Kerr Black Holes
April 8, 2010 4 Comments
The Schwarzschild reference frame is static outside the Black Hole, so even though space is curved, and time is slowed down close to the Black Hole, it is much like the absolute space of Newton. But we will need a generalized reference frame in the case of rotating Black Holes. Roy Kerr generalized the Schwarzschild geometry to include rotating stars, and especially rotating Black Holes. Most stars are rotating, so it is natural to expect newly formed Black Holes to process significant rotation too.
Features of a Kerr Black Hole
The Kerr black hole consists of a rotating mass at the center, surrounded by two event horizons. The outer event horizon marks the boundary within which an observer cannot resist being dragged around the black hole with space-time. The inner event horizon marks the boundary from within which an observer cannot escape. The volume between the event horizons is known as the ergosphere.
What is a Kerr black hole?
The usual idealised “static” black hole is stationary, unaccelerated, at an arbitarily-large distance from the observer, is perfectly spherical, and has a point-singularity at its centre.
When one of these idealised black holes rotates, it gets an extra property. It’s no longer spherically symmetrical , the receding and approaching edges have different pulling strengths and spectral shifts, and the central singularity is no longer supposed to be a dimensionless point.
The equatorial bulge in the event horizon can be deduced in several ways
- … as a sort of centrifugal forces effect. Since it’s possible to model the (distantly-observed) hole as having all its mass existing as an infinitely-thin film at the event horizon itself (i.e. where the mass is “seen” to be), you’d expect this virtual film to have a conventional-looking equatorial bulge, through centrifugal forces.
- … as a sort of mass-dilation effect. Viewed from the background frame, the “moving film” of matter ought to appear mass-dilated, and therefore ought to have a greater gravitational effect, producing an increase in the extent of the event horizon. Since the background universe sees the bh equator to be moving faster than the region near the bh poles, the equator should appear more mass-dilated, and should have a horizon that extends further.
- … as a shift effect. This tidy ellipsoidal shape isn’t necessarily what people actually see – it’s an idealised shape that’s designed to illustrate an aspect of the hole’s deduced geometry independent of the observer’s viewing angle. In fact, the receding and approaching sides of the hole (viewed from the equator) might appear to have different radii, because it’s easier for light to reach the observer from the approaching (blueshifted) side than the receding (redshifted) side (these shifts are superimposed on top of the normal Schwarzchild redshift).
If we calculate these motion shifts using either the SR shift assumptions f’/f = (flat spacetime propagation shift) × root[1 – v²/c²] or the plain fixed-emitter shift law f’/f = (c-v)/c, and then treat them as “gravitational”, then by multiplying the two opposing shifts together and rooting the result, we can get the same averaged dilation factor of f’/f=root(1 – v²/c²) in each case, and by applying the averaged value, we recreate the same sort of equatorially-dilated shape that we got in the other two arguments.
Of course, none of these “film” arguments work for a rotating point, which immediately tells us that the distribution of matter within a rotating black hole is important, and that the usual method of treating the actual extent of a body within the horizon as irrelevant (allowing the use of a point-singularity) no longer works when the hole is rotating (a rotating hole can’t be said to contain a point-singularity).
In the case of a rotating hole, the simplest state that we can claim is equivalent to the rotating film of matter for a distant observer is a ring-singularity.
- The idea of being able to treat a non-rotating black hole as either a point-singularity or a hollow infinitely-thin film is a consequence of the result that the actual mass-distribution is a “null” property for a black hole, as long as it is spherically symmetrical. If the mass fits into a Schwarzchild sphere, the usual static model of a black hole allows the hole’s mass to be point-sized, golfball-sized, or of any size up to the size of the event horizon.
It’s usual to treat all the matter as being compacted to a dimensionless point, but sometimes it’s useful to go to the other extreme and treat the matter as being at its “observed” position – as an infinitely-thin film at the event horizon (see Thorne’s membrane paradigm).
- The idea of being able to treat all shifts as being propagation effects is something that probably ought to be part of GR – in the context of black holes, the time-dilation effect comes out as a curved-space propagation effect due the enhanced gravitation due to kinetic energy. However, there’s a slight “political” problem here, in that GR is supposed to reduce to SR, and SR is usually interpreted as having Lorentz shifts which are supposed to be non-gravitational (because allowing the possibility of gravitational effects upsets the usual SR derivations). A GR-centred physicist might not have a problem with this approach of treating all shift effects as being equivalent, a SR-centred one probably would.
- The “bulginess” of a Kerr black hole is illustrated on p.293 of the Thorne book (fig 7.9). Thorne says that the effect of the spin on the horizon shape was discovered Larry Smarr in 1973.
Overview of Kerr Spacetime
Kerr spacetime is the unique explicitly defined model of the gravitational field of a rotating star. The spacetime is fully revealed only when the star collapses, leaving a black hole — otherwise the bulk of the star blocks exploration. The qualitative character of Kerr spacetime depends on its mass and its rate of rotation, the most interesting case being when the rotation is slow. (If the rotation stops completely, Kerr spacetime reduces to Schwarzschild spacetime.)
The existence of black holes in our universe is generally accepted — by now it would be hard for astronomers to run the universe without them. Everyone knows that no light can escape from a black hole, but convincing evidence for their existence is provided their effect on their visible neighbors, as when an observable star behaves like one of a binary pair but no companion is visible.
Suppose that, travelling our spacecraft, we approach an isolated, slowly rotating black hole. It can then be observed as a black disk against the stars of the background sky. Explorers familiar with the Schwarzschild black holes will refuse to cross its boundary horizon. First of all, return trips through a horizon are never possible, and in the Schwarzschild case, there is a more immediate objection: after the passage, any material object will, in a fraction of a second, be devoured by a singularity in spacetime.
