To explain the evolution of ideas and provide points of comparison, we will introduce the notion of dynamical horizons following a chronological order. Readers who are not familiar with causal structures can go directly to Definition 5 of dynamical horizons (for which a more direct motivation can be found in [30]).
As discussed in Section 1, while the notion of an event horizon has proved to be very convenient in mathematical relativity, it is too global and teleological to be directly useful in a number of physical contexts ranging from quantum gravity to numerical relativity to astrophysics. This limitation was recognized early on (see, e.g., [113], page 319) and alternate notions were introduced to capture the intuitive idea of a black hole in a quasilocal manner. In particular, to make the concept ‘local in time’, Hawking [111, 113] introduced the notions of a trapped region and an apparent horizon, both of which are associated to a spacelike 3surface representing ‘an instant of time’. Let us begin by recalling these ideas.
Hawking’s outer trapped surface is a compact, spacelike 2dimensional submanifold in such that the expansion of the outgoing null normal to is nonpositive. Hawking then defined the trapped region in a surface as the set of all points in through which there passes an outertrapped surface, lying entirely in . Finally, Hawking’s apparent horizon is the boundary of a connected component of . The idea then was to regard each apparent horizon as the instantaneous surface of a black hole. One can calculate the expansion of knowing only the intrinsic 3metric and the extrinsic curvature of . Hence, to find outer trapped surfaces and apparent horizons on , one does not need to evolve away from even locally. In this sense the notion is local to . However, this locality is achieved at the price of restricting to lie in . If we wiggle even slightly, new outer trapped surfaces can appear and older ones may disappear. In this sense, the notion is still very global. Initially, it was hoped that the laws of black hole mechanics can be extended to these apparent horizons. However, this has not been possible because the notion is so sensitive to the choice of .
To improve on this situation, in the early nineties Hayward proposed a novel modification of this framework [116]. The main idea is to free these notions from the complicated dependence on . He began with Penrose’s notion of a trapped surface. A trapped surface a la Penrose is a compact, spacelike 2dimensional submanifold of spacetime on which , where and are the two null normals to . We will focus on future trapped surfaces on which both expansions are negative. Hayward then defined a spacetime trapped region. A trapped region a la Hayward is a subset of spacetime through each point of which there passes a trapped surface. Finally, Hayward’s trapping boundary is a connected component of the boundary of an inextendible trapped region. Under certain assumptions (which appear to be natural intuitively but technically are quite strong), he was able to show that the trapping boundary is foliated by marginally trapped surfaces (MTSs), i.e., compact, spacelike 2dimensional submanifolds on which the expansion of one of the null normals, say , vanishes and that of the other, say , is everywhere nonpositive. Furthermore, is also everywhere of one sign. These general considerations led him to define a quasilocal analog of future event horizons as follows:
Definition 4: A future, outer, trapping horizon (FOTH) is a smooth 3dimensional submanifold of spacetime, foliated by closed 2manifolds , such that
In this definition, Condition 2 captures the idea that is a future horizon (i.e., of black hole rather than white hole type), and Condition 3 encodes the idea that it is ‘outer’ since infinitesimal motions along the ‘inward’ normal makes the 2surface trapped. (Condition 3 also serves to distinguish black hole type horizons from certain cosmological ones [116] which are not ruled out by Condition 2). Using the Raychaudhuri equation, it is easy to show that is either spacelike or null, being null if and only if the shear of as well as the matter flux across vanishes. Thus, when is null, it is a nonexpanding horizon introduced in Section 2.1. Intuitively, is spacelike in the dynamical region where gravitational radiation and matter fields are pouring into it and is null when it has reached equilibrium.
In truly dynamical situations, then, is expected to be spacelike. Furthermore, it turns out that most of the key results of physical interest [29, 30], such as the area increase law and generalization of black hole mechanics, do not require the condition on the sign of . It is therefore convenient to introduce a simpler and at the same time ‘tighter’ notion, that of a dynamical horizon, which is better suited to analyze how black holes grow in exact general relativity [29, 30]:
Definition 5: A smooth, threedimensional, spacelike submanifold (possibly with boundary) of spacetime is said to be a dynamical horizon (DH) if it can be foliated by a family of closed 2manifolds such that
Note first that, like FOTHs, dynamical horizons are ‘spacetime notions’, defined quasilocally. They are not defined relative to a spacelike surface as was the case with Hawking’s apparent horizons nor do they make any reference to infinity as is the case with event horizons. In particular, they are welldefined also in the spatially compact context. Being quasilocal, they are not teleological. Next, let us spell out the relation between FOTHs and DHs. A spacelike FOTH is a DH on which the additional condition holds. Similarly, a DH satisfying is a spacelike FOTH. Thus, while neither definition implies the other, the two are closely related. The advantage of Definition 5 is that it refers only to the intrinsic structure of , without any conditions on the evolution of fields in directions transverse to . Therefore, it is easier to verify in numerical simulations. More importantly, as we will see, this feature makes it natural to analyze the structure of using only the constraint (or initial value) equations on it. This analysis will lead to a wealth of information on black hole dynamics. Reciprocally, Definition 4 has the advantage that, since it permits to be spacelike or null, it is better suited to analyze the transition to equilibrium [30].
