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

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.


  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
  • [Snapshot]


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
For more information see T. S. Biró, P. Ván, "About the Temperature of Moving Bodies".
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+