The physics of isolated horizons

Daniel Sudarsky, ICN-UNAM, Mexico
sudarsky@nuclecu.unam.mx

In the last couple of years there has been substantial progress in various paths toward the elucidation of the deep relationship that emerges in the study of the classical dynamics of black holes, the behavior of quantum fields in background black hole spacetimes, and the ordinary laws of thermodynamics. One of the most successful is the String Theory program which has produced a detailed evaluation of the statistical mechanical entropy of some extremal and near extremal stationary black holes, through the relation of these objects with a certain class of states in the weak gravity sector of the theory. This last feature is what makes the approach somehow unsatisfying to some researchers approaching the question from the gravity point of view, who would like to know, for example, where do the degrees of freedom that account for the black hole entropy reside? i.e., the horizon's surface? its exterior? its interior?.

The second program that has met recently with substantial success is the nonperturbative quantum geometry program also known as "loop quantum gravity" [1].The first step has been to both, generalize and properly define the sector of the theory that is going to be treated. In doing so, Ashtekar and his colleagues [3][4], were guided by the need to start with a well defined action that would be differentiable in the sector under consideration. This leads to the specialization of the notion of Trapping Horizons [2] of Hayward's to that of Isolated Horizons [3] . Physically the idea is to represent "horizons in internal equilibrium and decoupled from what is outside". Essentially, an isolated horizon is defined to be a null 3 surface $\Delta$, with topology S² x R, with nonexpanding null tangent vector field l^a which is also a Killing field for the induced metric on $\Delta$, foliated by a preferred set of 2-spheres $\lbrace S \rbrace$ that are marginally trapped, and such that the induced metric on each S is spherically symmetric. The definition includes also the requirement that the horizon be nonrotating which demands among other things that the second null vector field that is normal to the 2-surfaces S of the isolated horizon n^a, be shear free and have a spherically symmetric expansion. Moreover, in defining the class of spacetimes containing isolated horizons one requires that the field equations be satisfied on the horizon, (not so elsewhere in the spacetime).

Here, we must point out that the spacetimes themselves are not assumed to have any Killing field, not even in a neighborhood of the horizon, and as such the class is extremely large, in contrast, say, with the 3 parameter class of stationary black hole solutions of Einstein Maxwell theory, as they include for example black hole spacetimes with nonstationary matter or gravitational waves in the exterior (in these cases the horizon will be isolated for as long as those nonstationary components have not crossed the horizon). Moreover, given that the definition of isolated horizon is semilocal and does not rely, for example, in the existence of asymptotic null infinity, the class of spacetimes containing isolated horizons is not limited to the asymptotically flat case and includes for example cosmological examples such as de Sitter spacetime. The crucial point of the definition is that it is possible to add a surface term to the usual bulk action of general relativity, possibly coupled with suitable matter fields like Maxwell and scalar fields such that the action is differentiable within the corresponding class of spacetimes. This is analogous to the usual addition of a surface term to the bulk action of general relativity on manifolds with boundary formulated in terms of the spacetime metric in such a way that it becomes differentiable on the class of spacetimes with a fixed value of the metric on the manifold boundary. This is a very important, and nontrivial point for the quantization program, because one needs to start with a classical action and configuration space from which one can extract not only the equations of motion but also the symplectic form. This can be achieved through a well defined procedure, once such a differentiable action is provided [5]. The surface term that must be added when the spacetimes under consideration are taken to have an isolated horizon as one of its boundaries and when the gravitational variables are taken to be the soldering form and the spin connection, turns out to be the Chern Simons action for a U(1) connection on $\Delta$. The term itself depends on the value of the horizon area, thus the action is appropriate for a class of spacetimes with fixed horizon area A.

Obviously, if the class of spacetimes under consideration allows other boundaries, one must add the corresponding surface terms to ensure differentiability of the action within the class. The fact that such a differentiable action exists seems to depend completely on the choice of the physical boundary conditions and not so much on the choice of variables, as long as the action is a first order action as in the Palatini formulation.

