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Velocity Transformation

Consider a particle P whose co-ordinates in Lorentz frames S and S’ are (x, y, z, t) and (x’, y’, z’, t’). The two sets of co-ordinates are related by

x’ = (x – v_t), y’ = y, z’ = z, t’ = (t – v_x/c²)

where v is the velocity of S’ relative to S.

The velocity of particle P as seen from S and S’ are given by

We want to find the transformation equation between U and U’. To do that, we write the Lorentz transformation equations in the infinitesimal form:

dx’ = (dx – vdt), d_y’ = d_y, d_z’ = d_z, dt’ = (dt – v/c² dx)

Hence, we get

Equations represent the relativistic transformation of velocities. These equations show that:

(i) In non-relativistic limit, i.e. v << c, U_x << c, we neglect vU_x/c² and get

U_x’ = U_x – v,      U_y’ = U_y,      U_z’ = U_z

which are velocity transformations in Galilean relativity.

Thus, in special relativity, if velocity of P relative to S is U_x and that of S’ relative to S is v, then the velocity U_x’ of P relative to S’ is not given by simple addition law, viz. U_x’ = U_x – v,


(ii) If the velocity of particle P in frame S’ is equal to velocity of light c, i.e. if P is a photon with U_x’ = c, U_y’ = U_z’ = 0, then its velocity in frame S is also c:

The above result is expected since the entire structure of standard Lorentz transformations is built up on the assumption that velocity of light is same from all Lorentz frames.

(iii) The reverse transformation gives the velocity U of the particle in frame S, if its velocity is U’ in S’. This is obtained by replacing v by – v, i.e. regarding frame S to be moving with velocity – v with respect to S’. Hence, we get