On the Explanation of Stellar Aberration in the Theory of Relativity

B. N. Gimmel'farb

1. On the Repetition of Old Errors in New Editions

The explanation given by the theory of relativity for the phenomenon of stellar aberration was subjected to detailed discussion in the 1920s, when opponents of the theory of relativity put forward this explanation as allegedly leading to contradictions with observations and therefore capable of serving to refute the theory (see, for example, ^1). In the course of the discussion it became clear that the explanation of stellar aberration presents no fundamental difficulties for the theory of relativity, and that the source of these apparent contradictions is erroneous reasoning in which, first, the actual conditions of astronomical observations leading to the detection and measurement of stellar aberration are not taken into account, and, second, the conclusions of the special theory of relativity are extended to cases of non-inertial motion, where they are essentially inapplicable.

The question was, in the main, clarified—and moreover in a sufficiently elementary form—in Emden’s article ^2, where it is considered within the framework of the special theory of relativity. However, even this article by Emden is not free of incorrect assertions. In addition, in certain new editions, including such authoritative ones as the Great Soviet Encyclopedia, 2nd ed. ^3, and others, the assertion is repeated that stellar aberration depends on the motion of the light source relative to its receiver, the erroneousness of which was clarified more than 25 years ago during the discussion mentioned above. The confusion that exists in the literature on this question still serves as a pretext for fruitless attempts to refute the theory of relativity*).

) See, for example, the article by S. B. Luk'yanov, Astron. zhurn. 30*, 302 (1953).

These circumstances, too, impel us once again to turn to the consideration of the question raised.

That stellar aberration does not depend on the motion of the light source is most clearly shown by an example that was also cited during the discussion in the 1920s. This example concerns phenomena observed in spectroscopic binary stars, and is based on the fact that the aberrational displacement of the apparent position of a star depends on the component of velocity perpendicular to the line of sight, while the Doppler displacement of the lines in the star’s spectrum depends on the radial component of its velocity.

If stellar aberration were determined by the velocity of the light source relative to the observer’s frame of reference, then in spectroscopic binary stars there would be observed alternately either a splitting of the spectral lines or a visual separation of the components of the binary star. Indeed, in position I (Fig. 1) the difference of the radial components of the velocities has its greatest value, and consequently the splitting of the spectral lines reaches its greatest magnitude, while in position II the difference of the components of velocity perpendicular to the line of sight has its greatest value and, according to the supposition, the apparent positions of the components of the binary star should undergo the greatest aberrational displacement in opposite directions. This effect could not have escaped observation, since at velocities of the order of tens and hundreds of km/sec, observed in spectroscopic binary stars, the apparent separation of the components of a binary star would have had to reach tens and hundreds of seconds of arc*), i.e. a magnitude noticeable even in small telescopes and, moreover, one not depending on the distance to the star (since distance does not enter into the formulae for stellar aberration and the Doppler effect, but only velocities do). Such an apparent separation of the components in spectroscopic binary stars could not be taken as real, because it would have to occur precisely at those moments when spectroscopic data indicate that the angular distance between the components of the binary star has its smallest value.

Fig. 1.

*) This displacement, expressed in seconds of arc, may approximately be found from the formula

\[ \alpha = \frac{v}{c \sin 1''} = 0.7v, \]

where \(v\) is the relative velocity of the components of the binary star, expressed in km/sec; \(c\) is the speed of light.

Lenard’s and others’ arguments⁴ amounted to the following. The theory of relativity recognizes only the mutual motion of material bodies; in the phenomenon of stellar aberration two bodies figure: the observed star and the Earth, and therefore what should manifest itself here is the motion of the observed star with respect to the Earth; but since the example cited directly testifies that the star’s motion is irrelevant here, stellar aberration, consequently, can serve to reveal the absolute motion of the Earth and thus constitutes a refutation of the basic postulate of the theory of relativity.

