Let's Sort Out This Gravity Thing - c And All That

[Wallis]

"General relativity predicts that disturbances in the gravitational field propagate as gravitational waves, and that low-amplitude gravitational waves travel at the speed of light." It seems high energy waves may travel slower, but certainly not faster.

So, a large star whilst going supernova can still emit gravitational waves. OK. But once the black hole forms, the escape velocity inside the Schwartzschild radius is greater than c, so Einstein says the gravitational waves cannot exit (they travel at c.) If this is the case, the gravitational effects of a black hole could never exceed the critical limit that made the Schwartzschild radius in the first place. So, why the supermassive black hole at the centre of galaxies with ~1 billion solar masses? Are we postulating that it all formed at once rather than accreting over billennia? Why is a super-massive black hole different from other black holes? It seems black holes have hair, and the hairstyle changes as they accrete further mass. So, gravitons (if they even exist) must travel faster than c. Discuss.

Is space-time far more fundamental than c? Maybe there are no gravitons. It seems space-time is a fundamental building-block of the universe, separate from radiation. Grand-unification is, after all, only a wish of some physicists. We should find that c is an effect of space-time and the amount of mass within it.

 

[PeterDonis]

once the black hole forms, the escape velocity inside the Schwartzschild radius is greater than c, so Einstein says the gravitational waves cannot exit (they travel at c.)

But that's only after the hole forms. More precisely, it is after the hole forms and "settles down" to a stationary state. In the process of getting to that point, gravitational waves can certainly be emitted.

If this is the case, the gravitational effects of a black hole could never exceed the critical limit that made the Schwartzschild radius in the first place.

This is not correct. The gravitational field that is customarily viewed as "coming from" the black hole actually doesn't propagate from the hole; it propagates from the collapsing matter that originally formed the hole. See this article for a quick overview:

http://math.ucr.edu/home/baez/physics/Relativity/BlackHoles/black_gravity.html

Are we postulating that it all formed at once rather than accreting over billennia?

No. When matter falls into an already existing black hole, the increased gravitational field propagates from the matter before it falls in; it doesn't have to propagate out of the hole.

 

[phinds]

You are overlooking the obvious fact that gravity is a field and does not "propagate" at all. What DOES propagate is changes in gravity. As matter comes together to form massive objects such as planets, stars, and black holes, the changing gravity during the formation propagates but once they are formed, there is no longer any need for propagation. If new matter is added, then that does represent a change and the change propagates.

EDIT: damn! Peter is just too fast for me.

 

[Wallis]

Peter's reference quotes a source that says gravitons probably don't exist in this form owing to the classic "infinity of infinities" causing the inability to renormalise-around singularities in graviton theory, so maybe the whole thing is not a suitable topic (yet) for the forums. Are there any references for this freezing of gravity waves and the convenient intergral over dilated time and distance adding up to the total gravitation of the infalling matter? I am keen to learn more about this explanation for the "apparent escape of gravity" from superluminal regions.

 

[PeterDonis]

Are there any references for this freezing of gravity waves and the convenient intergral over dilated time and distance adding up to the total gravitation of the infalling matter?

I'm not sure what you mean by this.

I am keen to learn more about this explanation for the "apparent escape of gravity" from superluminal regions.

There is no need to invoke quantum effects of any sort (gravitons or anything else) to explain this. The explanation I referred to is purely classical and doesn't depend on what particular version of quantum gravity ends up being correct. It only depends on the behavior of spacetime being classical in the regime we observe.

 

[pervect]

My $.02. We can draw a useful analogy of "gravity escaping a black hole" with "electric fields escaping a charged black hole". It's pretty clear from the classical results (in particular the Reisnner-nordstrom metric of a charged black hole) that electric fields not only can, but MUST, because of Gauss's law as generalized to curved space-time, be able to escape a black hole.

The electromagnetic case is a bit clearer, because we have a well-worked out and widely accepted theory of virtual photons, something that we can't claim for "gravitons", which are not on the same solid footing.

So any understanding of black holes that says that electromagnetic fields and/or gravitational fields "can't escape" must be flawed. It's a little harder to find a consensus on the best way to explain this flaw to laypeople, but it's really clear that it's just plain wrong to think that either electric fields or gravitational fields "can't escape" from a black holes.

My personal preference is to place the blame on an overly-physical interpretation of the significance of virtual particles. An overly-physical interpretation of virtual particles also causes other misunderstandings, such as the long-standing "speed of gravity" issues and/or the equivalent arguments about the "speed" of electrostatic forces. Rather than digress into the details of these other misunderstandings, which would likely derail the thread, I'll leave it at that.

