Photon Patterns in Space: Could They Exist?

[TerranIV]

Would it be possible for photons to form some sort of static structure, like a standing waveform, if space-time was curved enough?

It seems like a structure like this exists around black holes. Is there any other distortion or phenomenon that could cause this, such as a neutron star or a galaxy?

 

[PeterDonis]

Moderator's note: Thread moved to relativity forum.

 

[PeterDonis]

Would it be possible for photons to form some sort of static structure, like a standing waveform, if space-time was curved enough?

No. Maxwell's Equations have no solution that describes an EM wave (photon) that varies only in space, not in time.

It seems like a structure like this exists around black holes.

No, that's not correct. There is a place called the "photon sphere" at a radius of 3/2 the horizon radius of a black hole, where photons can make circular orbits around the hole. But that is neither a static structure nor a standing waveform.

Is there any other distortion or phenomenon that could cause this, such as a neutron star or a galaxy?

No.

 

[kimbyd]

No. Maxwell's Equations have no solution that describes an EM wave (photon) that varies only in space, not in time.

No, that's not correct. There is a place called the "photon sphere" at a radius of 3/2 the horizon radius of a black hole, where photons can make circular orbits around the hole. But that is neither a static structure nor a standing waveform.

I don't think that's applicable to the problem. Standing waves still vary in time. Standing EM waves occur all the time in resonant cavities, such as microwave ovens.

So I think the question is: could space-time be curved sufficiently that it could produce the equivalent of a resonant cavity?

I think the answer is a clear "sort of". The photon sphere around a black hole is just such a situation. The reason why the answer is "sort of" is because the photon orbits are not stable. In principle if you have photons orbiting a black hole within the photon sphere, they could form standing waves (these standing waves would be spherical harmonics). But because the orbits aren't stable it's not really possible in practice for standing waves to form.

Is it possible to come up with a space-time geometry which would permit stable photon orbits? Maybe, but probably not without using something like a universe that wraps back on itself.

Finally, even when you have a situation where stable orbits are possible, that doesn't guarantee the formation of standing waves. You need to have some kind of physical interaction which makes the standing wave states preferred over other possible states. In a resonant cavity like a microwave oven, the physical interaction is the sides of the microwave where the electric field is forced to be zero by the conductivity of the walls. If all you're working with is the curvature of space-time, I can't imagine there being a physical interaction which would prefer the standing states over traveling states.

 

[TerranIV]

I think the answer is a clear "sort of". The photon sphere around a black hole is just such a situation. The reason why the answer is "sort of" is because the photon orbits are not stable. In principle if you have photons orbiting a black hole within the photon sphere, they could form standing waves (these standing waves would be spherical harmonics). But because the orbits aren't stable it's not really possible in practice for standing waves to form.

Thank you for your detailed response!

While the photon sphere is not stable for most of the orbits wouldn't it be stable for some photons with a specific angle of incident?

 

[PeterDonis]

Standing waves still vary in time.

We might have an issue of definition here. I have always taken "standing wave" to be defined as a wave that only varies in space, not time. But the relevant question for this thread is not what definition I would use, but what definition the OP is using. If the OP is using your definition, then your comments are obviously more relevant than mine.

 

[PeterDonis]

While the photon sphere is not stable for most of the orbits wouldn't it be stable for some photons with a specific angle of incident?

No. All orbits, whether they are at the photon sphere or orbits of timelike objects outside the photon sphere, are unstable inside a radius equal to 3 times the hole's horizon radius. (I am assuming here a non-rotating Schwarzschild black hole.)

 

[PeterDonis]

In principle if you have photons orbiting a black hole within the photon sphere, they could form standing waves (these standing waves would be spherical harmonics).

Spherical harmonics don't vary in time, so I think we still have an issue of definition here.

Also, I don't know whether spherical harmonics are valid solutions of Maxwell's Equations with the appropriate constraint applied (that the EM field is only nonzero on the photon sphere). They would have to be for your statement here to be correct.

 

[TerranIV]

No. All orbits, whether they are at the photon sphere or orbits of timelike objects outside the photon sphere, are unstable inside a radius equal to 3 times the hole's horizon radius. (I am assuming here a non-rotating Schwarzschild black hole.)

Right, that makes sense. The only "stable" orbit would be have to be inside the event horizon where spacetime would warp enough for a "straight" line to be an inescapable circular path, correct? And at that point the photons would basically just become part of the black hole's mass.

 

[PeterDonis]

The only "stable" orbit would be have to be inside the event horizon

There are no orbits at all inside the horizon, if by "orbit" you mean a path with a constant radius $r$. Inside the horizon everything has to fall towards $r = 0$; nothing can stay at a constant radius $r$, not even light.

In fact, there are no orbits inside the photon sphere, if by "orbit" you mean a path which has a constant $r$ and which an object can follow in free fall. Inside the photon sphere but outside the horizon, the only way to maintain a constant $r$ is to fire rockets to maintain altitude.

 

[Ibix]

The only "stable" orbit would be have to be inside the event horizon where spacetime would warp enough for a "straight" line to be an inescapable circular path, correct?

No. Once you cross the event horizon, you hit the singularity in very short order. There's no stable orbit inside there.

The only available circular orbit for light is at 1.5 Schwarzschild radii (edit: 1.5 times the radius of the black hole, that is). The point about "unstable" is that at 1.5+0.0000...0001 Schwarzschild radii, the light will escape. At 1.5-0.0000...0001 Schwarzschild radii, it will fall in. So these circular orbits are like balancing a pencil on its point - it's theoretically possible to do, but a flea sneezing on the other side of the room will knock it over. Similarly, any disturbance will make light in a circular orbit either fall into the hole or escape.

 

[kimbyd]

Thank you for your detailed response!

While the photon sphere is not stable for most of the orbits wouldn't it be stable for some photons with a specific angle of incident?

Others have answered this, but I thought I might add my own wording to the pot.

The stability has nothing to do with the photon's trajectory, but with the curvature itself. An unstable orbit is one where any tiny perturbation will kick the photon out of the orbit. So if you have a photon that is in this orbit and is staying there, then any object passing near the orbit will push it a teeny tiny bit, causing it to either fall into the black hole or escape to infinity. So in realistic terms if there's an unstable orbit, nothing is going to stay there for terribly long.