Why does the second law of thermodynmics only occur in the time dimension? If I break an egg, how broken it is depends on "when" I am not "where" I am. Why is the time dimension special?
 A: First off, there are a couple of reasons why we don't have complete symmetry between space and time:


*

*We have 3 spatial dimensions and only one timelike one.

*The particles in the standard model have timelike or lightlike world-lines; as far as we know, there are no tachyons.
For these reasons, it only makes sense to talk about laws of physics that take initial conditions on one spacelike surface and predict the evolution of the system from there. We don't have laws of physics that can take data on a timelike surface and evolve that outward in space.
The other issue is that to get the second law, you need two ingredients: (1) some sort of state-counting that says there are more way to make some macrostates than others; (2) something that breaks time-reversal symmetry. (See this answer.) What breaks time-reversal symmetry in our universe is that, for reasons unknown to us, we had a low-entropy Big Bang. This necessary fact about cosmology also breaks the symmetry between space and time. The Big Bang was something that happened at a certain point in the past, everywhere; it's not something that happened at a certain point in space, at all times.
