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?
Answer
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...
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