I was wondering if one can extend the heat equation in order to make it covariant.
In other words, in there some kind of operator \(D_\mu\)and a vector field (or perhaps another representation of the Lorentz group) \(A^\mu\)such that one can derive the heat equation \((\partial_0 - \partial^i \partial_i)\phi = 0\) as a projection or a special case of \(D_\mu A^\mu = 0\)?
All I can find has a special treatment for time that I want to avoid.
Here are some random thoughts that did not actually help me :
- Perhaps I should add a length-dimensional factor \(\alpha\) to \(\partial^i \partial_i\) which has a certain transformation behaviour ;
- There is a problem with the number of Lorentz indices in the expression of the heat equation ;
- I somehow expect the heat/energy to diffuse with a wave-like front replacing the instant transmission of heat of classical solutions ;
- Heat is energy which is part of the 4-momentum : we should be able to find another 4-vector with heat as its timelike component ;
- Using the substitution \(t\mapsto i\tau\), one can transform the heat equation into a Schrödinger equation ; the covariant heat equation thus might be found using a covariant extension of the Schrödinger equation, such as the Dirac equation (but I do not expect the Dirac equation to be the right one : what would spinors do here ?!)