Lenard’s and his followers’ error lay in the very starting point of their reasoning, since they ascribed to the theory of relativity an incorrect assertion which in no way follows from it: that stellar aberration depends on the motion of the observed luminary with respect to the telescope. It must be said, however, that such an assertion is quite often encountered in the literature on the theory of relativity in a more or less explicit form (for examples see in Emdén’s cited article²). In the well-known popular book On the Special and General Theory of Relativity, A. Einstein wrote: “I shall mention here, as particularly important, the fact that the theory of relativity explains in a very simple way the influence of the Earth’s motion relative to the fixed stars on the light sent by these stars to us. Such are the annual motion of the apparent position of the stars, caused by the motion of the Earth around the Sun (aberration of light), and the influence of the radial component of the motion of the fixed stars relative to the Earth on the color of the light reaching us... (Doppler principle).”⁵ This passage was cited as evidence that the founder of the theory of relativity himself also adhered to the opinion that stellar aberration depends on the motion of the light source with respect to its receiver.⁶ However, in his time Thirring⁷ explained that what is meant here is the motion of the Earth not with respect to each given star, but with respect to the totality of all stars, and that the possibility of detecting by means of physical experiments the nonrectilinear motion of the Earth with respect to this inertial system of “fixed stars” does not contradict the theory of relativity.

In fact, it follows from the theory of relativity that a change in the frequency of light (the Doppler effect) and in the direction of a light ray (stellar aberration) occurs in passing from any reference system to any other moving with respect to the first. Which particular reference systems’ mutual motion appears in one case or another depends on the concrete conditions of the experiment (observations). Thus, in spectroscopic investigations of the motions of celestial bodies, the velocity of the luminary with respect to the Earth usually appears, since the position of lines in the spectrum of a given luminary is compared with the position of the corresponding lines in the spectrum of a terrestrial source, at rest with respect to the spectroscope. But already

In the example considered of observing a spectroscopic binary star, the motion of the Earth as a reference system is excluded, since the relative positions of lines in the spectra of the two components of the binary star are being compared, and thereby the velocity of motion of these components relative to one another is measured. Conversely, in observing stellar aberration the motion of the luminary is excluded, and what matters is the velocity of the Earth’s orbital motion (in the case of the annual aberration of stars), or the linear velocity of the telescope in the Earth’s daily rotation (in the case of diurnal aberration).

Here, of course, what is meant is not the “absolute” motion of the Earth, i.e., its motion with respect to a hypothetical luminiferous medium. The annual aberration of stars is the difference between the direction of a light ray in a reference system connected with the Earth’s center of gravity and the direction of the same ray in a reference system connected with the Sun’s center of gravity. The diurnal aberration is the difference between the direction of a light ray in the reference system in which the telescope is at rest and the direction of the same ray in a reference system connected with the center of the Earth. In both cases, stellar aberration is manifested in the change with time of the direction of the light ray and, correspondingly, in the change with time of the apparent position of the luminary, which in turn is caused by the change in the direction (and, in the case of annual aberration, also in the magnitude) of the velocity of motion of the reference system connected with the telescope.

This, naturally, follows from the fact that the aberration of stars always exists only as an effect of relative motion. Therefore it is observed only in those cases when the telescope passes from one inertial system into another, moving relative to the first. The change in the direction of the telescope’s motion is precisely equivalent to its transition from one instantaneous inertial system to another, which also accounts for the change in the direction of the light ray recorded by it. In those cases where the velocity of motion of the reference system remains unchanged in magnitude and direction for a very long time—as, for example, in the case of the motion of the Earth together with the entire solar system with respect to the system of “fixed stars”—the “secular aberration” caused by this motion can only be calculated, but cannot be measured directly.

From the considerations adduced it follows that the aberration of stars, in a certain respect, represents an optical analogue of Foucault’s pendulum experiment: both determine a certain plane in space which remains fixed in an inertial reference system and turns in a rotating reference system. In the mechanical experiment such a plane is the plane of oscillation of a freely suspended pendulum, while in the optical experiment it is the plane passing through the light ray and the vector of the instantaneous velocity of motion of its receiver (the telescope).

