Understanding the phrase "simultaneity convention"

[martinbn]

I agree with @PeterDonis, i.e., a useful physical definition of synchronizity should be given by a foliation.

And not just any foliation. For instance a foliation of timelike hypersurfaces may be of interest but I still think it is strange to call it a synchronization convention. In fact if it is not a family of Cauchy surfaces, so that you can study initial value problems, it would be a strange choice of words to call them surfaces of simultaneity.

ps To be fair the paper may be listing that convention just for completeness.

 

[PeterDonis]

In fact if it is not a family of Cauchy surfaces, so that you can study initial value problems, it would be a strange choice of words to call them surfaces of simultaneity.

While I don't disagree with this, I would point out that it implies that only globally hyperbolic spacetimes can even have "simultaneity conventions" defined in them in the first place. I personally don't think that's much of an issue, since all of the spacetimes we actually use in models of our actual universe or things in it are globally hyperbolic, but it is still a very restrictive condition mathematically.

 

[Ibix]

While I don't disagree with this, I would point out that it implies that only globally hyperbolic spacetimes can even have "simultaneity conventions" defined in them in the first place. I personally don't think that's much of an issue, since all of the spacetimes we actually use in models of our actual universe or things in it are globally hyperbolic, but it is still a very restrictive condition mathematically.

I think spacetimes that aren't globally hyperbolic have issues with causality anyway, don't they? Things like CTCs, or removed points? That would make "now" a slippery concept, so perhaps "now is a Cauchy surface" is appropriately restrictive from a physics perspective.

 

[martinbn]

While I don't disagree with this, I would point out that it implies that only globally hyperbolic spacetimes can even have "simultaneity conventions" defined in them in the first place. I personally don't think that's much of an issue, since all of the spacetimes we actually use in models of our actual universe or things in it are globally hyperbolic, but it is still a very restrictive condition mathematically.

Yes, but I meant it for the initial value problem. What are other reasons to have a simultaneity convention?

 

[martinbn]

I think spacetimes that aren't globally hyperbolic have issues with causality anyway, don't they? Things like CTCs, or removed points? That would make "now" a slippery concept, so perhaps "now is a Cauchy surface" is appropriately restrictive from a physics perspective.

If one believes in the strong cosmic censorship conjecture then the non globally hyperbolic ones are the exceptions.

 

[PeterDonis]

I think spacetimes that aren't globally hyperbolic have issues with causality anyway, don't they? Things like CTCs, or removed points?

Not quite, there are a number of conditions in between globally hyperbolic and spacetimes with those kinds of pathologies. The gory details are in Hawking & Ellis.

 

[Ibix]

The gory details are in Hawking & Ellis.

I think I'll finish Wald first...

 

[PeterDonis]

I meant it for the initial value problem. What are other reasons to have a simultaneity convention?

Clock synchronization conventions are useful for all kinds of things. For example, the GPS system has one that doesn't match Einstein clock synchronization (it's impossible to have a global Einstein clock synchronization convention for a rotating congruence of worldlines), but works fine for its intended purpose. Astronomers use a different synchronization convention for labeling events in the solar system, which also works fine for its intended purpose. One might say that part of the intended purpose of at least the latter is solving initial value problems, since the solar system convention is used for things like launching space probes and predicting when they will reach particular mission milestones, but it's certainly not limited to that. But the spacetime in question is globally hyperbolic in any case.

 

[PeterDonis]

I think I'll finish Wald first...

Wald Chapter 8 discusses causality conditions, though not to anything like the level of detail that Hawking & Ellis does.

 

[Ibix]

Wald Chapter 8 discusses causality conditions, though not to anything like the level of detail that Hawking & Ellis does.

Yeah. Chapter 8 is definitely one I need to revisit...

