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Einsteins theory of gravitation, general relativity, is a purely geometric theory.

In a recent question I wanted to know what the relation of Brownian Motion to the Helmholtz equation is and got a very thorough answer from George Lowther.

He pointed out that there is, roughly speaking, a very general relation of semi-elliptic second order differential operators of the form

$$Af = \frac12 a^{ij}f_{,ij} + b^i f_{,i} - cf = 0$$

to a "killed" Brownian motion. (I used some summation convention and $,i = \frac{\partial}{\partial x^i}$.)

Now, the Einstein field equations

$$R_{\mu\nu}-\frac12 g_{\mu\nu}R = \frac{8\pi G}{c^4}T_{\mu\nu}$$

are coupled hyperbolic-elliptic partial differential equations (I dropped the cosmological constant here). Can we somehow adopt the relation of a random process to this kind of equation, or

Is there a way to interprete the Einstein equations stochastically?

Thank you in advance
Sincerely

Robert

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The evolutionary Einstein's equation: probably not. The hyperbolic part of the equation does have the same interpretation, especially since the equation is non-linear and there is no easy way to perform separation of variables (or "Fourier transform in time") to reduce to elliptic problem. (Also, no preferred time function etc.) For the constraint problem, which is purely elliptic, there maybe something you can say. There's some recent work by Blaine Lawson on fully non-linear elliptic systems arising in geometrical contexts, which formally has many parallels with the analytic half of ... –  Willie Wong Jan 13 '11 at 13:51
    
... George's answer you linked to, so one may try to run the same argument and see if a random process pops out. I thought about it briefly in passing last year, but the situation seems rather complicated (to me) and I am not personally aware of anyone fully fleshing out the ideas just yet. –  Willie Wong Jan 13 '11 at 13:54
    
@Willie Wong: Thank you again for your nice comments! You have a lot of knowledge about the structure of general relativity and the underlying geometrical and functional concepts and I enjoy reading your thoughts. So, for this question, the situations seems to be arbitrarily complicated - even though there might exist a corresponding stochastic process we simply don't know it yet :) You may consider posting your comments as answer, I would be glad to accept. Sincerely –  Robert Filter Jan 13 '11 at 14:40

1 Answer 1

up vote 3 down vote accepted

The link between Einstein equations and a stochastic process can be achieved through Ricci flow. I can state the question very easily for the 2d case while, for higher dimensions things may become quite involved. The idea is that a stochastic process satisfy a diffusion equation

$$\partial_tP=\Delta_2P$$

and one can write down the solution through a Wiener integral

$$P=\int[dx(t)]e^{-\frac{1}{2}\int_0^td\tau{\dot x}^2(\tau)}.$$

When one extend this to a generic two-dimensional manifold, the diffusion equation, when applied to the metric, is that of the Ricci flow as one has just the Laplacian replaced by the Beltrami operator applied to metric. Then, the fixed point of this Ricci flow is just Einstein equations for the two-dimensional manifold at hand. I have given some considerations about, well-founded on a theorem by Baer and Pfaeffle (see here).

The exciting idea behind this is that a Ricci flow could be always derived from a stochastic process underlying a manifold. I think that this is material to be studied yet.

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Dear @Jon, thank you very much for your answer! Since I think the generalization to a four-dimensional Pseudo-Riemannian manifold is challenging, I appreciate your insightful two dimensional approach very much. Greets –  Robert Filter Dec 7 '11 at 14:49

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