2. STELLAR ABERRATION AND PLANETARY ABERRATION

From what was said in the preceding section it follows that, from the point of view of the theory of relativity, the aberration of stars in an inertial system exists not in itself, but only in relation to another inertial system taken as stationary; moreover, the motion of the luminary relative to each of the reference systems under consideration is of no significance. According to the established terminology, the position of a luminary on the celestial sphere, altered by stellar aberration, is called in astronomy the apparent place, while the one freed from the influence of stellar aberration, i.e. corresponding to the reference system taken as stationary, is called the true place. It should be noted that such terminology by no means reflects the actual state of affairs, since the “true” position of the luminary in the indicated sense is no less illusory than its apparent position, for it corresponds not to the direction of the straight line joining the simultaneous positions of the luminary and of the crosshairs in the telescope eyepiece, but only to the direction of the light ray in the reference system taken as stationary. During the interval of time in which light traverses the distance from the luminary to the telescope, i.e. during the so-called aberration time, the luminary has time to move away from the point at which it was located at the moment of emission of the light that reaches the telescope at the moment of observation. This displacement of the luminary during the aberration time is called planetary aberration. At the distances encountered in astronomy, measured in many light-years, this displacement may be very considerable.

Thus, the essence of stellar aberration consists in the difference in the direction of the light ray in different reference systems moving relative to one another, while the essence of planetary aberration consists in the difference between the direction of the light ray and the straight line joining the simultaneous positions of the luminary and the telescope. Therefore the sharp opposition drawn in S. N. Blazhko’s Course of Spherical Astronomy[^8] between the “true” position, as if true in the full sense of the word, and the apparent position, which the cited author proposes to call “seeming,” and the use there of the expression “direction from the observer to the luminary” without a clear indication that what is meant is the direction of the ray of light coming from the given luminary, and not the direction of the straight line joining the simultaneous positions of the luminary and the telescope, can only mislead the student. Such a danger was avoided by the author of the older analogous textbook S. A. Kazakov,[^9] who distinguishes the true position of a luminary (i.e. one freed from the influence of stellar aberration) from its actual position, obtained as a result of also taking planetary aberration into account.

In view of the fact that stellar aberration does not depend on the motion of the light source, it must also exist in a moving frame of reference relative to which the light source is at rest. In saying this, when we speak of a moving frame of reference, we mean its motion relative to some other inertial frame of reference that we take to be at rest.

Let us consider two frames of reference \(K\) and \(K'\), moving uniformly and rectilinearly toward each other along the straight line \(MN\) (Fig. 2).

Fig. 2.

Let the light source \(S\) be at rest with respect to the frame of reference \(K\), which moves with velocity \(v\) relative to the frame of reference \(K'\). At the moment when the origins of coordinates of both frames of reference coincide at the point \(P\), the position of the luminary \(S\) is measured in both systems. A light ray which in the frame of reference \(K\) has the direction \(SP\) will, in the frame of reference \(K'\), have the direction \(S'P\).

Since both frames of reference under consideration are inertial, they are physically equivalent. Between them there exists a relation of complete reciprocity: each of them may with equal right be regarded as at rest, and the motion of the other may be considered relative to it. Then stellar aberration will be absent in the frame of reference taken as at rest, and will exist in the second frame, moving relative to the first.

An observer located in the frame of reference \(K\) and regarding it as at rest sees the luminary at the point \(S\) and considers this point to be the true position of the luminary, since in a stationary frame of reference stellar aberration is absent. The position \(S'\), at which the same luminary is seen by an observer moving with velocity \(-v\) together with the frame of reference \(K'\), is regarded by the observer in the frame of reference \(K\) as apparent, displaced as a consequence of stellar aberration caused by the motion of the frame of reference \(K'\).

But an observer located in the frame of reference \(K'\), which is also inertial like the frame of reference \(K\), has the right to regard his own frame of reference as at rest, and the frame of reference \(K\) as moving. Taking his own frame of reference to be at rest, the observer in the system of reference \(K'\) ...

system \(K'\) takes the point \(S'\), at which he sees the luminary, to be the true position of this luminary, since in a stationary reference system stellar aberration is absent. The position \(S\), in which the same luminary is seen by an observer moving with velocity \(v\) together with the reference system \(K\), is regarded by an observer in the reference system \(K'\) as apparent, displaced as a consequence of the stellar aberration caused by the motion of the reference system \(K\).

