6. Jayant K. Patwardhan, A Probabilistic Formulation of the Special Theory of Relativity

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Volume 17: Pages 307-337, 2004

A Probabilistic Formulation of the Special Theory of Relativity

Jayant K. Patwardhan

1943 Bordeaux Terrace, Chula Vista, California 91913 U.S.A.

An attempt to derive an expression for electromagnetic momentum using the spacetime Lorentz transformation presents an anomaly, pointing to the need for a probabilistic formulation of the spacetime transformations. Einstein's postulates are suitably modified to accommodate this probabilistic formulation. Using the group property, we develop differential equations that are valid for sublimiting and superlimiting relative velocities. Two solutions emerge from these equations. One coincides with the Lorentz transformation, while the other displays a probabilistic character. Sublimiting relative velocity solutions are obtained using the boundary conditions at zero velocity. The sublimiting velocity solutions facilitate superlimiting velocity solutions. An invariant spacetime metric for the transformation is derived. For the Lorentz transformation the invariant metric valid for sublight relative velocities does not carry over into the superlight domain. For the probabilistic solution the invariant metric remains valid over the full range and provides a better foundation for general relativity, which assumes the spacetime metric to remain unchanged for spacelike worldlines. Compliance with the Lorentz transformation in expected value is demonstrated using a simple probability distribution. Other consistent probability distributions are left for separate investigations. The mathematics of the solution introduces two vector entities with random direction and integer indices n, with an expected value of −1/2. A short section is devoted to developing the kinematics aspects, including an expression for mass and momentum.

Keywords: probabilistic, relativity, transformation, metric, sublimiting, superlimiting, group property

Received: October 15, 2003; Published online: December 15, 2008