Dynamical horizons

2.2 Dynamical horizons

This section is divided into three parts. In the first, we discuss basic definitions, in the second we introduce an explicit example, and in the third we analyze the issue of uniqueness of dynamical horizons and their role in numerical relativity.

2.2.1 Definitions

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 quasi-local 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 space-like 3-surface $M$ representing ‘an instant of time’. Let us begin by recalling these ideas.

Hawking’s outer trapped surface $S$ is a compact, space-like 2-dimensional sub-manifold in $(ℳ, g ) ab$ such that the expansion $Θ (ℓ)$ of the outgoing null normal $a ℓ$ to $S$ is non-positive. Hawking then defined the trapped region $𝒯 (M )$ in a surface $M$ as the set of all points in $M$ through which there passes an outer-trapped surface, lying entirely in $M$ . Finally, Hawking’s apparent horizon $∂ 𝒯 (M )$ is the boundary of a connected component of $𝒯 (M )$ . The idea then was to regard each apparent horizon as the instantaneous surface of a black hole. One can calculate the expansion $Θ (ℓ)$ of $S$ knowing only the intrinsic 3-metric $qab$ and the extrinsic curvature $Kab$ of $M$ . Hence, to find outer trapped surfaces and apparent horizons on $M$ , one does not need to evolve $(qab,Kab )$ away from $M$ even locally. In this sense the notion is local to $M$ . However, this locality is achieved at the price of restricting $S$ to lie in $M$ . If we wiggle $M$ 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 $M$ .

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 $M$ . He began with Penrose’s notion of a trapped surface. A trapped surface $S$ a la Penrose is a compact, space-like 2-dimensional sub-manifold of space-time on which $Θ Θ > 0 (ℓ) (n)$ , where $ℓa$ and $na$ are the two null normals to $S$ . We will focus on future trapped surfaces on which both expansions are negative. Hayward then defined a space-time trapped region. A trapped region $𝒯$ a la Hayward is a subset of space-time through each point of which there passes a trapped surface. Finally, Hayward’s trapping boundary $∂𝒯$ 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, space-like 2-dimensional sub-manifolds on which the expansion of one of the null normals, say $ℓa$ , vanishes and that of the other, say $na$ , is everywhere non-positive. Furthermore, $ℒ Θ n (ℓ)$ is also everywhere of one sign. These general considerations led him to define a quasi-local analog of future event horizons as follows:

Definition 4: A future, outer, trapping horizon (FOTH) is a smooth 3-dimensional sub-manifold $H-$ of space-time, foliated by closed 2-manifolds $S --$ , such that

the expansion of one future directed null normal to the foliation, say $a ℓ$ , vanishes, $Θ (ℓ) = 0$ ;
the expansion of the other future directed null normal $na$ is negative, $Θ (n) < 0$ ; and
the directional derivative of $Θ (ℓ)$ along $a n$ is negative, $ℒn Θ(ℓ) < 0$ .

In this definition, Condition 2 captures the idea that $H-$ 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 $na$ makes the 2-surface 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 $H --$ is either space-like or null, being null if and only if the shear $σab$ of $a ℓ$ as well as the matter flux $a b Tabℓ ℓ$ across $H$ vanishes. Thus, when $H-$ is null, it is a non-expanding horizon introduced in Section 2.1. Intuitively, $H-$ is space-like 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, $H-$ is expected to be space-like. 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 $ℒnΘ (ℓ)$ . 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, three-dimensional, space-like sub-manifold (possibly with boundary) $H$ of space-time is said to be a dynamical horizon (DH) if it can be foliated by a family of closed 2-manifolds such that

on each leaf $S$ the expansion $Θ (ℓ)$ of one null normal $a ℓ$ vanishes; and
the expansion $Θ (n)$ of the other null normal $na$ is negative.

Note first that, like FOTHs, dynamical horizons are ‘space-time notions’, defined quasi-locally. They are not defined relative to a space-like 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 well-defined also in the spatially compact context. Being quasi-local, they are not teleological. Next, let us spell out the relation between FOTHs and DHs. A space-like FOTH is a DH on which the additional condition $ℒn Θ (ℓ) < 0$ holds. Similarly, a DH satisfying $ℒn Θ (ℓ) < 0$ is a space-like 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 $H$ , without any conditions on the evolution of fields in directions transverse to $H$ . 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 $H$ 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 $H-$ to be space-like 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 space-like 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 space-times [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 non-equilibrium situations³ .

2.2.2 Examples

Let us begin with the simplest examples of space-times 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 4-metric of the Vaidya space-time is given by

where $M (v)$ is any smooth, non-decreasing function of $v$ . Thus, $(v,r,θ,φ)$ are the ingoing Eddington–Finkelstein coordinates. This is a solution of Einstein’s equations, the stress-energy tensor $Tab$ being given by

where $˙ M = dM ∕dv$ . Clearly, $Tab$ satisfies the dominant energy condition if $˙ M ≥ 0$ , and vanishes if and only if $˙ M = 0$ . Of special interest to us are the cases illustrated in Figure 4

: $M (v)$ is non-zero until a certain finite retarded time, say $v = 0$ , and then grows monotonically, either reaching an asymptotic value $M0$ as $v$ tends to infinity (panel a), or, reaching this value at a finite retarded time, say $v = v0$ , and then remaining constant (panel b). In either case, the space-time region $v ≤ 0$ is flat.