But with respect to the reference system \(K\) the luminary \(S\) is at rest. Therefore its true position, determined in this reference system, is its actual position at the moment of observation: the planetary aberration of the luminary \(S\) with respect to the reference system \(K\) is absent. With respect to the reference system \(K'\), however, the luminary \(S\) moves with velocity \(v\). Therefore its “true” position \(S'\), determined in the reference system \(K'\), is not its actual position at the moment of observation: it indicates the direction in which, in the reference system \(K'\), the ray of light came from the luminary to the point \(P\), but not the direction of the straight line connecting the point \(P\) with the luminary at the moment of observation. During the interval of time in which light traversed the distance from the luminary to the point \(P\), i.e. during the aberration time, the luminary had time to move from the point \(S'\) to the point \(S\).

Consequently, from the point of view of the reference system \(K'\), the visible position of the luminary \(S\), determined from the reference system \(K\), which moves together with the luminary, coincides with the actual position of the luminary at the moment of observation, but this fact is explained not by the absence of stellar aberration in the reference system \(K\), but by planetary aberration, caused by the motion of the luminary itself and compensating for the influence of stellar aberration in the reference system that moves together with the luminary.

From the preceding considerations it follows that stellar aberration, which occurs if the reference system connected with the luminary is taken as stationary, turns into planetary aberration if the reference system connected with the telescope is taken as stationary. Such a reciprocity of stellar and planetary aberration exists only with respect to inertial reference systems. In the general case the fundamental difference between stellar and planetary aberrations consists in the fact that stellar aberration depends on the instantaneous velocity of the observer’s reference system and does not depend on the distance between the luminary and the telescope, whereas planetary aberration depends on the finite displacement of the luminary during the aberration time, which depends on this distance. Namely,

\[ \tau=\frac{r}{c}, \]

where \(\tau\) is the aberration time, \(r=S'P\), \(c\) is the speed of light.

However, a non-inertial system, strictly speaking, cannot be treated on the same footing as inertial systems as certainly stationary,\(^{10}\) and therefore with respect to non-inertial systems the reciprocity of the phenomena is absent. Forgetting this circumstance leads

to imaginary paradoxes of relativity, an example of which may be the well-known “clock paradox,” recently once again discussed in the pages of Nature^11, ^12.

Thus, the apparent position of a luminary is determined by stellar aberration (aberratio fixarum), which depends on the instantaneous velocity of the frame of reference connected with the telescope, while the actual position of the luminary at the moment of observation is determined by planetary aberration (aberratio planetarum), depending on the finite displacement of the luminary itself during the aberration time; moreover, the motion of both the luminary and the telescope is referred to some inertial frame of reference taken as stationary.

3. ON THE DIFFERENCE BETWEEN THE RELATIVISTIC TREATMENT OF STELLAR ABERRATION AND THE CLASSICAL ONE. RAY CONSIDERATION

If \(K'\) is an inertial frame of reference moving with velocity \(v\) relative to the inertial frame of reference \(K\), then a ray of light which in the frame \(K\) makes an angle \(\vartheta\) with the direction of the relative motion of these frames will make in the frame \(K'\) an angle \(\vartheta'\) with the same direction, the relation between the angles \(\vartheta\) and \(\vartheta'\) being given by the formula

\[ \operatorname{ctg}\vartheta' = \frac{\operatorname{ctg}\vartheta+\beta\operatorname{cosec}\vartheta} {\sqrt{1-\beta^{2}}}, \tag{1} \]

where, as usual, \(\beta=\dfrac{v}{c}\), \(c\) is the speed of light. Usually this formula is obtained by applying the Lorentz transformations to the equation of the trajectory of a light ray^13, or as a special case of Einstein’s velocity-addition theorem, when one of the velocities being added is the speed of light^14. We give here a simplified derivation of formula (1), based on the fact of contraction of the length scale in the direction of relative motion. Such elementary derivations are always of pedagogical interest, not only because they make the consequences of the theory accessible to the understanding of the less prepared reader, but also because in many cases they make it possible to reveal more clearly the physical meaning of the quantities entering into the discussion, a meaning that is often obscured by mathematical calculations in more rigorous derivations.

