the aberration of light, and to the dragging of light in moving water, yielding relativistically valid expressions for these phenomena are detailed below. It is also possible to use the velocity addition formula, assuming conservation of momentum (by appeal to ordinary rotational invariance), the correct form of the -vector part of the momentum four-vector, without resort to electromagnetism, or a priori not known to be valid, relativistic versions of the Lagrangian formalism. This involves experimentalist bouncing off relativistic billiard balls from each other. This is not detailed here, but see for reference Wikisource version (primary source) and . Fizeau experiment When light propagates in a medium, its speed is reduced, in the rest frame of the medium, to , where is the index of refraction of the medium . The speed of light in a medium uniformly moving with speed in the positive -direction as measured in the lab frame is given directly by the velocity addition formulas. For the forward direction (standard configuration, drop index on ) one gets, Collecting the largest contributions explicitly, Fizeau found the first three terms. The classical result is the first two terms. Aberration of light Another basic application is to consider the deviation of light, i.e. change of its direction, when transforming to a new reference frame with parallel axes, called aberration of light. In this case, , and insertion in the formula for yields For this case one may also compute and from the standard formulae, the trigonometric manipulations essentially being identical in the case to the manipulations in the case. Consider the difference, correct to order . Employ in order to make small angle approximations a trigonometric formula, where were used. Thus the quantity the classical aberration angle''', is obtained in the limit . Relativistic Doppler shift Here velocity components will be used as opposed to speed for greater generality, and in order to avoid perhaps seemingly ad hoc introductions of minus signs. Minus signs occurring here will instead serve to illuminate features when speeds less than that of light are considered. For light waves in vacuum, time dilation together with a simple geometrical observation alone suffices to calculate the Doppler shift in standard configuration (collinear relative velocity of emitter and observer as well of observed light wave). All velocities in what follows are parallel to the common positive , so subscripts on velocity components are dropped. In the observers frame,