Curvature Diffusions in General Relativity

Abstract

We define and study on Lorentz manifolds a family of covariant diffusions in which the quadratic variation is locally determined by the curvature. This allows the interpretation of the diffusion effect on a particle by its interaction with the ambient space-time. We will focus on the case of warped products, especially Robertson-Walker manifolds, and analyse their asymptotic behaviour in the case of Einstein-de Sitter-like manifolds.Comment: 34 page

Nonholonomic relativistic diffusion and exact solutions for stochastic Einstein spaces

“…7 A general d-connection is denoted D = (hD, vD), being distinguished into, respectively, h-and v-covariant derivatives, hD and vD. We proved [17,18] that the Einstein equations in various gravity theories decouple and became integrable in very general forms for the so-called canonical d-connection, D. With respect to N-adapted bases (5) and (6), D is computed to have the coefficients Γ γ αβ = ( L i jk , L a bk , C i jc , C a bc ), for h D = { L i jk , L a bk } and v D = { C i jc , C a bc }, with…”

“…of necessary signature (±, ±, ±, ±) when local coordinates are parametrized in the form u α = (x i , y a ), where x i = (x 1 , x 2 ) and y a = y 3 = v, y 4 = y . 5 Indices i, j, k, ... = 1, 2 and a, b, c, ... = 3, 4 are for a conventional (2 + 2)-splitting of dimension when the general (small Greek) abstract/coordinate indices when α, β, . .…”

“…In recent years stochastic methods and the theory of (relativistic) diffusion on Lorentz manifolds, and (for instance) Finsler/ supersymmetric generalizations, re-attracted considerable interest and applications in various directions of mathematics, gravity and modern cosmology and astrophysics. During the last decade a lot of papers have been devoted to such issues (there are many ways and approaches); for reviews and original results in physics and mathematics, see [2,3,4,5,6,8,11,12] and references therein.…”

A probabilistic view on singularities

“…Think for example of a diffusivity σ in (2.3) depending on the location of the particle (it may be the scalar curvature of the manifold at that point for instance). Consult [22] and [24] for more material on this subject, seen from a mathematical point of view, and [25] for a physical point of view on related matters. Let us repeat here that this diffusion is not to be thought of as a mathematical model for a physical diffusion phenomenon but rather as a mathematical object useful for studying some features of the spacetime geometry.…”

Trends to equilibrium for a class of relativistic diffusions

“…Since the late 1990s, there has been a renewed interest in the subject with the work of Debbasch and his co-authors on the Relativistic Ornstein-Uhlenbeck Process [9,10,16,17], then those of Dunkel and Hänggi on the so-called "relativistic Brownian motion" [20,21]. The notion of relativistic diffusion has been extented to the realm of general relativity in [22,15] and the literature on the topic is now thriving, both in Mathematics, see for example [23,6,4,3] and in Physics [24,25,14] etc.…”