Proving Anti-time Mathematically?

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.

The Anti-time/Anti-matter Controversy

"Anti-matter looks like matter going backwards in time," is a quote I've been hearing lately. Here's a question that popped into my head: What does matter going back in time look like? The typical response is it looks like a rewinding video. However, if matter truly goes back in time it would simply vanish or would exist in the past, not the present or future. It would be unobservable. All we can see is it going in the opposite direction in forward time.

Case and point: positrons have been experimentally trapped for as long as 16 minutes. How is that possible if they go back in time? To exist for 16 minutes, they have to go forward in time for 16 minutes. If you ask me, anti-matter looks like matter with an opposite charge--FULL STOP.

Reverse time comes with its own set of problems. Click here to read about those. Not withstanding these problems, a case can be made for reverse time that doesn't take you back to your high school reunion. I call it anti-time.

Anti-time does not take you to the past, but rather, it makes the past possible. If there was no anti-time there would be nothing to cancel the current moment in time. The moments would pile up. You would not only be living in the present moment, but all your past moments as well. All your memories would be all too real. Hopefully, for your sake, they are good memories.

To demonstrate how anti-time works, let's start with a time-moment represented by an arrow:

The plus sign indicates that it is a positive, forward time-moment. The time-line arrow to the right is how we normally think of time: just a straight line going up in this instance. Now let's add a second time-moment and see what happens:

The first time-moment is canceled by an anti-time-moment (arrow pointing down). That leaves us with the present moment. We get a similar result when another time-moment is added:

But why isn't there an anti-time arrow to cancel the current time-moment? Well, at the beginning of time, there was no past, so time could only go forward, but once it went forward a little, there was some room to go back and still room to go forward--so we get a forward arrow followed by a backward arrow and another forward arrow, etc. As a result, we get time that has a forward bias.

This is a pretty bizarre theory! Can it be tested? Sure. Ask yourself, "Is history history or is it still happening?" If history is history, obviously something is cancelling those time-moments that would otherwise pile up. In mathematics we use a minus sign to cancel a plus sign, so it stands to reason that -time cancels +time leaving us only with the present.

Where is the Anti-matter Hiding?

Around the time of the Big Bang there was, according to one theory, unequal amounts of matter and anti-matter. When matter met anti-matter, they annihilated each other, albeit a little bit of matter was left over--that matter is the matter of our universe. Below is a Feynman diagram illustrating the process of matter-anti-matter creation and destruction.

According the above diagram, a particle (+A) and an anti-particle (-A) destroy each other and produce a boson (B). The boson then goes on to produce another +A/-A pair. This is all done in time (+t) and minus time (-t). (Anti-particles allegedly have anti-time.)

To my knowledge, the above theory has yet to be experimentally verified. It would be great if some scientist could show that a photon, for example, could produce an electron without the positron. That would surely establish why our universe is predominately matter.

The next best thing is to demonstrate that an anti-particle always comes with a particle, but in spite this, a universe could still end up with mostly matter or anti-matter. That is what we are going to demonstrate below.

(Note: What we call matter could very well be anti-matter. However, we are biased and like to think positive--so we label what we have "matter" and its opposite "anti-matter.")

Let's kick things off with a diagram of two electrically charged plates. Let's pretend the positively charged plate is matter and the negatively charged plate is anti-matter. The arrows represent the field lines. Let's also assume the plates are identical in every way except for the charge--this will be analogous to equal amounts of matter and anti-matter.

The arrows originating from the positive plate point away from the source. One arrow goes up and another goes down and through the negative plate:

The arrows originating from the negative plate behave in the opposite fashion:

Now let's add the arrows. Arrows pointing in opposite directions shall cancel each other. Arrows pointing in the same direction shall enhance each other.

Well would you look at that! The arrows didn't cancel each other out completely. We end up with two left over. So it is possible to start with equal amounts of opposite charge, put them together and not end up with zero. However, what if we had a second pair of plates that are reversed?

As you can see the arrows left over are pointing in the opposite direction. If we add those arrows to our original left-over arrows, we get zero.

So whether we get zero or left-over arrows depends on whether we have an odd or even number of plate pairs. An odd number will always give us left-over arrows. Even-numbered pairs will sometimes give us left-over arrows and sometimes not.

We could ask, what is the probability we will get an odd or even number of pairs? I'd say .50 is a reasonable estimate. (P1(a) stands for probability of getting an odd or even number of pairs. The "a" stands for annihilation.)

Let's assume our luck is bad and there are two pairs of plates. What is the probability (P2(a)) that the left-over arrows will be opposite and cancel each other?

How about .50? Finally, what is the total probability (P(a)) that we will end up with zero arrows? What is the probability (P(!a)) we will have something left over when matter and anti-matter annihilate each other?

It should be obvious our universe had at least a .75 (or 75%) chance of having some matter (anti-matter) left over. The odds improve when bigger numbers are crunched. For example, four pairs of plates have a .375 probability of cancelling each other out and yielding zero. That raises the chance of left-over matter to .78 (or 78%) (1- (.5 * .375)= .81; [.81+.75]/2=.78).