A DH which is also a FOTH will be referred to as a spacelike future outer horizon (SFOTH). To fully capture the physical notion of a dynamical black hole, one should require both sets of conditions, i.e., restrict oneself to SFOTHs. For, stationary black holes admit FOTHS and there exist spacetimes [166] which admit dynamical horizons but no trapped surfaces; neither can be regarded as containing a dynamical black hole. However, it is important to keep track of precisely which assumptions are needed to establish specific results. Most of the results reported in this review require only those conditions which are satisfied on DHs. This fact may well play a role in conceptual issues that arise while generalizing black hole thermodynamics to nonequilibrium situations^{3}.
Let us begin with the simplest examples of spacetimes admitting DHs (and SFOTHs). These are provided by the spherically symmetric solution to Einstein’s equations with a null fluid as source, the Vaidya metric [180, 138, 186]. (Further details and the inclusion of a cosmological constant are discussed in [30].) Just as the Schwarzschild–Kruskal solution provides a great deal of intuition for general static black holes, the Vaidya metric furnishes some of the much needed intuition in the dynamical regime by bringing out the key differences between the static and dynamical situations. However, one should bear in mind that both Schwarzschild and Vaidya black holes are the simplest examples and certain aspects of geometry can be much more complicated in more general situations. The 4metric of the Vaidya spacetime is given by
where is any smooth, nondecreasing function of . Thus, are the ingoing Eddington–Finkelstein coordinates. This is a solution of Einstein’s equations, the stressenergy tensor being given by where . Clearly, satisfies the dominant energy condition if , and vanishes if and only if . Of special interest to us are the cases illustrated in Figure 4: is nonzero until a certain finite retarded time, say , and then grows monotonically, either reaching an asymptotic value as tends to infinity (panel a), or, reaching this value at a finite retarded time, say , and then remaining constant (panel b). In either case, the spacetime region is flat.Let us focus our attention on the metric 2spheres, which are all given by and . It is easy to verify that the expansion of the outgoing null normal vanishes if and only if ( and) . Thus, these are the only spherically symmetric marginally trapped surfaces MTSs. On each of them, the expansion of the ingoing normal is negative. By inspection, the 3metric on the world tube of these MTSs has signature when is nonzero and if is zero. Hence, in the left panel of Figure 4 the surface is the DH . In the right panel of Figure 4 the portion of this surface is the DH , while the portion is a nonexpanding horizon. (The general issue of transition of a DH to equilibrium is briefly discussed in Section 5.) Finally, note that at these MTSs, . Hence in both cases, the DH is an SFOTH. Furthermore, in the case depicted in the right panel of Figure 4 the entire surface is a FOTH, part of which is dynamical and part null.

This simple example also illustrates some interesting features which are absent in the stationary situations. First, by making explicit choices of , one can plot the event horizon using, say, Mathematica [189] and show that they originate in the flat spacetime region , in anticipation of the null fluid that is going to fall in after . The dynamical horizon, on the other hand, originates in the curved region of spacetime, where the metric is timedependent, and steadily expands until it reaches equilibrium. Finally, as Figures 4 illustrate, the dynamical and event horizons can be well separated. Recall that in the equilibrium situation depicted by the Schwarzschild spacetime, a spherically symmetric trapped surface passes through every point in the interior of the event horizon. In the dynamical situation depicted by the Vaidya spacetime, they all lie in the interior of the DH. However, in both cases, the event horizon is the boundary of . Thus, the numerous roles played by the event horizon in equilibrium situations get split in dynamical contexts, some taken up by the DH.
What is the situation in a more general gravitational collapse? As indicated in the beginning of this section, the geometric structure can be much more subtle. Consider 3manifolds which are foliated by marginally trapped compact 2surfaces . We denote by the normal whose expansion vanishes. If the expansion of the other null normal is negative, will be called a marginally trapped tube (MTT). If the tube is spacelike, it is a dynamical horizon. If it is timelike, it will be called timelike membrane. Since future directed causal curves can traverse timelike membranes in either direction, they are not good candidates to represent surfaces of black holes; therefore they are not referred to as horizons.
