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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    $\begingroup$ @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? $\endgroup$ Sep 23 '15 at 11:24
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    $\begingroup$ @CuriousOne: why does $e^- + e^+ \rightarrow 2g$ violate the equivalence principle? That's a genuine not a rhetorical question. $\endgroup$ Sep 23 '15 at 11:30
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    $\begingroup$ @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. $\endgroup$ Sep 23 '15 at 11:33
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    $\begingroup$ @CuriousOne: Is electron/positron annihilation to two gravitons forbidden? $\endgroup$ Sep 23 '15 at 12:39
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    $\begingroup$ @Ballistics: Then you may be interested in this question and the answers there physics.stackexchange.com/questions/4015/… $\endgroup$
    – MBN
    Sep 24 '15 at 7:03

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.

  • $\begingroup$ So general relativity alone is still insufficient to explain it. A more comprehensive quantum gravity theory is needed. $\endgroup$
    – Ballistics
    Sep 23 '15 at 12:51
  • $\begingroup$ 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. $\endgroup$
    – Ballistics
    Sep 23 '15 at 12:58
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    $\begingroup$ Hawking 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. $\endgroup$
    – Slereah
    Sep 23 '15 at 13:04

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