Departamento de Fisica, Facultad de Ciencias, Universidad de Los Andes Mérida, Venezuela (ricevuto lТ11 Marzo 1985; manoscritto revisionato ricevuto il 10 Ottobre 1985)

Summary. Classical ether theories and special relativity predict different results for the experiment of Arago. This consists in observing deviations of starlight, when refracting through a prism and then through air. Apparently, AragoТs experiment indicates a nonnull result since he observed, for different stars, different deviation differences. If these differences are due to experimental errors, only special relativity is confirmed; if they represent a real effect, its magnitude is such that it can be predicted only by a Stokes-Planck-like ether theory, while it rules out the other ether theories and special relativity.

PACS. 03.30. - Special relativity.

The most general co-ordinate transformations between two inertial reference frames S _{ 0 } and S in relative motion with velocity v, can be expressed ( ^{ 1 } ) in terms of the parameters a, b, d and the synchronization parameter ε:

(1)

Assuming implicitly that space is isotropic everywhere in the frame S _{ 0 } and taking into account the well-known classical optical experiments, Mansouri and Sexl ( ^{ 2 } ) found that agreement with experiments is achieved by

independently of the arbitrary synchronization parameter ε.

This point has been considered also by Podlaha ( ^{ 3 } ), Sjodin and Podlaha ( ^{ 4 } ), Flidrzinski and Nowicki ( ^{ 5 } ), Cavalleri and Spinelli ( ^{ 6 } ).

There are, however, two points which have to be clarified with respect to the Mansouri and Sexl inquiry. The first is that there are other co-ordinate transformations, based on absolute synchronization of clocks, here called generalized Galileo transformations (GGT), i.e. transformations (1) with ε = 0, which account for experimental data if the space is not isotropic everywhere in the frame S _{ 0 } .

Notice that a particular case of GGT is that of Tangherlini ( ^{ 7 } ) which, having and b = 1, may lead to the same predictions of special relativity (SE) if space is considered as isotropic. This interesting problem of the equivalence between the Lorentz (LT) and the Tangherlini transformations will be considered in detail in a future paper.

A nonisotropic space could be provided by a modern version ( ^{ 8 } ) of the Stokes-Planck ether theory ( ^{ 9 } ), according to which the space is isotropic in S _{ 0 } save for the region surrounding moving material bodies like stars and planets. The ether around such a body is dragged along in its motion so that the ether, or physical space surrounding it, is at rest with the body itself ( ^{ 10 } ).

The second point is that Mansouri and Sexl have not considered explicitly in their inquiry the experiment of Arago ( ^{ 11 } ). We will show in this paper that this experiment allows us to discriminate between the Stokes- Planck-like ether theories and SE. It also discriminates the classical ether theories which are already disproved by the Michelson-Morley experiment.

Yet Mansouri and Sexl point out that, for experiments involving additional assumptions about the electrodynamic properties of bodies moving through the ether, an ether theory based on the Tangherlini transformations may lead to predictions different from those of SE. This could be the case of AragoТs experiment.

In 1810 Arago carried out an experiment, the first step of which was to place a prism in front of the telescope and observe a star through it. In this case the direction of the ray emerging from the prism forms an angle with respect to the normal to the equatorial plane. After removing the prism, Arago observed the same star with the telescope moved to point in the actual direction of the ray. Arago measured the variation for a large number of bright stars. The precision of his measurements was high enough to allow him to rule out the Newtonian theory of refraction of light. Arago found that is not constant and in his last, most precise set of measurements could vary as much as 10″ of arc from one star to another. Since he was looking for variations of the order of 28″, he attributed the comparatively small observed variations to possible experimental errors. Regardless to the fact that the observed variations might or might not be due to experimental errors, AragoТs experiment is important because, in principle, it allows us to differentiate between ether theories and SB.

We have shown in a previous paper ( ^{ 8 } ) that the wave equation and its plane-wave solution is invariant in form under the general co-ordinate transformations (1). Since in the Arago experiment we are looking for effects of the order we might well consider here the Galileo transformations.

In this case the transformation properties of the plane-wave solution

indicate that the wave velocity transforms as

(2) where с is the wave velocity in the frame S _{ 0 } where space is isotropic and θ and φ are the spherical polar co-ordinates of the light direction. The frame S(x, у, z) can be taken at rest with the Earth, the x-axis being taken in the direction of v and the z-axis perpendicular to the orbital plane.