Let us focus our attention on the metric 2-spheres, which are all given by $v = const.$ and $r = const.$ . It is easy to verify that the expansion of the outgoing null normal $ℓa$ vanishes if and only if ( $v = const.$ and) $r = 2GM (v)$ . Thus, these are the only spherically symmetric marginally trapped surfaces MTSs. On each of them, the expansion $Θ (n)$ of the ingoing normal $na$ is negative. By inspection, the 3-metric on the world tube $r = 2GM (v)$ of these MTSs has signature $(+, +,+ )$ when $M˙(v)$ is non-zero and $(0,+, + )$ if $M˙(v)$ is zero. Hence, in the left panel of Figure 4 the surface $r = 2GM (v)$ is the DH $H$ . In the right panel of Figure 4 the portion of this surface $v ≤ v0$ is the DH $H$ , while the portion $v ≥ v0$ is a non-expanding horizon. (The general issue of transition of a DH to equilibrium is briefly discussed in Section 5.) Finally, note that at these MTSs, $ℒn θ(ℓ) = − 2 ∕r2 < 0$ . Hence in both cases, the DH is an SFOTH. Furthermore, in the case depicted in the right panel of Figure 4 the entire surface $r = 2GM (v)$ is a FOTH, part of which is dynamical and part null.

Figure 4:

Penrose diagrams of Schwarzschild–Vaidya metrics for which the mass function $M (v )$ vanishes for $v ≤ 0$ [138 ]. The space-time metric is flat in the past of $v = 0$ (i.e., in the shaded region). In the left panel, as $v$ tends to infinity, $M˙$ vanishes and $M$ tends to a constant value $M0$ . The space-like dynamical horizon $H$ , the null event horizon $E$ , and the time-like surface $r = 2M0$ (represented by the dashed line) all meet tangentially at $+ i$ . In the right panel, for $v ≥ v0$ we have $M˙ = 0$ . Space-time in the future of $v = v0$ is isometric with a portion of the Schwarzschild space-time. The dynamical horizon $H$ and the event horizon $E$ meet tangentially at $v = v 0$ . In both figures, the event horizon originates in the shaded flat region, while the dynamical horizon exists only in the curved region.

This simple example also illustrates some interesting features which are absent in the stationary situations. First, by making explicit choices of $M (v)$ , one can plot the event horizon using, say, Mathematica [189 ] and show that they originate in the flat space-time region $v < 0$ , in anticipation of the null fluid that is going to fall in after $v = 0$ . The dynamical horizon, on the other hand, originates in the curved region of space-time, where the metric is time-dependent, 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 space-time, a spherically symmetric trapped surface passes through every point in the interior of the event horizon. In the dynamical situation depicted by the Vaidya space-time, they all lie in the interior of the DH. However, in both cases, the event horizon is the boundary of $J − (ℐ+ )$ . 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 3-manifolds $τ$ which are foliated by marginally trapped compact 2-surfaces $S$ . We denote by $ℓa$ the normal whose expansion vanishes. If the expansion of the other null normal $a n$ is negative, $τ$ will be called a marginally trapped tube (MTT). If the tube $τ$ is space-like, it is a dynamical horizon. If it is time-like, it will be called time-like membrane. Since future directed causal curves can traverse time-like 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 $H$ . In the Oppenheimer–Volkoff dust collapse, however, the situation is just the opposite; the unique MTT on which each MTS $S$ is spherical is time-like [181 , 45 ]. Thus we have a time-like 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 time-like character of $τ$ 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 space-like 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 space-like everywhere [102 ]. Finally, the geometry of the numerically evolved MTTs has been examined in two types of non-spherical situations: the axi-symmetric collapse of a neutron star to a Kerr black hole and in the head-on collision of two non-rotating black holes [46 ]. In both cases, in the initial phase the MTT is neither space-like nor time-like all the way around its cross-sections $S$ . However, it quickly becomes space-like and has a long space-like 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 space-time which asymptotically approaches the event horizon will become space-like 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 time-like membranes evolve (see Appendix A of [30 ]).

2.2.3 Uniqueness

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 space-time, very little is known analytically about non-spherically symmetric marginally trapped surfaces. It is then natural to ask if the Vaidya metric admits other, non-spherical dynamical horizons which also asymptote to the non-expanding one. Indeed, even if we restrict ourselves to the 3-manifold $r = 2GM (v)$ , can we find another foliation by non-spherical, 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 space-time $(ℳ, gab)$ , can a space-like 3-manifold $H$ 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 $H$ admits a dynamical horizon structure, it is unique.

Second, we can ask the following question: How many DHs can a space-time admit? Since a space-time 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 (space-time) 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 non-intersecting SFOTHs $H$ and $′ H$ become tangential to the same non-expanding 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 $H$ and $H ′$ 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 non-trivially constrained. In particular, none of the marginally trapped surfaces on $H$ can lie entirely to the past of $H ′$ .

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 $τ$ is generated by a foliation of a region of space-time by partial Cauchy surfaces $Mt$ such that each MTS $St$ is the outermost MTS in $Mt$ . Then $τ$ can not be a time-like membrane. Note however that this does not imply that $τ$ is necessarily a dynamical horizon because $τ$ may be partially time-like and partially space-like on each of its marginally trapped surfaces $S$ . The requirement that $τ$ be space-like – i.e., be a dynamical horizon – would restrict the choice of the foliation $M t$ of space-time 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 space-time $(ℳ, gab)$ admit a DH $H$ which asymptotes to the event horizon. Let $M0$ be any partial Cauchy surface in $(ℳ, gab)$ which intersects $H$ in one of the marginally trapped surfaces, say $S0$ . Then, $S0$ is the outermost marginally trapped surface – i.e., apparent horizon in the numerical relativity terminology – on $M0$ .