Why is the future and the past so different?

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The "directionality" of time can be easily explained in terms of entropy, as you point out --and not vice-versa. It's also easy to understand, if you consider the statistical mechanics' definition of entropy.

Slightly formal: Systems evolve from the past, where they find themselves in an improbable state, to the future, where they find themselves in a more probable state than the previous one. If you see water steam entering a manhole in NYC,  you know you are watching a reversed movie.

Less formal: Immagine you have a bookshop, with all books ordered alphabetically. Take a picture in the morning. Then customers arrive,  leave a few books around,  put it back in the wrong order etc. At the end of the day you take one more picture. If there is no one tidying the bookshop, you could identify immediately which picture was taken in the past, and which in the future. Boltzmann (the physicist which formalized the statistical definition of entropy) would say that the entropy in the morning was lower than entropy in the evening.

The entropy of a system in a certain state is a measure of the probability of that system to be in that state. From that, it follows that a system will naturally evolve from a state of low entropy to a state of high entropy, because (again, this is how entropy was defined by Boltzmann) the latter is more probable. Clearly, if you put energy in the system, you can go towards a state of lower entropy, but if you took a bigger picture, you would see other systems going towards states of even higher entropy (your body while tidying up the bookshop, for instance).

In this sense we say that time is a dimension with direction. It is different from space, and this is why, to be technical, in the space-time, time has a different metric from space.

There are two interesting points following from that. One is that you can imagine a universe where time is like space, not-directional. This is what Augustine of Hippo said 1,000 years ago to answer the question: why did God create the universe, considering that he's perfect and should not need a universe to feel even more perfect? Answer: there was not time before the universe was created, because is the universe which defines time, as Boltzmann explained. A similar argument is in "The Large Scale Structure of Space-Time" by Hawking and Ellis, when they say that there is a singularity in the past which "constitutes, in some sense, a beginning to the universe".

PS Note that things like a person jumping from ground floor to the 10th floor of a building are not time-defining. It appears to go against the flow past-to-future, but only because we know it's difficult for someone to have that strength. If the person is wonderwoman, we don't notice the movie is actually reversed, as the trajectory makes perfect sense.

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