If c _{ r } is the velocity of a photon in S, or ray velocity, c _{ r } and c* are not coincident in S. In fact, the direction of the ray and that of the normal to the wave front form an angle a given by the relation This angle α, which turns out to be given by (3) is precisely the aberration angle and expression (3) gives the experimentally observed variation of arc corresponding to the variation for a star with θ = 0 over a period of six months due to the orbital motion of the Earth around the Sun. Thus, according to ether theories, light coming from stars reaches the Earth in the form of aberrated plane waves, i.e. plane waves with In SR, where the observed variations of aberration are explained ( ^{ 12 } ) in the wave theory in terms of nonconservation of simultaneity, we have instead and Because of this difference between

SB and ether theories, we expect that the phenomenon of refraction of the aberrated ray of light through a prism will lead to different predictions.

Fig. 1. - An aberrated plane wave, incident on the surface of separation between two media of index n and n', is refracted according to SnellТs law. The direction of the ray ¬ forms an angle a with the incident wave vector k. After refraction, R' forms an angle α′ with the refracted wave vector k'.

Let us now consider in the frame S the refraction of an aberrated plane wave when it passes from a moving medium of refraction index n to a medium of index n′ at rest in S, as shown in fig. 1. The wave front normal lies in the direction of the incident wave vector k, forming the angle of incidence i and the angle of refraction r with respect to the normal N to the surface separating the two media. Let us suppose that the medium of index n is the ether at rest with frame S _{ 0 } and, therefore, moving with respect to S. We must deal here with the Maxwell equations for moving media and, as mentioned above, the result is that the equation of propagation of a wave in the moving medium is formally the same as if the medium were at rest ( ^{ 8 } ), provided that the velocity of propagation с* is given by (2). Thus, in frame S, such a medium can be considered at rest but the index n is given now by

(4)

indicating that the medium is anisotropic in S. Furthermore, this medium appears to be polarized in S in such a way that the Poynting vector, which gives the direction of propagation of the ray R, forms an angle α with respect to k. Because of the boundary conditions, the medium of index n′, which is at rest in S, will be similarly polarized, with the consequence that after refraction the ray will now form an angle α′ with respect to wave vector Since, on account of condition (4), the two media can be thought of as being both at rest in S, SnellТs law applies for the refraction of the wave. In order to determine α′ and thus obtain the angle of refraction of the ray, we have to apply the boundary conditions using the Maxwell equations for moving media obtained by transforming the usual Maxwell equations from S _{ 0 } to S with the Galileo transformations. Taking for simplicity the plane of incidence coincident with the (x, y)-plane, i.e. for we obtain from the boundary conditions (5) where

Thus α′ is of the same order of α, i.e. of order and, for we have

In the case of a modern version of the Stokes-Planck ether theory the medium of index at rest in the frame S of the Earth can be the ether, or physical space surrounding the Earth. We will show now how, according to this theory, the speed of light, is constant and isotropic within this medium, as required by the Michelson-Morley null result.

Let us consider a universe composed by three massive bodies. One of the bodies, of mass m, is the Earth, the second body of mass m _{ s } represents the matter nearest to the Earth like the Sun, and the third body of mass M stands for the average effect of the far-away matter localized at the centre of mass of the Universe or of the local galaxy. According to the Stokes-Planck theory each body drags the ether, or physical space surrounding it, and carries it along as if it were its own atmosphere. Let the function indicate the effect of dragging. Then, with respect to a reference frame S, the velocity of the ether at a point r in space can be defined as

** (6) ** where the velocities and corresponding dragging functions refer in an obvious way to the above-mentioned bodies.

Unfortunately, the theory does not give a quantitative expression for the dragging. A physical choice would be to assume a dragging proportional to mass/distance as in the case of the gravitational field. However, in this case there would be, at the surface of the Earth, an ether wind, due to the ether of the Sun which would be detectable with the modern version of the Michelson- Morley experiment. In fact, this improved experiment ( ^{ 13 } ) gives a limit of 5 cm/s for the ether wind at the surface of the Earth.

