The theory of relativity on the Finsler spacetime

According to some local properties of Lorentz transformation, Einstein stated: “Velocities greater than that of light have no possibility of existence.” He neglected to point out the applicable range of the special theory of relativity. In fact, it could only be applied to the subluminal-speeds. This paper shows that if ones think of the possibility of the existence of the superluminal-speeds and redescribe the special theory of relativity following Einstein's way, it could be supposed that the physical spacetime is a Finsler spacetime, characterized by themetric ds
⁴
=g
ijkl
dx
ⁱ
dx
^j
dx
^k
dx
^l
. If so, a new spactime transformation could be found by invariant ds
⁴
and the theory of relativity is discussed on this transformation. It is possible that the Finsler spacetime F(x, y) may be endowed with a catastrophic nature. Based on the different properties between the ds
²
and ds
⁴
, it is discussed that the flat spacetime will also have the catastrophic nature on the Finsler metric ds
⁴
. The spacetime transformations and the physical quantities will suddenly change at the catastrophe set of the spacetime, the light cone. It will be supposed that only the dual velocity of the superluminal-speeds could be observed. If so, a particle with the superluminal-speed ν> c could be regarded as its anti-particle with the dual velocity ν
1
= c
²
/ ν< c. On the other hand, it could be assumed that the horizon of the field of the general relativity is also a catastrophic set. If so, a particle with the superluminal-speeds could be projected near the horizon of these fields, and the particle will move on the space-like curves. It is very interesting that, in the Schwarzschild fields, the theoretical calculation for the space-like curves should be in agreement with the data of the superluminal expansion of extragalactic radio sources observed year after year. (see Cao, 1992b). The catastrophe of spacetime has some deep cosmological means. According to the some interested subjects in the process of evolution of the universe the catastrophe nature of the Finsler spacetime and its cosmological implications are discussed. It is shown that the nature of the universal evolution could be attributed to the geometric features of the Finsler spacetime. (see Cao, 1993)