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8 Outlook

In the last six sections, we summarized the isolated and dynamical horizon frameworks and their applications. These provide a quasi-local and more physical paradigm for describing black holes both in the equilibrium and dynamical regimes. One of the most pleasing aspects of the paradigm is that it provides a unified approach to a variety of problems involving black holes, ranging from entropy calculations in quantum gravity, to analytical issues related to numerical simulations, to properties of hairy black holes, to laws of black hole mechanics. More importantly, as summarized in Section 1, these frameworks enable one to significantly extend the known results based on Killing and event horizons, and provide brand new laws in the dynamical regime.

In this section, we will discuss some of the open issues whose resolution would lead to significant progress in various areas.

Isolated horizons
This is the best understood part of the new paradigm. Nonetheless, several important issues still remain. We will illustrate these with a few examples:

Black hole mechanics
Throughout, we assumed that the space-time metric is k C (with k ≥ 3) and the topology of Δ is S2 × R. These assumptions rule out, by fiat, the presence of a NUT charge. To incorporate a non-zero NUT charge in black hole mechanics, one must either allow Δ to be topologically S3, or allow for ‘wire singularities’ in the rotation 1-form ωa on Δ. The zeroth law goes through in these more general situations. What about the first law? Arguments based on Euclidean methods [11414796] show that entropy is no longer given by the horizon area but there is also contribution due to ‘Misner strings’. However, to our knowledge, a systematic derivation in the Lorentzian regime is not yet available. Such a derivation would provide a better understanding of the physical origin of the extra terms. The covariant phase space methods used in the isolated horizon framework should be applicable in this case.

Application to numerical relativity
We saw in Section 5.4 that, in the IH framework, one can introduce an approximate analog of future null infinity ℐ+ and invariant coordinate systems and tetrads in its neighborhood. With this structure, it is feasible to extract waveforms and energy fluxes in a reliable manner within the standard 3 + 1 Cauchy evolution of numerical relativity, without having to do a Cauchy characteristic matching or use conformal field equations. The challenge here is to develop, on the approximate ℐ+, the analog of the basics of the Bondi framework [52Jump To The Next Citation Point32185] at null infinity.

Colored black holes
As discussed in Section 6, black hole uniqueness theorems of the Einstein–Maxwell theory fail once non-Abelian fields are included. For example, there are black hole solutions to the Einstein–Yang–Mills equations with non-trivial Yang–Mills fields whose only non-zero charge at infinity is the ADM mass. Thus, from the perspective of infinity, they are indistinguishable from the Schwarzschild solution. However, their horizon properties are quite different from those of the Schwarzschild horizon. This example suggests that perhaps the uniqueness theorems fail because of the insistence on evaluating charges at infinity. Corichi, Nucamendi, and Sudarsky [74] have conjectured that the uniqueness theorems could be restored if they are formulated in terms of all the relevant horizon charges. This is a fascinating idea and it has been pursued numerically. However, care is needed because the list of all relevant charges may not be obvious a priori. For example, in the case of static but not necessarily spherical Yang–Mills black holes, the conjecture seemed to fail [133] until one realized that, in addition to the standard Yang–Mills charges at the horizon, one must also include a topological, ‘winding charge’ in the list [73]. Once this charge is included, uniqueness is restored not only in the static sector of the Einstein–Yang–Mills theory, but also when Higgs fields [109] and dilatons [132] are included. The existence of these semi-numerical proofs suggests that it should be possible to establish uniqueness completely analytically. A second set of problems involves the surprising relations, e.g., between the soliton masses and horizon properties of colored black holes, obtained using isolated horizons. Extensions of these results to situations with non-zero angular momentum should be possible and may well provide yet new insights.

