Can a Photon Orbit a Black Hole?

[RobtO]

In MTW it is derived that a photon can orbit a black hole at a radius r=3M. However, the surface at the Schwartzschild radius r=2M is lightlike, isn't it? So can a photon orbit at both radii? What am I missing?

 

[pervect]

Circular orbits occur at r=3M. There are plenty of other possible general orbits. The photon "hanging" at r=2M on an outgoing geodesic is just one interesting and somewhat extreme example of a non-circular orbit.

 

[MeJennifer]

The photon "hanging" at r=2M on an outgoing geodesic is just one interesting and somewhat extreme example of a non-circular orbit.

But these orbits are not stable right? At least I think that is what you claimed a month or two ago.

 

[pervect]

The circular orbit of a photon at R=3m is not stable, and the photon hanging at r=2m is also not stable. There are a lot of different possible orbits for a photon. If the photon doesn't pass too close to the black hole, most photon orbits will tend to be nearly hyperbolic. Close to R=3m, one will see orbits that either spiral into the black hole, or spiral out. MTW has some pictures of some of the spiral orbits, I believe.

 

[RobtO]

But what about circular orbits at r=2M? Are they possible, too?

 

[Jonny_trigonometry]

does the wavelength of the photon have anything to do with its orbital stibility? like if you have something like bohr orbits only with light waves.

$$n \lambda = 2 \pi r$$

Are there any stable orbits? is there a continuous or discrete spectrum of them if there are any?

 

[pervect]

But what about circular orbits at r=2M? Are they possible, too?

The "orbit" of an outgoing photon at R=2m in Schwarzschild coordinates (r, theta, phi, t) is a single point:

r=2M
theta=phi=constant

This is not a circular orbit, because the angles are constant. There are other orbits that start at R=2m and propagate in different directions - all of these orbits wind up at the central singularity. Only the outgoing photon escapes this fate, if the photon starts off at even a small angle away from outgoing, it will get sucked into the black hole.

A circular orbit, by comparison is

r=3M
theta=constant
phi = w t

where w is the angular frequency

 

[Jonny_trigonometry]

Does the wavelength/energy of the photon have anything to do with its orbital stability? What if a photon had the same energy as the black hole itself... hypothetically? could a photon have so much energy that it itself is a black hole?

 

[Chris Hillman]

Stable and unstable photon orbits

Hi, Jonny,

Looks like pervect already answered your original question, but you then asked:

Does the wavelength/energy of the photon have anything to do with its orbital stability?

A fundamental principle in gtr is that the "path" taken by a monochromatic light beam does NOT depend upon its wavelength. (If this weren't true, gtr would be in serious trouble!) See for example section 15.4 of D'Inverno, Introducting Einstein's Relativity.

could a photon have so much energy that it itself is a black hole?

Well, the notion of a "photon" belongs to the domain quantum theory, but gtr is a classical field theory, so this question doesn't quite make sense in the context of gtr. So let's modify it: "can a laser pulse have so much mass-energy that it becomes a black hole?" You can probably now figure out what is the answer from gtr :-/

Chris Hillman