Go to previous page Go up Go to next page

3.1 Preliminaries

The first law of black hole mechanics (1View Equation) tells us how the area of the black hole increases when it makes a transition from an initial equilibrium state to a nearby equilibrium state. The question we want to address is: Can one obtain an integral generalization to incorporate fully dynamical situations? Attractive as this possibility seems, one immediately encounters a serious conceptual and technical problem. For, the generalization requires, in particular, a precise notion of the flux of gravitational energy across the horizon. Already at null infinity, the expression of the gravitational energy flux is subtle: One needs the framework developed by Bondi, Sachs, Newman, Penrose, and others to introduce a viable, gauge invariant expression of this flux [52Jump To The Next Citation Point32Jump To The Next Citation Point185Jump To The Next Citation Point]. In the strong field regime, there is no satisfactory generalization of this framework and, beyond perturbation theory, no viable, gauge invariant notion of the flux of gravitational energy across a general surface.

Yet, there are at least two general considerations that suggest that something special may happen on DHs. Consider a stellar collapse leading to the formation of a black hole. At the end of the process, one has a black hole and, from general physical considerations, one expects that the energy in the final black hole should equal the total matter plus gravitational energy that fell across the horizon. Thus, at least the total integrated flux across the horizon should be well defined. Indeed, it should equal the depletion of the energy in the asymptotic region, i.e., the difference between the ADM energy and the energy radiated across future null infinity. The second consideration involves the Penrose inequality [157] introduced in Section 1. Heuristically, the inequality leads us to think of the radius of a marginally trapped surface as a measure of the mass in its interior, whence one is led to conclude that the change in the area is due to influx of energy. Since a DH is foliated by marginally trapped surfaces, it is tempting to hope that something special may happen, enabling one to define the flux of energy and angular momentum across it. This hope is borne out.

In the discussion of DHs (Sections 3 and 4.2) we will use the following conventions (see Figure 5View Image). The DH is denoted by H and marginally trapped surfaces that foliate it are referred to as cross-sections. The unit, time-like normal to H is denoted by a ^τ with a b gab^τ ^τ = − 1. The intrinsic metric and the extrinsic curvature of H are denoted by qab := gab + ^τa^τb and Kab := qacqbd∇c ^τd, respectively. D is the derivative operator on H compatible with qab, ℛ ab its Ricci tensor, and ℛ its scalar curvature. The unit space-like vector orthogonal to S and tangent to H is denoted by a ^r. Quantities intrinsic to S are generally written with a tilde. Thus, the two-metric on S is ^qab and the extrinsic curvature of S ⊂ H is K^ab := q^a c^qb dDc ^rd; the derivative operator on (S,^qab) is ^D and its Ricci tensor is ℛ^ab. Finally, we fix the rescaling freedom in the choice of null normals to cross-sections via ℓa := ^τa + ^ra and na := ^τa − ^ra (so that ℓan = − 2 a). To keep the discussion reasonably focused, we will not consider gauge fields with non-zero charges on the horizon. Inclusion of these fields is not difficult but introduces a number of subtleties and complications which are irrelevant for numerical relativity and astrophysics.

View Image

Figure 5: H is a dynamical horizon, foliated by marginally trapped surfaces S. a ^τ is the unit time-like normal to H and ^ra the unit space-like normal within H to the foliations. Although H is space-like, motions along ^ra can be regarded as ‘time evolution with respect to observers at infinity’. In this respect, one can think of H as a hyperboloid in Minkowski space and S as the intersection of the hyperboloid with space-like planes. In the figure, H joins on to a weakly isolated horizon Δ with null normal a ℓ¯ at a cross-section S0.

  Go to previous page Go up Go to next page