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According to wikipedia, various symmetries in natural laws result in conservation laws. What conservation law would arise if temperature were translation symmetric, and how would this affect the universe?

Edit: To clarify I'm asking what a universe would be like if there were there were a property such that for any temperature $T$, the transformation $T\rightarrow T+\delta T$ would not create a meaningful difference in any physical laws, and if there were no absolute zero temperature.

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    $\begingroup$ I'm not clear if the big-bang could even happen under such circumstances. Is this a whole universe you're referring to or a localised phenomenon in this one? $\endgroup$ – We are Monica. Sep 6 at 7:07
  • $\begingroup$ Seems like it would mean that entropy could be reversible under some conditions, and therefore perpetual-motion machines could also work under some conditions. Oops, the sun just crisped the Earth, too bad for us. $\endgroup$ – user535733 Sep 6 at 8:19
  • $\begingroup$ There is no guarantee a conservation law would arise even if temperature-translation symmetry existed. What might happen is that the laws of thermodynamics would be different. If that happened, some odd and interesting things would be the case. $\endgroup$ – a4android Sep 6 at 8:24
  • $\begingroup$ @user535733 That may be more possible than we would normally think. I came upon a remark that entropy should be covered by a conservation rule, but somehow this doesn't happen. So in the OP's scenario reversible entropy might be a thing. This might not guarantee perpetual motion machines. Biology might be quite remarkable, in fact, very, very remarkable. $\endgroup$ – a4android Sep 6 at 8:29
  • $\begingroup$ @user535733 I did note your "under some conditions" and was making, what was basically the same point, in different words. I suspect there would be really weird stuff if entropy was reversible. Working out what was weird wouldn't be easy. Time & lots of thinking required. Don't have enough of either. The Sun might freeze Earth instead of crisping it. $\endgroup$ – a4android Sep 6 at 9:15
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Temperature is not a coordinate like position, velocity, time, or rotation. Symmetries on those coordinates lead to conservation of momentum, special relativity, conservation of energy, and conservation of angular momentum. Symmetries of those values are aspects of the entire system, while temperature is an aspect of each part of the system.

Absolute zero is a real thing, a specific temperature below which things cannot go. Its existence is dictated by the math of statistical dynamics. You can't just move all the temperatures up or down.

Also, addressing some of the comments about "entropy being reversible": At its most fundamental level, the laws of entropy translate to "things that are more likely to happen tend to happen more often." If you calculate the entropies of different states, that's the math behind the calculations.

So saying "entropy tends to increase" is essentially a tautology. It really can't go backward.

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I think that temperature is translationally symmetric already -- short of absolute zero.

You can add and scale temperature any way you want in a system, and then reduce the temperature back to the original value. Now, the physical matter in that system may not return to its original state since it may undergo structural, physical, and chemical changes that are not reversible with temperature.

For example, if your system is a volume of H2 and O2 at standard temperature and pressure, and you raise the temperature high enough, you'll end up with H20 + OH + H2O2 vapor. When you cool it down, it won't change back to diatomic gases.

But, if your system was a mass of Fe and you heated it up high enough, it would melt, then vaporize. When you cooled it back down, you'd end up with an equal mass of Fe as you started, but it wouldn't likely have the same shape it as originally had.

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