Does $E=mc^²$ mean there is a maximum attainable temperature?

Question

If energy is proportional to mass, and temperature is proportional to energy density, does it mean there is a certain absolute maximum temperature which can never be exceeded? With further energy added to the system simply increasing its mass without affecting the temperature anymore?

Answer 1

Does e=mc² mean there is a maximum attainable temperature?

First lets clear up the misunderstanding implied on what $m$ is in the $E=mc^2$ . The formula comes from the early days of special relativity studies, where this $m$ is the relativistic mass, it is the mass a particle would have in Neutonian physics, the inertial mass, the resistance to increase momentum of a particle with velocities close to the velocity of light. At present one uses four vectors to do the kinematics of Lorentz transformations and the mass of a particle,( or a system of particles), is the length of the four vector called invariant mass, or rest mass because

The two formulae coincide in the frame where the momentum is zero. It is energy and momentum that are conserved within an inertial frame.

and temperature is proportional to energy density

Temperature is not proportional to energy density, it is related to the average kinetic energy in ideal gases, and gets more complicated for solids or fluids.

With further energy added to the system simply increasing its mass without affecting the temperature anymore?

Adding energy would increase the four vectors defining a system and accordingly, if the energy comes with additional four vectors, the invariant mass could grow, in principle without bounds if it were not for general relativity and the gravitational effects due to the large masses. When quantum mechanical theories are applied in cosmological models there are the Planck units , and for energy it is expected that for larger energies nothing can be known, so in that sense at present one considers a limit to the possibility of calculating energy beyond some value. Note that the mass of the proton is used in the definition of the Planck constants .This limit of energy introduced in an ideal gas model will also give limit to the knowledge of temperature.