# Transformations of Relativistic Temperature: Planck-Einstein, Ott, Landsberg, and Doppler Formulas

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This Demonstration illustrates spacetime velocity vectors representing internal energy currents and their equilibration for a system of two relativistic thermodynamic bodies in 1+1 dimensions. The velocities of the two bodies are and in units with . The corresponding spacelike internal energy current vectors are and . The equilibration vectors are the sums for . In thermal equilibrium their directions become parallel.

Contributed by: P. Ván and T. S. Bíró (March 2011)
Open content licensed under CC BY-NC-SA

## Details

This Demonstration proposes a clarification of contradictory theoretical approaches to relativistic thermodynamics. According to our suggestion, a new concept of internal energy leads to thermodynamic equilibrium conditions that incorporate the existing propositions. You can manipulate spacetime vectors related to the intensive parameter representing energy-momentum equilibration.

In equilibrium, two moving and interacting thermodynamic bodies obey . In the coordinate system of the first body this condition can be expressed by the spacetime velocity vectors of body centers and energy currents as and , respectively. The snapshots show famous particular cases for the relative speed .

Snapshot 1: the Planck–Einstein equilibrium, when the energy current is carried by the moving body, ; in this case the moving body appears cooler

Snapshot 2: the Blanusa-Ott equilibrium, when the energy current stays with the observing body, ; in this case the moving body appears hotter

Snapshot 3: the Landsberg equilibrium, when the energy current velocities compensate each other inside the two bodies, ; in this case the temperatures of the two moving bodies are equal

Snapshots 4 and 5: the Doppler blue shift and red shift () equilibria, when the energy transfer is due to radiation, , ; in these cases the temperatures of the two moving bodies are related by a Doppler shift