While there may be no clear consensus on the best way of explaining the issue at a popular level - for instance the sci.physics.faq offers a couple of different "explanations", the fact that gravity and electric fields can and must be able to escape from a black hole is not really open to debate,.

 

[PeterDonis]

It's pretty clear from the classical results (in particular the Reisnner-nordstrom metric of a charged black hole) that electric fields not only can, but MUST, because of Gauss's law as generalized to curved space-time, be able to escape a black hole.

Actually, it isn't. What is clear is that electric fields can be detected outside of a charged black hole. But, as the Baez article I linked to makes clear, there is a perfectly good classical explanation for this; you don't need virtual photons. The classical explanation is that the electric field does not propagate from the hole: it propagates from the charged matter that originally collapsed to form the hole. A similar classical explanation works for the hole's gravity. On this view, electric fields (and gravity) do not have to "escape" from the hole, because the sources of the fields we observe outside the hole are (or more precisely were) also outside the hole.

the sci.physics.faq offers a couple of different "explanations"

Yes, it's unfortunate, IMO, that, after giving a perfectly good classical explanation, the FAQ article then muddies the water by talking about virtual particles--which aren't even very good heuristics for many quantum effects (since they are only present in perturbation theory, and many important quantum effects can't be described by perturbation theory).

 

[Wallis]

I seem to need books. It seems, from what Peter says, that neither the "field" escapes, nor the "changes in the field" escape. So, nothing escapes, but "everything we need is conveniently frozen in extreme time-dilation at the Scwartzschild radius." I need a book that can explain how this convenient situation arises.

Searching bookshops results in many many hits, many of which have been disappointing. I have some general-public books on relativity, but I have insufficient information in these to understand what is being discussed here. I probably need a maths book on del, nabla, etc as I never got "generic field operators." I could never understand how it was possible for "del cross x" to mean anything. It seems I need two books.

As a guide, I could once do contour integration, got the idea of, but never quite succeeded with PDEs and can understand space-time diagrams. I understand the concept of scalar and vector fields and their derivatives. Can anyone suggest some useful books that would help explain this concept of freezing gravity at the Swartzschild radius?

 

[stevendaryl]

"General relativity predicts that disturbances in the gravitational field propagate as gravitational waves, and that low-amplitude gravitational waves travel at the speed of light." It seems high energy waves may travel slower, but certainly not faster.

Just for clarification, the equations of General Relativity are NOT wave equations. When gravity is weak, the equations become approximately wave equations, so we can talk about gravitational waves in that limit. But for strong gravity, the full equations don't describe wave propagation.

So it's not so much that strong gravity has high-amplitude waves that propagate differently than the low-amplitude waves of weak gravity. Rather, it's only appropriate to describe gravity in terms of "wave propagation" in the case of weak gravity.

 

[WannabeNewton]

So it's not so much that strong gravity has high-amplitude waves that propagate differently than the low-amplitude waves of weak gravity. Rather, it's only appropriate to describe gravity in terms of "wave propagation" in the case of weak gravity.

This is incorrect. Gravitational waves can perfectly well be defined on strongly curved backgrounds; the statement quoted by the OP is valid. One simply looks at linear perturbations of the curved background.

One then gets new phenomena such as back-scattering wherein modes whose wavelengths are comparable to the background radius of curvature will scatter off of said background and cause the gravitational wave to have a tail that propagates slower than the head.

In the geometric optics limit one recovers the fact that gravitational waves travel on null geodesics as this approximation is just WKB in which case the frequency is too high to "see" the background curvature.

 

[stevendaryl]

This is incorrect. Gravitational waves can perfectly well be defined on strongly curved backgrounds; the statement quoted by the OP is valid. One simply looks at linear perturbations of the curved background.

But he's talking about large and small amplitudes. A linear approximation only makes sense for small amplitudes. If the amplitude is large, then the field equations cannot accurately be described as a wave equation.

You're right, that what I said was wrong about strong versus weak gravity. The background gravity can be strong or weak--it's only the strength of the departure from the background that is relevant to the issue of whether the propagation can be described accurately by a wave equation.

 

[PeterDonis]

nothing escapes, but "everything we need is conveniently frozen in extreme time-dilation at the Scwartzschild radius."

