# On Ricci flow and 'nonlinear relativistic heat equation

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?

Note:

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,

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