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Take a light ray approaching a black hole from infinity which goes out again to infinity. What is the maximum finite rotation it can describe? (I know it can loop around indefinitely in the photosphere)

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There are three kinds of orbits that encompass all cases: bounded $E < E_{crit}$, extreme $E = E_{crit}$ and unbounded orbits $E > E_{crit}$. As Akano has said, the extreme case can circulate around the `photon sphere' for ever, because the energy is tuned to exactly repel the inward pull.

You can still have an unbounded orbit circulate around as many times as you like because you can take any $\epsilon >0$ and consider a photon with energy $E = E_{crit} + \epsilon$. The orbit will verrry nearly circulate the critical sphere exactly but eventually will escape. So, in theory, with infinite precision to tune the energy you could have any number of orbits ($n \in \mathbb{N}$). Of course, in reality, $\epsilon$ will have some maximum resolution (number of decimal points) it can contain, so the number of orbits will be strictly bounded.

See here (http://jetp.ac.ru/cgi-bin/dn/e_064_01_0001.pdf) for a more mathematical exposition.

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    $\begingroup$ Ah settles the question, thanks a lot! (I've won a bet! :), I'm sorry I can't upvote you) $\endgroup$ – Real Mar 10 '15 at 22:17
  • $\begingroup$ @Real what was the wager on this bet? just out of curiosity :) $\endgroup$ – Adsy Mar 11 '15 at 10:17
  • $\begingroup$ What's the maximum resolution of $\epsilon$ that you speak of? (Or at least, how would one go about finding it?) $\endgroup$ – David Z Mar 11 '15 at 12:12
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    $\begingroup$ That's a good question @DavidZ. All I meant was that, to get an arbitrarily large number of orbits, you would need an enormous number of decimal points, like $E = E_{crit} + 0.00 \cdots 1$. Thinking of the energy like $E = h \nu$ would mean we would need to tune the frequency very precisely like $\nu = \frac {1} {h} (E_{crit} + 0.0\cdots 1)$. Now, Given some physical process for altering a photons frequency you could proceed to think about how finely tuned you could make it. For scattering or something, I find it hard to believe you could set the problem up so precisely, leading to limited res $\endgroup$ – Arthur Suvorov Mar 12 '15 at 5:12
  • $\begingroup$ Yeah, certainly to get a large number of orbits requires very fine control over the initial conditions, but from the point of view of the theory itself that's not a problem. There's no fundamental reason in GR that you couldn't fix the setup as precisely as you want. So I object to the statement that there is some maximum resolution (or I think you mean maximum number of orbits). Given a particular experimental apparatus, sure there is, but it comes from the limits of your equipment, not the theory itself. $\endgroup$ – David Z Mar 12 '15 at 5:23
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The photon sphere lies outside the event horizon ($r_p = 3GM/c^2 > r_s = 2GM/c^2$), so a noncaptured photon can orbit a black hole as many times as it wants. Since at the photon sphere the gravitational potential of the black hole is at a peak, it is unstable, so a photon coming in from infinity and having just the right trajectory can spiral inward towards the photon sphere, but never reach it, indefinitely.

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    $\begingroup$ Thanks! But I was actually wondering about trajectories which escape to infinity again (I guess I didn't word it correctly -- I'll edit the question). Can light perform large rotations and escape the black hole again? What's the maximum? $\endgroup$ – Real Mar 10 '15 at 21:52
  • $\begingroup$ Large rotations, like an elliptical orbit? If I understand you correctly the answer is no because light always travels at the speed of light. Matter can have a long elliptical orbit but that's dependent on a velocity changing due to gravitational effects. Light can't enter an elliptical orbit, which is why light wouldn't travel around a photonsphere. It's a theoretical limit, not a real situation. Plus, in theory, an elliptical orbit couldn't enter the photonsphere cause once inside, the matter wouldn't escape on it's own. . . . not sure I answered your question though. $\endgroup$ – userLTK Mar 10 '15 at 22:40
  • $\begingroup$ @userLTK it came from a hypothesis of mine that a light ray could approach a black hole, describe e.g. a 720 degree rotation and escape, which is true as Arthur pointed out. This is in contrast with classical mechanics, where an object can rotate at most 180 degrees around another object (parabolic trajectory), before it assumes a low enough energy and instead rotates around the other object. $\endgroup$ – Real Mar 10 '15 at 23:04
  • $\begingroup$ Ah, that makes sense. In theory, I think a ray of light could travel several times around a black hole if it was positioned just right but the further from the photonsphere it got the faster it would escape so there's probobly a practical limit. Certainly well over 180. Here's a diagram. One of the light rays drawn goes over 360. rantonels.github.io/starless/pics/bhscattersmall.png and source: rantonels.github.io/starless . . . Not really an answer, but I learned that the "black" we kinda see is the photonsphere, not the event horizon reading up on this. Cool. $\endgroup$ – userLTK Mar 10 '15 at 23:25
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Probably only once or twice, as a very small gravitational disturbance (a nearby electron) would send the photon into an orbit with a necessary velocity not = to C.

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