the speed of light, and in 1870, George Airy tested the hypothesis using a water-filled telescope, finding that, against expectation, the measured aberration was identical to the aberration measured with an air-filled telescope. A "cumbrous" attempt to explain these results used the hypothesis of partial aether-drag, but was incompatible with the results of the Michelson–Morley experiment, which apparently demanded complete aether-drag. Assuming inertial frames, the relativistic expression for the aberration of light is applicable to both the receiver moving and source moving cases. A variety of trigonometrically equivalent formulas have been published. Expressed in terms of the variables in Fig. 5-2, these include OR OR Relativistic Doppler effect Relativistic longitudinal Doppler effect The classical Doppler effect depends on whether the source, receiver, or both are in motion with respect to the medium. The relativistic Doppler effect is independent of any medium. Nevertheless, relativistic Doppler shift for the longitudinal case, with source and receiver moving directly towards or away from each other, can be derived as if it were the classical phenomenon, but modified by the addition of a time dilation term, and that is the treatment described here. Assume the receiver and the source are moving away from each other with a relative speed as measured by an observer on the receiver or the source (The sign convention adopted here is that is negative if the receiver and the source are moving towards each other). Assume that the source is stationary in the medium. Then where is the speed of sound. For light, and with the receiver moving at relativistic speeds, clocks on the receiver are time dilated relative to clocks at the source. The receiver will measure the received frequency to be where and is the Lorentz factor. An identical expression for relativistic Doppler shift is obtained when performing the analysis in the reference frame of the receiver with a moving source. Transverse Doppler effect The transverse Doppler effect is one of the main novel predictions of the special theory of relativity. Classically, one might expect that if source and receiver are moving transversely with respect to each other with no longitudinal component to their relative motions, that there should be no Doppler shift in the light arriving at the receiver. Special relativity predicts otherwise. Fig. 5-3 illustrates two common variants of this scenario. Both variants can be analyzed using simple time dilation arguments. In Fig. 5-3a, the receiver