Time reversal is a fundamental property in physics. Consider any simple interaction in physics. A reversal in the 'sign' of time will give you an equation that is also true. There is one exception to this and this is that the entropy of a system always increases. This has led a number of physicists to agonize over the idea of time reversal; is it possible or even meaningful? Could time ever be reversed in reality? Consider a box with all the atoms of gas in it confined by a partition. Now remove the partition and in a short while the atoms will fill all the available space resulting in an increase in entropy of the system. Now reverse the direction of time. As we have already asserted, we know of no physical law which would be broken if we did this. So would the gas after a short period be crowded into a smaller space, thus decreasing the entropy, thus breaking the first law of thermodynamics?

The answer to this is really quite simple as it follows directly from Einstein's equivalence principle. Whenever we talk of using a different reference frame, we could equally be talking about a similar reference frame but measured in different units. Changing our unit of distance from km to light years will not affect the underlying physics, which will still be true. So we have to include, not just arbitrary coordinate systems but the same system represented in different units. Think in terms of a (real) scale factor $$S_\alpha$$ for each ($$\alpha$$) axis. Now if this factor is arbitrary, it could be -1 for one axis. Apply this to the x-axis and all we are saying is that distances to the left will be considered positive instead of the normally used convention, that they would be negative. Nobody would raise an eyebrow at this suggestion. Now apply this to the time axis. There is nothing to stop us from measuring time in the opposite direction, and nothing in the basic laws of physics will change. In case you think that this is just not reasonable, remember that we measure time from an event in the past (the birth of Christ); we could equally be measuring it to an event in the future - maybe the second coming, or some cataclysmic event in the future. (The millennium bug springs to mind, but that has now past.) Whatever the reasoning, it is clear that it will never affect the underlying physics.

The only remaining question is "has this any relevance to black holes"? Well, time invariance is assumed in deriving the Schwarzschild solution, so it is at least significant.