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I realize that Special Relativity is more of a physics concept than a math one, but I figured that since I learned the heat equation, $$\frac{\partial u}{\partial t} = \kappa\frac{\partial^2 u}{\partial x^2} $$ in a PDE math class, I would ask it here. $$$$ If the heat equation does, in fact, violate SR, is there a relativistic heat equation?

Physics_Plasma
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    The top hit on Google for "relativistic heat equation" appears to be the Wikipedia article on relativistic heat conduction, which gives the equation $\dfrac{\partial u}{\partial t}=\kappa\left(\nabla^2u-\dfrac1{c^2}\dfrac{\partial^2u}{\partial t^2}\right)$. –  Feb 17 '13 at 00:24
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    And if you are not quite ready for the heat equation to become hyperbolic, see my answer here. There the equation is still parabolic but nonlinear. –  Feb 17 '13 at 00:35
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    Never mind relativity, it also violates the atomic theory of matter... – GEdgar Feb 17 '13 at 00:45
  • A note regarding the comment of Rahul: $\nabla^2-\frac{1}{c^2}\frac{\partial^2}{\partial t^2}$ is in fact the Laplacian in $\mathbb{R}^4$ with the Minowski metric. – Daniel Robert-Nicoud Aug 06 '13 at 21:08
  • @Rahul But that relativistic heat equation is not Lorentz Covariant, is it? Because it contains derivatives of different order: i.e. the time derivative appears once as order 1 and once as order 2. Isn't that annoying? Shouldn't a relativistic heat conduction equation be lorentz covariant? – onephys Jul 29 '16 at 11:03

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