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 \frac { \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 = \frac { m } { r } , is needed ( in addition to velocity ) , and assumed to be bounded by 0 and \frac { c ^ { 2 } } { G } = 1 , in geometrized units .
Several results are derived : The equation E = mc ^ { 2 } apparently needs to be modified to E ^ { 2 } = \frac { s ^ { 2 } c ^ { 10 } } { G ^ { 2 } } + m ^ { 2 } c ^ { 4 } , but a simpler ( invariant ) parameter is the “ energy to length ” ratio , which is \frac { 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 \rightarrow 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 .