Gravitational waves as not "proximal"?

[ComradeConrad]

TL;DR Summary

General Relativity is local theory even though there is "flat"(er) space in between gravitational waves and their source. Is there a term for this state of affairs? Non-proximal?

Usually spacetime curvature is localized/proximal to what is "causing" it, right? I'm wondering whether there is a term for the situation seen with gravitational waves where there is some relatively flat space between observable gravitational effects and the mass(es) that "caused" them? I'm calling this "non-proximal," because I don't know anything and it sounds like it could be a thing.

I was looking around for an idea like "non-local gravity" and found Bahram Mashhoon (student of Wheeler) who is apparently working on a theory of gravity that modifies the principle of locality in a way that, like most things, is way over my head, but I understood enough to know that's not what I'm getting at.

Go easy on me, this is my first ever post!

 

[Ibix]

General relativity is a local theory in that the curvature of spacetime at any event can be deduced from curvature nearby in a 4d sense. Roughly speaking, if I know the curvature in some small region of space now, I can propagate it forwards to deduce the curvature in a similar small region a short time later. (This is somewhat imprecise, but can be stated formally with maths.) That's what's meant by "local" here - that I don't need to know where a gravitational wave came from or why it's here now to predict it's behaviour. I only need to know about what's going on here now.

I hadn't heard the word proximal in this context. It would help if you linked to whatever you've been reading about this theory by Mashhoon.

 

[ComradeConrad]

Thanks for the response! I was trying to make clear that I understand locality and that it has nothing to do with the question I'm doing my best to ask...

I'm talking about curvature being distanced from its source with flat space in between, as it is with gravitational waves. What I'm wondering is if the math has anything to say about this kind of situation. Are there other observed or predicted examples of such a thing? Or could it be shown that it is only possible in the case of radiating gravitational waves? It's an interesting topic for me! Not sure how or who else to ask all this

 

[Ibix]

I'm talking about curvature being distanced from its source with flat space in between, as it is with gravitational waves.

If you think about it four dimensionally, gravitational waves form a cone-like structure in spacetime, so they are connected to their source event.

Are there other observed or predicted examples of such a thing?

Electromagnetic waves are the obvious similar phenomenon. But if you mean in terms of gravity, not that I'm aware of. Any change in a gravitational field propagates as a gravitational wave. I'd be surprised if you can have a static field disconnected from its source, since there'd be no reason for the difference in curvature. I could be wrong, though. See what others say.

 

[Dale]

I'm talking about curvature being distanced from its source with flat space in between, as it is with gravitational waves.

This doesn’t sound right. Curvature gets monotonically higher the closer you are to a gravitating body.

 

[ComradeConrad]

Any change in a gravitational field propagates as a gravitational wave. I'd be surprised if you can have a static field disconnected from its source, since there'd be no reason for the difference in curvature. I could be wrong, though. See what others say.

Ya I'm definitely wondering about a "static" field situation. Or how about static from a certain reference frame? There's the idea of "geons" that Wheeler came up with (https://en.wikipedia.org/wiki/Geon_(physics)) the gravitational version of which seems at least related. If such a thing isn't stable on its own, could it get "transfered" to a typical mass-containing gravitational potential? It's really interesting to me and I'm wondering what people who can do the math think. Is there anything in there that prohibits a static field? Are these all still open questions? Do I HAVE to learn the math to study it more at this point?!

This doesn’t sound right. Curvature gets monotonically higher the closer you are to a gravitating body.

Gravitational waves don't "sound right" to me at all in the first place! But the fact of them compels me understand.

 

[PeterDonis]

I'm definitely wondering about a "static" field situation.

In a static situation, there cannot be any gravitational waves. There cannot be anything that changes. That's what "static" means. So there is nothing that has to "propagate" in a static situation. That makes it a very bad concept to use if you want to understand gravitational waves.

how about static from a certain reference frame?

Whether or not a particular spacetime is static does not depend on your choice of reference frame; it's an invariant fact about the spacetime geometry.