Therefore, considering the analogy between the dragging of the ether and the dragging of the atmosphere by a body, we make here the ad hoc assumption that the dragging function f is analogous to that describing, as a function of the distance from the body, the atmosphere density dragged by the gravitational field. Since this function is roughly represented by an exponential, we take

(7)

In expression (7), and represent the position of the three bodies with respect to the origin of the frame S, while and are arbitrary parameters which can be fixed requiring the experimental agreement.

Near body M or far away both bodies and m, with we obtain from (6) and (7) if and

Let us take the frame S at rest with the centre of mass of the Universe, or local galaxy. Then we obtain i.e. the ether is at rest with the centre-of-mass system in the region of space far away from the solar system.

Near the Sun the ether has a velocity governed by the Sun velocity if we take However, at the surface of the Earth we have from (6) and (7) with where is the ether wind velocity with respect to the Earth,

km is the Earth-Sun distance and If we require that cm/s, then and which is the relative velocity Earth-Sun. This condition gives and, for the Earth, we can assume, e.g., Thus, if , we obtain on the Earth. At a distance greater than from the Earth, the ether wind velocity is essentially equal to the Sun-Earth velocity.

Before considering the Arago experiment in detail, there is another important aspect, concerning the propagation of aberrated waves in a Stokes-Planck- like ether, which should be considered. When the wave passes from the medium of index n to the one of index n', the speed of light changes from The change of speed is immediate, in agreement with the extinction theorem, and takes place in the relatively thin boundary region, frequently represented by a separation surface. Also the change of the aberration angle, from to , takes place immediately at the separation surface. After the change, the wave might procede indefinitely in the medium of index n' keeping its state of aberration unvaried, i.e. with = const. However, the value of , or the degree of aberration might change as the aberrated wave propagates. In fact, the natural way of propagation for an electromagnetic wave generated in this medium corresponds to a state of no aberration. Thus, as a second ad hoc hypothesis, we assume here that in the region of space containing nonaberrated electromagnetic radiation, the ether displays the tendency to keep its unpolarized state. Therefore, an incoming aberrated wave might reduce, or lose completely, its degree of aberration. In the case of the solar system, for example, the predominant electromagnetic radiation is that emitted by the Sun itself. This is true even for the region of space within the ether dragged by the planets. Then, with respect to an observer at rest with the Sun, light proceeding from other stars may lose its original aberration and propagate within the solar system as nonaberratcd wave. However, this wave will still be aberrated, with α' given by expression (3), for an observer moving with velocity v with respect to the Sun. Then the phenomenon of aberration discovered by Bradley can be interpreted in the following simple way. If the star light reaches the Earth with the ray R forming the angle with respect to N as described in fig. 1, where after six months it will form an angle because the Earth has changed its velocity from v to Цv. Therefore, the telescope has to be moved by an angle

On account of (4) and (2) we have so that and and

and differ by terms of order . With and in (5), the difference contributing only with negligible second-order corrections to the variation Thus we obtain , which is the usual expression for the aberration.

For the old Stokes-Planck theory, which makes use of a material ether, the interpretation of the aberration phenomenon is somewhat more complicated ( ^{ 10 } ).

Considering that, to first order, in the following sections we will assume that for any ether theory the rays of starlight incident on the Earth form an angle with the wave front normal.

Let us consider then the refraction of aberrated light through a prism of refraction index , as shown in fig. 2. The cross-section of the prism containing the normals to the prism surfaces lies on an azimuthal plane with respect to the reference frame of the laboratory. We will take now the new laboratory reference frame at rest with with the axes coincident, but with the -axis in the direction of EarthТs polar axis. The wave front normal will form an angle with respect to and an angle i with respect to the normal to the prism surface. The wave will be refracted through the prism according to SnellТs lawand will emerge from it with the wave front normal forming with an angle and an angle with respect to the prism normal. The incident ray will enter the prism at an angle and leave it forming an angle

(8)

If the prism is fixed to the telescope, as in AragoТs experiment, all the rays leaving the prism must enter the telescope along its axis. Therefore, δ must be constant, the observational conditions of the apparatus being the same for any star observed. Since the principal angle of the prism is and , the ray direction suffers a variation