Quantum black holes
In the approach based on isolated horizons, the microscopic degrees of freedom responsible for the statistical mechanical entropy of black holes are directly related to the quantum geometry of horizons. Therefore, their relation to the curved space-time geometry is clearer than in, say, the string theory calculations based on D-branes. Therefore, one can now attempt to calculate the Hawking radiation directly in the physical space-time as a process in which quanta of area are converted to quanta of matter. However, such a calculation is yet to be undertaken. A direct approach requires quantum field theory (of matter fields) on a quantum geometry state which is approximated by the classical black hole space-time. Elements of this theory are now in place. The concrete open problem is the calculation of the absorption cross-section for quantum matter fields propagating on this ‘background state’. If this can be shown to equal the classical absorption cross-section to the leading order, it would follow [28] that the spectrum of the outgoing particles would be thermal at the Hawking temperature. Another, perhaps more fruitful, avenue is to introduce an effective model whose Hamiltonian captures the process by which quanta of horizon area are converted to quanta of matter. Both these approaches are geared to large black holes which can be regarded as being in equilibrium for the process under consideration, i.e., when a Δ ≫ ℓ2Pl. However, this approximation would fail in the Planck regime whence the approaches can not address issues related to ‘information loss’. Using ideas first developed in the context of quantum cosmology, effects of the quantum nature of geometry on the black hole singularity have recently been analyzed [16]. As in the earlier analysis of the big-bang singularity, it is found that the black hole singularity is resolved, but the classical space-time dissolves in the Planck regime. Therefore, the familiar Penrose diagrams are no longer faithful representations of the physical situation. Suppose that, evolving through what was singularity in the classical theory, the quantum state becomes semi-classical again on the ‘other side’. Then, the indications are that information would not be lost: It would be recovered on full + ℐ, although observers restricted to lie in the part of space-time which is completely determined by the data on ℐ − would see an approximate Hawking radiation. If on the other hand the evolved state on the ‘other side’ never becomes semi-classical, information would not be recovered on the available ℐ+. An outstanding open issue is which of this possibility is actually realized.

Dynamical horizons
The DH framework is less developed and the number of open issues is correspondingly higher. At least for the classical applications of the framework, these problems are more important.

Free data and multipoles
Since H is space-like, to find the fields (qab,Kab ) which constitute the DH geometry, one has to solve just the initial value equations, subject to the condition that H admits a foliation by marginally trapped surfaces. A general solution of this problem would provide the ‘free data’ on DHs. In the case when the marginally trapped surfaces are round spheres, this problem has been analyzed by Bartnik and Isenberg [36]. As noted in Section 2.2, in this case there are no DHs in absence of matter sources. In presence of matter, one can freely specify the trace K of the extrinsic curvature and the (radial component of the) momentum density a b Tabˆτ ˆr, and determine the geometry by solving a non-linear ordinary differential equation whose solutions always exist locally. It would be interesting to analyze the necessary and sufficient conditions on the free data which guarantee that global solutions exist and the DH approaches the Schwarzschild horizon asymptotically. From the point of view of numerical relativity, a more pressing challenge is to solve the constraint equations in the vacuum case, assuming only that the marginally trapped surfaces are axi-symmetric. Using the free data, as in the case of isolated horizons [23Jump To The Next Citation Point], one could define multipoles. Since Ψ 2 is again defined unambiguously, a natural starting point is to use it as the key geometrical object as in the IH case. However, just as the Bondi mass aspect acquires shear terms in presence of gravitational radiation [52], it is likely that, in the transition from isolated to dynamical horizons, Ψ2 would have to be supplemented with terms involving, e.g., σab (and perhaps also ζa). For instance, by adding a suitable combination of such terms, one may be able to relate the rate of change of the mass quadrupole moment with the flux of energy across H.