 

[vanhees71]

Clock synchronization conventions are useful for all kinds of things. For example, the GPS system has one that doesn't match Einstein clock synchronization (it's impossible to have a global Einstein clock synchronization convention for a rotating congruence of worldlines), but works fine for its intended purpose. Astronomers use a different synchronization convention for labeling events in the solar system, which also works fine for its intended purpose. One might say that part of the intended purpose of at least the latter is solving initial value problems, since the solar system convention is used for things like launching space probes and predicting when they will reach particular mission milestones, but it's certainly not limited to that. But the spacetime in question is globally hyperbolic in any case.

Well, sometimes pretty local synchronization conventions are sufficient, which should be the case for the GPS and the astronomer's one.

 

[PeterDonis]

Well, sometimes pretty local synchronization conventions are sufficient, which should be the case for the GPS and the astronomer's one.

Yes, it all depends on what "pretty local" means. GPS is useful in or near the Earth. The astronomers' convention is useful in the solar system. Those are large regions for us humans, but of course they're extremely small when compared to the universe as a whole.

 

[vanhees71]

Sure, but it shows that "FAPP" we only need local concepts like simultaneity conventions, etc.

 

[cianfa72]

A "simultaneity convention" is a way of breaking up the spacetime into disjoint 3-dimensional subsets, such that all of the events in each subset are defined to happen "at the same time". This requires that, for each subset, all of the events in the subset are spacelike separated from each other (meaning that no two events can be connected by a timelike or null curve).

So there is actually a restriction on the type of "constant coordinate time" hypersurfaces allowed to define "a simultaneity convention" i.e. events "at the same time" (i.e. they must be spacelike).

On the other hand any other type of set of 3d hypersurfaces foliating spacetime is good from the point of view of defining a coordinate chart.

 

[PeterDonis]

So there is actually a restriction on the type of "constant coordinate time" hypersurfaces allowed to define "a simultaneity convention" i.e. events "at the same time" (i.e. they must be spacelike).

Yes.

On the other hand any other type of set of 3d hypersurfaces foliating spacetime is good from the point of view of defining a coordinate chart.

Strictly speaking, you don't even need a foliation to define a coordinate chart.

 

[cianfa72]

Strictly speaking, you don't even need a foliation to define a coordinate chart.

So in case that coordinate chart has just a local/finite extension in spacetime.

 

[Ibix]

Strictly speaking, you don't even need a foliation to define a coordinate chart.

Doesn't a coordinate chart imply a foliation, by picking one coordinate and holding it constant? Or does "foliation" imply spacelike planes?

 

[PeterDonis]

Doesn't a coordinate chart imply a foliation, by picking one coordinate and holding it constant?

Except in some pathological cases (spacetimes with "holes" in them, etc.--such examples are discussed in Hawking & Ellis), yes, I believe so. But you don't need to define a foliation first in order to define a coordinate chart.

Or does "foliation" imply spacelike planes?

In some contexts, yes, but I don't believe the general definition does.

 

[PeterDonis]

does "foliation" imply spacelike planes?

It's worth noting that in globally hyperbolic spacetimes, a foliation by spacelike 3-surfaces always exists--in fact these surfaces are Cauchy surfaces, which means every timelike or null curve intersects the surface exactly once. Many physicists believe that all spacetimes that are actually realizable physically are globally hyperbolic.

 

[vanhees71]

Doesn't a coordinate chart imply a foliation, by picking one coordinate and holding it constant? Or does "foliation" imply spacelike planes?

A coordinate chart is just a mathematical description of some neighborhood of a differentiable manifold, i.e., a continuous bijective map between an open subset of a Hausdorff point manifold and ##\mathbb{R}^n## with the standard topology (e.g., induced by the Euclidean metric).

It think to do physics you need a bit more, i.e., some notion of a local reference frame and some notion of causal time ordering. This means to do physics you need in fact at least in some local neighborhood a "foliation". That's in order to be able to describe some physical system like "point particles" and "fields" as an initial-value problem of equations of motion describing the "dynamics" of this system.