thinking covariantly about time (mathy and potentially confusing)

[Metric]

Okay -- the more I think about this, and the more posting I see on whether or not the universe "always existed" etc. the more I see how many misconceptions result from not "thinking covariantly." When you begin to think covariantly, hopefully it will feel like getting a new brain.

First, let's think of physics and the universe non-covariantly. as I'm sure 99.99% of the population already does.

There is a state of the universe at time t (now). Call it |psi(t)>. To predict the future (time t'), we evolve this thing into a new state using the "dynamical laws of physics" represented by U(t',t).

I.E. at time t' in the future, the universe will be in new state |psi(t')> = U(t',t) |psi(t)>

It should also be mentioned that since |psi> represents the whole universe, it is really a product of a bunch of subsystems, x1, x2, etc. so |psi> = |x1>|x2>|x3>|x4>... and each one of these variables also evolves in the same way with time.

The goal of physics, then, is to figure out exactly what "U" is, and to figure out in detail what |psi> is like "right now." Time in this framework is an external parameter that we use to describe "when the universe is."

This works well enough for some things, but it tends to lead one to ask certain questions that might not be defined. Suppose, for example, we conclude that there was a "big bang" at time t'' -- then what was the universe like before t''? Or, how can this possibly work if I evolve a "time traveller" forward in time and he winds up in the past? Doesn't this create a paradox?

Thinking covariantly (which is kind of forced on relativists) is different. Instead of time as an external parameter, it is incorporated into the definition of the state itself. So when we write down a "covariant state of the universe" we write down something like this:

|PSI> (capital letters will indicate a "covariant state")

And that's it. It represents all times. But let's break it down into subsystems again to see where the "illusion of time evolution" comes from:

|PSI> = |psi1>|t1> + |psi2>|t2> + |psi3>|t3> + ...

i.e. |PSI> is a superposition of a bunch of different non-covariant psi-states entangled with another subsystem representing time.

Time is not an external parameter any more -- it is a dynamical quantity treated on the same footing as all the rest of the dynamical variables. It can be thought of as a "subsystem of PSI" -- time is simply a part of the universe. Now, the lower-case psi's are still related to each other as they were before, but the lower-case psi's aren't "everything" now -- we can't ignore the "t" part of the state. The way relativists express this, is that |PSI> is a solution to the "Wheeler-DeWitt equation." H|PSI> = 0 Physics, in this covariant picture, is finding out what "H" is, and finding out which solution |PSI> the universe corresponds to. So we treat the universe, including the past present and future as a singular "thing" rather than a sequence of things ordered by an external, pre-existing time variable.

Now the questions that seemed to give us trouble before don't seem quite so troublesome. We realize that |PSI> might not contain a piece representing a "before the big bang" -- there's no reason to expect it to. "Wait!" you object, "Does this mean the universe didn't 'always exist'?" But there is no meaning to "always exist" -- the universe, |PSI>, just simply exists! All of it, at all "times!"

"But what about time travel??? How can you 'evolve a person into the future' and then have them exist in the past without creating paradoxes? What if I kill my grandfather?" All time travel is taken into account in the solution of H|PSI>=0 -- it doesn't bother me at all if |psi3> contains a guy walking into a time machine and |psi1> contains a guy trying to kill his grandfather, as long as it solves the Wheeler-DeWitt equation (and it probably simply won't be a solution if he actually succeeds in killing his grandfather in |psi1&gt. There is no evolution -- |psi1> doesn't "change" based on what happens in |psi3>. Either it is part of the full state |PSI> or it isn't.