    Next: Relativistic Energy and Momentum Up: Special Relativity Previous: Proper Time and Time   Contents If we form the infinitesimal version of the Lorentz transformation of coordinates:   (15.66)   (15.67)   (15.68)   (15.69)

Point is moving at velocity in frame , which is in turn moving at velocity with respect to the rest'' frame . We need to determine (the velocity of in ). We will express the problem, as usual, in coordinates and to the direction of motion, exploiting the obvious azimuthal symmetry of the transformation about the direction.

Note that (15.70)

for . Then       (15.71)

Similarly, (e.g. -- ) is given by     (15.72)

or (15.73)

We see, then, that the velocity changes in both the and the directions.

Note also that if and , then (15.74)

and (15.75)

so that we recover the Gallilean result,   (15.76)   (15.77)

What about the other limit? If , then (15.78)

as you should verify on your own. This is Einstein's second postulate! We have thus proven explicitly that the speed of light (and the speed of anything else travelling at the speed of light) is invariant under Lorentz coordinate transformations. This is their entire motivation.

We observe that the three spatial components of velocity'' do not seem to transform like a four vector. Both the and the components are mixed by a boost. We can, however, make the velocity into a four vector that does. We define     (15.79)     (15.80)

where is evaluated using the magnitude of u. It is an exercise to show that this transforms like the coordinate 4-vector .

Now we can guess'' that the 4-momentum of a particle will be . To prepare us for this, observe that (15.81)

are just the -scaled velocities'' of the particle:     Next: Relativistic Energy and Momentum Up: Special Relativity Previous: Proper Time and Time   Contents
Robert G. Brown 2007-12-28