# Are temperature and pressure related by relativity?

Gerald Kaiser
Since the temperature of a gas is related to its average molecular energy and the pressure to the average molecular momentum, it would seem that a Lorentz transformation would somehow relate the two. Does anyone know of related work?

Staff Emeritus
Homework Helper
Gold Member
2021 Award
The temperature and pressure of a fluid is related to those properties in the fluid's rest frame. The temperature and pressure are related via an equation of state such as the ideal gas law. Lorentz transformations mix pressure and energy density with the off diagonal components of the energy-momentum tensor (of which kinetic energy of course gives a contribution to the energy density).

Gerald Kaiser
Gerald Kaiser
Thanks. I should have added that I meant the LOCAL temperature and pressure in a fluid that's not necessarily in (global) equilibrium. Would those be computed in the local rest frame? I understand the the relativistic character of temperature has been a controversial topic. Has some consensus developed around it?

Gerald Kaiser
OK, I figured it out. Due to its isotropy, p depends only on the average of v^2 and is, in fact, proportional to the average energy. The expression for the temperature in terms of the average energy is not derived independently but uses the equation of state to show that it, too is proportional to the average energy (at least for an ideal gas). So I was wrong to think Lorentz transformations would relate the two. Thanks for the help.