# What--is the Ultimate Limit of heat based off c? [duplicate]

As heat goes up, molecules start moving at a faster rate. A gas molecule, if unhindered, could speed across the United States in three hours. I don't even want to know about plasma. But if heat consists of the speed of the molecule (which is an if) then shouldn't there be an Absolute Infinity as well as an Absolute Zero? And if so, why isn't this mentioned in textbooks? Shouldn't "Absolute Infinity" be important in thermodynamics? And are there quantum repercussions if some plasma approaches the speed of light through the individual speeds of its molecules? Does this also mean that the process of heating objects up is not linear anymore (because as you approach c, it becomes harder to accelerate the molecules)?

Thanks guys! More questions coming :)

EDIT: In light of the fact that there are many contending theories for the Ultimate Heat Limit (Hagedorn temperature, Planck temperature, even 0 K) I will leave this open to a wide variety of answers. :)

## marked as duplicate by John Rennie, Emilio Pisanty, ACuriousMind♦, Ryan Unger, Kyle KanosApr 23 '15 at 1:42

But if heat consists of the speed of the molecule (which is an if) then shouldn't there be an Absolute Infinity as well as an Absolute Zero?

This question's "if" is not correct. Temperature (not "heat", as we use this word in a specific technical way) consists of the energy, not the speed, of particles. While these two are obviously related, it's important to note that while speed has a limit in special relativity, energy is unbounded. Energy approaches infinity (that is, increases without bound) as speed approaches $c$, so as you heat up the gas it can gain limitless energy without running into any problems due to the speed limit.

And are there quantum repercussions if some plasma approaches the speed of light through the individual speeds of its molecules?

You've got some answers about the Planck scale, but I tend to be pretty skeptical of these numbers--Planck unit sometimes smell of numerology to me, and it should be noted that the "Planck mass" is macroscopic and quite ordinary, so not everything that comes from mashing constants together gives you some amazing new physics. However, there are repercussions from having a gas at such high energy, and that is that new high-energy physics may become important, and we don't necessarily know what this is yet. Whenever you read about the LHC "recreating the Big Bang" or similar, this is because, in the early universe, temperatures were very high. Thus, particles that are now exotic or rare could be created from vacuum, such as heavy quarks, etc. As the universe cooled, there wasn't enough "excess" energy flying around to create these things. But conceivably, a really hot gas would have (for instance) Higgs bosons flying around in it. Until we have a complete theory of all particles, then, it's hard to say what starts to show up in the gas at high temperatures--but this is essentially what particle accelerators are trying to do. You do not run into a limit due to relativity on the temperature, however.

Does this also mean that the process of heating objects up is not linear anymore (because as you approach c, it becomes harder to accelerate the molecules)?

It wasn't necessarily linear to start with. The specific heat of an ideal gas is constant, but this isn't necessarily true for other models. For instance, at low temperatures (below what we call the Debye temperature), solids have a $C_v \propto T^3$.

Addendum: you're interested in the behavior of a gas of particles at high speeds. But all this really means is that the gas is ultrarelativistic, that is, $E \approx pc$ instead of $E \approx \frac{p^2}{2 m}$. Particles like this exist--after all, we study (extensively) the photon gas, and the model actually describes all particles that go fast enough to have the $E = pc$ dispersion relation.

There is a point of temperature called Planck temperature where are understanding starts to break down. Advances in quantum gravity will help us understand this incredibly high temperature and its effects on molecules.

• How exactly does our understanding break down? According to Wikipedia the reason it is beyond our understanding is that the wavelength would be smaller than the Planck length, meriting a quantum representation of gravity, instead of preventing the molecules from achieving hyperluminal speeds. @Jimmy360 – HyperLuminal Apr 22 '15 at 12:19
• @HyperLuminal That is part of the problem: we don't quantum gravity completely in our grasps yet. We have not even observed the quantum of gravity itself. – Jimmy360 Apr 22 '15 at 12:21
• Also, some say that it might become a black hole at this temperature. – Jimmy360 Apr 22 '15 at 12:21
• What would be the 'speed limit' of heat, so to speak, if we left off the unknown? Also why would it be a black hole? Seems kinda counter-intuitive because if there is a ton of molecules jiggling around crazily, I would expect it to do something like explode instead of implode. @Jimmy360 – HyperLuminal Apr 22 '15 at 12:24
• @HyperLuminal because $m = E/c^2$, though the idea that it will become a black hole is not universally agreed upon – Jimmy360 Apr 22 '15 at 12:26