No, that's not what I said, and not what the article I linked to says. The matter that collapses to form a black hole is not "frozen" at the Schwarzschild radius; it falls right on through. But all during the process of collapse before it reaches that radius, the matter is leaving behind an "imprint" of its presence on spacetime. That imprint is what we detect as the "gravity" of the black hole.

In fact, even before the matter collapsed, it was leaving an imprint of its presence on spacetime. Suppose, for example, that the Sun suddenly collapsed to a black hole. (This will never actually happen, but for a thought experiment we can suppose it.) If the collapse were spherically symmetric, we on Earth would detect no change in the Sun's gravity at all; the Earth's orbit would remain the same. (We're assuming that the process of collapse does not involve any change in the Sun's mass, i.e., that no matter is ejected and no radiation is emitted.) So the "gravity" that is determining the Earth's orbit is not coming from inside the black hole that used to be the Sun; it's coming from the past, from the Sun before it ever collapsed. It's the imprint on spacetime that the Sun made when it originally formed.

 

[stevendaryl]

About the issue of how electric fields or gravity escapes from a black hole, I think that the image of electric fields "propagating" from the source is probably a little misleading. It's not entirely accurate, but for some purposes, you can think of electric field lines as strings coming out of the charged particle stretching out to infinity. Those strings can only begin or end on a charged particle. If you take a bunch of charged particles and squeeze them together, you will make a change to the strings close to the particles, but far, far way, there is no change to the strings. There is nothing you can do to the charges that will cause the strings far away to change. Not even dropping the particles into a black hole.

 

[pervect]

Electromagnetism, henceforth abbreviated as E&M is well known to be described by a wave equation, whose solutions are waves that travel at "c". However, going from this general fact to the specific notion that the coulomb force between charges "propagates at c like light" - to be more precise and give a specific experimental test, to say that the coulomb force aberrates just like light does, is unfortunately wrong.

Describing gravity as a wave equation is a bit more difficult, but with the right coordinate / gauge choices (the de Donder gauge / harmonic coordinates), you can regard gravity as being the solution of a wave equation, called the "relaxed wave equation. It's not quite the same as the E&M case, as the source terms include an effective part, but it still falls under the same general umbrella.

However, the idea that the fact that gravity can be described by some sort of wave equation therefore implies that the gravitational coulomb force of gravity "travels at c" is just as wrong as it the E&M case.

While the original question was about "escaping" rather than aberration, the point is that treating the coulomb force as if it were light just isn't a good idea, and will cause multiple problems.

 

[PeterDonis]

going from this general fact to the specific notion that the coulomb force between charges "propagates at c like light" - to be more precise and give a specific experimental test, to say that the coulomb force aberrates just like light does, is unfortunately wrong.

I agree with this as you state it, but I'm not sure this is a helpful way of stating it. Remember that the "coulomb force" is just another word for "static electric field", and the whole point of "static" is that nothing is changing, so there is no "propagation" of anything. So questions like "how fast does the coulomb force propagate?" don't even make sense.

However, we can ask a related question that does make sense: if there is a sudden change in the charge-current distribution, how fast does information about that change propagate to distant locations? The answer to that question is, it propagates at $c$. More precisely: if there is a sudden change in the charge-current distribution in some localized region of spacetime, the effects of that change on the fields are only felt in the future light cone of that region. Or, turning it around, if we want to know the fields in some localized region of spacetime, we only need to look at the charge-current distribution in the past light cone of that region.

Similar remarks apply to gravity; the notion of a static gravitational field "propagating" doesn't make sense, but the more general question about how the effects of a changing stress-energy distribution propagate has the same answer as for EM: they propagate at $c$.

It's also worth noting that, despite what I've said above, it is true that you can't use observed aberration of forces to deduce their speed of propagation, in either the EM or the gravity case. The classic treatment of this issue is Carlip's paper:

http://arxiv.org/abs/gr-qc/9909087

The focus is on the speed of gravity, but he treats the EM case as a "warmup" for the gravity case, since it raises the same basic issues. The key point Carlip makes is that, even though EM and gravity propagate at $c$, in the sense I described above, both forces, when treated relativistically, have velocity-dependent terms in their interactions that compensate, at least partially, for the time delay due to the finite speed of propagation. So, in a situation that is nearly static, the forces look like they are nearly instantaneous, i.e., the observed aberration is much smaller than the "naive" expectation based on just looking at the "coulomb" part of the force (which is only a function of the inverse square of the distance), because of the additional velocity-dependent terms.