements relating to low pressures (approximately up to 400 mm) (O. Hülburt. Phys. Rev. 22, 24, 1923).
³) One might also try to suppose that the absorption of the Balmer lines occurs not in the same layer as the absorption of the other lines. Since, however, the shift of lines under the influence of pressure is under all conditions associated with their broadening, this possibility is ruled out if only the line width was the same in both spectra (hydrogen and non-hydrogen. Translator’s note). Unfortunately, Adams’s work contains no indications on this question. As for the relations between broadening and displacement under the influence of pressure, neither in Rossi’s cited work nor in Hülburt’s work are there any indications concerning the displacement of the maximum of the lines.
NEW DATA FOR AND AGAINST THE THEORY OF RELATIVITY
Aberration.
Lenard1 considered that the following circumstance contradicts the theory of relativity. If aberration depends only on the relative velocity of the observer and the light source, then spectroscopic binary stars should appear separate precisely when they are in opposition, for at that moment their velocities are oppositely directed and perpendicular to the line of sight. Since this does not occur, and since, on the other hand, the motion of the earth is manifested in aberration, it would therefore seem possible to detect absolute motion in this way. Thirring2 explained this question as follows. The theory of relativity does not deny the possibility of detecting accelerated motion relative to the inertial system of the fixed stars, which can be detected at least in Foucault’s pendulum experiment. Aberration in fact depends on motion with respect to this inertial system. For the consistent application of the general theory of relativity leads to the following result: for an observer at rest relative to the inertial system, the phenomena of aberration reduce only to the fact that he sees a moving luminous point at the place where it was \(\frac{r}{c}\) seconds ago (\(r\) is the effective distance of the point from the observer). Therefore, when observing binary stars, he simply sees each of them in an earlier phase. Conversely, with accelerated motion of the observer relative to the inertial system of the fixed stars, there occurs a curvature of the light rays which, in motion along the earth’s orbit, leads to aberrational phenomena coinciding with the requirements of the elementary theory.
In the same sense should also be interpreted the results of the interesting experiments of J. Stark3. Stark caused canal rays of hydrogen to fall upon a flat aluminum surface. Casting the image of the aluminum surface onto the slit of a spectrograph, Stark observed luminescence in the plane of this surface perpendicular to the direction of the canal rays. By observations in the longitudinal direction it was established that the hydrogen lines are emitted chiefly by moving particles. In addition to these particles, however, along the surface of the aluminum there were also mercury vapors, excited by the hydrogen canal rays and luminous in a state of rest. It turned out that in the light of the moving hydrogen rays the boundary of the aluminum surface was visible in exactly the same place as in the light
of the resting mercury atoms. If the normal to the wave surface of the moving H-rays, owing to the motion of their carriers, made with the normal to the wave surface of the light of the immobile atoms an angle of the order of \(\frac{v}{c}\), then, judging from the dimensions of the apparatus used for the observations, one would have had to expect a displacement of \(1.2\ \mathrm{mm}\). In reality, however, this displacement was in any case less than \(5 \cdot 10^{-2}\ \mathrm{mm}\). On the contrary, according to the considerations given above, there should be no displacement at all. The observer is at rest in an inertial system and sees the boundary of luminescence of the moving particles at the very place where these particles actually cease to emit, i.e. on the surface of the aluminum. At that very same place he also sees the boundary of luminescence of the mercury atoms.
Repetition of Michelson’s Experiment
Although all the facts set forth are subject to interpretation in favor of the theory of relativity, nevertheless, if the results obtained in the repetition of Michelson’s experiment on Mount Wilson are confirmed, this theory will still have to be regarded as disproved. The essence of this experiment is well known, so that we shall not dwell on its description. If one of the arms of the interferometer makes, with the horizontal component of the earth’s velocity (or the velocity of the “ether wind”\(^4\)), an angle \(\beta\), and if \(a\) is the angle between the direction of the ether wind in space and the normal to the plane of the apparatus, then according to the theory of the immobile ether, when the apparatus is rotated through \(90^\circ\), one should expect the following displacement of the interference fringes:
\[ \Delta \nu = \frac{2l}{\lambda}\left(\frac{v}{c}\right)^2 \sin^2 a \cos 2\beta \]
where \(l\) is the length of the arm, \(\lambda\) is the wavelength of light, and \(v\) is the velocity of the earth relative to the ether. As is known, the absence of the expected displacement in the experiment performed by Michelson and Morley in 1887 led to the creation of the theory of relativity. In 1904 and 1905 Morley and Miller\(^1\) repeated this experiment, using an apparatus of considerably larger dimensions, in which the length of the arm reached 32 meters (thanks to the use of multiple reflection, the effective length of the arm was reduced to 4 m). In this experiment too, the displacement of the fringes upon rotation of the apparatus lay within the limits of measurement error and was considerably less than the expected displacement, so that the result of the experiment was recognized as negative. Using the very same appara—
\(^1\) E. W. Morley and D. C. Miller. Phil. Mag. 9, 669 and 680, 1905.
