Lorentz transformation of temperature and effective temperature

We derive the Lorentz transformation of temperature by using holography. The Lorentz transformation of temperature has been under debate for more than a century, and three different transformations have been proposed. Here we show that the correct transformation is that proposed by Einstein and Planck. We obtain a relativistic thermal distribution function from holography as well. The distribution function is Lorentz invariant and is consistent with the covariant first law of thermodynamics proposed by Einstein and Planck. The same method is applicable to the effective temperature in nonequilibrium steady states. We derive the boost transformation of the effective temperature. This enables us to distinguish the boost effect from the many-body effect in the effective temperature of a test particle dragged in a heat bath.