Kerr black hole

dropped ‘straight in’ (pφ = 0) from infinity is ‘dragged’ just by the influence of gravity so that it acquires an angular velocity in the same sense as that of the source of the metric (we’ll see below that, for the Kerr metric, ω has the same sign as a). This effect weakens with distance (roughly as 1/r3; see Eq. (11.90) below for the Kerr metric), and it makes the angular momentum of the source measurable in principle, although in most situations the effect is small, as we have seen in Exer. 19, § 8.6. This effect is often called the dragging of inertial frames.

This effect has a close analogy with magnetism. Newtonian gravity is, of course, very similar to electrostatics, with the sign change that ensures that ‘charges’ of the same sign attract one another. In electromagnetism, a spinning charge creates additional effects which we call magnetism. Here we have the gravitational analog, the gravitational effects due to a spinning mass. For that reason these effects are called gravitomagnetism. The analogy between gravitomagnetism and standard magnetism is perhaps easier to see in the Lense– Thirring effect: a gyroscope placed in orbit around a rotating star will precess by a small amount that is proportional to the angular momentum of the star, just as a spinning electron precesses if it orbits through a magnetic field.

Small as it is, the Lense–Thirring effect created by the spin of the Earth has been measured. Detailed studies of the orbits of two geodesy satellites have so far been able to verify that the effect is as predicted by general relativity, with an accuracy at the 20% level (Ciufolini et al., 2006). An order of magnitude improvement could be made with an additional satellite in a specially chosen orbit. In addition, a satellite experiment called Gravity Probe B (GP-B), which has measured the precession of on-board gyroscopes, is currently (2008) analyzing its data and is expected to report soon, hopefully with errors below the 10% level. GP-B is one of the most sensitive and high-precision experiments ever performed in orbit. Gravitomagnetic effects are regularly taken into account in modeling the details of the emission of X-rays near black holes (e.g. Brenneman and Reynolds 2006), and they may also soon be seen in the double pulsar system PSR J0737-3039 referred to earlier in this chapter.

Ergoregion

Consider photons emitted in the equatorial plane (θ = π/2) at some given r. In particular, consider those initially going in the ±φ-direction, that is tangent to a circle of constant r. Then they generally have only dt and dφ nonzero on the path at first and since ds2 = 0, we have

We will see below that for the Kerr metric the second solution gives dφ/dt the same sign as the parameter a, and so represents the photon sent off in the same direction as the hole’s rotating. The other solution means that the other photon – the one sent ‘backwards’– initially doesn’t move at all. The dragging of orbits has become so strong that this photon cannot move in the direction opposite the rotation. Clearly, any particle, which must move slower than a photon, will therefore have to rotate with the hole, even if it has an angular momentum arbitrarily large in the opposite sense to the hole’s!

We shall see that the surface where gtt = 0 lies outside the horizon; it is called the ergosphere. It is sometimes also called the ‘static limit’, since inside it no particle can remain at fixed r, θ , φ. From Eq. (11.71) we conclude that it occurs at

Inside this radius, since gtt > 0, all particles and photons must rotate with the hole. Again, this effect can occur in other situations. Models for certain rotating stars are

known where there are toroidal regions of space in which gtt > 0 (Butterworth and Ipser 1976). These will have these super-strong frame-dragging effects. They are called ergoregions, and their boundaries are ergotoroids. They can exist in solutions which have no horizon at all. But the stars are extremely compact (relativistic) and very rapidly rotating. It seems unlikely that real neutron stars would have ergoregions.

The Kerr horizon

In the Schwarzschild solution, the horizon was the place where gtt = 0 and grr = ∞. In the Kerr solution, the ergosphere occurs at gtt = 0 and the horizon is at grr = ∞, i.e. where

It is clear that the Kerr ergosphere lies outside the horizon except at the poles, where it is tangent to it. The full proof that this is the horizon is beyond our scope here: we need to verify that no null lines can escape from inside r+. We shall simply take it as given (see the next section below for a partial justification). Since the area of the horizon is important (Hawking’s area theorem), we shall calculate it.

The horizon is a surface of constant r and t, by Eq. (11.81) and the fact that the metric is stationary. Any surface of constant r and t has an intrinsic metric whose line element comes from Eq. (11.71) with dt = dr = 0:

The proper area of this surface is given by integrating the square root of the determinant of this metric over all θ and φ:

0 0