Equipped with the above structure the procedure to get a Hamiltonian Formulation is straight forward, i.e., consider a foliation of spacetime (which is assumed to have the topology $\Sigma \times R$), introduce lapse and shift and identify the canonical variables, which in this case include, in addition to the ordinary Hamiltonian variables of bulk gravitational sector and possibly the matter fields, the projection of the U(1) connection on the intersection on the isolated horizon with the hypersurface $\Sigma$. Thus one has an adequate formulation corresponding to a phase space $\Gamma_A$ associated with configurations with a fixed value A for the area of the isolated horizon.

The stage is set to look at the "thermodynamics" of these isolated horizons. The first step is to define a notion of surface gravity which at first sight seems to be straight forward given the existence of a null Killing field for the metric of the isolated horizon, however one immediately faces the problem of choice of normalization for this vector field. In the case of stationary black holes, this task is accomplished through the normalization of the Killing fields at asymptotic infinity, and thus, the same strategy is unavailable in the isolated horizon case, because no such Killing field is in general available for the whole spacetime (in general, there is not even an asymptotically flat boundary). The problem is solved by fixing the expansion n^a to coincide with the value it takes in Reissner Nordstrom, and then fixing the normalization of l^a by l^a n_a =-1. It is noteworthy fact that such a simple recipe exists which results in the correct value for the surface gravity of the static black holes in Einstein Maxwell Dilaton Theory.

Not so surprisingly, given the high degree of symmetry of the horizon itself, the surface gravity thus defined turns out to be constant on the isolated horizon. Therefore, the zeroth law of "thermodynamics" of isolated horizons holds. Nevertheless, we must note that the identification of the surface gravity of general isolated horizons with a physical temperature is not clear since there is so far no analog to the analysis that established the phenomena of Hawking radiation.

The next step is to define a notion of mass associated with the isolated horizon, which given the general absence of an asymptotically flat boundary can not be taken to be the ADM mass, and, moreover, even when there is such a boundary the fact that there is general matter and gravitational waves in the exterior spacetime the ADM mass would depend on those fields and not only on the isolated horizon itself. Notably the answer lies in the construction of an appropriate hamiltonian, i.e. one that would give the correct equations of motion upon considerations of arbitrary variations within the phase space (i.e. variations that not necessarily vanished at the boundaries and, in particular, at the isolated horizon). Moreover, it will be necessary to consider a new phase space $\Gamma'$ constructed by taking the union of the phase spaces of isolated horizons for all possible values of the horizon area. This requires the addition of a surface term associated with the isolated horizon boundary, and a choice of lapse and shift corresponding to l/a at the isolated horizon boundary, that is completely analogous to the usual addition of the ADM mass term in association with asymptotically flat boundaries and evolution with a choice of lapse and shift that correspond to a time translation at infinity. The fact that such a boundary term making the Hamiltonian differentiable exists is highly nontrivial, in particular, no such term is known to exist in the case of general internal boundaries with general choices of lapse and shift. This boundary term in the Hamiltonian is naturally identified with the mass of the isolated horizon and coincides with the ADM mass in the case of static black holes in Einstein Maxwel Dilaton theory.

One is then in the position of considering the first law of thermodynamics of Isolated Horizons. A straight forward calculation yields

$$\delta M_\Delta = {\kappa \over {8 \pi}} \delta A + work terms$$

Thus establishing the validity of the first law. We note that the physical process version of the first law is not fully satisfactory because, strictly speaking, the intermediate stages of the process need not contain isolated horizons, and, therefore, do not correspond to points in $\Gamma'$ Related concerns can be raised about the usual analysis, as well. We note, for example, that through a physical process the ADM mass can not change, so, in this last respect, the Isolated Horizon approach seems more satisfactory.

