The CPT theorem consists of three symmetries: charge conjugation (C), parity reversal (P), and time reversal (T). Taken together they provide the symmetries needed to make the laws of physics invariant. However, time reversal causes paradoxes that mess up the laws of physics. To find out more details, click here.
When we think of reverse time, we think of going back in time or time travel and the paradoxes that come with it. That's why I prefer the term anti-time. Anti-time is symmetrical with time, but it's not about going back in time to the past. It makes the past, so it doesn't create the paradoxes as reverse time does. (To learn more about anti-time, click here.)
Let's see if we can at least mathematically prove that anti-time exists. We begin our proof with a very famous equation (E=energy; p=momentum; c=light speed; m=mass):
Notice the plus and minus signs in front of the radical. Einstein's equation predicts both matter and anti-matter, since the square root of a positive number can be negative or positive. Anti-particles allegedly have negative mass and energy--so the negative root corresponds with anti-matter.
With a little algebra we can derive the following:
We can take the square roots of the last equation and fit them in a Lorentz right triangle (t and t' = time; v = velocity). Such a triangle is used to derive the Lorentz Factor. (Click here for more details.)
By matching the Einstein equation terms with the Lorentz terms we can derive something equivalent to the Lorentz Factor.
It looks like we ended up with a Lorentz-type equation with an extra v^2 added to the c^2. We can fix that when we realize c is the top speed limit and that added velocity (v) has no effect. Just to be sure, we do the math below (where 0 <= epsilon <= 1):
In the steps above we take the square root of v^2 + c^2 to be equal to c + (epsilon)c. We then plug it into the velocity-addition formula. We get c. Good! Let's square it and put it back into our Lorentz equation:
Since we were able to derive the Lorentz time equation from an equation that predicts matter and anti-matter, does it then follow that the time equation predicts time and anti-time? Note the plus and minus sign in front of the radical. A negative square root is just as mathematically valid as a positive one.
One could assume that matter corresponds with time and anti-matter corresponds with anti-time, but let's look at this issue from another angle. We go back a few steps and derive the following:
The last equation shows we can only get anti-time if we either have a positive numerator divided by a negative denominator or vice versa. If we divide negative energy by negative energy or positive energy by positive energy, we get positive time!
Imagine two universes: one made entirely of matter and the other entirely of anti-matter. They both would have positive time and no anti-time to put the present moment into the past. Now imagine a universe with both matter and anti-matter. There we can have both time and anti-time. To get anti-time, we need to be able to divide matter by anti-matter (or vice versa) in the last equation above. All this implies that positive time is caused by matter or anti-matter, and anti-time is caused by a combination of matter and anti-matter.