Can light "orbit" a black hole?

[Mel Odious]

I'm currently teaching myself basic physics, and a strange thought occured to me while I was casually looking ahead in my textbook this afternoon.

Please forgive me if what follows is a bit simplistic. I haven't studied the chapters on light yet! Okay, here goes ...

If light passes by at a sufficiently large distance from a black hole, it will be bent by the black hole's gravity and then continue off into space.

But if the light passes too close to the black hole, it will be pulled into the black hole.

Therefore, It seemed to me that there should be a distance at which light will neither be pulled into the black hole nor pass by after being bent by the black hole's gravity. In other words, the light should "orbit" the black hole for some arbitrary length of time. Or at least, there should be photons of light in orbit around the black hole. Is that better?

I don't know if I've reasoned this out correctly or not, but the idea fascinates me. Am I correct in my thinking?

 

[Yllanes]

Therefore, It seemed to me that there should be a distance at which light will neither be pulled into the black hole nor pass by after being bent by the black hole's gravity. In other words, the light should "orbit" the black hole for some arbitrary length of time. Or at least, there should be photons of light in orbit around the black hole. Is that better?

Yes. For a Schwarzschild black hole, light can follow a circular orbit at a distance several times the Schwarzschild radius from the singularity (I forgot the precise distance). The orbit is unstable though.

 

[asmodean]

<yoda voice>Photon sphere! You seek Photon Sphere!</yoda voice>

http://en.wikipedia.org/wiki/Photon_sphere

 

[jmercer]

Yes. For a Schwarzschild black hole, light can follow a circular orbit at a distance several times the Schwarzschild radius from the singularity (I forgot the precise distance). The orbit is unstable though.

Why is it unstable?

 

[SirPhilip]

Why is it unstable?

Probably because it's orbiting a black hole.

 

[AgeGap]

Why is it unstable?

That is what i thought but I also thought the event horizon would be where it orbited. Any higher it would escape and lower it would be captured. Damn Newtonion Physics.

 

[Yllanes]

Why is it unstable?

The equation of motion for a massless particle around a Schwarszchild black hole is

[latex]
\footnotesize
\[
\left(\frac{\mathrm{d} r}{\mathrm{d} \lambda}\right)^2 + \frac{L^2}{r^2} \left(1-\frac{2GM}{r}\right)=E^2
\]
[/latex]

which can be written as a particle moving in a potential V₀

[latex]
\footnotesize
\[\frac12 \left(\frac{\mathrm{d}r}{\mathrm{d} \lambda}\right)^2 + V_0(r)=\frac12 E^2,\qquad V_0(r)=\underbrace{\frac{L^2}{2r^2}}_{\textsf{centrifugal barrier}} - \underbrace{\frac{L^2 GM}{r^3}}_\substack{\textsf{attractive potential}\\ \textsf{dominates at small $r$}}
\]
[/latex]

So it is just as the OP said. At large r, the centrifugal term dominates, while at small r the attractive term is bigger (unlike the Newtonian case). If you solve V₀'(r₀) = 0 for r₀, you can find the positions for which there is a circular orbit. Doing this you get r₀ = 3GM. But you can also see that V₀(3GM) is a maximum, not a minimum, of the potential, so that the equilibrium is unstable and any small perturbation destroys the orbit.

For massive particles the picture is a little different and depends on the angualr momentum L. Now there are both stable and unstable circular orbits.

 

[sol invictus]

Why is it unstable?

Yllanes answered that mathematically. It's essentially because there is only one speed light can move, which means one possible radius for an orbit. Any closer and it will get pulled in; further away, and it will escape - so it's unstable.

Imagine sound waves propagating in a toilet bowl (which is not a bad analogy for a black hole!) as it continuously flushes. There will be a radius where sound can propagate in a circle (at least, I think so). Any closer and it will get flushed, any higher and it will escape.

That is what i thought but I also thought the event horizon would be where it orbited. Any higher it would escape and lower it would be captured. Damn Newtonion Physics.

At the event horizon light which is propagating directly out of the black hole (so radially, rather than circumferentially as in an orbit) will "hover" at fixed radius. Since it's not orbiting, that's not usually considered an orbit.

 

[shadron]

Why is it unstable?

It's unstable because if it is microscopically less than the exact radius of the ideal it will degrade and eventually go lower; if its just slightly higher, it will escape. For something to be in true orbit it has to be exactly at that radius, which is only microscopically possible. It is unstable in the same way that the first, second and third Lagrangian points are unstable; think of them as the peaks on a hill (close, not perfect analogy); any small displacement away from the perfect position will drag them off. A stable orbit would drag them back towards the orbit when they deviated, or at least accommodate deviance as a different possible orbit.

The best way to think about them is that a stable orbit can be used forever (sans perturbations) with no active assist, while staying in an unstable orbit requires restoration (by rocket motor, perhaps) from time to time.

See http://en.wikipedia.org/wiki/Lagrangian_point.

 

[Ziggurat]

Trivia: if you built a circular space station surrounding a black hole at this radius and stood inside it, it would appear to be straight. If there were no obstructions and you have a good enough telescope, you could look at the back of your own head.

Freaky!

 

[shadron]

Trivia: if you built a circular space station surrounding a black hole at this radius and stood inside it, it would appear to be straight. If there were no obstructions and you have a good enough telescope, you could look at the back of your own head.

Freaky!

Yeah, and if you walked over to look at the hole, you'd never make it back.

 

[Mel Odious]

I just wanted to thank everyone who posted replies here; your responses were all very helpful. It was gratifying for me to see that even though my attempted explanation may have left something to be desired, at least my basic understanding of the situation was accurate (super-important in physics, as I'm learning). Even what Ziggurat said makes sense to me now.

Thanks again everyone!

 

[Soapy Sam]

Thank you for an excellent question! Welcome in.