Fig. 3.

Let us denote (Fig. 3) the segments \(AD\) and \(CD\), by means of which the angle \(\vartheta'\) is determined, respectively by \(x\) and \(y\), if they are measured in the scales of the frame of reference \(K\), and by \(x'\), \(y'\)—in the scales of the frame

Toward an Explanation of the Aberration of Stars in the Theory of Relativity

of reference \(K'\). In view of the fact that the angle \(\vartheta'\) is measured in the system \(K'\), the segments by means of which it is determined must be expressed in the scales of this system of reference:

\[ \operatorname{ctg}\vartheta'=\frac{x'}{y'} . \tag{2} \]

The system of reference \(K'\) moves with velocity \(v\) relative to the system of reference \(K\). Therefore, between the unit scale of length \(\xi\) in the direction of the relative motion of the systems of reference and \(\eta\) in the direction perpendicular to it in the system of reference \(K\), and the corresponding unit scales of length \(\xi'\) and \(\eta'\) in the system of reference \(K'\), there exist the relations

\[ \xi'=\xi\sqrt{1-\beta^2}; \qquad \eta'=\eta . \tag{3} \]

In an elementary exposition these relations may be presented as an experimental fact, following, for example, from the negative result of the Michelson experiment. Further, we have the obvious relations

\[ x\xi=x'\xi'; \qquad y\eta=y'\eta', \tag{4} \]

whence, taking into account relations (3),

\[ x'=\frac{x}{\sqrt{1-\beta^2}}; \qquad y'=y . \tag{5} \]

Substituting from here \(x'\) and \(y'\) into (2), we have:

\[ \operatorname{ctg}\vartheta'=\frac{x}{y\sqrt{1-\beta^2}} . \tag{6} \]

Let us now express \(x\) and \(y\) through functions of the angle \(\vartheta\):

\[ x=c\cos\vartheta+v; \qquad y=c\sin\vartheta \tag{7} \]

and substitute into (6). Then we obtain:

\[ \operatorname{ctg}\vartheta'=\frac{c\cos\vartheta+v}{c\sin\vartheta\sqrt{1-\beta^2}}, \]

or

\[ \operatorname{ctg}\vartheta'=\frac{\operatorname{ctg}\vartheta+\beta\operatorname{cosec}\vartheta}{\sqrt{1-\beta^2}} . \]

This is the relativistic formula for the aberration of stars (1).

Expressing the segments by means of which the angle \(\vartheta'\) is determined in the scales of the system of reference \(K\) (the angle \(\vartheta'\) so determined will be denoted by \(\vartheta'_0\)), we obtain the classical formula for the aberration of stars:

\[ \operatorname{ctg}\vartheta'_0=\frac{x}{y}=\frac{c\cos\vartheta+v}{c\sin\vartheta}, \]

whence

\[ \operatorname{ctg}\vartheta'_0=\operatorname{ctg}\vartheta+\beta\operatorname{cosec}\vartheta . \tag{8} \]

Formula (8) differs from formula (1) only by the absence of the factor \(\sqrt{1-\beta^2}\) in the denominator of the right-hand side. As is evident from the derivation given, this factor appears as a result of the contraction of the length scale in the direction of motion of the reference system, with the scale unchanged in the perpendicular direction.

Introducing the angle of aberration \(\alpha=\vartheta-\vartheta'_0\), formula (8) is easily transformed into the usual form:

\[ \sin\alpha=\beta\sin\vartheta'_0 . \tag{9} \]

This formula can be obtained from the drawing (Fig. 3) directly.

Consequently, the classical formula of aberration corresponds to a description of the phenomenon as it occurs from the point of view of a reference system assumed to be stationary.

