So I was thinking about an analogy, that could potentially be used for an explanation or at least to take a different perspective on thermodynamics as it is. But I don't want to abuse the analogy, so my question is if it is meaningful to make this comparison or I'm twisting things too much.
With Galilean relativity we know that any experiment will yield the same results regardless of the inertial reference frame from witch it is described. If, for example, in one reference frame I saw two objects, $A$ and $B$, moving at the same constant speed $\vec{v}_A = \vec{v}_B = 100$ m/s I could describe the situation from the reference frame co-moving with the objects (let's choose $A$) and say that $\vec{v}_A^A = \vec{v}_B^A = \vec{v}_B-\vec{v}_A= 0$ m/s. In the first case the objects have a lot of kinetic energy that vanishes when viewed from the frame of reference of the objects, since they have no relative velocity. Even if the values of the velocity and the energy are different, the relevant physics remain the same; In both cases they are not approaching each other, and thus would not collide or interact in any way. On one reference frame we would interpret this as the objects are not transferring the kinetic energy they have to one another because of their trajectories don't intersect, but in the other the interpretation would simply be that the objects have no energy at all so no interaction is expected either.
This reminded me of how temperature behaves (temperature will be analogous to velocity here), because there are many situations where what matters is not the absolute value of temperature but the relative value (just like with velocity). So let me define a new concept, a "thermal frame of reference". A "thermal frame" would be a temperature of reference that is used to measure all other temperatures. For example, object $A$ is at $T_A = 100$ $^{\circ}$C and object $B$ is at $T_B = 150$ $^{\circ}$C. From the "thermal frame" of object $A$ we would see that $A$ has a temperature of $T_A^A = 0$ $^{\circ}$C and B has a temperature of $T_B^A= 50$ $^{\circ}$C. Here I'm using some kind of Galilean transformation for temperature so that $T_B^A = T_B-T_A$ (just like velocity addition). From the temperature frame of reference of B the temperatures would be $T_B^B = 0$ $^{\circ}$C and $T_A^B = -50$ $^{\circ}$C. We can see that:
- The difference in temperature would be the exact same in any "frame", $50$ $^{\circ}$C.
- $B$ is hotter than $A$ in all "frames". Thus heat flows from $B$ to $A$ according to the laws of thermodynamics.
- Heat flows even at the exact same rate (according to Fourier's law of heat conduction), regardless of the "thermal frame of reference". So if the objects were to have their temperatures suddenly increased by $100.000$ $^{\circ}$C they would both still behave in the same way.
If the temperatures were the same in some "thermal reference frame" then they would be the same in any "thermal reference frame". In this case there wouldn't be any heat transfer and, like in the example I gave with Galilean relativity, we would have a situation that could be viewed as if the objects have no temperature, thus no internal energy and thus no heat to transfer between each other (from a thermal frame where $T_A = T_B = 0$ $^{\circ}$C) or as if there is no heat flow because there is no temperature gradient (from the point of view of a thermal frame where $T_A' = T_B' = 500$ $^{\circ}$C).
We know that no machine can operate without a thermal gradient; no matter how hot or cold the thermal bath is, if the temperature of the machine is the same as the temperature of the bath it will not produce work.
There are many more examples, but the idea is that one can see a connection between how velocity behaves in Galilean relativity and how temperature behaves in thermodynamics.
Would this concept of a "thermal frame of reference" be a meaningful way of understanding all the relevant physics? Or will the analogy break after a while? If that is the case, then why?
Trying to answer my own question I came to realize that this might not work, because temperature has some absolute value in the Kelvin scale. There is an absolute zero! So there would be thermal reference frames that would allow for objects to have negative kelvin temperatures, which contradicts the laws of thermodynamics (the laws of thermodynamics are not valid for any "thermal frame of reference" doesn't sound like Galilean relativity anymore). This could mean that there are actually experiments that would break the analogy; situations where you would know the temperature of the object without having to reference the temperature of any other object. Situations where the temperature is the same for all observers regardless of their "thermal frame of reference". Right?
In that case, another analogy comes to mind. Maybe the zero kelvin behaves the same way as the speed of light in special relativity, as an absolute quantity that is measured the same by all observers and that acts as a limit (with velocity is an upper bound and with temperature is a lower bound). And thus the analogy can be rescued, but with a deeper meaning and more complicated framework, so that "thermal reference frames" are subjected to some thermodynamic equivalent of the Lorentz transformations in relativistic kinematics.
