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Roy Kerr, a native New Zealander, discovered a solution to Einstein's field equations that represents the spacetime around a rotating black hole. It was in 1963, while he was at the University of Texas.
It is thought that most black holes that formed from stellar collapse must be rotating, simply because most, if not all, stars do rotate. Like all rotating stellar objects, rotating black holes also bulge out around their equators. Actually, in this case it's not quite a "bulge" that forms, but a "flattening" at the poles, as shown in the figure below - a bit exaggerated for effect!
The broken line circle represents the event horizon radius of a non-rotating (i.e., a Schwarzschild) black hole with radius 2GM/c^2. When the black hole is spinning, three distinct radii develop: an inner event horizon (Rinner, black region), an outer event horizon (Router, brown region) and a so-called static limit (Rstatic, green region). Before describing the different radii of interest, I will give the equations. This should be to the liking of some of my friends on this forum, e.g., "The Engineer"!
The first thing to do is to normalize as far as is practical - it helps to reduce unnecessarily complicated-looking equations.
The first two equations in the list above are: bar(M), the normalized mass of the black hole and a, the "spin parameter" of the black hole, both expressed in the SI units of meters - if you do not believe me, check it out by inserting the SI units and cancel them out! We will see below that the value of a can range form 0 (no spin) to bar(M) (extreme spin). The parameter J is the angular momentum of the black hole in kg m2 s-1 and G is Newton's gravitational constant in m3 kg-1 s-2. The spin parameter a is also sometimes expressed as the specific angular momentum (normalized angular momentum per unit mass), with a value 0 ≤ a ≤ 1.
Note that equations (3) and (4), apart from the sign difference before the square root, are the same. They represent the inner and the outer event horizons of a spinning black hole respectively. The outer event horizon is a radius at which all matter and light will just go inward - the "one-way membrane", similar to the event horizon of a Schwarzschild black hole. The inner event horizon theoretically reverses this and matter and light can possibly move in and out of there. This gives rise to the "wormhole" hypothesis - more about that in a future post.
The static limit (R_static) is a distance where no amount of force can keep an object static. Loosely speaking, it is where the spacetime rotates around the black hole at the speed of light. To prevent being dragged around, you must move faster than light relative to this rotating spacetime. This is called dragging of inertial frames or just frame dragging for short. It is also called the Lense-Thirring effect, after its discovers.
The region between the outer horizon and the static limit is called the ergo-sphere of the Kerr black hole - this is where frame dragging rules. Outside of the static limit, frame dragging still occurs, but it diminishes rather rapidly with distance. It took very, very sensitive equipment to detect it aboard a spacecraft (Gravity Probe B) in Earth orbit. See Gravity Probe B link at end of post.
As a sanity check on equations (3) to (5), set spin parameter a = 0, representing the non-rotating Schwarzschild black hole. Then R_inner = 0 and R_outer = R_static = 2bar(M), as expected. Objects cannot remain static at R=2bar(M) from a Schwarzschild black hole - they will not be rotating around, but just be dragged radially inward, irrespective of the amount of thrust that they have available.
Equations (3) and (4) shows that as a black hole rotates faster, R_inner moves outward and R_outer moves inward, until at spin a = bar(M), they are both equal to bar(M). A black hole with this angular momentum is called an extreme Kerr black hole. It is theorized that no black hole can spin faster than this rate without destroying its event horizons. In fact the values of R_inner and R_outer go imaginary there (due to the square root of a negative number being taken).
However, imaginary numbers are valid in mathematics and it can be used to state that, should a black hole spin faster and faster, both event horizons will eventually shrink to zero, revealing the central singularity of the black hole. This is called a naked singularity, which may or may not exist. It is not even certain that singularities, whether naked or not, exist. There are possibly quantum mechanical effects that prevent such bizarre things to exist in nature.
It is also not sure if the Kerr solution to Einstein's field equations describes the inner workings of a rotating black hole at all. From what can be cosmologically observed, it appears that it describes the outer spacetime adequately. The inner spacetime may forever be hidden behind the "clothing" called event horizons and we may never know…
You can read more about Gravity Probe B (mentioned above) on my web page Tests of Relativity. In my next Blog post, I will discuss the phenomenon of frame dragging a bit more fully.
Till then, regards, Jorrie
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