In view of the fact that in all cases actually encountered in astronomy \(\alpha\) is a small angle, usually in formula (9) its sine is replaced by its arc, and in the right-hand side the difference between \(\vartheta\) and \(\vartheta'_0\) is neglected. Then

\[ \alpha''=k\sin\vartheta, \tag{10} \]

where \(\alpha''\) is the aberration angle, expressed in seconds of arc, and \(k=\dfrac{\beta}{\sin 1''}\) is the so-called aberration constant.

Substituting \(\vartheta=90^\circ\) into formula (10), we see that the aberration constant is equal to the aberrational displacement of a star whose ray comes at right angles to the direction of motion of the telescope. Setting \(\vartheta=90^\circ\) in formula (1), we obtain:

\[ \operatorname{ctg}\vartheta'=\frac{\beta}{\sqrt{1-\beta^2}} . \]

This quantity, which also characterizes the aberrational displacement of a star situated at an angular distance \(\vartheta=90^\circ\) from the observer’s apex, may be called the aberration constant with relativistic correction. However, the aberrational displacement of a star situated at an arbitrary angular distance \(\vartheta\) from the observer’s apex is not expressed by the formulas of the theory of relativity as simply through the aberration constant as by the classical formula (10).

It should be mentioned that, in a note printed in the journal Priroda,\(^{15}\) we gave an extremely elementary derivation of the aberration constant with the relativistic correction directly from the fundamental propositions of the special theory of relativity, without references to any formulas, since the necessary transforma-

the time interval is obtained there along the way from considering the phenomenon of stellar aberration itself*).

Expanding formula (1) in a series in powers of \(\beta\) and comparing it with formula (8), we see that the aberrational displacement required by the theory of relativity differs from the classical one in terms of order \(\beta^2\):

\[ \operatorname{ctg}\vartheta'=\operatorname{ctg}\vartheta+\beta\operatorname{cosec}\vartheta+\frac{1}{2}\beta^2\operatorname{ctg}\vartheta+\cdots, \tag{11} \]

while the relativistic correction to the aberration constant has a magnitude smaller by yet another order:

\[ \frac{\beta}{\sqrt{1-\beta^2}}=\beta+\frac{1}{2}\beta^3-\cdots \tag{12} \]

In the case of annual aberration of stars, \(\beta=10^{-4}\); consequently,

\[ \frac{\beta^2}{\sin 1''}\sim 0''.001, \]

which lies beyond the limits of accuracy of astrometric measurements. Moreover, the determination of the aberration constant is connected with special difficulties, as a result of which even the most accurate determinations of it are not free from systematic errors. Therefore, at the international conference on astronomical constants held in Paris in 1950, it was decided to exclude it from the number of fundamental astronomical constants determined directly from observations, and in the future to derive its numerical value from the parallax of the Sun\(^{16}\).

4. ON THE DIFFERENCE BETWEEN THE RELATIVISTIC TREATMENT OF STELLAR ABERRATION AND THE CLASSICAL ONE. WAVE CONSIDERATION

Up to now we have considered only the kinematic (ray) aspect of the phenomenon. The distinction between the classical and relativistic explanations of stellar aberration appears more distinctly when the wave aspect of the phenomenon is considered.

From the point of view of the classical theory (ether theory), aberration serves as evidence of the motion of the telescope relative to a hypothetical light-carrying medium. Such motion leads to a lag of the light waves that have passed through the entrance aperture of the instrument,

*) It should be noted that in the indicated note\(^{15}\) the inaccuracy mentioned in the first section of the present article is repeated: in the reduction presented there, the system of reference associated with the Earth and the system of reference associated with a star are considered, whereas in reality the second of the systems of reference under consideration must be the system associated with the Sun. Moreover, the derivation given there implicitly includes the assumption that length scales do not undergo Lorentz contraction in the direction perpendicular to the direction of motion of the system of reference.

in the direction opposite to the direction of motion of the instrument (“ether wind”), as a result of which there is produced an inclination of the wave normal n to the ray r, equal to the aberrational rotation of the ray (Fig. 4). Such an inclination is a natural consequence of the fact that the wave surfaces in all reference systems have the same position, whereas the direction of the light ray depends on the velocity of motion of the telescope receiving it. The presence of an inclination of the wave normal to the direction of the light ray distinguishes a moving reference system from a resting one and, consequently, may serve as evidence of the absolute motion of the reference system in which the observations are made.

