thinking covariantly about time (mathy and potentially confusing)
 

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>). 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.

Re: thinking covariantly about time (mathy and potentially confusing)
 

So, Metric, would it be fair to say that, contrary to popular belief, theories of physics/cosmology, specifically those including a Big Bang, do <font color="red">NOT</font> require that something comes from nothing?

Re: thinking covariantly about time (mathy and potentially confusing)
 

Forgive me if I miss your point...
I have 2 'problems' with your approach -

First, it doesn't really seem to account for anything. We are still left with the need to explain the arrow of time, and all the associated issues (the 'ordering' of t1, t2, etc...). I guess that comes down to figuring out U, but isn't that kind of the whole deal?

Secondly, you appear to be saying something that isn't neccesarily 'true'. It looks to me like you are taking a metaphysical position that doesn't have solid experimental grounding. Why can't I just continue to think of time as an outside parameter? (within the bounds of relativity, of course).

Did those points make sense? I'm ashamed to say that they were kind of stream of consciousness....I'll check back when this thread has some traffic [img]/images/graemlins/smile.gif[/img]

Re: thinking covariantly about time (mathy and potentially confusing)
 

Would such a unitary state (with time as an internal parameter) exist in some other sort of time?

Re: thinking covariantly about time (mathy and potentially confusing)
 

[ QUOTE ]
So, Metric, would it be fair to say that, contrary to popular belief, theories of physics/cosmology, specifically those including a Big Bang, do <font color="red">NOT</font> require that something comes from nothing?

[/ QUOTE ]
Well, there still is the problem of the whole universe |PSI&gt; existing at all, but as far as something "causing the big bang to happen" -- yeah, there is nothing really special about it, except that classical theories are singular there (but no more so than at the center of black holes, etc.).

Re: thinking covariantly about time (mathy and potentially confusing)
 

The trump card that once you incorporate GR, you are forced to think covariantly. There is no Schrodinger equation with an "external, pre-existing t" -- there is only the Wheeler-DeWitt equation. Finding the "arrow of time" internally actually becomes a very non-trivial problem in and of itself.

However, most physicists don't do GR-related stuff, so an external time seems like an obvious way to think about things.

Re: thinking covariantly about time (mathy and potentially confusing)
 

That is one hellaciously awesome post.

You get a <font color="orange">Gold Star</font>.

Re: thinking covariantly about time (mathy and potentially confusing)
 

[ QUOTE ]
The trump card that once you incorporate GR, you are forced to think covariantly. There is no Schrodinger equation with an "external, pre-existing t" -- there is only the Wheeler-DeWitt equation. Finding the "arrow of time" internally actually becomes a very non-trivial problem in and of itself.

However, most physicists don't do GR-related stuff, so an external time seems like an obvious way to think about things.



[/ QUOTE ]

Please pardon my butchery of thermodynamics. I actually did take a college course in it, replete with calculus, all for the fun of it, but I’ve forgotten everything but the analogies.

Do the schrodinger equations not take into account thermodynamics? If I drop a coffee cup, I see it break, but I never see the pieces put themselves back in order. Because of this observation, it would seem like we are stuck with a definite arrow of time, based on the progression of thermodynamics. Why can’t anyone ever remember the future?

Re: thinking covariantly about time (mathy and potentially confusing)
 

[ QUOTE ]
Do the schrodinger equations not take into account thermodynamics? If I drop a coffee cup, I see it break, but I never see the pieces put themselves back in order. Because of this observation, it would seem like we are stuck with a definite arrow of time, based on the progression of thermodynamics. Why can’t anyone ever remember the future?

[/ QUOTE ]
The Schrodinger equation specifies an external mechanical time variable in which things evolve. But it does not tell you which direction is the past, and which direction is the future. The 2nd law of thermodynamics does tell you which direction is the past (low entropy), and which direction is the future (high entropy), but it assumes you already are working with a well-defined time variable (which is trivial in the case of the Schrodinger equation since you refer to a pre-defined external one, but it can be very non-trivial in certain covariant theories if you haven't identified which variable happens to be "playing the role of time"). There actually is no fully defined and completely general theory of "covariant statistical mechanics" -- it remains as mysterious in some ways as quantum gravity itself.

Re: thinking covariantly about time (mathy and potentially confusing)
 

[ QUOTE ]
The Schrodinger equation specifies an external mechanical time variable in which things evolve. But it does not tell you which direction is the past, and which direction is the future. The 2nd law of thermodynamics does tell you which direction is the past (low entropy), and which direction is the future (high entropy), but it assumes you already are working with a well-defined time variable (which is trivial in the case of the Schrodinger equation since you refer to a pre-defined external one, but it can be very non-trivial in certain covariant theories if you haven't identified which variable happens to be "playing the role of time"). There actually is no fully defined and completely general theory of "covariant statistical mechanics" -- it remains as mysterious in some ways as quantum gravity itself.



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Metric
I was just wondering. Could you please hurry up and come up with a complete theory of covariant statistical mechanics and while you’re at it, come up with an explanation for quantum gravity so that the rest of us can breathe a sigh of relief that the mysteries of the universe have been explained. [img]/images/graemlins/smile.gif[/img]

Is there an ETA as to when these ideas will be explained? Or are they simply beyond the human capacity for understanding.

I’m also wondering Metric, what do you think happens to us when we die? You’ve said that we will always exist in space time, but what does that mean to you? Is there any philosophy that can be gleaned from your knowledge? Should we have as much sex and drugs as possible so that throughout existence we will always be in a state of euphoria? What are your thoughts.

