Our summary of the mechanics of DHs is divided in to three parts. In the first, we begin with some preliminaries on angular momentum. In the second, we extend the area balance law (25) by allowing more general lapse and shift functions, which leads to the integral version of the first law. In the third, we introduce the notion of horizon mass.

As one might expect, the angular momentum balance law results from the momentum constraint (15) on the
DH . Fix any vector field on which is tangential to all the cross-sections of , contract
both sides of Equation (15) with , and integrate the resulting equation over the region to
obtain^{4}

Equation (42) is a balance law; the right side provides expressions of fluxes of the generalized angular momentum across . The contributions due to matter and gravitational waves are cleanly separated and given by

with , so that As expected, if is a Killing vector of the three-metric , then the gravitational angular momentum flux vanishes: .As with the area balance law, here we worked directly with the constraint equations rather than with a Hamiltonian framework. However, we could also have used, e.g., the standard ADM phase space framework based on a Cauchy surface with internal boundary and the outer boundary at infinity. If is a vector field on which tends to on and to an asymptotic rotational symmetry at infinity, we can ask for the phase space function which generates the canonical transformation corresponding to the rotation generated by . When the constraints are satisfied, as usual the value of this generating function is given by just surface terms. The term at infinity provides the total angular momentum and, as in Section 4.1.2, it is natural to interpret the surface term at as the -angular momentum of . This term can be expressed in terms of the Cauchy data on as

where is the unit normal to within . However, since the right side involves the extrinsic curvature of , in general the value of the integral is sensitive to the choice of . Hence, the notion of the -angular momentum associated with an arbitrary cross-section is ambiguous. This ambiguity disappears if is divergence-free on . In particular, in this case, one has . Thus, although the balance law (42) holds for more general vector fields , it is robust only when is divergence-free on . (These considerations shed some light on the interpretation of the field in the area balance law (25). For, the form of the right side of Equation (43) implies that the field vanishes identically on if and only if vanishes for every divergence-free on . In particular then, if the horizon has non-zero angular momentum, the -contribution to the energy flux can not vanish.)Finally, for to be interpreted as ‘the’ angular momentum, has to be a symmetry. An obvious possibility is that it be a Killing field of on . A more general scenario is discussed in Section 8.

To obtain the area balance law, in Section 3.2 we restricted ourselves to vector fields , i.e., to lapse functions and shifts . We were then led to a conservation law for the Hawking mass. In the spherically symmetric context, the Hawking mass can be taken to be the physical mass of the horizon. However, as the Kerr space-time already illustrates, in presence of rotation this interpretation is physically incorrect. Therefore, although Equation (25) continues to dictate the dynamics of the Hawking mass even in presence of rotation, a more general procedure is needed to obtain physically interesting conservation laws in this case. In the case of WIHs, the first law incorporating rotations required us to consider suitable linear combinations of and the rotational symmetry field on the horizon. In the same spirit, on DHs, one has to consider more general vector fields than , i.e., more general choices of lapses and shifts.

As on WIHs, one first restricts oneself to situations in which the metric on admits a Killing field so that can be unambiguously interpreted as the angular momentum associated with each . In the case of a WIH, was given by and the freedom was in the choice of constants and . On a DH, one must allow the corresponding coefficients to be ‘time-dependent’. The simplest generalization is to choose, in place of , vector fields

where is an arbitrary function of , and the lapse is given by for any function of . Note that one is free to rescale and by functions of so that on each cross-section (‘instant of time’) one has the same rescaling freedom as on a WIH. One can consider even more general lapse-shift pairs to allow, e.g., for differential rotation (see [30]).Using in place of , one obtains the following generalization of the area balance equation [30]:

Note that there is one balance equation for every vector field of the form (47); as in Section 4.1, we have an infinite number of relations, now ensured by the constraint part of Einstein’s equations.The right side of Equation (48) can be naturally interpreted as the flux of the ‘energy’ associated with the vector field across . Hence, we can rewrite the equation as

If and are only infinitesimally separated, this integral equation reduces to the differential condition Thus, the infinitesimal form of Equation (48) is a familiar first law, provided is identified as an effective surface gravity on the cross-section . This identification can be motivated as follows. First, on a spherically symmetric DH, it is natural to choose . Then the surface gravity reduces to , just as one would hope from one’s experience with the Schwarzschild metric and more generally with static but possibly distorted horizons (See Appendix A of [13]). Under the change , we have , which is the natural generalization of the transformation property of surface gravity of WIHs under the change . Finally, can also be regarded as a 2-sphere average of a geometrically defined surface gravity associated with certain vector fields on [53, 30]; hence the adjective ‘effective’.To summarize, Equation (48) represents an integral generalization of the first law of mechanics of weakly isolated horizons to dynamical situations in which the horizon is permitted to make a transition from a given state to one far away, not just nearby. The left side represents the flux of the energy associated with the vector field , analogous to the flux of Bondi energy across a portion of null infinity. A natural question therefore arises: Can one integrate this flux to obtain an energy which depends only on fields defined locally on the cross-section, as is possible at null infinity? As discussed in the next section, the answer is in the affirmative and the procedure leads to a canonical notion of horizon mass.

In general relativity, the notion of energy always refers to a vector field. On DHs, the vector field is . Therefore, to obtain an unambiguous notion of horizon mass, we need to make a canonical choice of , i.e., of functions and on . As we saw in Section 4.1.3, on WIHs of 4-dimensional Einstein–Maxwell theory, the pair suffices to pick a canonical time translation field on . The associated horizon energy is then interpreted as the mass . This suggests that the pair () be similarly used to make canonical choices and on . Thanks to the black hole uniqueness theorems of the 4-dimensional Einstein–Maxwell theory, this strategy is again viable.

Recall that the horizon surface gravity and the horizon angular velocity in a Kerr solution can be expressed as a function only of the horizon radius and angular momentum :

Given a cross section of , the idea is to consider the unique Kerr solution in which the horizon area is given by and angular momentum by , and assign to effective surface gravity and angular velocity through Repeating this procedure on every cross-section, one obtains functions and on , since is a function of alone. The definition of the effective surface gravity then determines a function of and hence uniquely. Thus, using Equation (51), one can select a canonical vector field and Equation (49) then provides a canonical balance law: The key question is whether this equation is integrable, i.e., if for some which depends locally on fields defined on . The answer is in the affirmative. Furthermore, the expression of is remarkably simple and is identified with the horizon mass: Thus, on any cross-section , is just the mass of the Kerr space-time which has horizon area and angular momentum : As far as the mass is concerned, one can regard the DH as an evolution through ‘a sequence of Kerr horizons’. The non-triviality of the result lies in the fact that, although this definition of mass is so ‘elementary’, thanks to the balance law (48) it obeys a Bondi-type flux formula, for a specific vector field , where each term on the right has a well-defined physical meaning. Thus, DHs admit a locally-defined notion of mass and an associated, canonical conservation law (56). The availability of a mass formula also provides a canonical integral version of the first law through Equation (48): The infinitesimal version of this equation yields the familiar first law .On weakly isolated as well as dynamical horizons, area and angular momentum arise as the fundamental quantities and mass is expressed in terms of them. The fact that the horizon mass is the same function of and in both dynamical and equilibrium situations is extremely convenient for applications to numerical relativity [33]. Conceptually, this simplicity is a direct consequence of the first law and the non-triviality lies in the existence of a balance equation (48), which makes it possible to integrate the first law.

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