Geometric analysis
The dynamical horizon framework provides new inputs for the proof of Penrose inequalities which, when applied to time symmetric data (i.e., when the extrinsic curvature vanishes), say that the total (ADM) mass of space-time must be greater than half the radius of the apparent horizon on any Cauchy slice. This conjecture was recently proved by Bray [57], and by Huisken and Ilmamen [122]. Recently, for the non-time symmetric case, Ben–Dov has constructed an example where the apparent horizon does not satisfy this inequality [45]. This is not a contradiction with the original Penrose inequality which referred to the area of cross-sections of the event horizon, however it does show that extending the results beyond time-symmetry would be quite non-trivial. The ‘flows’ which led to the area law in Section 3 and balance equations in Section 4.2.2 may be potentially useful for this purpose. This approach could lead to an inequality relating the area of certain marginally trapped surfaces (the ones connected to future time like infinity via a dynamical horizon) to the future limit of the Bondi mass. The framework also suggests a program which could shed much light on what John Wheeler called ‘the issue of the final state’: what are the final equilibrium states of a dynamical black hole and how, in detail, is this equilibrium reached? From a mathematical perspective an important step in addressing this issue is to analyze the non-linear stability of Kerr black holes. Consider, then, a neighborhood of the initial data of the Kerr solution in an appropriate Sobolev space. One would expect the space-time resulting from evolution of this data to admit a dynamical horizon which, in the distant future, tends to an isolated horizon with geometry that is isomorphic to that of a Kerr horizon. Can one establish that this is what happens? Can one estimate the ‘rate’ with which the Kerr geometry is approached in the asymptotic future? One avenue is to first establish that the solution admits a Kerr–Schild type foliation, each leaf of which admits an apparent horizon, then show that the world tube of these apparent horizons is a DH, and finally study the decay rates of fields along this DH.

Angular momentum
As we saw in Section 5, most of the current work on IHs and DHs assumes the presence of an axial symmetry φa on the horizon. A natural question arises: Can one weaken this requirement to incorporate situations in which there is only an approximate – rather than an exact – symmetry of the horizon geometry? The answer is in the affirmative in the following sense. Recall first that the Newman–Penrose component Ψ 2 is gauge invariant on IHs and DHs. Let 2 F = |Ψ2|. (While any geometric field could be used here, F is the most natural candidate because, for IHs, Ψ2 encodes the horizon geometry.) If the horizon geometry admits a symmetry, the orbits of the symmetry field φa are the level surfaces of F. More generally, let us suppose that the level surfaces of F provide a foliation of each good cut S (minus two points) of the horizon. Then, using the procedure of section 2.1 of [23], one can introduce on S a vector field a φ0, tangential to the foliation, which has the property that it agrees with the symmetry vector field φa whenever the horizon geometry admits a symmetry. The procedure fails if the metric on S is at of a round sphere but should work generically. One can then use φa 0 to define angular momentum. On IHs this angular momentum is conserved; on DHs it satisfies the balance law of Section 4.2.2; and in terms of the initial data, angular momentum J(φ0) has the familiar form (59View Equation). IS this proposal viable for numerical simulations? In cases ready analyzed, it would be interesting to construct φa0 d compare it with the symmetry vector field obtained via Killing transport. A better test would be provided by non-axisymmetric Brill waves. A second issue associated with angular momentum is ether the Kerr inequality 2 J ≤ GM can be violated in the early stages of black hole formation or merger, particularly in a non-axisymmetric context. Equations that must hold on DHs provide no obvious obstruction [30Jump To The Next Citation Point]. Note that such an occurrence is not incompatible with the DH finally settling down to a Kerr horizon. For, there is likely to be radiation trapped between the DH and the ‘peak’ of the effective gravitational potential that could fall into the DH as time elapses, reducing its angular momentum and increasing its mass. The issue of whether a black hole can violate the Kerr inequality when it is first formed is of considerable interest to astrophysics [88]. Again, numerical simulations involving, say, Brill waves would shed considerable light on this possibility.