Re: thinking covariantly about time (mathy and potentially confusing)
 

[ QUOTE ]
[ QUOTE ]
So, Metric, would it be fair to say that, contrary to popular belief, theories of physics/cosmology, specifically those including a Big Bang, do <font color="red">NOT</font> require that something comes from nothing?

[/ QUOTE ]
Well, there still is the problem of the whole universe |PSI&gt; existing at all, but as far as something "causing the big bang to happen" -- yeah, there is nothing really special about it, except that classical theories are singular there (but no more so than at the center of black holes, etc.).

[/ QUOTE ]

I find the question of why does something exist instead of nothing to be really easy. There is exactly 1 way for nothing to exist and infinitely many ways for something to exist, so it seems infintessimally likely for nothing to exist by any reasonable measure.

BTW I had been planning to start a thread entitled `Something <font color="red">CANNOT</font> come from Nothing' but I think I can make the point in your thread now. So, Metric, you seem to agree with the statements that `Theories of physics/cosmology, specifically those including a Big Bang, do <font color="red">NOT</font> <u>require</u> that something comes from nothing' and moreover `Theories of physics/cosmology, specifically those including a Big Bang, do <font color="red">NOT</font> <u>claim</u> that something comes from nothing'. Do you also agree with the statement that `Something <font color="red">CANNOT</font> come from Nothing'?

Also, on the spectrum ranging from a specific, fully understood, tested, coherent, consistent equation, to a vague and nebulous concept or principle, where does "Wheeler-DeWitt equation" sit (and how does it compare to other theories in this regard)? What otherwise-normally-assumed physics concepts does it dispense with, and what does it retain and/or require?

Re: thinking covariantly about time (mathy and potentially confusing)
 

[ QUOTE ]
So, Metric, you seem to agree with the statements that `Theories of physics/cosmology, specifically those including a Big Bang, do NOT require that something comes from nothing' and moreover `Theories of physics/cosmology, specifically those including a Big Bang, do NOT claim that something comes from nothing'. Do you also agree with the statement that `Something CANNOT come from Nothing'?

[/ QUOTE ]
I agree that the big bang does not "come from nothing" any more than the rest of the universe. I also agree that cosmology makes no claim as to something arising from nothing -- its job, like any other physical theory, is just to describe what we see. As for the absolute statement "something cannot come from nothing" -- it seems to make a great appeal to my intuition. Certainly any physical theory requiring "something" to follow from "nothing" I would be extremely skeptical of.

[ QUOTE ]
Also, on the spectrum ranging from a specific, fully understood, tested, coherent, consistent equation, to a vague and nebulous concept or principle, where does "Wheeler-DeWitt equation" sit (and how does it compare to other theories in this regard)? What otherwise-normally-assumed physics concepts does it dispense with, and what does it retain and/or require?

[/ QUOTE ]
A Wheeler-DeWitt equation is basically guaranteed to show up as soon as you incorporate general covariance into your (quantum) theory. So quantum general relativity, then, obeys a Wheeler-DeWitt equation. So as far as you consider general covariance established and tested, so follows the WDW equation in quantum theory.

One caveat -- if you do perturbative (graviton) quantum gravity, you may still have a Schrodinger-like equation, because you are working on a background spacetime that defines a "fixed, pre-existing" time variable for you. But if you really want to get to the full nonperturbative thing, you're stuck with a Wheeler-DeWitt equation. BTW, wikipedia has a nice little entry.

Re: thinking covariantly about time (mathy and potentially confusing)
 

I'm not sure how much statistics you know so maybe someone else can answer this for me.

I feel like this is analogous to the difference between a logistic regression model and a loglinear model. In a logistic regression, there are defined dependent variable and independent variables, whereas the loglinear model variables are not treated as independent or dependent, but as a whole 'system' that simply exists.

I wish I had some of my stat books with me so I could actually look at the equations. I'll try and dig something up online. I also have a vague feeling there might be similar things in time series. I think when you said 'covariate' it triggered something in my brain.

Re: thinking covariantly about time (mathy and potentially confusing)
 

Well I guess thats one way of putting it.

Re: thinking covariantly about time (mathy and potentially confusing)
 

For the record I don't know that much statistics, so I can't be of much help here -- but the way you describe it, it does seem rather analogous.

Re: thinking covariantly about time (mathy and potentially confusing)
 

Great post (almost missed it).

A few questions though that popped in my head while reading it:

Is it ever possible to accurately describe the state of a singularity? So, in other words, are singularities incorporated in the state function of the universe?
If so, how can they be described?
And if not, how can one evolve a state that is not properly described? (And thus one cannot evolve the state of the universe, because one component of the state function should be the state function of the Big Bang for instance).

Maybe these questions dont make sense; havent thought it through yet.

Regards

Re: thinking covariantly about time (mathy and potentially confusing)
 

Those are good questions. Singularities are problems in both pictures, and they arise from the use of a specific singularity-prone model under consideration -- not from the use of covariant dynamics, or the lack of the use of it.

In the non-covariant picture, singularities mean that there is some kind of breakdown of "U" -- it fails to give you sensible answers when you evolve certain states sufficiently far forward or back in time.

In the covariant picture, the problem manifests itself in your inability to adequately describe the whole state |PSI&gt;. Certain pieces of it are just "bad" and sometimes get arbitrarily "thrown out" by people who want to keep the part of the state that makes sense (but of course this doesn't really solve the problem -- if singularities show up, it's an indication that the model you are using simply goes bad under certain conditions).