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Assume (for the sake of simplicity) a Schwarzschild black hole (non-rotating, non charged). This black hole has a photon sphere in $r=1.5r_s$, where photons may travel in a circular orbit. Will a gravitational wave propagating tangentially on the boundary of this orbit possibly be caught in a stable or an unstable orbit around the black hole?

I would naively assume the answer is yes, as both light and gravitational waves are supposed to behave the same (namely, propagate along null geodesics).

If so, will such a rotating gravitational wave, a continuously accelerated energy density, produce by itself a gravitational wave (albeit weaker by many magnitudes)? The same question stands for the photons in the photon sphere- will they emit gravitational radiation?

In short: Can a gravitational wave be caught in orbit around a black hole? Will it emit gravitational radiation?

Thank you!

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    $\begingroup$ Gravitational waves need quadruples afaik, and the photosphere is a symmetric locus. en.wikipedia.org/wiki/Gravitational_wave#Sources . $\endgroup$ – anna v Nov 13 '17 at 15:23
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    $\begingroup$ It can be caught, it will not emit gravitional radiation, nor the photon sphere, because they are spherically symmetric. $\endgroup$ – anna v Nov 13 '17 at 15:28
  • $\begingroup$ @annav How about gravitational waves/photons falling (spiralling) into a BH? Would they emit gravitational waves, as they have quadrupole moment? How about GW around a Kerr BH, falling from the outer photon sphere to the inner photon sphere? $\endgroup$ – A. Ok Nov 15 '17 at 9:09
  • $\begingroup$ a massless particle has to be spherically symmetric whether falling in or not $\endgroup$ – anna v Nov 15 '17 at 9:13
  • $\begingroup$ So a photon or a GW (or any other massless energy density) can never emit gravitational waves? $\endgroup$ – A. Ok Nov 15 '17 at 9:16
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This is partly a guess, but yes it seems a gravitational wave could orbit. For simplicity let's assume it has much less energy than the black hole, so Schwarzschild spacetime is a good approximation. Then we would expect an unstable circular orbit at the photon sphere $r=1.5r_s$ as you suggest. Now, suppose it is a plane wave of smallish extent: we want to avoid symmetry, so don't want the waves spread evenly over the photon sphere! Then by analogy with binary black hole systems, this is a binary-ish system of black hole plus localised-ish energy, so will indeed emit new gravitational waves.

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