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A bit tangential, but I'm surprised no one mentioned time in general relativity (Ken Knox discussed special relativity, but the case in general relativity is subtly different). This discusses relativity from a perspective closer to a physicist's, since it's a bit more elementary (and hence easier to understand) in my view. As discussed at the end, relativity in general is somewhat incompatible with statistical mechanics, at least under the standard approximations, so this is almost surely not what you're looking for, but it may be of use to someone.

In GR, space-time is a 4-manifold which is endowed with a Lorentzian metric $g_{ab}$, which is a rank 2 covariant tensor. The scalar product of two vectors $V$ and $W$ is then $g_{ab}V^aW^b$, from which we can compute things as in special relativity (where the metric is $\eta_{ab}$, which is $0$ if $a \ne b$, -1 if $a=b$ and $x^a$ is a spacial coordinate (i.e. x,y,z), and 1 if $a=b$ and $x^a=t$ is the time coordinate).

If a vector (field) V satisfies $g_{ab}V^aV^b >0$, we call it time-like, and similarly for trajectories based on their tangent vectors. These are the possible trajectories for particles with positive mass. Any time-like trajectory corresponds in the limit of low mass and low speed to a local frame of reference in which it is the 'time', and if the trajectory is a geodesic then the frame is inertial. Particles with 0 mass (i.e. light) have null trajectories, with $g_{ab}V^aV^b =0$. If $g_{ab}V^aV^b <0$, the vector is space-like. Particles with no forces (other than gravity) acting on them travel on geodesics. The trajectory of a massive particle is called it's world line. The sign conventions here are often reversed, so care is advised. For a time-like path, we can compute its length by $ds^2 = g_{ab} x^a x^b$, where $x$ are your coordinates.

For any observer, they observe themself as stationary, and traveling forward in time with unit speed. Time for that observer corresponds exactly to the length of their trajectory. That observer can even set up a local set of coordinates in which the metric is approximately $\eta_{ab}$, provided all masses are sufficiently far away. Locally, then, time behaves like in special relativity. The big difference between special and general relativity is that the latter has no inertial reference frames in general, so observers can only measure times in at their own location.

To summarize, time is another coordinate in space-time, just like space, but it isn't universal (it's observer dependent), and the only real restrictions on it is that it must be the part of the metric that has positive signature (in the sense above).

Since this was no-doubt useless in explaining the difference between time in special and general relativity, I recommend Hughston & Tod's An introduction to general relativity. It's fairly light reading and has an introduction to special relativity, but it's rigorous enough for mathematicians reading it. A number of other books at a higher level are available, of which Hawking & Ellis, Wald, and Misner, Thorne, & Wheeler are all good references.

However, if you're looking for a formulation in which the second law of thermodynamics is provable, or even where entropy is defined, relativity is the wrong place to look. Even in special relativity, concepts like thermal equilibrium depend on a particular reference frame, so entropy may be defined in one inertial frame but not another. There has historically been great debate over how temperature (which is the thermodynamic conjugate of entropy) should transform under Lorentz transformations, and it's still not totally resolved.

2 deleted 144 characters in body

A bit tangential, but I'm surprised no one mentioned time in general relativity (Ken Knox discussed special relativity, but the case in general relativity is subtly different). This discusses relativity from a perspective closer to a physicist's, since it's a bit more elementary (and hence easier to understand) in my view. As discussed at the end, relativity in general is somewhat incompatible with statistical mechanics, at least under the standard approximations, so this is almost surely not what you're looking for, but it may be of use to someone.

In GR, space-time is a 4-manifold which is endowed with a Riemannian Lorentzian metric $g_{ab}$, which is a rank 2 covariant tensor. The scalar product of two vectors $V$ and $W$ is then $g_{ab}V^aW^b$, from which we can compute things as in special relativity (where the metric is $\eta_{ab}$, which is $0$ if $a \ne b$, -1 if $a=b$ and $x^a$ is a spacial coordinate (i.e. x,y,z), and 1 if $a=b$ and $x^a=t$ is the time coordinate).

The length element can then be expressed by $ds^2 = g_{ab} dx^a dx^b$ where $x$ are your coordinates. Technically we should have an absolute value, since the right hand side can be negative, but it will be useful to ignore this momentarily.

If a vector (field) V satisfies $g_{ab}V^aV^b >0$, we call it time-like, and similarly for trajectories based on their tangent vectors. These are the possible trajectories for particles with positive mass. Any time-like trajectory corresponds in the limit of low mass and low speed to a local frame of reference in which it is the 'time', and if the trajectory is a geodesic then the frame is inertial. Particles with 0 mass (i.e. light) have null trajectories, with $g_{ab}V^aV^b =0$. If $g_{ab}V^aV^b <0$, the vector is space-like. Particles with no forces (other than gravity) acting on them travel on geodesics. The trajectory of a massive particle is called it's world line. The sign conventions here are often reversed, so care is advised. For a time-like path, we can compute its length by $ds^2 = g_{ab} x^a x^b$, where $x$ are your coordinates.

