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I've seen a lot of questions about maximum temperature and “absolute hot” — several ask if special relativity places any limits on temperature (clearly not). (Also this discussion of absolute hot on a NOVA blog post.)

But I haven't seen general relativity addressed in any of these discussions — shouldn't there be a point where increasing the temperature of a given system will cause it to exceed some critical threshold of energy density and consequently cause it to collapse into a black hole? And wouldn't that bound the upper limit of temperature?

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    $\begingroup$ Heat is just how much particles move, that threshold should be speed of light. $\endgroup$
    – Matas Vaitkevicius
    Commented Jan 28, 2016 at 13:59
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    $\begingroup$ No, that's incorrect, if you read some of the answers I linked you'll see that you're thinking of the kinetic energy of the particles, and in special relativity there is no limit to the kinetic energy a particle can have, it grows asymptotically towards infinity as the velocity approaches the speed of light. But that's a limited understanding of heat anyways — in one of those questions someone mentions a photon gas, which has the property of temperature but is made of photons which travel at the speed of light. (I don't know much about that stuff outside of that though.) $\endgroup$
    – Yrast
    Commented Jan 28, 2016 at 14:13
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    $\begingroup$ In any case, it is a mistake to think of temperature as the kinetic energy of the constituent particles (which is only true when the equipartition theorem holds). Temperature is, fundamentally, the rate of change of energy with respect to entropy, i.e. a measure of how much bigger the available phase space gets per unit of added energy. This need not be bounded even if the energy density is. $\endgroup$
    – Emilio Pisanty
    Commented Jan 28, 2016 at 14:22
  • $\begingroup$ I thought about mentioning thermodynamic beta and asking if that changed anything, but decided against it. I guess to alter my question a little bit I could say the limit I'm expecting would really just be a limit before it collapses into a black hole, at which point the details of what exactly thermodynamic beta/temperature may or may not entail becomes less clear. $\endgroup$
    – Yrast
    Commented Jan 28, 2016 at 14:25
  • $\begingroup$ do you mean that the radius of a black hole , which depends on its mass/energy , is a kind of upper bound for the energy density ? $\endgroup$
    – user46925
    Commented Jan 29, 2016 at 16:52

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To continue Matas' comment: the definition of temperature is a quantity dependent on mean energy. It's sort of ugly when the particles all go near-relativistic, but there is no upper bound.

Interestingly, you can extend the definition to things like magnetic particle orientation states. In this case, beyond a certain magnetic field strength, all the particles are forced to align, and you get negative temperatures!

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In principle yes, though the situation isn't as clear cut as you describe.

If you could confine a volume of matter within some volume then gradually heat it by adding energy to it then at some point the total energy density would exceed the density required to form a black hole and at that point the matter would start to collapse into a black hole.

However the density of a black hole depends on its size, so the maximum temperature would be dependent on the size of your assemblage of matter.

I suppose you could argue that the smallest volume measurable would be a Planck volume, and you could base your calculation on this. However at such fantastically high energy densities it isn't obvious that temperature has much meaning.

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  • $\begingroup$ So the maximum temperature limit goes up as the size of the hot stuff gets smaller. Most hot stuff still has most of the energy located in the mass of the particles, but its still a neat question. (e.g. - think of an electron - positron gas and the temp you need to go to see over 511keV of kinetic energy per particle). $\endgroup$
    – Tom Andersen
    Commented Jan 31, 2016 at 22:12

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