On Fresnel's convection coefficient in general relativity

It is well known that a ray of light travelling in any moving medium has only a fraction of the velocity of the medium added to its own, which is usually called Fresnel’s convection coefficient. A satisfactory explanation of this phenomenon is readily furnished by Einstein’s Addition Theorem for two velocities. In this explanation the velocities of light and the medium are supposed to be independent of one another, in so far as the velocity of the ray alone, in accordance with the First Postulate of Relativity, is considered to be the same as when the medium is at rest, and to this is added, according to the Relativity law, the small velocity of the medium. But, it may be argued that the light phenomenon is a distinct one governed by Maxwell’s electromagnetic equations, and other auxiliary relations in case of material bodies, and the velocity of light in vacuum or in any other material medium should follow immediately from these equations alone. It appears, then, that the more direct way of obtaining the velocity of light in a moving medium would be to appeal to Maxwell’s electromagnetic equations in the medium, and this woidd also furnish means for examining the applicability of the Addition Theorem in the present case. The two processes, however, should be mutually compatible, since Maxwell’s equations, as written in accordance with restricted Relativity, are covariant with regard to Lorentz transformation in which, however, are embodied the Postulates of Relativity, and the Addition Theorem is only a consequence of this transformation. It is, therefore, more usual and also easier to deduce Fresnel’s convection coefficient and Doppler effect by taking a single light-wave, say, a sine vibration, in a co-ordinate system in which the medium rests and then to subject it to Lorentz transformation, than to start from Maxwell’s equations and the auxiliary relations for the moving medium and obtain the velocity from them. But in General Relativity the latter is the only course open to us ; but the question now is far more complicated, since the G-field within a material body is unknown, depending on its inner dynamical conditions. Here an attempt is made to determine the velocity inside such bodies under certain hypotheses, which enable us to obtain Fresnel’s coefficient in the usual form.