If such a thing isn't stable on its own, could it get "transfered" to a typical mass-containing gravitational potential?

If such a thing could happen, it wouldn't be static.

Is there anything in there that prohibits a static field?

There are certainly known solutions in GR that are static spacetimes; the Schwarzschild solution that describes the vacuum surrounding a spherically symmetric massive object is the best known one.

What these spacetimes do not describe, as above, is situations where something changes. Emission of gravitational waves is a change.

 

[Dale]

But the fact of them compels me understand.

Sure, but I think you have the wrong facts. I think you should get the right facts first before seeking to understand those facts.

Curvature in the Schwarzschild spacetime, at least, gets monotonically greater as you get closer to the center. So it does not exhibit this “non-proximal” behavior. I am not sure if there are spacetimes that do, so I am concerned about establishing the validity of the “fact” before explaining it. Invalid facts are notoriously difficult to explain.

 

[Ibix]

There are certainly known solutions in GR that are static spacetimes

I think the OP is looking for a static spacetime that has a flat region inside a curved region. Kind of like a Dyson sphere spacetime (flat inside the sphere, Schwarzschild outside), but without the shell. I suspect it isn't possible.

I think the gravitational waves are just a (non-static, obviously) example of the kind of layout he's looking for.

 

[PeterDonis]

I think the OP is looking for a static spacetime that has a flat region inside a curved region.

No static spacetime can have any gravitational waves in it, so if the OP is trying to understand gravitational waves, static spacetimes are the wrong ones to look at.

Kind of like a Dyson sphere spacetime (flat inside the sphere, Schwarzschild outside), but without the shell. I suspect it isn't possible.

For the two geometries you mention, flat Minkowski and Schwarzschild, it is impossible to match those geometries at a boundary at all; that's true whether the spacetime as a whole is static or not static. There must be a region in between that contains nonzero stress-energy. In the static case, that region would have to be some kind of "shell" made of stress-energy in hydrostatic equilibrium. In the non-static case, the region in between could be, for example, a thin "shell" of radiation, either ingoing or outgoing. But either way there has to be something in between.

 

[pervect]

I'm not too sure I understand the question, but there are some comments I can make.

GR can be formulated as a "well-posed initial value problem", as can electromagnetism. Unrleated to this is the issue of the causal structure of GR, a separate topic. Locally, the causal structure of general relativity is that of special relativity, cause and effect is determined by light cones. Globally, though, issues with causality, such as closed timelike curves, can arise under some solutions.

I'm not sure which aspect (the initial value aspect, or the causal aspect) is more closely related to your question, but you could do some research on both. I haven't ever seen a formal discussion of "proximal", so I'm not sure which of these might be applicable to your question.

As far as "flat space between gravitational waves" goes, if you consider a small enough region of space-time, you can ignore the curvature, just as you can ignore the curvature of the Earth in a small region.

As a practical matter, for instance, if you get a map of your local area, you can accurately draw a scale map on a flat sheet of paper of a small region of the Earth. And you can use this map to navigate short distances , for instance to walk to the store. But you can't use such a map to plan a longer journey without problems, for instance if you wanted to make a long ocean voyage. You can make a map that covers the oceans, but it won't be and can't be at a 1:1 scale everywhere if it's drawn on a flat sheet of paper.

In a related issue, you can't gift-wrap a sphere, say a baseball, with flat paper, assuming the paper is not "stretchy". You'll find that you have excess material if you try.

It used to be common to get a book of such local maps for local navigation. Nowadays maps are more and more going online and not being printed in this way, though you might still find some.

However, if you want to consider large regions, you'll really need a globe to get a good accurate depiction of the geometry of the surface of the Earth.

This associated technical languate here is "tangent space". The tangent space exists for the Earth and for GR, and is always flat.

In the case of gravitational waves, the general test is that if the 3-d region under consideration is much smaller than the wavelength of the gravitational wave, you can treat it as if it were flat. This is sufficient, though perhaps not necessary - you might be able to find regions larger than a wavelength in some directions if you make the extent in the other directions small.

