Here's a question on how to overcome the contradiction and incompatibility between Fourier's equation (Fick's Law) and the theory of relativity (in the latter case speed of heat propagation is infinite while in the latter case nothing is faster than light).

A fix for this conundrum is to stick a 2nd order time derivative to Fourier PDE and hey presto resulting in the Hyperbolic Heat Conduction (HHC) equation, a bit like the telegrapher's equation. But we need to introduce the

*second sound* into the PDE.

IMO it is the same tale as with negative probabilities.

Questions

1. This is not the mathematical approach ... too many assumptions (in fact, it violates the 2nd law of thermodynamics).

2. Second sound is fiction (it has no physical reality)

3. HHC is a weak description of relativity but it is still Newtonian.

4. You can't just stick bits and pieces to a PDE and hope it will work. (disclaimer: I did QM/SR/GR at undergrad but I don't remember 1/2 of it)

At which step in inventing HHC did it go wrong? What is the major (implicit) assumption here?