### Re: Speed of light and maximum temperature

Area: Physics
Posted By: Georg Hager, Grad student Theorie III
Date: Wed Aug 28 05:01:31 1996
Message ID: 833499031.Ph

Dear Chris!

As you surely know, we define temperature in thermodynamics as the derivative of energy with respect to entropy. There's no physical reason why this derivative should be somehow limited. Although the velocity of all particles in a gas is bounded by the velocity of light, there is no limit to their kinetic energy, which really constitutes the internal energy of a gas and therefore its temperature.

Maybe you are confused by the fact that in ideal gas calculations one usually uses the nonrelativistic energy-momentum relation E=p^2/2m, and with p=mv energy should be limited due to the limitation in v. But as you heat a gas to higher and higher temperatures, this relation is no longer valid but must be replaced by the relativistic equation E^2=m^2c^4+p^2c^2 with p=mv\gamma. Here \gamma is the well-known gamma factor 1/\sqrt{1-\beta^2}. So there is no limit to the kinetic energy.

Adopting the relativistic energy-momentum relation makes the formulae look slightly different, but a glance into any good textbook on thermodynamics and statistics will tell you at once that nothing fundamental changes.

Hope that helps,
Georg

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