Black hole thermodynamics
The fact that an integral version of the first law is valid even for non-equilibrium processes, during which the horizon makes a transition from a given state to one which is far removed, has interesting thermodynamic ramifications. In non-equilibrium thermodynamical processes, in general the system does not have time to come to equilibrium, whence there is no canonical notion of its temperature. Therefore, while one can still interpret the difference E2 − E1 − (work) as the heat ΔQ absorbed by the system, in general there is no longer a clean split ΔQ = T ΔS of this term into a temperature part and a change in entropy part. If the process is such that the system remains close to equilibrium throughout the process, i.e., can be thought of as making continuous transitions between a series of equilibrium states, then the difference can be expressed as ∫ T dS, where the temperature T varies slowly during the transition. The situation on dynamical horizons is analogous. It is only when the horizon geometry is changing slowly that the effective surface gravity ¯κ of Section 4.2.2 would be a good measure of temperature, and the horizon area a good measure of entropy (see Section 5.3 of [30]). These restricted situations are nonetheless very interesting. Can the black hole entropy derivations based on counting of micro-states, such as those of [9], be extended to such DHs? In the case of event horizons one would not expect such a procedure to be meaningful because, as we saw in Equation 2.2.2, an event horizon can be formed and grow in a flat space region in anticipation of a future gravitational collapse. It is difficult to imagine how a quasi-local counting of micro-states can account for this phenomenon.

Perhaps the most surprising aspect of the current status of the theory of black holes is that so little is known about their properties in the fully dynamical and non-linear regime of general relativity. Indeed, we do not even have a fully satisfactory definition of a dynamical black hole. Traditionally, one uses event horizons. But as we discussed in detail, they have several undesirable features. First, they are defined only in space-times where one can unambiguously identify infinity. Even in these restricted contexts, event horizons are teleological, can form in a flat region of space-time and grow even though there is no flux of energy of any kind across them. When astronomers tell us that there is a black hole in the center of Milky Way, they are certainly not referring to event horizons.

Numerical simulations [46102] suggest that the outermost marginally trapped world-tubes become dynamical horizons soon after they are formed. As we saw, dynamical horizons have a number of attractive properties that overcome the limitations of event horizons: they are defined quasi-locally, can not be formed in flat space-time, and their growth is dictated by balance laws with direct physical interpretation. Physically, then, dynamical horizons satisfying the additional physical condition ℒnΘ (ℓ) < 0 (i.e., SFOTHs) appear to be good candidates to represent the surface of a black hole. But so far, our understanding of their uniqueness is rather limited. If a canonical dynamical horizon could be singled out by imposing physically reasonable conditions, one could use it as the physical representation of an evolving black hole.

A plausibility argument for the existence of a canonical dynamical horizon was given by Hayward. Note first that on physical grounds it seems natural to associate a black hole with a connected, trapped region 𝒯 in space-time (see Section 2.2.1). Hayward [116] sketched a proof that, under seemingly natural but technically strong conditions, the dynamical portion of its boundary, ∂𝒯, would be a dynamical horizon H. This H could serve as the canonical representation of the surface of an evolving black hole. However, it is not clear whether Hayward’s assumptions are not too strong. To illustrate the concern, let us consider a single black hole. Then, Hayward’s argument implies that there are no trapped surfaces outside H. On the other hand there has been a general expectation in the community that, given any point in the interior of the event horizon, there passes a (marginally) trapped surface through it (see, e.g., [85]). This would imply that the boundary of the trapped region is the event horizon which, being null, can not qualify as a dynamical horizon. However, to our knowledge, this result has not been firmly established. But it is clear that this expectation contradicts the conclusion based on Hayward’s arguments. Which of these two expectations is correct? It is surprising that such a basic issue is still unresolved. The primary reason is that very little is known about trapped and marginally trapped surfaces which fail to be spherical symmetric. Because of this, we do not know the boundary of the trapped region even in the Vaidya solution.

If it should turn out that the second expectation is correct, one would conclude that Hayward’s assumptions on the properties of the boundary of the trapped region are not met in physically interesting situations. However, this would not rule out the possibility of singling out a canonical dynamical horizon through some other conditions (as, e.g., in the Vaidya solution). But since this dynamical horizon would not be the boundary of the trapped region, one would be led to conclude that in the dynamical and fully non-linear regime, one has to give up the idea that there is a single 3-manifold that can be interpreted as the black hole surface without further qualifications. For certain questions and in certain situations, the dynamical horizon may be the appropriate concept, while for other questions and in different situations, the boundary of the trapped region (which may be the event horizon) may be more appropriate.

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