 

[PeterDonis]

GR can be formulated as a "well-posed initial value problem", as can electromagnetism. Unrleated to this is the issue of the causal structure of GR, a separate topic.

Not really. In the solutions you refer to in which there are issues with causality, such as closed timelike curves, the initial value problem is not well posed. Or, to put it the other way around, in order to formulate the initial value problem for GR, you must start with a spacelike 3-surface that satisfies certain constraint equations; and such a 3-surface can only produce, once the initial value problem is solved, a spacetime that is globally hyperbolic, i.e, that cannot have any of the causality issues you describe. So causal structure and the initial value problem being well posed are not independent of each other.

 

[pervect]

Not really. In the solutions you refer to in which there are issues with causality, such as closed timelike curves, the initial value problem is not well posed. Or, to put it the other way around, in order to formulate the initial value problem for GR, you must start with a spacelike 3-surface that satisfies certain constraint equations; and such a 3-surface can only produce, once the initial value problem is solved, a spacetime that is globally hyperbolic, i.e, that cannot have any of the causality issues you describe. So causal structure and the initial value problem being well posed are not independent of each other.

Ah, I wasn't aware - Wald has a section on treating GR as a well-posed initial value problem, but I haven't really read the fine print.

So, to take an example, would the Kerr black hole be well-posed at the inner horizon? And am I correct in remembering that closed timelike curves exist there?

 

[PeterDonis]

would the Kerr black hole be well-posed at the inner horizon?

The inner horizon of Kerr spacetime is a Cauchy horizon, so the initial value solution doesn't cover it or the region inside it. That doesn't mean Kerr spacetime as a whole isn't well-posed (at least, as I understand it); it just means the initial value solution from a Cauchy surface for Kerr spacetime (more precisely, from a Cauchy surface that includes the Kerr exterior and asymptotically flat region) is not identical to its own maximal analytic extension.

 

[PeterDonis]

am I correct in remembering that closed timelike curves exist there?

Yes.

 

[ComradeConrad]

Thank you all for the responses. I think I might be getting closer to being able to ask intelligible questions!

I think the OP is looking for a static spacetime that has a flat region inside a curved region...

I think the gravitational waves are just a (non-static, obviously) example of the kind of layout he's looking for.

This is indeed what I'm getting at. I was trying to generalize from the gravitational wave situation. I was calling a curved region being outside a flat region "non-proximal" to its source. If that is all valid enough, then the question is whether this example is the only way that such a state could occur or whether others have been theorized?

[curvature] ---- flat space ----- [source]

Starting to wonder if this is hard or impossible to answer because the field equations are nonlinear and so what's possible is just sort of ¯\_(ツ)_/¯

 

[Ibix]

It's definitely not possible in the case of spherical symmetry, by Birkhoff's theorem. I can't see simply changing the mass distribution to cubical (or whatever) really changing that, not least since at any significant distance the gravitational field will be indistinguishable from spherically symmetric. More generally, if you've got more-or-less flat spacetime in a static situation then it's weak field gravity and you can use Newton. And then it's straightforward to see that gravity can't increase again at a larger distance. You can, of course, have two masses separated by a large distance such that their gravity is negligible for some region between them, but I don't think that's what you are after.

Basically, I think the issue is that there has to be a reason for the curvature to start increasing. In the hollow sphere example I mentioned, it's the mass of the sphere wall. In the (non-static) gravitational wave spacetime, it's the presence of the gravitational wave at a slightly different place a short time ago (and you can iterate that argument backwards to whatever event caused the wave in the first place). But you want vacuum so you can't appeal to mass, and you want static so you can't appeal to an earlier state.

 

[Dale]

[curvature] ---- flat space ----- [source]

I don’t think such solutions exist.

 

[PeterDonis]

[curvature] ---- flat space ----- [source]

Not possible. Any region of spacetime that is flat cannot have a source inside it.

