Imagine two rocks in one-dimensional newtonian space. They come at each other, collide elastically and then bounce away. We can draw the situation with time on the vertical axis and space on the horizontal axis.
We don't have to point the arrow of time straight up. We can rotate it up to 45 degrees clockwise of counterclockwise and newtonian mechanics still works.
We can even reverse it.
We can point the arrow of time many different directions. Half of the unit circle is valid.
But the real world has more than two rocks in it. Suppose we had multiple collisions at different angles. We must point the arrow of time in a direction that is allowed by all collisions. The only direction allowed by every possible collision is straight up, right?
Wrong. When we're playing with spacetime like this we need to use special relativity instead of classical relativity. In special relativity, space and time have the same units where one second of time (one second) equals one light-second of space ( meters).
Our two rocks can't be travelling at 45 degrees because that's the speed of light and matter can't move at the speed of light. Instead, the rocks must be travelling steeper angles in our diagram.
Applying special relativity also means we can't literally rotate our image a certain number of degrees. What we were previously calling a "rotation" is more robustly defined as a "change in reference frame". Instead of literally rotating the diagram we have to perform a Lorentz transformation. With Lorentz transformations the same 180 degrees are valid for all collisions. Everything works out. This solves the problem of multiple collisions at different angles.
We've now proved that time can point in an infinite number of different directions. In the real world time feels like it goes one direction. How do we know which direction this is in our universe of floating rocks?
It depends on your perspective. If you're one of the rocks then your perception of time is perpendicular to your trajectory in space. Your spacetime trajectory is a line. It's possible to follow this line in two directions. We still must figure out which end is forward in time and which one is backward in time.
The past is distinguished from the future via differences in entropy. One direction has greater entropy than the other. The direction toward increasing entropy is the future and the direction towards decreasing entropy is the past.
The region of allowable time arrows is separated into two non-adjacent regions separated by the region of disallowed time arrows. One of the two allowable regions has higher entropy than the other. So "future" refers to the set of green arrows with higher entropy and "past" refers to set of green arrows with lower entropy.
"Entropy" means multiplicy of possible microstates for a particular macrostate.
Suppose a universe is made up of quantum particles each of which has two states: 0 and 1. In this sense, the entire universe is just a number like 0b0000000000000000. Our microstate is the entire 16-digit binary number. Our macrostate is the total number of 1s in the string. I will use to denote the macrostate with particles in the 1 state. So the macrostate corresponding to microstate 0b0001001001001000 is because there are 4 ones in the binary microstate.
Let's suppose the lifetime of a quantum particle is 1/16 instants. To simplify the math let's also assume that every instant exactly one quantum particle changes its state.
Our universe now consists of possible states. We can plot them out in a 16-dimensional hypercube. Every point in the hypercube is connected to exactly 16 other universe microstates. Every edge corresponds to a possible past and future of each instant. From this perspective there is no arrow of time. To a microstate, each of the 16 adjacent microstates is equally likely.
That was looking at the universe in terms of microscates. If we look at the universe in terms of macrostates then we get a one-dimensional line from to . Each macrostate can change into a macrostate one greater or one lesser than itself. (Except and which must move to and , respectively.) So the macrostate can become or in the next instant. But they're not equally likely. The macrostate has microstates. The macrostate has microstates so the macrostate has a 97% chance of entering the macrostate and a 3% chance of entering the macrostate. That's another way of saying the macrostate has greater entropy than the macrostate.
The heat death of our universe is macrostate , which has maximum entropy. But there are two beginnings of minimal entropy: and . If the real world is like this then there could be another big bang (or more) on the other side of the heat death of the universe.