General relativistic heat conduction equation

Question

I read this wikipedia article https://en.wikipedia.org/wiki/Relativistic_heat_conduction It mentions the relativistic heat conduction equation: $$\frac{\partial \theta}{\partial t}=\Box\, \theta$$ Where $\theta$ is the Temperature and $\Box$ is the D'Alembert Operator How can one generalize this to situations in GR? As far as I know, it is clear that the D'Alembert Operator is invariant under Lorentz transformations and so one just has to replace the derivatives with the covariant derivatives. However what happens to the term on the LHS? Do I just replace it with the covariant derivative as well?

Answer 1

Note that there is a distinction between being covariant in the sense that it is not coordinate dependent and time and space are treated on the same footing, and between being hyperbolic in the sense that changes in temperature propagate at a finite speed. Your equation with the D'Alembertian is hyperbolic but your question is really how to make it covariant.

The problem you are running into is that time and space appear in fundamentally different ways in the equation. To represent this you need to incorporate some kind of four-vector $u^\mu$ to keep track of the rest frame. The heat equation naturally appears in relativistic descriptions of fluids, and this four-vector $u^\mu$ will have a natural interpretation as the velocity of the fluid. (There is a subtlety that the energy and charge in the fluid might flow with slightly different velocities, but let's ignore that and just choose a $u$.)

Then if you are in the rest frame of the fluid at a given point, a time derivative can be written as $$\partial_0 \rightarrow u^\mu \nabla_\mu$$

To get a space derivative you need to project out the time component $$\partial_i\rightarrow (\delta^{\mu}_{\nu}-u^\mu u_\nu)\nabla_\mu.$$ In the rest frame, this vanishes if the index $\nu=0$ (I am using +--- signature). You can contract this projected covariant derivative with itself to make a covariant version of the spatial Laplacian.

So you can just by hand generalize the ordinary heat equation in a covariant way that can be used in curved spacetime and with non-trivial fluid velocity. In practice the heat equation coming from fluids in general relativity will look almost exactly like what you would naively do, but there may be extra terms involving things like pressure gradients.