 

[ComradeConrad]

I was assuming curvature around ("proximal" to) the source. So, like this, where the curvatures share the same source:

[curvature] --- flat space ---- [curvature] ---- [source]

Is this not patently the case with gravitational waves? The curvature on the left would be... "non-proximal?"

 

[Dale]

Is this not patently the case with gravitational waves?

Gravitational waves are not static. Also, in 4 dimensions the GW is directly connected to the source by a light like path.

 

[ComradeConrad]

I agree with you on both points, Dale. I'm trying to ask about the state of an evolving system at a point in time... or trying to learn how to ask about that.

 

[Dale]

I'm trying to ask about the state of an evolving system

I thought you wanted to talk about static solutions.

 

[DrGreg]

I was assuming curvature around ("proximal" to) the source. So, like this, where the curvatures share the same source:

[curvature] --- flat space ---- [curvature] ---- [source]

Is this not patently the case with gravitational waves? The curvature on the left would be... "non-proximal?"

I think what you are trying to describe is something like this sketch of a spacetime diagram:

White respresents an almost-flat region of spacetime and increasing intensity of colour represents increasing curvature. The vertical red line is the source and the diagonal lines are a brief burst of gravitational waves emitted by the source.

The dotted black line represents a moment in time that you were trying to plot.

While it is true that the there's a triangular region of almost-flat spacetime separating the wave from the source, the wave is nevertheless connected to the source in spacetime.

 

[PeterDonis]

I was assuming curvature around ("proximal" to) the source. So, like this, where the curvatures share the same source:

[curvature] --- flat space ---- [curvature] ---- [source]

Is this not patently the case with gravitational waves?

No. The gravitational waves are not propagating through flat spacetime. (It should be "spacetime", not 'space", in all of this.) They are propagating through a curved spacetime that is curved because of other properties of the source besides those that cause gravitational waves. It is impossible to have a source that emits gravitational waves but produces no other spacetime curvature of any kind. (There are spacetimes that don't have any other curvature besides gravitational waves in them, but those spacetimes don't have any "source" at all; the gravitational waves are just propagating eternally with no source, because they are "built in" to the spacetime geometry from the start.)

 

[PeterDonis]

there's a triangular region of almost-flat spacetime separating the wave from the source

"Almost" flat, yes (assuming there are no other gravitating bodies). But "almost flat" is not the same as "flat". The OP appears to have been insisting on "flat". If he's ok with just "almost flat", that's a different discussion than the one I thought we had been having.

 

[ComradeConrad]

While it is true that the there's a triangular region of almost-flat spacetime separating the wave from the source, the wave is nevertheless connected to the source in spacetime.

Now we're getting somewhere! (Even though light cone diagrams make me anxious). Follow up question: How is this "connectedness" in spacetime related to the principle of locality? In my mind they are equivalent, but my mind is a scary place... When I started this thread I noted that general relativity was a local theory which was supposed to show that I understood that essential connectedness. It blows my mind that there could ever be a large almost-flat spacetime separating curvature and its source, and so my next thought the past few years has been whether this situation could ONLY ever occur with gravitational waves. Is that even a meaningful question? No idea! Trying to find out. It sort of vaguely seems like it could be provable one way or another by someone smarter than me.

I have been assuming "almost flat" this whole time because my understanding is that some kind of "absolute flatness" only emerges on the inter-galactic scale, if not larger. But it's helpful for me to know how much more precise I need to be with the language in order to communicate, so I very much appreciate everyone's thoughts!

 

[PeterDonis]

I have been assuming "almost flat" this whole time because my understanding is that some kind of "absolute flatness" only emerges on the inter-galactic scale, if not larger.

There is no "absolute flatness" anywhere in our actual universe. The best you'll get anywhere is "almost flat".

 

[PeterDonis]

How is this "connectedness" in spacetime related to the principle of locality?

What do you think the "principle of locality" means? Note that "locality" means locality in spacetime, not in "space".