3.4 The Horizon and the Ergosphere |
53 |
Fig. 3.3 Value of the angular velocity ω of a locally non-rotating observer plotted against the radius r and the declination angle θ . In the planes at θ = 0, π namely at the North and South poles of the hole, we have ω = 0, i.e. there is no rotational dragging of the inertial frames. On the other hand ω is maximal on the equatorial plane θ = π2 . On the other hand ω decreases with the distance r from the hole and vanishes at r = ∞ where it is uniformly zero for all values of θ
implies that the 4-velocity coincides with the time-translation Killing vector k. In asymptotic geometry this Killing vector is time-like, but as we get closer to the hole its norm (k, k) shrinks and there is a surface where it vanishes, namely (k, k) = 0. This equation defines the static limit:
ΣSL : 0 = (k, k) ≡ gtt
0 = q2 + r2 + α2 − α2 sin2(θ ) − 2mr (3.3.21)
r = r±(θ ) ≡ m ± m2 − q2 − α2 + α2 sin2(θ )
Corresponding to the two roots of the quadratic equation there are two vanishing surfaces for the norm of Killing vector k: one outer r = r+(θ ) and one inner r = r−(θ ). The static limit corresponds to the outer surface r = r+(θ ). An image of the static limit surface is displayed in Fig. 3.4.
3.4 The Horizon and the Ergosphere
As we have seen in the previous section, a physical observer has an angular velocity Ω falling in the range (3.3.9) comprised between the two roots Ω± of the quadratic
54 |
3 Rotating Black Holes and Thermodynamics |
Fig. 3.4 The static limit surface ΣSL defined by
r = r+(θ ) is an ellipsoid and contains inside itself the
spherical surface
r = rH = r+( π2 ) which, as we discuss in the main text, is the event horizon ΣH . The static limit surface is tangential to the horizon at the North and South poles of the hole. The region contained between ΣSL and ΣH is named the ergosphere
form (3.3.7). When the discriminant of that quadratic form vanishes, the two-roots coincide and we have:
Ωmax = Ωmin = ΩH |
(3.4.1) |
Inspecting (3.3.7) we see that its discriminant is given by the expression: |
|
= gt2φ − gtt gφφ = r2 + α2 − 2mr + q2 |
(3.4.2) |
which is indeed the building block function Δ(r) introduced in (3.2.2). The reason for the choice of its name becomes now apparent.
We claim that the bigger root of the quadratic equation Δ(r) = 0 is the eventhorizon of the black-hole. Let us first spell out the two roots and then motivate our
statement. We have |
|
= 0 → r = r± = m ± m2 − q2 + α2 |
(3.4.3) |
Let us now argue in the following way. Given the two Killing vectors (3.3.1) let us define the family of Killing fields:
χ (Ω) |
= k + Ω ˜ |
(3.4.4) |
k |
which, as we know from (3.3.6), correspond to the 4-velocities of test-bodies having angular velocities Ω with respect to the fixed stars. For each χ (Ω) let us consider the light-like radial curves that admit χ (Ω) as the tangent vector field. Explicitly we set:
dt = dp; dφ = Ω dp; dθ |
(3.4.5) |
and we obtain the equation:
0 = gtt dp2 + 2Ωgtφ dp2 + Ω2gφφ dp2 + grr dr2 |
(3.4.6) |
so that for each Ω we have an effective 2-dimensional metric:
ds2 = gpp(r, Ω) dp2 + grr dr2 |
(3.4.7) |