However, the right definition of the past and the future is independent of these sign conventions for spacetime coordinates. The right definition says that
The future is evolving from the past (and the present).
Correspondingly, the calculations that are designed to theoretically mimic this evolution have the same arrow:
The future is calculated from the past (and the present) as long as we use the usual calculations that resemble the evolution.
It’s important that you can’t exchange the words “future” and “past” in the sentence above.
That doesn’t mean that science can never say anything about the past, by manipulating with the present data or the data from a closer past. But this type of calculation is different from predictions of the future. It follows different formulae, too. They’re the formulae of logical inference, e.g. Bayesian inference.
And as we have explained many times, the results of this inference – the retrodictions – always depend on our priors. So the knowledge of the present is enough to calculate the future (classically) or to predict the unique probabilities of various states in the future (quantum mechanically). But it is simply never enough to calculate the unique state or unique probabilities of various states in the past.
— Logical arrow of time and terminology
— Lubos Motl
2012.02.28 Tuesday ACHK