# Temperature Required to give Particles of Matter Speed comparable to the speed of light

We know that at 0 K, particles of matter almost stop vibrating/moving completely. Also, as we increase temperature of the system, the particles gain energy (in the form of the heat supplied) which gets converted to kinetic energy, and the motion of the particles become faster.

Since it is practically impossible for an object to get to the speed of light, it is just possible for it to have a speed near the speed of light.

My question is,

What is the temperature, at which, particles of matter gain so much energy that they start vibrating/moving with a speed near the speed of light?

Also, what would our observation be, on looking at such a piece of matter? What would it look like?

• How would you define "near the speed of light"? – enumaris Jul 20 '18 at 16:34
• @enumaris maybe, as something, which is almost the speed of light, but just slightly slower. – Mrigank Pawagi Jul 20 '18 at 16:36
• Have you searched for "relativistic plasma"? – JEB Jul 20 '18 at 16:40
• @JEB No! What is it about? Is it related to my question? – Mrigank Pawagi Jul 20 '18 at 16:43
• That still doesn't answer the question. What do you mean by "almost" and "just slightly lower"? Is $v=0.9c$ "near" the speed of light? or does it need to be $v=0.999999c$ to count? What about $v=0.5c$ or $v=0.1c$? – Emilio Pisanty Jul 20 '18 at 17:17

Since you don't provide an actual speed, I can only give an order of magnitude type estimate for you. A particle is said to become "relativistic" if it's kinetic energy becomes comparable (or greater than) its rest mass-energy $mc^2$. Particles in thermal equilibrium have kinetic energies of order $kT$ where $k$ is the Boltzmann constant and $T$ is the temperature. This gives us, for matter of mass $m$ the temperature at which the individual particles would become relativistic at a temperature of $T\approx mc^2/k$. If $m$ is the electron mass (the electrons in the matter will go relativistic first), this corresponds to temperatures of order $10^{10}K$. If $m$ is an atomic mass unit, this corresponds to temperatures of order $10^{13}K$.
For a particle which has kinetic energy equal to the rest mass, the gamma factor would be $\gamma=2$ which corresponds to a velocity of $\sqrt{3}c/2\approx .87c.$