F.1.1. Lorentz transformation

Lorentz transformation is a transformation of space-time from one frame F to another F ? moving with a relative space-velocity U. When the relative velocity U = U is toward the x-direction, the transformation law is expressed as follows (where ? = U/c, and c is the light speed):

In the Lorentz transformation with the variables ( x ^?) = ( ct, x), ^[1] the length d s of a line-element d s = ( cd t, d x) is a scalar, namely a Lorentz-invariant. With the Minkowski metric g _?? = diag ( ?1 , 1 , 1 , 1), ^[2] its square is given by

i.e. invariant under the transformation between ( t, x) and ( t ? , x ?).

Scalar product of a 4-momentum P = ( E/c, p) of a particle of mass m with the line-element d s is also an invariant (by the definition of scalar),

where m ₀ is the rest mass, v and p = m v are 3-velocity and 3-momentum of the particle respectively [Fr97; LL75], and

The expression on the left-most side ( P, d s) of (F.3) is a scalar product so that it is invariant with respect to the...