Fig. 4.

From the standpoint of the special theory of relativity, all inertial reference systems are physically equivalent, and no inclination of the wave normal to the ray is observed in all such systems. This is explained by the fact that a wave surface is a surface of simultaneous equal phases, and events simultaneous in one reference system are not simultaneous in other reference systems moving relative to this system. Therefore, in different reference systems moving relative to one another, the wave surfaces occupy different positions, just as the direction of the light ray also differs, depending on the velocity of motion of the reference system.

Thus, one may say that in the classical explanation of stellar aberration the inclination of the wave normal to the ray arises because of the absolute character of simultaneity, as a result of which events simultaneous in any one reference system are also simultaneous in all other reference systems; whereas in the relativistic explanation of the phenomenon such an inclination is absent because of the relative character of simultaneity, as a result of which events simultaneous in one reference system are not simultaneous in other reference systems moving with respect to the first.

Proceeding from the fact that, as was said above, in an inertial system stellar aberration exists not in itself, but only with respect to other similar systems, Emden, in the cited article², says that according to the special theory of relativity there is in general no stellar aberration. We see that this assertion is confirmed by considering the wave aspect of the phenomenon. But in the same article Emden asserts that in cases of uniform and rectilinear motion of the reference system stellar aberration

in it cannot be detected by observations from the point of view of ether theory to the same extent as from the point of view of the theory of relativity, since “only mechanically, and not optically, does the given standard direction make possible the measurement of aberration.” The latter assertion is true only insofar as the detection and measurement of the aberration of stars by means of astrometry is presupposed, which is possible only under the condition of a change with time in the direction or velocity of motion of the reference system. But this assertion is in general incorrect, since the classical explanation of the aberration of stars leads to the existence of an inclination of the wave normal to the direction of the light ray and thereby can serve as the basis for an interference experiment in which the expected effect lies within the limits of accuracy attainable experimentally. In such an experiment, from the classical point of view, the wave normal appears precisely as an “optically given standard direction,” common to all reference systems and coinciding with the light ray in the reference system that is at rest relative to the luminiferous medium.

Although the inclination of the wave normal to the ray is of order \(\beta\), it is easy to show that here too, as in general in all cases where differences between the classical and relativistic treatments of phenomena are manifested, only quantities of order \(\beta^2\) prove accessible to observation. Let us consider a scheme of a possible experiment for determining this inclination at the maximum angle of aberration, corresponding to an angular distance of the luminary from the observer’s apex \(\vartheta = 90^\circ\) (Fig. 5).

Fig. 5.

The straight line \(AB\) is perpendicular to a bundle of parallel rays inclined, as a consequence of aberration, by an angle \(\alpha\) to the perpendicular to the direction of motion of the apparatus. In view of the fact that the wave planes are parallel to the direction of motion of the apparatus, between the waves at the points \(A\) and \(B\) there exists a path difference equal to \(BD = b \operatorname{tg}\alpha\), where \(b = AB\).

But \(\operatorname{tg}\alpha = \dfrac{v}{c}\), where \(v\) is the velocity of motion of the apparatus. Consequently, the oscillation at the point \(B\) lags behind the oscillation at the point \(A\) by the time \(t_1 = \dfrac{bv}{c^2}\).

To detect this path difference, a mirror is placed at the point \(A\), which directs ray \(1\) to the point \(B\), where it is combined with ray \(2\), reflected from a semitransparent mirror placed at this point, and then both rays enter the tube \(L\),

by means of which observations are made of the interference pattern that arises. If at points \(A\) and \(B\) there were no difference in path between rays 1 and 2, then, when the rays are combined at point \(B\), it would arise because, after reflection from mirror \(A\), ray 1 must spend the interval of time

\[ t_2=\frac{b}{c}, \]

in order to reach point \(B\), under the condition that the instrument is at rest with respect to the light-bearing medium. The motion of the instrument, in addition to leading to the appearance of a phase difference at points \(A\) and \(B\), must also cause a change in the interval of time spent by the light in traversing the segment \(AB\). When the instrument moves with velocity \(v\), the corresponding interval of time is

\[ t_3=\frac{b}{c-v\cos\alpha}. \]

