# If the speed of light is the maximum speed anything can travel, and temperature describes vibration or motion, is there a maximum temperature? [duplicate]

Apologies for everything. This is my first question.

Basically the title. If the speed of light is the maximum speed anything can travel, and temperature describes vibration or motion, is there a maximum temperature? If I understand correctly,

1. The speed of light in a vacuum is the fastest speed anything can travel
2. Temperature describes the vibration/collision rate of matter
3. Vibrations and collisions can be measured as speed
4. Therefore there must be a maximum temperature

The idea of a maximum temperature intrigues me, but I must be going wrong somewhere. Help me out here?

Temperature relates to the kinetic energy of the constituents of a substance. (When very hot any substance will be in the form of a plasma, of course)

In terms of relativistic physics: while there is an upper limit to velocity, there is no upper limit to the kinetic energy of some physical entity.

An extreme example of that is high energy cosmic rays.
The highest-energy cosmic ray ever observed was an event recorded in 1991.

In the wikipedia article the following is stated about the inferred energy of that cosmic ray:
"[...] equivalent to a 142-gram (5 oz) baseball travelling at about 28 m/s (100 km/h; 63 mph)"

Temperature is linearly related to energy via the Boltzmann constant k. As a particle approaches the (finite) speed of light its energy diverges (becomes infinite). Thus, there isn't a maximum temperature.

Though,to be clear, this is only a matter of definition.

This is an enormously complicated topic because at the temperatures and/or pressures required to get particles moving that fast, all the classical equations of motion begin to be overwhelmed by the rules of quantum mechanics in such a way that the "ordinary" relationships between temperature, pressure, density and particle velocity no longer apply.

The matter being crushed together hard enough and hot enough to enter this regime is called degenerate matter and to get an idea of the temperatures involved you need to invoke the physics concepts of Fermi energy and Fermi temperature.