Why is time special? In Special Relativity, the spacetime interval between two events is $s^2 = -(c{\Delta}t)^2+({\Delta}x)^2+({\Delta}y)^2+({\Delta}z)^2$ giving the Minkowski metric $\eta_{\mu\nu}=\text{diag}(-1, 1, 1, 1)$. What is the justification for making time have a negative coefficient, and how closely is that related to the 2nd law of thermodynamics? Sure, by letting $\eta = \text{diag}(1, 1, 1, 1)$, we get a pretty boring spacetime, and the boosts in the Poincaré group become trig instead of hyperbolic functions, but what's the physical reasoning behind this?
 A: There is a direct link between the minus sign in the metric and thermodynamics. Because the sign is negative, positive energies cannot be rotated to negative energies, and it makes sense to say that the energy of a physical system is always positive. This gives rise to thermodynamic partitioning.
Unlike energy, spatial momentum randomizes with signs, so there is little point in considering the partitioning of momentum. The momentum in a thermal environment will partition around the mean CM velocity of the environment, which can be taken to be zero. But the energy partitions with an extra parameter, the temperature, controlling equilibrium, and there is no change of coordinates which zeroes out the equilibrium energy.
Of course, the same holds in the Galilean space-time, the $c\rightarrow\infty$ version of Minkowski spacetime, so it does not force the issue in any way.
A: I think this is a case of the mathematics being designed to model reality. As you say, making the time component of the metric positive would give a space that doesn't match what we observe. In particular, the negative component for time allows us to disconnect regions of space that aren't causally linked. In other words, the fact that the speed of light is finite and a maximum means that we must describe space-time with a shape that keeps causally disconnected regions separate. The necessary shape is reflected in the choice of the sign of $\eta_{00}$.
That's how I understand anyway...
A: Intuitively here’s how you can imagine it: if time had the same signature as the spatial components, there wouldn’t be a T-variance. That is, there wouldn’t be a distinction between non-space like curves, and we already know that in a sense entropy has to be monotonically increasing in a certain direction of time, i.e. the future direction. This is called “time-orientability” of spacetime, and we require that all causal curves form a causal set distinguishing between space like and causal curves in spacetime, so as to form boundaries. You could ask if spacetime is space-orientable, or if spatial components require a distinction as well — no, because we have the simple basis that I could say that left and right are not the same, or in some sense there is a “preference” in nature, such as in particle decays and PT symmetry or CP symmetry. But spatially we require no causal distinction, therefore the uniform signature.
