Behavior of a tangent light beam to an event horizon?

[SW VandeCarr]

Given present theory, how would a laser photon stream behave at a tangent point to an event horizon (EH)? Is it possible for a photon stream to orbit a black hole? Could the beam be split at the EH with one branch spiraling into the black hole while one branch follows some geodesic (perhaps not a stable orbit) outside the EH?

 

[Jonathan Scott]

Given present theory, how would a laser photon stream behave at a tangent point to an event horizon (EH)? Is it possible for a photon stream to orbit a black hole? Could the beam be split at the EH with one branch spiraling into the black hole while one branch follows some geodesic (perhaps not a stable orbit) outside the EH?

From the viewpoint of an observer who is at rest anywhere outside the EH, time slows towards a standstill at points approaching the event horizon, and is completely at a standstill at the event horizon itself. Close to the horizon, tangential light will curve inwards more sharply than the horizon and will fall in. Further out, at radius 3GM/c² in Schwarzschild coordinates, there is a phenomenon known as the "photon sphere" (look it up) which is where the curvature of tangential light paths means that they can in theory travel round in circular orbits.

 

[George Jones]

Consider spherical coordinates $r$, $\theta$, and $\phi$. For something in circular orbit (photon or massive particle), $r$ is constant, but at least one of $\theta$ and $\phi$ changes as the particle orbits.

Circular orbits for photons lie above the event horizon; see

https://www.physicsforums.com/showthread.php?p=1812881#post1812881.

The event horizon is a lightlike hypersurface in spacetime, but photons that stay on the event horizon have constant, $r$, $\theta$, and $\phi$, and consequently aren't "in orbit".

If you want, I can fill in some of the mathematics later (going out with my family soon, so I don't have time now).

 

[SW VandeCarr]

From the viewpoint of an observer who is at rest anywhere outside the EH, time slows towards a standstill at points approaching the event horizon, and is completely at a standstill at the event horizon itself. Close to the horizon, tangential light will curve inwards more sharply than the horizon and will fall in. Further out, at radius 3GM/c² in Schwarzschild coordinates, there is a phenomenon known as the "photon sphere" (look it up) which is where the curvature of tangential light paths means that they can in theory travel round in circular orbits.

By "at rest" I assume you mean with respect to the black hole. (Excuse me if this is a given. This is not my field.) Also you're saying that a laser beam cannot contact the EH at a tangent point but must fall into the black hole at some angle to the tangent?

I've heard of the Schwarzschild radius. Can I take it that the radius of the black hole at the EH is also is also some function of 3GM/c^2?

EDIT: I've looked up the photon sphere. It appears the situation depends on whether the black hole is rotating or not. Fascinating stuff. Would this have anything to do with closed time loops? Thanks.

 

[SW VandeCarr]

The event horizon is a lightlike hypersurface in spacetime, but photons that stay on the event horizon have constant, $r$, $\theta$, and $\phi$, and consequently aren't "in orbit".

If you want, I can fill in some of the mathematics later (going out with my family soon, so I don't have time now).

Yes, I understand that all the information contained in a black hole is on the EH. Is this a consequence of our limits of observation or can we meaningfully talk about a photon (or anything else) falling toward the singularity inside the EH?

If you can provide some mathematics I would appreciate it. I would guess we could model the behavior of particles trapped inside the EH although this behavior is inherently unobservable. Thanks in advance.

EDIT: I've found the mathematics I want. It appears the situation with the photon sphere is dependent on whether the black hole is rotating or not.

 

[Jonathan Scott]

By "at rest" I assume you mean with respect to the black hole. (Excuse me if this is a given. This is not my field.) Also you're saying that a laser beam cannot contact the EH at a tangent point but must fall into the black hole at some angle?

I've heard of the Schwarzschild radius. Can I take it that the radius of the BH at the EH is also is also some function of 3GM/c^2?

I've looked up the photon sphere. Fascinating stuff. Would this have anything to do with closed time loops? Thanks.

From the point of view of an outside observer, things slow down to a standstill as they get close to the EH, including light, so any description of motion of any sort is not consistent with that point of view. If you shine a light tangentially anywhere close to the EH, it will curve down, so it will turn towards the EH, although it takes forever to reach it.

The radial coordinate of the EH in Schwarzschild coordinates is 2GM/c², known as the Schwarzschild radius, and the location of the black hole central singularity in the same coordinates is radial coordinate 0. Note that a "radial coordinate" is not really the same as a "radius"; as space is so far from flat, we have to adopt particular conventions for labelling points, in a similar way to the way in which we map the curved surface of the Earth to flat maps using a variety of projections.

As far as I know, photon spheres have nothing at all to do with "closed time loops" nor even closed timelike curves, which might be what you were suggesting.