Spacetime manifold: initial condition or result of GR?

[JustinLevy]

I apologize for the poorly worded title. Let me try to explain my question better.

A scientific theory must be predictive to be useful. Since we only know what happened in the past, the global topology of spacetime cannot be an input to the theory.

Given space-like slices/"chunk" of the manifold, and the metric on that chunk along with the fields + physics of how they evolve, GR appears to let us solve for not just the metric on the manifold outside this 'initial chunk' ... but "for the manifold itself".

For example, let's start with an initial condition in which a wormhole exists. Run the equations and we might find that the wormhole closes off in the future. We did not know the topology ahead of time ... in some sense we are solving for the manifold itself.

How can Einstein's field equations (or really any local theory) result in changes to the global structure?

I have a feeling I'm really approaching this wrong, and would like to have a discussion about it. I would appreciate your insight.

 

[Naty1]

My first reaction was that when we integrate, for example in either time or space, say from from 0 to infinity we grow our local causality to 'global' size...My other thought is that local theories work because we seem to live in a causal universe. but I have the feeling that's too simple, so I think I'm approaching your question wrong.

Here are a few insights that might help focus a subsequent discussion if they do address what is of concern to you. (I happen to be reading these currently and their comments came to mind regarding your question.)

James Hartle, in discussing a "theory of everything":

Dynamical laws predict regularities in time. It is a fortunate empirical fact that the fundamental dynamical laws are local- both in space and time. The trajectory of a tennis ball depends only on the conditions that are nearby in space and time, and not, for example, either on what is going on in distant parts of the universe or a long time ago. This is fortunate because that means the dynamical laws can be discovered and studied in laboratories on Earth and extrapolated, assuming locality, to the rest of the universe.

The initial condition of a classical theory that did not allow ignorance would have to be as detailed as the present description of the universe...

and would require more information than we could ever develop.
This means we can likely predict with near certainty only a few of the many regularities in the universe.

In another talk Roger Penrose makes these observations:

Either we do physics on a large scale, in which we use classical level physics: the equations of Newton, Maxwell or Einstein, and these equations are deterministic, time symmetrical and local. Or we may do quantum theory, if we are looking at small things; then we tend to use a different framework, where time evolution is described by what is called unitary evolution. ...according to the Schrödinger equation... deterministic, time symmetrical and local.

He goes on to note that

...the 'reduction of the state vector' or 'collapse of the wave function' (describes) the procedure that is adopted when an effect is magnified from the quantum to the classical level...Some words for this procedure are non deterministic, time assymetrical and non-local.

So Penrose says we have no non local theory with which to work.

I'll be interested to see what experts here have to say in response to your question...

 

[JustinLevy]

My question is more basic than that.

Essentially, on one hand it seems that we would need a manifold with coordinate charts before we could even talk about solving for the metric on it (how could we even do the math without this?), but on the other hand we don't even know the global topology ahead of time so it seems in some sense we learn what the manifold looks like / topology is as we 'solve' the equations ... the equations give us a new 'chunk' of the very manifold we need to work them!?

I think I need to understand that before I can even enter into logical discussion of the 'local theory dictating global properties' question.

 

[hamster143]

We have a 3d topology as part of our initial conditions, right? We evolve our 3d topology + matter distribution and that gives us a partial 4d topology. Emphasis on "partial" (generally speaking, even the complete knowledge of the space-like slice may be insufficient to solve the equations for all observers and all times.)

 

[atyy]

I think many solutions of Einstein's equations do not have a global Cauchy surface - maybe even the Schwarzschild solution doesn't have one.