Brownian Motion and General Relativity

TL;DR

This paper models Brownian motion within the framework of general relativity on pseudo-Riemannian manifolds, introducing a novel dual-time process and deriving related distributions for space-time events and velocities.

Contribution

It presents a new stochastic model of Brownian motion in relativistic spacetime using two independent proper time variables, leading to novel distributional solutions.

Findings

Space-time event statistics follow a temperature-dependent Gaussian distribution.

4-velocity statistics obey a Maxwell-Juttner distribution.

The model involves partial differential equations with derivatives in two proper time variables.

Abstract

We construct a model of Brownian Motion on a pseudo-Riemannian manifold associated with general relativity. There are two aspects of the problem: The first is to define a sequence of stopping times associated with the Brownian "kicks" or impulses. The second is to define the dynamics of the particle along geodesics in between the Brownian kicks. When these two aspects are taken together, we can associate various distributions with the motion. We will find that the statistics of space-time events will obey a temperature dependent four dimensional Gaussian distribution defined over the quaternions which locally can be identified with Minkowski space. Analogously, the statistics of the 4-velocities will obey a kind of Maxwell-Juttner distribution. In contrast to previous work, our processes are characterized by two independent proper time variables defined with respect to the laboratory frame: a discrete one corresponding to the stopping times when the impulses take place and a continuous one corresponding to the geodesic motion in-between impulses. The subsequent distributions are then solutions of partial differential equations which contain derivatives with respect to both time variables.