This is somewhat related to a previous question, but is different at the core.

I proposed a Relativistic Ricci flow equation that takes the form

$$\frac{\partial R}{\partial t} = \alpha \Box^2 R = -\frac{\alpha}{c^2}\frac{\partial^2 R}{\partial t^2} + \alpha \nabla^2 R$$

(where we have used the squared notation to make sure we know we are talking about the operator in terms of squared derivatives, as related to a previous question I posted as linked: Squared d'Alembert Operator)

The Ricci flow is the heat equation for a Riemannian manifold. This equation is a modification of a suggestion made by Arun and Sivaram who suggested that curvature flows in the sense of the Ricci flow

$$\frac{\partial R}{\partial t} = \alpha \nabla^2 R$$

The heat flow,

$$Q = -k\nabla T = -k \frac{\partial T}{\partial x}$$

must also be modified in such a way that:

$$Q = -k\Box T = - k \nabla T + \frac{ik}{c} \frac{\partial T}{\partial t} =- k \nabla T + ik \frac{\partial T}{\partial \tau} $$

where $\tau=ct$ is the spacelike time.

It was also explained to me in another thread, (see: Physical Interpretation of the Diffusion Constant $k$) that I really can think of a diffusion as dispersion of matter from some point source - which is really just another way to say the distribution of a ''thing'' like say, a particle. One application of understanding curvature in terms of the heat flow came in the form of replacing the divergence with the respective christoffel symbol (or connection)

$$Q = -k \Gamma T$$

This is not such an unusual thing to do, since it seems that such approaches where already in literature, one such example was the renormalized Ricci flow

$$\frac{\partial R}{\partial t} = \nabla^2 + 2R$$

Suggested by another author - here you can see the additional term has the Ricci scalar curvature playing the role of the squared space derivatives.

First question: In terms of what I was told in the diffusion thread, is the Ricci flow in this case related to the diffusion constant in much the same sense that the diffusion of a thing can be thought of dispersion of a system from one place to another?

Second question: Ive noticed these equations have a temperature with derivatives in time. It suggests that temperature does not need to remain constant, in such cases I have been led to believe this always leads to irreversible thermodynamic processes. Is this true?


It was also possible to construct an entropy equation which directly involves temperature variations. For instance

$$Q = -k \Gamma T$$

$$S = \frac{Q}{T} = -k \Gamma \frac{T}{T_0}$$

From it I speculated also

$$\Delta S = -k\Box \ln \frac{T}{T_0},$$

and the time derivative gives the rate of change of the 4D entropy gradient $\Box$, but on the RHS it also measures a temperature variation,

$$\dot{S} = -k\Box \frac{\dot{T}}{T}.$$


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