For any observer, they observe themself as stationary, and traveling forward in time with unit speed. Time for that observer corresponds exactly to the length of their trajectory. That observer can even set up a local set of coordinates in which the metric is approximately $\eta_{ab}$, which will be valid so long as provided all masses are sufficiently far away. Locally, then, time behaves like in special relativity. The big difference between special and general relativity is that the latter has no inertial reference frames in general, so observers can only measure times in at their own location.

To summarize, time is another coordinate in space-time, just like space, but it isn't universal (it's observer dependent), and the only real restrictions on it is that it must be the part of the metric that has positive signature (in the sense above).

Since this was no-doubt useless in explaining the difference between time in special and general relativity, I recommend Hughston & Tod's An introduction to general relativity. It's fairly light reading and has an introduction to special relativity, but it's rigorous enough for mathematicians reading it.

However, if you're looking for a formulation in which the second law of thermodynamics is provable, or even where entropy is defined, relativity is the wrong place to look. Even in special relativity, concepts like thermal equilibrium depend on a particular reference frame, so entropy may be defined in one inertial frame but not another. There has historically been great debate over how temperature (which is the thermodynamic conjugate of entropy) should transform under Lorentz transformations, and it's still not totally resolved.

1

A bit tangential, but I'm surprised no one mentioned time in general relativity (Ken Knox discussed special relativity, but the case in general relativity is subtly different). This discusses relativity from a perspective closer to a physicist's, since it's a bit more elementary (and hence easier to understand) in my view. As discussed at the end, relativity in general is somewhat incompatible with statistical mechanics, at least under the standard approximations, so this is almost surely not what you're looking for, but it may be of use to someone.

In GR, space-time is a 4-manifold which is endowed with a Riemannian metric $g_{ab}$, which is a rank 2 covariant tensor. The scalar product of two vectors $V$ and $W$ is then $g_{ab}V^aW^b$, from which we can compute things as in special relativity (where the metric is $\eta_{ab}$, which is $0$ if $a \ne b$, -1 if $a=b$ and $x^a$ is a spacial coordinate (i.e. x,y,z), and 1 if $a=b$ and $x^a=t$ is the time coordinate).

The length element can then be expressed by $ds^2 = g_{ab} dx^a dx^b$ where $x$ are your coordinates. Technically we should have an absolute value, since the right hand side can be negative, but it will be useful to ignore this momentarily.

If a vector (field) V satisfies $g_{ab}V^aV^b >0$, we call it time-like, and similarly for trajectories based on their tangent vectors. These are the possible trajectories for particles with positive mass. Any time-like trajectory corresponds in the limit of low mass and low speed to a local frame of reference in which it is the 'time', and if the trajectory is a geodesic then the frame is inertial. Particles with 0 mass (i.e. light) have null trajectories, with $g_{ab}V^aV^b =0$. If $g_{ab}V^aV^b <0$, the vector is space-like. Particles with no forces (other than gravity) acting on them travel on geodesics. The trajectory of a massive particle is called it's world line. The sign conventions here are often reversed, so care is advised.

For any observer, they observe themself as stationary, and traveling forward in time with unit speed. Time for that observer corresponds exactly to the length of their trajectory. That observer can even set up a local set of coordinates in which the metric is $\eta_{ab}$, which will be valid so long as all masses are sufficiently far away. Locally, then, time behaves like in special relativity. The big difference between special and general relativity is that the latter has no inertial reference frames in general, so observers can only measure times in at their own location.

To summarize, time is another coordinate in space-time, just like space, but it isn't universal (it's observer dependent), and the only real restrictions on it is that it must be the part of the metric that has positive signature (in the sense above).

Since this was no-doubt useless in explaining the difference between time in special and general relativity, I recommend Hughston & Tod's An introduction to general relativity. It's fairly light reading and has an introduction to special relativity, but it's rigorous enough for mathematicians reading it.

However, if you're looking for a formulation in which the second law of thermodynamics is provable, or even where entropy is defined, relativity is the wrong place to look. Even in special relativity, concepts like thermal equilibrium depend on a particular reference frame, so entropy may be defined in one inertial frame but not another. There has historically been great debate over how temperature (which is the thermodynamic conjugate of entropy) should transform under Lorentz transformations, and it's still not totally resolved.
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