When I saw the questions why matter-anti matter annihilation produces photons not gravitons, it suddenly occured to me that if the latter really happens, it means the stress energy tensor vanishes completely because gravitational wave has zero stress energy tensor. So is it allowed when the ordinary matter(particle matter or radiation matter) convert to gravitational wave in general relativity just as particle matter convert to radiation matter in special relativity? Does such process violate the Einstein's equation?
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Matter can't convert to gravitons. That would violate any number of conservation laws. General relativity doesn't have anything to say about these things, though, since gravitational waves are perfectly classical, there are no quanta involved. – CuriousOne Sep 23 '15 at 11:17
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@CuriousOne: what conservation laws would $e^- + e^+ \rightarrow 2g$ violate? – John Rennie Sep 23 '15 at 11:19
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@JohnRennie: That means a free electron gas can not cause gravitational waves because of lepton number conservation, but a hydrogen cloud can? Cool. There goes the equivalence principle down the drain. Or am I supposed to sacrifice lepton number? – CuriousOne Sep 23 '15 at 11:21
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Radiation does not contribute much to the stress energy tensor but it does produce gravitational waves expressed in the weyl tensor. It is called an electrovacuum. – Horus Sep 23 '15 at 11:23
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1@CuriousOne: oops, I meant electron + positron not electron + proton - I've corrected my comment. My question still stands: what conservation laws are violated by an electron and positron annihilating to two gravitons? – John Rennie Sep 23 '15 at 11:24
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@JohnRennie It would depend on how much energy that reaction can produce wouldn't it? – Horus Sep 23 '15 at 11:25
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@JohnRennie: That's a little better, but you are still sacrificing the equivalence principle and I still don't know how ordinary matter makes gravitons. – CuriousOne Sep 23 '15 at 11:25
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4@CuriousOne: why does $e^- + e^+ \rightarrow 2g$ violate the equivalence principle? That's a genuine not a rhetorical question. – John Rennie Sep 23 '15 at 11:30
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@JohnRennie: Because that annihilation process should produce stronger gravitational waves than the interaction of cold gas which does not have that process. Am I missing something? Wouldn't that violate the equivalence principle? How would nature balance these things? – CuriousOne Sep 23 '15 at 11:32
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2@CuriousOne: I have to confess that your reasoning escapes me. But we'll get shouted at by a moderator if we continue here, and more importantly I want my lunch now. – John Rennie Sep 23 '15 at 11:33
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@JohnRennie: I am simply asking why the interaction of an electron positron gas should produce more gravitons than the interaction of a free electron gas or cold hydrogen. I don't see that being plausible without giving up the foundation of GR or lepton number conservation. I don't want to interfere with your lunch, though. – CuriousOne Sep 23 '15 at 11:35
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2@CuriousOne: Is electron/positron annihilation to two gravitons forbidden? – John Rennie Sep 23 '15 at 12:39
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They way I understand the question is: is it possible to have a solution of the EFE such that for a particular 3+1 split it is non-vacuum at an instant and vacuum at a later instant. And mathematically there is no problem. – MBN Sep 23 '15 at 15:52
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@MBN When you say mathematically do you mean if you had exotic matter? – Timaeus Sep 23 '15 at 21:41
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@MBN Yes, that's exactly what I mean. – Ballistics Sep 24 '15 at 05:49
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Obviously you can, yes. Just solve the EFE for the stress energy tensor $T \propto \theta(t)$, or any other function starting at 0 before a time t. – Slereah Sep 24 '15 at 06:33
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@Timaeus: Yes, I mean finding a Lorenzian manifold without any restriction on the stress-energy. – MBN Sep 24 '15 at 07:01
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1@Ballistics: Then you may be interested in this question and the answers there http://physics.stackexchange.com/questions/4015/empty-universe-in-the-past-non-empty-in-the-future – MBN Sep 24 '15 at 07:03
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@MBN Thank you for your comment. But there is a slight difference beween our questions. My question doesn't require a "flat space" in the future, instead I need the gravitational wave to carry the energy and momentum. – Ballistics Sep 24 '15 at 10:44
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@Ballistics: I didn't mean that it was a duplicate, just that it might be of interest as it has answers that may be adapted for your question. – MBN Sep 24 '15 at 15:00
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Something like this happens in Hawking radiation. Matter (particle-antiparticle pairs) is produced with the energy coming from the gravitational field. This implies a change in the stress-energy tensor.
It isn't clear to me whether annihilation to two gravitons is allowed, but assuming it is then yes the stress-energy tensor will change as a result.
Your description is implicitly semi-classical because you are equating a classical description to the expectation value of a quantum system. This is a perfectly reasonable approximation, and indeed it's how Hawking radiation was first described. But I doubt we can get any real understanding of the processes involved from what is an effective theory. To understand exactly what's happening will require a theory of quantum gravity that doesn't currently exist.
John Rennie
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So general relativity alone is still insufficient to explain it. A more comprehensive quantum gravity theory is needed. – Ballistics Sep 23 '15 at 12:51
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I came up with this question because it puzzled me whether we should regard gravitational wave as "real matter". Although I know gravitational wave can create black hole theoretically, the conversion between gravitational field and "real matter" is more direct to link them together. – Ballistics Sep 23 '15 at 12:58
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1Hawking radiation does not happen in a Schwarzschild metric, though, by its very nature. If you consider the vacuum backreaction, it is never actually a vacuum solution. You could turn it on and off, I suppose, but then that would simply be a metric with a discontinuity in it. – Slereah Sep 23 '15 at 13:04