For a nonaberrated wave, , we would have

Then, according to ether theories, the variations observable in an AragoТs

Fig. 2. - An aberrated plane wave of incident wave vector is refracted through a prism fixed to a telescope, according to SnellТs law. The incident ray forms an angle with and emerges from the prism forming an angle with respect to the surface normal . Since the emerging ray must enter the telescope along its axis, must be constant for all the observed stars. Thus the incident angle must vary if varies. type experiment as functions of should be given by (9)

In order to find the value of , we must express it in terms of the constant angle by means of Snell's law. If is the refraction index of the ether and that of the prism, we can write

Since and differ for terms of order , we set

Substituting (8) in (10) and neglecting terms of higher order yields

(11)

Finally on account of (11) we obtain from (8)

(12)

In expression (12) and differ from the indices of refraction at rest, and , by terms of order . Consequently the deviation D is a function

The explicit expressions of and depend on the specific hypothesis concerning the particular ether theory under consideration. In the case of SR we have Therefore, from (9) and (12) we obtain for any star. According to SR, no variation should have been observed in the Arago experiment.

In the case of an ether theory of the Stokes-Planck type the starlight penetrates the isotropic space of index surrounding the Earth before being refracted by the prism of index at rest in this space.

Therefore, and are constant and the contribution to D in (12) would come from the first two terms only. In order to compare result (12) with Arago's data, we have to consider the North-South component of and also a double or quadruple achromatic prism as used by Arago and not the simple prism we have considered. Without going into details, we would like to observe that the experimental results of Arago of October 8, 1810, could be accounted for by an expression analogous to (12) written for a double prism, such that The correct expression for , the North-South component of the aberration, can be deduced from the expression

where is the component of the ray velocity in the azimuthal plane and is its -component. Taking into account the transformations from to (and the motion of the Earth around the Sun), we obtain (13)

In expression (13) is the angle between the -axis and the -axis, i.e. the inclination of the Earth's axis, while is the angle formed by the -axis at the time of the observation with respect to its position on March 21. The position of the stars (declination , right ascension ) is available in a star catalog ( ^{ 14 } ).

The calculated deviations range from to and approximately match Arago's data. However, for one of the stars (Rigel), the discrepancy is of the order of , which would be the value of the average error. For the same prism, the deviations corresponding to the Newtonian theory of aberration amount to about of arc as expected by Arago.

For an ether theory with no dragging by the Earth the customary expression for the refraction index is given by the inverse of the wave velocity multiplied by :

(14a)

Allowing for the possibility of partial dragging by the prism, we set the refraction index for the prism:

(14ft)

Here F can take only values up to the Fresnel dragging coefficient in the case of Fresnel theory.

With and given by expressions (14), result (12), still of order v/c, gives the value of the theoretical deviations forseen by the theories with no dragging by massive bodies. These theories have to specify the value of v to be used in (12). In the case of a Stokes-Planck theory we have taken km/s to be the velocity of revolution of the Earth around the Sun. In fact, in this theory, the Earth moves in the ether at rest with the Sun and carries along in its motion the portion of the ether immediately surrounding it as it does with its own atmosphere. On the other hand, for theories with no dragging, there is no reason to conceive that the ether is at rest with a massive body like the Sun. Therefore, we can suppose here that is the velocity of the Earth with respect to the observed star or with respect to the centre of mass of the Universe or the local galaxies. In either case, if we keep in eq. (2), we must add to (12) terms of the order where is the declination velocity of the Earth with respect to the star or the privileged frame . For some stars observed by Arago (Arcturus, -Baleine), we have km/s, which gives deviations of the arc. Moreover, the observed cosmic background radiation anisotropy indicates ( ^{ 15 } ) that the presumed velocity of the Earth with respect to is of the order of 300 km/s, which would lead to even greater deviations. Even if the sensitivity of Arago's apparatus is taken to be of arc, such deviations would be easily observed.

We have shown that the experiment of Arago allows us to discriminate between GGT and LT, i.e. it discriminates between ether theories and SE. Arago's apparatus intrinsically detects effects related to the absolute value of the angle of aberration of light, while the phenomenon of aberration discovered by Bradley refers to relative variation in time of

In order to account for Arago's experiment, an ether theory has to forsee deviations of the order of at most . Expression (12) leads to those values when = 30 km/s, i.e. must be of the order of the velocity of revolution of the Earth. This fact suggests that the privileged frame of reference has to be at rest with the Sun. This would give to the solar system a privileged position in the Universe which is hardly justifiable for an ether theory with no dragging. The above assumption is, however, acceptable for an ether theory allowing for the dragging of the other by massive bodies, such as the Stokes- Planck theory.