NEW DATA FOR AND AGAINST THE THEORY OF RELATIVITY
... D. Miller1 repeated the experiment at an altitude of 1800 m on Mount Wilson, and, to the greatest amazement, found a displacement of the fringes reaching \(1/3\) of the expected value. This result was at first ascribed to errors of observation. Therefore the whole apparatus was rebuilt, and, in order to exclude magnetic effects, the iron parts were removed with the greatest care. But even with the new interferometer, a displacement of the fringes on Mount Wilson was again detected. This apparatus was then transferred to the plain, to Cleveland, where in 1922 and 1923 numerous observations were made, all of which gave a negative result. Beginning in September 1924, observations were again made on Mount Wilson, the apparatus being installed at a new site. Again a displacement was found of approximately the same magnitude as in 1921. The longest series of measurements was carried out in March and April 1925, when in all 1600 separate measurements were made. In Fig. 1 are plotted the speed and direction of the ether wind at various hours of sidereal time (from observations in April 1921 and April 1925). In Fig. 2 is plotted the azimuth of the ether wind observed in April 1925 as a function of sidereal time. With some skill, a sinusoidal curve can be drawn through the observed points. The final result of the experiment comes down, according to Miller2, to the conclusion that on Mount Wilson there is a relative motion of the earth and the ether with a speed of \(10 \frac{\mathrm{km}}{\mathrm{sec}}\).
Fig. 1.
Fig. 2.
What can be said about this result? Judging from the report on the experiments, it is difficult to suppose that there could have been gross sources of error, for, apparently, constancy of temperature and immobility of the installation were controlled by every conceivable means. Nevertheless, it is as yet impossible to form a reliable judgment. Further, one may ask whether the results are
1 D. C. Miller. Phys. Rev. 19, 407, 1922.
2 D. C. Miller. Proc. Nat. Ac. Washington 11, 314, 1925. D. C. Miller. “Ether Wind.” — Uspekhi Fizich. Nauk 5, p. 177, 1925.
G. I O O S
Miller’s, as such, clear and intelligible (einleuchtend)? In this respect we first of all encounter two doubts. 1) If on the earth’s surface at sea level the effect is absent, then, since the full magnitude of the relative velocity of the earth and the ether is equal to \(30 \frac{\text{km}}{\text{sec}}\), even from the point of view of a partially entrained ether it is difficult to understand how the relative velocity can increase so monstrously rapidly with height (up to \(1/3\) of the full value on Mount Wilson). If, however, as will be said below, the full velocity must be taken to be equal to several hundred kilometers per second, then this increase becomes not so incredibly rapid as at a velocity of \(30 \frac{\text{km}}{\text{sec}}\).
