# Trapped surfaces, horizons and exact solutions in higher dimensions

###### Abstract

A very simple criterion to ascertain if -surfaces are trapped in arbitrary -dimensional Lorentzian manifolds is given. The result is purely geometric, independent of the particular gravitational theory, of any field equations or of any other conditions. Many physical applications arise, a few shown here: a definition of general horizon, which reduces to the standard one in black holes/rings and other known cases; the classification of solutions with a -dimensional abelian group of motions and the invariance of the trapping under simple dimensional reductions of the Kaluza-Klein/string/M-theory type. Finally, a stronger result involving closed trapped surfaces is presented. It provides in particular a simple sufficient condition for their absence.

PACS Numbers: 04.50.+h, 04.20.Cv, 04.20.Jb, 02.40.Ky

In 1965 Penrose [1] introduced in General Relativity (GR) the concept of closed trapped surface, which was crucial for the development of the singularity theorems and the study of gravitational collapse, black holes, cosmological expansion and several types of horizons, see e.g. [2, 3]. Trapped surfaces (closed or not) are 2-dimensional imbedded spatial surfaces such that any portion of them has, at least initially, a decreasing area along any future evolution direction. The term “closed” is used if the surfaces are compact without boundary [1, 2, 3].

This concept carries over to general Lorentzian manifolds of any dimension [4]. To fix ideas and notation, let be a -dimensional surface with intrinsic coordinates () imbedded into the spacetime by the parametric equations

(1) |

is alternatively locally defined by two independent relations and . The tangent vectors of are

so that the first fundamental form of in reads

(2) |

which gives the scalar products of the in . Assume that is positive definite so that is spacelike. Then, the two linearly independent normal one-forms to can be chosen to be null and future directed everywhere on , so they satisfy

(3) |

where the last equality is a condition of normalization. Obviously, there still remains the freedom

(4) |

where is a positive function defined on .

The two null (future) second fundamental forms of are given by

(5) |

and their traces are

(6) |

where is the contravariant metric on : . The scalar defining the trapping of is then

(7) |

where is the mean curvature vector of [4]. Clearly, and are invariant under (4). is said to be trapped (respectively marginally trapped, absolutely non-trapped) if is positive, (resp. zero, negative) everywhere on . is called untrapped otherwise. See [2] and section 4 in [3] for details and examples. Notice that is trapped (resp. absolutely non-trapped) when is timelike (resp. spacelike). If is null at a point, then at least one of the traces vanishes there, so that a necessary condition for to be marginally trapped is that be null.

The meaning of the trapping is simple: the traces (6) are in fact the expansions of the two families of null geodesics emerging orthogonally from , which are tangent to at [2, 3]. Thus, is trapped if both null geodesics families are converging, or diverging, all over .

In this letter, a very simple way to check the trapping of surfaces is found, and thereby a definition of horizon will also arise naturally. Without loss of generality, the family of -dimensional spacelike surfaces can be described by , with , where are arbitrary constants and are local coordinates in . The line-element can be written as

(8) |

where and . There remains the freedom

(9) |

keeping the form (8) and the chosen family of surfaces. Using coordinate conditions one can try to achieve or other similar simplifications, which are in fact useful in many applications, but I prefer to keep the full generality.

Let us calculate the scalar of (7) for the surfaces . Clearly, the imbedding (1) for these surfaces is given locally by

and therefore the first fundamental form (2) for each is . The null normals to each satisfying (3) can be chosen as

(10) |

( is not necessarily the inverse of !). Now the calculation of the in (5) is straightforward:

where are the Christoffel symbols. From its definition so that, by setting

(11) |

and using , the two traces (6) can be obtained ()

(12) |

Hence, the mean curvature one-form reads

(13) |

where is the divergence operator on vectors at each , so that the scalar (7) for each is finally

(14) |

(13-14) are the desired formulae, which are invariant under changes of type (4) and (9). Observe that one only needs to compute the norm of as if it were a one-form in the 2-dimensional metric . The function , which from (11) gives the canonical -volume element of the surfaces (their area in ), arises as a fundamental object. As (13) shows, has a pure divergence term in general. However, as we are going to see presently, in many situations , in which case and only the normal variation of volume is relevant. Let us stress that (13-14) are purely geometric, independent of any matter contents, of energy or causality conditions [2, 3], and of any field equations. They hold in general dimension , including, in particular, the case of GR for .

In general, will change its causal character at different regions. The hypersurface(s) of separation , defined locally by the vanishing of , is a fundamental place in that I call the -horizon. This contains (i) the regions with marginally trapped , and (ii) the parts of each where one of the traces vanishes. coincides in many cases with the classical horizons, as shown in the examples that follow.

Many interesting applications can be derived from (13-14). Let us start with a simple illustrative example, the Kerr metric in Boyer-Lindquist coordinates (notation as in [2]). The case of physical interest arises for , . It is immediate to obtain , and

so , , and using (14) one easily derives (for ) , with . This is the standard result, which identifies the classical event and Cauchy horizons at as well as the closed trapped surfaces at .

Let us consider now the general spherically symmetric line-element in arbitrary

(15) |

where is the round metric on the -sphere and . Here , and is the classical apparent horizon [2, 3], which in particular becomes an event/Cauchy horizon in symmetric cases. The former case includes -dimensional Robertson-Walker cosmologies, and the latter the Reissner-Nordström-Tangherlini black holes [5], among many others. A -generalization of the standard “mass function” in spherical symmetry (see in GR, e.g., [6]) arises

so that for trapped -spheres , which in this case are obviously closed. This agrees (up to a constant -volume factor) with [7].

