Light orbiting a black hole

[md65536]

Is it possible trap light in a stable circular orbit around a tiny Schwarzschild black hole such that the energy of the light is greater than that of the black hole?

[Genady]

Is there a stable circular orbit for a massless test particle around a Schwarzschild black hole? Photon sphere is unstable, AFAIK.

[Markus Hanke]

8 hours ago, Genady said:

Is there a stable circular orbit for a massless test particle around a Schwarzschild black hole?

No. The potential energy function of a photon in Schwarzschild spacetime has only a local maximum (photon sphere), but no minimum.

[md65536]

I shouldn't have used the word "stable", what I meant was just "circular orbit" for some time (several orbits or so) because apparently circular already implies it's on the photon sphere.

 

A bit of a digression on this: I see in https://en.wikipedia.org/wiki/Photon_sphere , "all circular orbits have the same radius". At the event horizon, light aimed directly outward will have a constant r, and at the photon sphere, light aimed tangentially will have a constant r. Is that correct? Then, everywhere in between, there is some direction that will let light have a constant r. These photons would circle the black hole, but they're not called circular orbits?

[Markus Hanke]

40 minutes ago, md65536 said:

At the event horizon, light aimed directly outward will have a constant r, and at the photon sphere, light aimed tangentially will have a constant r. Is that correct?

Yes this sounds correct, for Schwarzschild BH.

41 minutes ago, md65536 said:

what I meant was just "circular orbit" for some time (several orbits or so)

In that case the answer is yes, such orbits exist, at least in principle.

[Genady]

If the light orbiting the BH is not a test particle but rather has the energy as described in the OP, then I think it is not on a photon sphere anymore. The "photon sphere" of the original black hole would be inside the new black hole.

Edited February 19 by Genady

[Markus Hanke]

20 hours ago, Genady said:

If the light orbiting the BH is not a test particle but rather has the energy as described in the OP, then I think it is not on a photon sphere anymore. The "photon sphere" of the original black hole would be inside the new black hole.

Technically quite correct. But I think we’re considering an idealised situation here, or else the Schwarzschild metric can’t be used, and everything gets more complicated.

[md65536]

I'm interested in any situation or metric, or any simplification (or complication) involving a system of trapped light and a minimum of anything else. (Now that I say that, I have a vague memory of well known physicists speculating on astronomical objects made of light itself, gravitationally bound to itself but not collapsed, but I can't remember what they're called and I think that might be harder to reason about.)

It does seem like if you think of the system of a black hole and a photon at 1.5 r_s with at least as much energy as the black hole, and consider it inside a sphere of size 2 r_s, it should collapse, but that assumes all the energy is contained within that radius, but it is not spherically symmetric, and it should have angular momentum (unless you contrived it not to by giving the BH itself the right angular momentum, but that just further complicates things), and like you say Markus, the Schwarzschild metric can’t be used.

It also seems like all these complications are just more "stress" than a Schwarzschild case, and more certain to collapse. But are there ways to remove stress from the system so you could increase energy without collapse. eg. a cosmological constant.

On 2/19/2024 at 1:19 AM, Genady said:

If the light orbiting the BH is not a test particle but rather has the energy as described in the OP, then I think it is not on a photon sphere anymore. The "photon sphere" of the original black hole would be inside the new black hole.

What might happen if the original photon was moved farther away to avoid collapse, such as at the photon sphere of the new black hole you describe? Or to make it symmetric, many uniformly distributed photons in a photon sphere.

It seems like in general, for a real black hole photon sphere of a given size, if you add more energy to the photon sphere, you could get away with a smaller black hole, to the point that you don't need the black hole at all (which sounds reasonable now that I remember the idea of objects made of gravitationally bound light).

[Genady]

6 minutes ago, md65536 said:

if you add more energy to the photon sphere

Schwarzschild metric and all related derivations such as photon sphere, are valid in vacuum. If we start adding a significant amount of energy there, we need to consider a different metric and different effects.

[Markus Hanke]

7 hours ago, md65536 said:

Now that I say that, I have a vague memory of well known physicists speculating on astronomical objects made of light itself, gravitationally bound to itself but not collapsed, but I can't remember what they're called and I think that might be harder to reason about.

The fully collapsed version of this is called a kugelblitz, whereas the non-collapsed version would be some form of gravitational  geon.

7 hours ago, md65536 said:

But are there ways to remove stress from the system so you could increase energy without collapse. eg. a cosmological constant.

The equivalent of Schwarzschild spacetime in the presence of a positive cosmological constant is called deSitter-Schwarzschild spacetime. There’s an upper limit here to how large such BH can be, which is called a Nariai black hole.

[md65536]

On 2/20/2024 at 10:16 PM, Markus Hanke said:

the non-collapsed version would be some form of gravitational  geon.

Yes, what I described, at the limit where the BH can be removed leaving just the photon sphere, fits the definition of a geon. I can't imagine that such a spherically symmetric shape wouldn't work, and that a geon needs a different shape.* It would be unstable because if any photon deviated slightly outward, its orbit would be wider and it would escape, leaving less energy, reducing the photon sphere radius and letting other photons escape. I imagine a photon deviating inward, without a central BH, would cross the photon sphere again and escape, but I'm not sure (maybe it could collapse the geon?).

* Actually, I see Wheeler's 1955 paper "Geons" describes this and interesting complications I hadn't thought of. PDF: https://blackholes.tecnico.ulisboa.pt/gritting/pdf/gravity_and_general_relativity/Wheeler_Geons.pdf

To make it stable, it would have to be a quantum geon. Those are theoretical only and seem to require quantum gravity. I guess the basic idea is that if energy can only leak in specific amounts, one could be coherent enough to prevent that. There are papers on them but I haven't yet found anything I can make sense of.

Edited February 24 by md65536