Some Further Notes on Black Holes

Anatomy of a Schwarzschild (non-rotating) black hole. The event horizon is spherical, whose radius is the Schwarzschild radius. Outside of this is the photon sphere, where photons can move around the black hole in perfectly circular (albeit unstable) orbits. Note the singularity is a single point (so I've drawn it too big!).

I have received a couple of good questions with regard to my latest Scientific American post “What We Know About Black Holes”, whose answers require going a little more into the technical side of black hole science. I’m going to try to answer these questions while keeping things still fairly non-technical, but be warned: I will use a little more math and physics jargon than I usually do on this blog! (I’ve thought for a while about starting a second blog to talk about higher-level subjects, since, you know, I don’t have nearly enough going on in my life right now.)

Most discussions about black holes focus on non-rotating or Schwarzschild black holes. There’s a good reason for this: Schwarzschild black holes are about as simple as they can be. They have one parameter that determines everything: their mass. The mass dictates the size of the event horizon, known as the Schwarzschild radius. The singularity — the place where all the mass of a black hole is concentrated — is a single point; the event horizon itself is a sphere. Since there is nothing actually at the event horizon, it’s perfectly featurless, so there’s no axis, no direction that looks any different from any other.

(Karl Schwarzschild was a German physicist who studied an early version of Einstein’s general theory of relativity. He found the solution to Einstein’s equations that bear his name, though the solution itself ended up being modified when the final version of general relativity was published. Schwarzschild died in 1916 while fighting on the Russian front of World War I.)

"Interior" of a rotating black hole. Note that the singularity -- the place where the mass is concentrated -- is depicted as a ring. While this is strictly true, it is also somewhat misleading, since the ring has no size!

Rotating or Kerr black holes are more realistic, and that’s what I wrote about in my Scientific American post. If collapsing stars do form black holes, which seems the most likely explanation, then the black hole formed should also be rotating. (This is why neutron stars rotate so quickly, turning them into pulsars.) Now we get the ergosphere I wrote about, and the shape of the event horizon is an oblate spheroid: a sphere that has been sat upon so that it bulges at the equator. While a featureless sphere can be rotated around any axis you draw through the center (spherical symmetry or SO(3)), a Kerr black hole has a special axis of rotation, so the angle at which you observe or approach it will dictate what you see and measure.

A rotating black hole has two parameters that dictate the size and shape of the event horizon: mass and spin. Mathematically, the spin breaks the perfect spherical shape; if you set that parameter to zero, you get back a Schwarzschild black hole. (I could also discuss Kerr-Newman black holes, which are Kerr black holes with a net electric charge; it’s unlikely that realistic black holes will have too much imbalance between positive and negative charges, so I’ll skip that discussion here.) Despite adding only one parameter, the mathematics get a lot wilder, to the point where none of the general relativity books on my shelf (including graduate-level texts) derive the Kerr solution from scratch.

Now we reach the technical part: a perfectly spherical singularity produces a perfectly spherical event horizon. What kind of singularity produces a rotating black hole? The answer is a ring, as I show in the figure: the axis of rotation goes right through the hole of the donut, and the ring lies in the plane perpendicular to that axis. However, the figure (which is similar to the one you’ll find in most standard textbooks on general relativity or astrophysics) is a bit misleading too, because the ring has no size. It’s not easy to describe it, but let me try anyway.

How big the ring is depends on the spin: the faster the spin, the wider the ring. The ring itself has no width (you can think of a donut where the hole is the size of the whole thing, which is probably the most cruel image I could summon), so particles can pass right into the middle of the “donut hole” without hitting the singularity itself. Because of rotation, if you are traveling straight along the axis of rotation, you won’t hit the singularity — you’ll pass right through. Only very special trajectories will land right on the singularity: those that are traveling in the equatorial plane of the black hole. The circle bounded by the ring singularity has its own bizarre properties, which look kind of like the exterior of the original black hole… but without an event horizon.

If we ever needed a demonstration that physics is geometry in general relativity, this is it.

Updated: My first attempt at writing this post suffered from incoherence. I apologize for that, so this is a somewhat modified version. I also apologize that I failed to answer the primary questioner’s query: I have been trying to avoid using math on this blog, so I still owe him a mathematical explanation.

4 Responses to “Some Further Notes on Black Holes”

  1. 1 Robert L. Oldershaw September 2, 2011 at 20:54

    Long ago E.R. Harrison (“Electrified black holes”, Nature, about 1976?)pointed out that stars have a net positive charge.

    This is because electrons are so much lighter than protons and nuclei.

    Electrons are ejected more efficiently by the hot star, and this gives the star a net positive charge.

    I think he also argued that black holes could have large net charges.

    If magnetars can have a magnetic field of 10^15 Gauss, then there has to be a rather large electric current somewhere. As Einstein showed us: no electric current – no magnetic field.

    Bottom line: if we assume net neutrality for astrophysical objects, we may be missing critical physics that is needed to explain the many enigmas of astrophysics, like the dark matter, the Sun’s amazing magnetic cycle, what powers very-high-speed pencil-beam jets in many stellar and galactic systems, pulsars, gamma-ray bursters, cosmic ray accelerators, etc.

    It is possible that Kerr-Newman objects dominate nature on many scales.

    Fractal Cosmology; Discrete Scale Relativity

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