Title:Phase Spaces in Special Relativity: Towards Eliminating Gravitational Singularities

Abstract: This paper shows one way to construct phase spaces in special relativity by expanding Minkowski Space. These spaces appear to indicate that we can dispense with gravitational singularities. The key mathematical ideas in the present approach are to include a complex phase factor, such as, e^{i\phi} in the Lorentz transformation and to use both the proper time and the proper mass as parameters. To develop the most general case, a complex parameter \sigma=s+im, is introduced, where s is the proper time, and m is the proper mass, and \sigma and {\sigma}/{|\sigma|} are used to parameterize the position of a particle (or reference frame) in space-time-matter phase space. A new reference variable, u={m}/{r}, is needed (in addition to velocity), and assumed to be bounded by 0 and {c^{2}}/{G}=1, in geometrized units. Several results are derived: The equation E=mc^2 apparently needs to be modified to E^2={s^2c^{10}}/{G^2}+m^2c^4, but a simpler (invariant) parameter is the "energy to length" ratio, which is {c^4}/{G} for any spherical region of space-time-matter. The generalized "momentum vector" becomes completely "masslike" for u\approx 0.79, which we think indicates the existence of a maximal gravity field. Thus, gravitational singularities do not occur. Instead, as u approaches 1 matter is apparently simply crushed into free space. In the last section of this paper we attempt some further generalizations of the phase space ideas developed in this paper.