However, the agreement of the calculated deviations (12) with Arago's data, in the case of a Stokes-Planck theory, is not reliable because of the discrepancy of more than for one star (Rigel). Thus it should be acceptable to interpret the deviations observed by Arago as experimental errors.

However, given the importance of the experiment in discriminating ether theories vs. SB, it is highly to be recommended that this be performed with our modern telescopes which can reach precisions up to

* * *

We wish to thank Profs. H. Poblete (ULA, Mérida) for useful comments and G. Cavalleri (Università Cattolica, Brescia, Italy) for helpful criticisms of the manuscript. Moreover, we wish to thank the Consejo de Desarrollo Científico у Humanístico (Grant c-89-77, CDCH), ULA, Mérida, Venezuela, for sponsoring this research.

Эксперимент Араго как проверка современных теорий эфира и специальной теории относительности.

Резюме (переведено редакцией). Классические теории эфира и специальная теория относительности предсказывают разные результаты эксперимента Араго. Этот эксперимент заключается в наблюдении отклонений света звезд в результате преломления при прохождении через призму, а затем через воздух. Очевидно, что эксперимент Араго приводит к ненулевому результату, т.к. наблюдались различные величины отклонений для различных звезд. Если эти отличия обусловлены погрешностями эксперимента, то он подтверждает только специальную теорию относительности. Если отличия представляют реальный эффект, их величина такова, что они могут объясняться только теорией эфира типа Стокса-Планка, тогда как другие теории эфира и специальная теория относительности исключаются.

( ^{ 1 } ) T. S. Sjodin: Nuovo Cimento B, 51, 229 (1979).

( ^{ 2 } ) R. Mansouri and R. U. Sexl: Gen. Rel. Grav., 8, 497, 515, 804 (1977).

( ^{ 3 } ) † M. F. Podlaha: Lett. Nuovo Cimento, 28, 216 (1980); Indian J. Theor. Phys., 28, 19 (1980).

( ^{ 4 } ) T. S. Sjodin and M. F. Podlaha: Lett. Nuovo Cimento, 31, 433 (1982).

( ^{ 5 } ) A. Flidrzynski and A. Nowicki: J. Phys. A, 15, 1051 (1982).

( ^{ 6 } ) G. Cavalleri and G. Spinelli: Found. Phys., 13, 1221 (1983).

( ^{ 7 } ) F. R. Tangherlini: Nuovo Cimento, Suppl., 20, 1 (1961).

( ^{ 8 } ) † G. Spavieri: Nuovo Cimento B, 86, No. 2, 177 (1985).

( ^{ 9 } ) G. G. Stokes: Philos. Mag., 27, 9 (1845); 28, 76 (1846); 29, 6 (1846). PlanckТs computations are reported by H. A. Lorentz: in Collected Papers, Vol. 4 (H. Nijoff, The Hague, 1939), p. 245.

( ^{ 10 } ) † For the interpretation of the classical optics experiments by the Stokes-Planck theory see G. Cavalleri, L. Galgani, G. Spavieri and G. Spinelli: Scientia (Italy), 3, 675 (1976).

( ^{ 11 } ) F. Arago: Comp. Rend., 36, No. 2, 38 (1853).

( ^{ 12 } ) See on this point Gr. Cavalleri, L. Gtalgani, Gr. Spavieri and Gr. Spinelli: Lett. Nuovo Cimento, 17, 25 (1976) on the clarification of a misunderstanding about aberration of plane waves.

( ^{ 13 } ) G. R. Isaak: Phys. Bull., 21, 255 (1970).

( ^{ 14 } ) University Observatory: Catalog of Bright Stars, 3rd Edition (Yale Connecticut, 1964).

( ^{ 15 } ) † G. F. Smoot, ћ. V. Gorenstein and R. A. Muller: Phys. Rev. Lett., 39, 898 (1977); R. B. Partridge and D. T. Wilkinson: Phys. Rev. Lett., 18, 557 (1967); R. Muller: Sci. Am., 238, 5, 64 (1978); S. Marinov: Gen. Rel. Grav., 12, No. 1, 57 (1980).