- The author’s remark raises doubt that no noticeable change in the magnitude of the effect as a function of the time of year was detected, and that, on the contrary, the velocity and direction of the ether wind are apparently determined by the sidereal time of observation. If the effect depended only on the velocity of the earth in its orbit around the sun, then the point of the sky from which the ether wind blows could easily be determined by the following considerations. The direction of the relative velocity of the earth and the ether must in any case lie in the plane of the orbit; in view of the approximately circular form of the orbit, the direction of the velocity must be perpendicular to the direction toward the sun. Since the projection of the earth’s orbit onto the sky coincides with the ecliptic, the sought point must lie on the ecliptic at a distance of \(90^\circ\) from the sun. It follows from this that this point, at different times of the year, lies in different regions of the starry sky. Thus it is completely incomprehensible how the direction of the ether wind can be a function of sidereal time, independent of the time of year. Matters will be otherwise if, upon the motion of the earth in its orbit, there is superposed also the general motion of the whole solar system, whose direction, of course, does not change in the course of the year. The resultant of the two velocities will depend on the time of year the less, the greater the general velocity of the solar system is in relation to the velocity of the earth in its orbit. Therefore, in order to explain the insignificance of the influence of the time of year, it is necessary to assume that the velocity of the solar system reaches several hundred kilometers per second. As for the velocity of the sun with respect to the fixed stars, it is comparatively reliably known that this velocity is equal to \(20 \frac{\text{km}}{\text{sec}}\), and, it would seem, is in no case sufficient for explaining the results of Miller’s experiment. However, at the present time in astronomy it is accepted that spiral nebulae are the most distant worlds, coordinated with our Milky Way system. Proceeding from the magnitude of the Doppler effect established in the observation of these nebulae, one must reckon with the motion of our Milky Way system
the paths relative to other Milky Way systems, the speed of which reaches approximately \(200 \frac{\mathrm{km}}{\mathrm{sec}}\)¹).
Repetition in the Mountains of the Trouton–Noble Experiment.
In view of the fundamental significance of Miller’s results, it is extremely necessary to verify them. For a new repetition of Michelson’s experiment, large resources and much time are required. On the other hand, Miller’s results make it desirable to repeat, at high altitude, a number of other experiments bearing on the theory of relativity and hitherto carried out only at sea level. First of all one should mention the experiment of Trouton and Noble²), which was intended to establish the existence of a rotating moment of forces, acting, according to the ether theory, on a freely suspended charged condenser and proportional to
\[ \left(\frac{v}{c}\right)^2 . \]
This experiment was prepared in Jena and, as R. Tomaschek kindly reports, was carried out this autumn on the Jungfrau Pass (Jungfraujoch). Although the sensitivity of the apparatus was 20 times greater than the sensitivity of the apparatus used by Trouton and Noble, and although with its aid it would easily have been possible to detect a “relative” velocity of \(3 \frac{\mathrm{km}}{\mathrm{sec}}\), nevertheless no effect was observed. Since the negative result of this experiment follows from the law of inertia of all forms of energy, this law must therefore in any case be considered valid, even if Miller’s results are confirmed.
Conclusion. Since, in order to refute the theory of relativity, it is sufficient to detect the translational motion of the earth in one single experiment, then although the other works set forth testify against the probability of the results obtained by Miller, they nevertheless can never eliminate the significance of these results, provided only that the latter are reliably established. In that case, however, not only the theory of relativity would be refuted, but also Lorentz’s theory of the immobile ether, which excludes all first-order effects with respect to \(\frac{v}{c}\). Therefore it seems not superfluous, before repeating
¹) J. Weber, in Phys. Zeitschr. 27, 5, 1926, points also to the following contradiction. From whatever point of the sky the ether wind may blow, the extreme values that its azimuth assumes in the course of a day, owing to the rotation of the earth about its axis, must be symmetrical with respect to the direction of the axis of rotation, i.e. with respect to the south–north line. Meanwhile, as is clear from Fig. 2, the azimuth values measured by Miller do not satisfy this requirement (a sharply expressed displacement of the azimuth curve downward, i.e. toward the west). Translator’s note.
²) F. T. Trouton and H. R. Noble. Phyl. Trans. A. 202, 165.
of Michelson’s experiment to repeat in the mountains at least one of the very easily feasible first-order experiments, for example, Ketteler’s interference experiment1. We shall not dwell on the picture of the chaos into which, if Miller’s results are confirmed, all theories of the electrodynamics of moving bodies will be plunged—except, it is true, for the Stokes–Planck theory2 of a compressible ether free from vortices. One may perhaps even doubt whether in that case a unified interpretation of electrodynamic and optical phenomena in moving bodies would be possible; but, on the other hand, in that case there would open up the unexpected possibility of measuring, under laboratory conditions, the motion of the Earth relative to the distant worlds of the spiral nebulae—a motion whose order of magnitude alone can be determined by astronomical methods.