There is, at present, no analog to the second law for isolated horizons. This is due in part to a problem similar to that mentioned above, namely the fact that in the definition of the notion of isolated horizons one leaves no room for a situation in which the area of the horizon changes. Furthermore, in this case the problem can not be sidestepped even through an approximation because the second law is supposed to have a validity that goes quite beyond the quasistationary regime.

Finally, the program turns out to be very successful in evaluating the statistical mechanical value of the entropy of an isolated horizon. This part starts with the quantization of the appropriate sector, namely the theory associated with the phase space with the standard Ashtekar bulk variables on an hypersurface $\Sigma$ with boundary S corresponding to the isolated horizon, together with the U(1) connection of the Chern Simons theory on the boundary. The theory is as usual, subject to the constraints which in this case involve the not only the Hamiltonian, diffeomorphism and Gauss constraints, but an additional one, inherited from the conditions defining the isolated horizon, that links the behavior on the U(1) connection on S with that of the soldering form in the bulk, evaluated at S. Having constructed the quantum version of the sector of isolated horizons one looks at the states corresponding to eigenvalues of the area operator for the boundary S (the area of the isolated horizon) lying within the range A-l_P² and A + l_P². Then, one takes the maximal entropy density matrix made out of these states, namely the equally weighed totally uncorrelated density matrix constructed from these states, and construct the density matrix describing the horizon degrees of freedom by tracing over the bulk degrees of freedom. Then, the evaluation of the entropy through the standard formula $-tr \rho \ln \rho$ yields, in the case of a large enough horizon area, the result to lowest order is

$$S = { \ln 2 \over {4 \pi \sqrt 3 l_P^2}} \gamma A$$

Thus, by selecting the value $\gamma = \ln 2 / (\pi \sqrt 3 )$ for the Immirzi parameter $\gamma$, (which amounts to selecting one among a continuous choice of unitarily unequivalent quantum theories corresponding to the same classical theory), the standard result S = A/4l_P² is obtained. It is worth to point out that this choice can be made only once, and that the number of different situations that must be accounted by such a choice is infinite so it is a highly nontrivial fact that such a choice exits. For example, it is conceivable that say the choice needed in the case of Reissner Nordstrom black holes would have been different than the choice needed for Schwarzschild black holes.

The program has therefore met with tremendous success so far, and the task of generalizing the setting to account also for rotating black holes is currently under way. More into the future, there is the hope of eventually treating fully dynamical horizons, and of replacing the effective analysis described above with a more fundamental one, in which, starting with the full quantum theory one could single out the appropriate sector of states and carry out the appropriate counting for the evaluation from first principles of the horizon entropy. We look forward to these and other developments resulting from this exciting program, as well as for new insights from other sources, in the hope that eventually we would be in a position to treat even more vexing questions such as those related to the ultimate fate of an evaporating black hole and the issue of information loss.

References:

[1] A. Ashtekar, Quantum Mechanics of Geometry, The Narlikar Festschrift, ed. N. Dadhich and A. Kemhavi 1999, gr-qc/990123. C. Rovelli, "Loop Quantum Gravity", Living reviews in Relativity, No. 1998-1. gr-qc/9710008. Ashtekar, Rovelli, Smolin, "Weaving a classical geometry with quantum threads", Phys.Rev.Lett.69:237-240,1992

[2] S. Hayward, General laws of black hole dynamics, Phys. Rev. D 49, 6467 (1994).

[3]A. Ashtekar, A. Corichi and K. Krasnov, Isolated Horizons: The classical phase space, Adv. Theor. Math. Phys., in press, gr-qc/9905089. A Ashtekar, C. Beetle and S. Fairhurst, Mechanics of Isolated Horizons gr-qc/9907068

[4] A Ashtekar, J. Baez, A. Corichi and K Krasnov, Quantum Geometry and Black Hole Entropy, Phys Rev. Lett 80, 904 (1998).

[5] See for instance Sec. II in J. Lee and R. Wald, Local Symmetries and Constraints, J. Math Phys. 31, 725, (1990).