Light orbiting black holes

Enola Straight · Post by **Enola Straight** » 2004-11-09 02:11pm

Let's start with the commonly accepted premise that the singularity has such a spacetime warping gravitational field so strong that nothing, not even light, can escape.

Furthermore, the gravity can bend incoming light in a form of gravitational lens.

For a given singularity of a certain mass, is there not an orbital radius where incoming light rays do not simply get absorbed by the singularity, not just deflected, but actually settle into a stable orbit?

If so, wouldn't this orbit eventually become saturated with EM energy?

Admiral Valdemar · Post by **Admiral Valdemar** » 2004-11-09 02:23pm

You mean the event horizon. And yes, a lot of EM energy builds up, but it is also shot out of most rotating Schwarzschild radii.

Sriad · Post by **Sriad** » 2004-11-10 04:19am

Actually the radius where light has a "stable" orbit is around 1.5 Schwarzschild radii. Light traveling tangentally to the event horizon orbits at this distance; when you're closer there are no stable orbits because space-time is flowing into the BH. To maintain position you need to accelerate away from it, and as you near the event horizon, the acceleration needed to maintain position approaches infinity. This is why the light from, say, a flashlight, would appear to become infinately red-shifted as the flashlight fell into the black hole.

Admiral V. is right that it doesn't build up to absurd levels; the orbit is fairly easily disturbed by passing matter. Also as energy collects, it alters the mass of the black hole system, which changes the stable light orbit radius. If you passed 1.5 Schwarzschild, though, you would see a brilliant flash as you passed through all the orbiting light, as well as many other wacky effects.

Kuroneko · Post by **Kuroneko** » 2004-11-10 10:31pm

The Schwarszschild Metric is dλ² = (1-2m/r)dt² - dr²/(1-2m/r) - r²[dθ²-sin²θ dφ²], which if restricted to the equatorial plane (θ = π/2, dθ = 0) and if the orbit is circular (dr = 0), then it simplifies to dλ² = [1-2m/r]dt² - r²dφ², and by Kepler's third law r²dφ² = (m/r)dt². Substituting back gives dλ² = (1-3m/r)dt², which becomes singular at r = 3m. Hence, there are no circular orbits with r < 3m, with r = 3m reserved for light. (Note that this orbit is unstable, so that any pertubation will destroy it.)

That the Schwarzchild metric preserves Kepler's third law for circular orbits is easy to verify. The geodesics satisfy [d²x^m]/dλ² + Γ^m_{ij} [dx^i dx^j]/dλ² = 0, where Γ^m_{ij} is the Christoffel Symbol of the Second Kind, where (x^0,x^1,x^2,x^3) = (t,r,θ,φ). With this knowledge, the rest is just taking partial derivatives. If the orbit is circular and in the equatorial plane, then it should also satisfy d²r/dλ² = d²φ/dλ² = 0, in which case the geodesic equation for x^0 = r simplifies to [2m/r²][dt/dλ]² = 2r[dφ/dλ]². That is equivalent to Kepler's third law Ω²r³ = m.

The non-existence of orbits with eccentricity 0<e<1 under the Schwarzschild metric is due to symmetry.

Admiral Valdemar · Post by **Admiral Valdemar** » 2004-11-11 05:35am

Kuroneko is what I almost became before I took biology up instead.

Kazuaki Shimazaki · Post by **Kazuaki Shimazaki** » 2004-11-11 08:49am

Kuronenko, how did you make Superscript characters in this board?

Admiral Valdemar · Post by **Admiral Valdemar** » 2004-11-11 08:52am

Kazuaki Shimazaki wrote:Kuronenko, how did you make Superscript characters in this board?

Don't do them in MS Word. It doesn't work. I just find something already in HTML such as the Trademark Symbol™.

Kuroneko · Post by **Kuroneko** » 2004-11-11 09:03am

Kazuaki Shimazaki wrote:Kuronenko, how did you make Superscript characters in this board?

The ² and ³ characters are part of both are Unicode characers 178 and 179, respectively. You can paste them directly or do "&# 178;" (without space). They're very portable (other superscripts aren't). I type Greek by setting my keyboard to a Greek locale.

phongn · Post by **phongn** » 2004-11-11 01:09pm

Maybe we should bug the phpBB developers to include MathML support at this rate

Kuroneko · Post by **Kuroneko** » 2004-11-11 11:27pm

phongn wrote:Maybe we should bug the phpBB developers to include MathML support at this rate ;)

Actually, it's very simple to include support for the <sup></sup> and <sub></sub> tags in phpBB. Some time ago I did this for another board (which is unfortunately now dead), as well as having Greek letters as simply [letter name] (which is even simpler, because it does not require deconversion in cases like the quoting of posts).

Admiral Valdemar · Post by **Admiral Valdemar** » 2004-11-12 04:09am

That'd be a good addition for science threads at least. Save having to post LaTeX stuff instead.

Durandal · Post by **Durandal** » 2004-11-12 11:25am

phongn wrote:Maybe we should bug the phpBB developers to include MathML support at this rate

Screw it, just have them include a server-side LaTeX renderer that outputs to PNG-24.