In view of the smallness of the angle \(\alpha\), one may, with sufficient accuracy, put \(\cos\alpha=1\); then

\[ t_3-t_2=\frac{bv}{c(c-v)}. \]

Thus, the delay of ray 1 relative to ray 2 when they are combined at point \(B\), as compared with that which should be obtained with the instrument at rest, is equal to

\[ t=(t_3-t_2)-t_1=\frac{bv}{c(c-v)}-\frac{bv}{c^2}, \]

whence, to accuracy up to terms of order \(\beta^2\),

\[ t=\frac{b}{c}\,\beta^2 . \tag{13} \]

Dividing \(t\) by the period of oscillation \(T\), we obtain the magnitude of the expected shift of the interference fringes:

\[ \Delta=\frac{b}{cT}\,\beta^2 \]

or, taking into account that \(cT=\lambda\),

\[ \Delta=\frac{b}{\lambda}\,\beta^2 . \tag{14} \]

Substituting, for visible light, \(\lambda=5\cdot10^{-5}\ \mathrm{cm}\) and, for the orbital motion of the Earth, \(\beta=10^{-4}\), we have, for \(b=500\ \mathrm{cm}\), \(\Delta=0.1\), i.e. a displacement by \(1/10\) of a fringe—a quantity quite accessible to measurements.

To discuss in detail the possibilities of carrying out such an experiment*) is hardly worthwhile, since its negative result can be predicted with confidence on the basis of the existing data of other experiments. In particular, on the basis of the negative result of Michelson’s experiment, the calculation of the difference of the time intervals \(t_2\) and \(t_3\) cannot withstand criticism. Having accepted this consequence of Michelson’s experiment, one must also accept the further consequences from it, and consequently abandon also the supposition of the existence of an inclination of the wave normal to the direction of the ray in moving reference systems and thereby—of the existence of the delay \(t_1\).

We have given this example only to confirm that, from the point of view of ether theory, stellar aberration (in the form of an inclination of the wave normal to the direction of the ray) could be detected experimentally also in a reference system moving rectilinearly and uniformly, and also in order to refute the erroneous opposite assertion in Emden’s article. From the point of view of the theory of relativity, the aberration of stars really exists in itself—as a change with time of the direction of the light ray and, correspondingly, of the apparent position of the star**) only in reference systems whose velocity changes with time in magnitude or direction, i.e. in non-inertial systems.

*) A difficulty here may arise in connection with the impossibility of observing interference at large path differences (with an insufficiently monochromatic light source). It can be overcome if, instead of the direct ray, one uses a ray reflected in a direction perpendicular to the direction of motion of the apparatus. In such a scheme (Fig. 6, \(AB = BC\)) the order of magnitude of the expected effect remains the same; only the numerical coefficient changes (as in the corresponding calculation connected with Michelson’s experiment). In addition, such a scheme makes it possible, by rotating the whole setup through \(90^\circ\) about the direction of the ray, to interchange the roles of mirrors \(A\) and \(C\) (this is shown by dashed lines in Fig. 6), as a result of which the path difference should change sign, i.e. one could expect a real displacement of the interference fringes.

Fig. 6.

**) Let us note that in the Great Soviet Encyclopedia aberration is defined precisely as “the change in the apparent position of a star on the celestial sphere, caused by the fact that the Earth moves around the Sun and continuously changes the direction of its motion relative to the star.”

Since in the classical theory stellar aberration is a consequence of the absolute motion of the frame of reference, it must also exist when the light source and the apparatus receiving it move jointly with respect to the hypothetical luminiferous medium. In this case as well it must be expressed in the inclination of the wave normal to the direction of the ray^17. According to the theory of relativity, such an inclination is absent in all cases. As in the Michelson experiment, for the phenomenon of stellar aberration it is immaterial whether one uses a terrestrial or a cosmic light source, since the motion of the light source itself is of no significance here.

REFERENCES

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