From what I've read, the usual heat equation is not relativistic. This is because the usual heat equation takes the information of temperature differences to be spread instantaneously. This is obviously not compatible with special relativity.
The Wikipedia article mentions a hyperbolic version of the heat equation which it claims to be consistent with relativity. The equation they give is in local coordinates so its covariance is not manifest.
Q. Is there a geometric description of the heat equation that is manifestly generally covariant?
