Fundamentals of Time and Relativity

Axiomatic Relativity

  • In principle, generalized Lorentz transformations characterized by an arbitrary limiting velocity, can be derived from the global spacetime symmetries.
  • It is a misconception that velocity reciprocity is a consequence only of the relativity principle.

The view of relativity as a universal theory instigated attempts to derive the Lorentz (Poincaré) transformations only from the relativity principle and the spacetime symmetries of homogeneity and isotropy, without invoking Einstein’s second postulate that the speed of light in vacuo is constant and the same for all inertial observers. The first attempt is attributed to Wladimir von Ignatowsky who in 1910 established a tight connection between the group structure implied by the relativity principle and the rules for the transformation of spacetime coordinates.

The derivation of von Ignatowsky. and many similar later derivations, makes use of the principle of reciprocity. In the case of the Galilean and Lorentz transformations, velocity reciprocity just amounts to the fact that the inverse transformation relating a given inertial system to another one, is obtained by simply replacing the relative velocity by its negative; see (5.6). However, this need not always be the case, and it is a misconception that velocity reciprocity is a consequence only of the relativity principle.

Still, the argument that the Lorentz transformations can in principle be derived from spacetime symmetries has some basic validity. Under hypotheses of causality, homogeneity of spacetime and isotropy of space, the relativity principle almost uniquely leads to a Lorentz group characterized by the invariant quadratic expression:

7.1

τ 2 = t 2 −a( x 2 + y 2 + z 2 )

The undetermined constant a associated with the structure of spacetime has the dimension of an inverse velocity squared.

Three cases are to be considered:

  1. a>0 : the transformation formulae are then recognized as proper orthochronous Lorentz transformations with unknown limiting velocity c=1/ a .
  2. a=0 : the transformations are the Galilean transformations of classical mechanics.
  3. a<0 : the transformations are ordinary rotations characterized by an invariant imaginary velocity c ; in contrast, the proper Lorentz transformations are hyperbolic rotations; see eq. (5.7).
  • It is necessary to set the unknown velocity c equal to the speed of light if the Lorentz transformations are to leave Maxwell’s equations invariant.
  • It seems that “the only function of the light postulate is to fix the invariant velocity” (Wolfgang Rindler, 2001). However, it also uniquely fixes the time parameter as Einstein time.