Energy–momentum density and pressure relations for a relativistic ideal gas with a bulk motion.

We derive here, from first principles, the energy–momentum density of a perfect fluid in the form of an ideal molecular gas, in an inertial frame where the fluid possesses a bulk motion. We begin from the simple expressions for the energy density and pressure of a perfect fluid in the rest frame of the fluid, where the fluid constituents (gas molecules) may possess a random motion, but no bulk motion. From a Lorentz transformation of the velocity vectors of the molecules moving in different directions in the rest frame of the fluid, we compute their energy–momentum vectors and number densities in an inertial frame moving with respect to the rest frame of the liquid. From that, we arrive at the energy–momentum density of the fluid in a frame where it has a bulk motion. This way, we explicitly demonstrate how a couple of curious pressure-dependent terms make an appearance in the energy–momentum density of a perfect fluid having a bulk motion. In addition to an ideal molecular gas, we compute the energy–momentum density for a photon gas also, which of course matches the energy–momentum density expression obtained for a molecular gas having an ultra-relativistic random motion. [ABSTRACT FROM AUTHOR]