If we dare to penetrate the horizon of this Kerr black hole we will find … another horizon. Behind this, the singularity in spacetime now appears, not as a central focus, but as a ring — a circle of infinite gravitational forces. Fortunately, this ring singularity is not quite as dangerous as the Schwarzschild one — it is possible to avoid it and enter a new region of spacetime, by passing through either of two “throats” bounded by the ring (see The Big Picture).
In the new region, escape from the ring singularity is easy because the gravitational effect of the black hole is reversed — it now repels rather than attracts. As distance increases, this negative gravity weakens, just as on the positive side, until its effect becomes negligible.
A quick departure may be prudent, but will prevent discovery of something strange: the ring singularity is the outer equator of a spatial solid torus that is, quite simply, a time machine. Travelling within it, one can reach arbitrarily far back into the past of any entity inside the double horizons. In principle you can arrange a bridge game, with all four players being you yourself, at different ages. But there is no way to meet Julius Caesar or your (predeparture) childhood self since these lie on the other side of two impassable horizons.
This rough description is reasonably accurate within its limits, but its apparent completeness is deceptive. Kerr spacetime is vaster — and more symmetrical. Outside the horizons, it turns out that the model described above lacks a distant past, and, on the negative gravity side, a distant future. Harder to imagine are the deficiencies of the spacetime region between the two horizons. This region definitely does not resemble the Newtonian 3-spacebetween two bounding spheres, furnished with a clock to tell time. In it, space and time are turbulently mixed. Pebbles dropped experimentally there can simply vanish in finite time — and new objects can magically appear.
Kerr-Newman Black Hole
A rotating charged black hole. An exact, unique, and complete solution to the Einstein field equations in the exterior of such a black hole was found by Newman et al. (1965), although its connection to black holes was not realized until later (Shapiro and Teukolsky 1983, p. 338).
Rotating (Kerr) Black Holes, Charged and Uncharged
Most stars spin on an axis. In 1963, Roy Kerr reasoned that when rotating stars shrink, they would continue to rotate. Kip Thorne calculated that most black holes would rotate at a speed 99.8% of their mass. Unlike the static black holes, rotating black holes are oblate and spheroidal. The lines of constant distance here are ellipses, and lines of constant angle are hyperbolas.
Unlike static black holes, rotating black holes have two photon spheres. In a sense, this results in a more stable orbit of photons. The collapsing star “drags” the space around it into rotating with it, kind of like a whirlpool drags the water around it into rotating. As in the diagram above, there would be two different distances for photons. The outer sphere would be composed of photons orbiting in the opposite direction as the black hole. Photons in this sphere travel slower than the photons in the inner sphere. In a sense, since they are orbiting in the opposite direction, they have to deal with more resistance, hence they are “slowed down”. Similarly, photons in the inner ring travel faster since they are not going against the flow. It is because the photon sphere in agreement with the rotation can travel “faster” that it is on the inside. The closer one gets to the event horizon, the faster one has to travel to avoid falling into the singularity – hence the “slower” moving photons travel on the outer sphere to lessen the gravitational hold the black hole has.
The rotating black hole has an axis of rotation. This, however, is not spherically symmetric. The structure depends on the angle at which one approaches the black hole. If one approaches from the equator, then one would see the cross-section as in the diagram above, with two photon spheres. However, if one approached at angles to the equator, then one would only see a single photon sphere.
The position of the photon spheres also depend on the speed at which the black hole rotates. The faster the black hole rotates, the further apart the two photon spheres would be. For that matter, a black hole with a speed equal to its mass would have the greatest possible distance between the two photon spheres. This is because of greater difference in the speed between the photon spheres. As the speed of rotation increases, the outer sphere of photons would slow down as it meets greater resistance, even as the inner sphere would travel “faster” as it is pushed along by the centripetal forces.
Next, we move on to look at the ergosphere. The ergosphere is unique to the rotating black hole. Unlike the event horizon, the ergosphere is a region, and not a mathematical distance. It is a solid ellipsoid (or a 3-dimensional ellipse). The ergosphere billows out from the black hole above the outer event horizon of a charged black hole (a.k.a. Kerr-Newman), and above the event horizon of an uncharged black hole (a.k.a. Kerr). This distance is known as the static limit of a rotating black hole. At this distance, it is no longer possible to stay still even if one travels at the speed of light. One would inevitably be drawn towards the singularity. The faster the rotation, the further out it billows. When the ergosphere’s radius is half the Schwarzschild radius along the axis of rotation, it experiences the greatest distance it can billow out. At this point, even light rays are dragged along in the direction of rotation. Strangely enough, it is postulated that one can enter and leave as one likes since technically, you have not hit the event horizon yet.
For a rotating black hole, the outer event horizon switches time and space as we know it. The inner event horizon, in turn, returns it to the way we know it. Singularity then becomes a place rather than a time, and can technically be avoided. When angular velocity increases, both the outer and the inner event horizon move closer together.
In the diagram, you would have noticed that the singularity here is drawn as a ring, and not a point, as it was for the static black hole. In the case of a rotating black hole, the gravity around the ringed singularity is repulsive. In other words, it actually pushes one away, allowing you to actually leave the black hole. The only way to approach the ring singularity would be to come in from the equatorial plane. Other trajectories would be repelled with greater strength, proportional to the closer the angle is to the axis of rotation.
In addition, there would be a third photon sphere about the ring singularity. If light is parallel to the axis of rotation, the gravity and the anti-gravity of the singularity are balanced out. Light then traces out the path of constant distance (which, in the case is an ellipsoid). Technically, this might lead the light into another universe through the singularity, and then back out again. At this point within the black hole, we may see three types of light: the light reflected from our universe behind us; the light from other universes; and the light from the singularity.