In Vaidya metrics, there is precisely one MTT to which all three rotational Killing fields are tangential and this is the DH . In the Oppenheimer–Volkoff dust collapse, however, the situation is just the opposite; the unique MTT on which each MTS is spherical is timelike [181, 45]. Thus we have a timelike membrane rather than a dynamical horizon. However, in this case the metric does not satisfy the smoothness conditions spelled out at the end of Section 1 and the global timelike character of is an artifact of the lack of this smoothness. In the general perfect fluid spherical collapse, if the solution is smooth, one can show analytically that the spherical MTT is spacelike at sufficiently late times, i.e., in a neighborhood of its intersection with the event horizon [102]. For the spherical scalar field collapse, numerical simulations show that, as in the Vaidya solutions, the spherical MTT is spacelike everywhere [102]. Finally, the geometry of the numerically evolved MTTs has been examined in two types of nonspherical situations: the axisymmetric collapse of a neutron star to a Kerr black hole and in the headon collision of two nonrotating black holes [46]. In both cases, in the initial phase the MTT is neither spacelike nor timelike all the way around its crosssections . However, it quickly becomes spacelike and has a long spacelike portion which approaches the event horizon. This portion is then a dynamical horizon. There are no hard results on what would happen in general, physically interesting situations. The current expectation is that the MTT of a numerically evolved black hole spacetime which asymptotically approaches the event horizon will become spacelike rather soon after its formation. Therefore most of the ongoing detailed work focuses on this portion, although basic analytical results are available also on how the timelike membranes evolve (see Appendix A of [30]).
Even in the simplest, Vaidya example discussed above, our explicit calculations were restricted to spherically symmetric marginally trapped surfaces. Indeed, already in the case of the Schwarzschild spacetime, very little is known analytically about nonspherically symmetric marginally trapped surfaces. It is then natural to ask if the Vaidya metric admits other, nonspherical dynamical horizons which also asymptote to the nonexpanding one. Indeed, even if we restrict ourselves to the 3manifold , can we find another foliation by nonspherical, marginally trapped surfaces which endows it with another dynamical horizon structure? These considerations illustrate that in general there are two uniqueness issues that must be addressed.
First, in a general spacetime , can a spacelike 3manifold be foliated by two distinct families of marginally trapped surfaces, each endowing it with the structure of a dynamical horizon? Using the maximum principle, one can show that this is not possible [92]. Thus, if admits a dynamical horizon structure, it is unique.
Second, we can ask the following question: How many DHs can a spacetime admit? Since a spacetime may contain several distinct black holes, there may well be several distinct DHs. The relevant question is if distinct DHs can exist within each connected component of the (spacetime) trapped region. On this issue there are several technically different uniqueness results [26]. It is simplest to summarize them in terms of SFOTHs. First, if two nonintersecting SFOTHs and become tangential to the same nonexpanding horizon at a finite time (see the right panel in Figure 4), then they coincide (or one is contained in the other). Physically, a more interesting possibility, associated with the late stages of collapse or mergers, is that and become asymptotic to the event horizon. Again, they must coincide in this case. At present, one can not rule out the existence of more than one SFOTHs which asymptote to the event horizon if they intersect each other repeatedly. However, even if this were to occur, the two horizon geometries would be nontrivially constrained. In particular, none of the marginally trapped surfaces on can lie entirely to the past of .
A better control on uniqueness is perhaps the most important open issue in the basic framework for dynamical horizons and there is ongoing work to improve the existing results. Note however that all results of Sections 3 and 5, including the area increase law and the generalization of black hole mechanics, apply to all DHs (including the ‘transient ones’ which may not asymptote to the event horizon). This makes the framework much more useful in practice.
The existing results also provide some new insights for numerical relativity [26]. First, suppose that a MTT is generated by a foliation of a region of spacetime by partial Cauchy surfaces such that each MTS is the outermost MTS in . Then can not be a timelike membrane. Note however that this does not imply that is necessarily a dynamical horizon because may be partially timelike and partially spacelike on each of its marginally trapped surfaces . The requirement that be spacelike – i.e., be a dynamical horizon – would restrict the choice of the foliation of spacetime and reduce the unruly freedom in the choice of gauge conditions that numerical simulations currently face. A second result of interest to numerical relativity is the following. Let a spacetime admit a DH which asymptotes to the event horizon. Let be any partial Cauchy surface in which intersects in one of the marginally trapped surfaces, say . Then, is the outermost marginally trapped surface – i.e., apparent horizon in the numerical relativity terminology – on .
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