# 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"? Jul 20, 2018 at 16:34
• @enumaris maybe, as something, which is almost the speed of light, but just slightly slower. Jul 20, 2018 at 16:36
• Have you searched for "relativistic plasma"?
– JEB
Jul 20, 2018 at 16:40
• @JEB No! What is it about? Is it related to my question? Jul 20, 2018 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 $v=0.1c$? Jul 20, 2018 at 17:17

## 1 Answer

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 $\sqrt{3}c/2\approx .87c.$

EDIT (per user request): Only the most massive of stars will ever achieve temperatures close to these (and even then, only in their cores). And those stars will run into pair production instabilities due to the spontaneous production of electron-positron pairs in their cores from the high energy gamma radiation around. As an isolated object (i.e. not massive like a star), an object of this temperature wouldn't be bound in any way. One would have to design apparatus to contain such an object. It would look essentially like a very hot plasma. (Since that's exactly what this object would be - a very hot plasma)

• Thanks for that Answer. But what would be our observation on looking at such an object? Jul 20, 2018 at 17:48
• Only the most massive of stars will ever achieve temperatures close to these (and even then, only in their cores). And those stars will run into pair production instabilities due to the spontaneous production of electron-positron pairs in their cores from the high energy gamma radiation around. As an isolated object (i.e. not massive like a star), an object of this temperature wouldn't be bound in any way. One would have to design apparatus to contain such an object. It would look essentially like a very hot plasma. (Since that's exactly what this object would be - a very hot plasma) Jul 20, 2018 at 17:59
• Thank You enumaris. You should Add that to your answer so that I can accept your answer! Thanks once again! Jul 21, 2018 at 5:42
• simple and elegant reasoning, clear explanation, thanks! Jul 23, 2018 at 8:48
• @MrigankPawagi done. Jul 23, 2018 at 20:37