For more up-to-date matters, let us apply the above results to the 5-dimensional rotating black rings/holes recently presented in [8] (containing a subset of the rotating black holes in [9]). Using the notation in [8] for the metric appearing in their formula (13), the physically relevant case arises for , . By simple inspection one reads off

and a very simple calculation gives (ergo once again) and, for the scalar (14)

Recalling that one gets except at , where . This is due to the vanishing of at the 2-surfaces . One can check that is located at . Thus, there are closed trapped 3-surfaces for some ( changes sign at ), while they are non-trapped for . The horizon formed by closed marginally trapped surfaces is located at , which is the event horizon described in [8]. In general, .

Another interesting application arises from the “generalized Weyl solutions” constructed recently in [10]. The main aim in [10] was to obtain the -generalization of the static and axisymmetric solutions of vacuum Einstein’s equations. However, many other solutions not of Weyl-type were implicitly, maybe inadvertently, found. The general metric of [10] (for the “non-Weyl” case characterized by having real coordinates in the notation of [10]) can be written in the form (8) by putting , diag, and letting and to depend only on . As proved with a particular coordinate choice in [10], the Ricci-flat condition for (8) implies then that satisfies (; is covariant derivative for )

(16) |

These two expressions are conformally invariant with respect to . The first is simply the wave equation in the 2-metric for , easily solvable in appropriate coordinates. The second relation (16) is identical with the equation in , so that in what follows one could always write down for each of the the widely known solutions found in GR. Notice, though, that the proper choice of coordinates depends on the particular physical situation to be tackled. For instance, a simple possibility would be

(17) |

However, from the previous analysis, this immediately implies that
, so that all the surfaces const. are
absolutely non-trapped. ^{3}^{3}3Whether or not these surfaces are closed
is an open question at this stage, depending on the specific topology of
the coordinates .
In other words, the choice (17) is adequate only
for the regions of the spacetime with absolutely
non-trapped , and without -horizon .
Analogous cases are given for instance by .
These situations are appropriate to describe cylindrical or plane symmetric
spacetimes, providing -generalizations of this type of solutions in GR.
This case is mentioned in [10]. There are, however,
other physically inequivalent situations depending on the causal
character of . These are essentially the following (keeping always
the form of in (17) for simplicity):

1. . Now all surfaces are trapped and again there is no . This case describes cosmological solutions, as for instance the -dimensional Kasner metric [11], given by with . Analogous cases are given by .

2. (or ). In this case and all the surfaces are marginally trapped. This kind of metrics include the plane waves subset of the “pp-waves” (see e.g. [12, 13, 14, 15] and references therein), although in they are surprisingly richer as shown in Appendix B of [10]. Still, there is no and are generically non-closed. The case with =const. is included here.

3. with periodic. These cases allow for topologies , , etc in the -part of , and are -generalizations of the Gowdy models [16], including some Robertson-Walker cosmologies. In this case there is a non-trivial -horizon , with two connected components, which splits the spacetime into 4 regions, two of them with trapped surfaces (which may be closed), the other two without them. A similar but open-universe case arises by setting .

4. Of course, one can use the general solution , with arbitrary functions . This is specially appropriate, with adequate choices of , to describe the collision of plane waves (see e.g. [13] and references therein for the GR case; -generalizations were given in [15, 17]). The standard procedure to build the colliding-wave spacetime is to replace and by , where is the Heaviside step function. Then, the two regions with are two plane waves, and the two zones with correspond to their interaction region and the flat background.

From the above one can derive yet another application. As is known, the previous vacuum solutions can be seen as Kaluza-Klein, or string/M-theory spacetimes which under dimensional reduction become 4-dimensional spacetimes with a number of scalar fields. The scalar fields are given by a subset of the or appropriate linear combinations of them, see e.g [18, 19]. Actually, there is a (apparently overlooked) one-to-one corespondence between the solutions found in [10] and the solutions generated by the technique explained in section 2 of [19], the of the former corresponding to the of the latter, as can be easily proved. One of the simplest dimensional reductions [18, 19] starts with a line-element of type

where is a 4-dimensional line-element and are coordinates on a -torus. As is clear, the physically observable -surfaces are those reduced to 2-surfaces in . For these, and become simply ()

so that the trapping properties of the surfaces remain unchanged whether they are seen as 2-surfaces in or as -surfaces in full . Hence, the -horizon is lifted (or reduced!) from 4 to ( to 4) dimensions. The considered surfaces are closed in dimensions if and only if they are closed in .

Let us come back to the theoretical approach. The results can be strengthened to the case of closed trapped surfaces. Take any closed spacelike -surface and adapt the coordinates such that where is timelike everywhere (apart from this is arbitrary). This can be done in many different ways. The imbedding (1) for is given by

(18) |

where are intrinsic coordinates on . As is compact without boundary, must reach their maximum and minimum somewhere on . From (18) it follows that so that, at any point where there is an extreme of (), must have a critical point (). This implies, firstly, that the two future-directed null normals of at are given by of (10), and secondly, that (so that the imbedding (18) has rank there.) A way to visualize this is that must be bi-tangent to some (here the assumption that the surfaces do not intersect is needed, so that are differentiable.) Consequently, we can choose in a neighbourhood of such that at . A straightforward calculation for the traces (6) of leads now to

(19) |

where are given in (12) and is the contravariant first fundamental form of . The analysis of the second term in the righthand side of (19) can be done as follows. Due to (18)

and given that the gradient of is timelike everywhere, it follows that is spacelike with respect to . Hence, as the inverse of is , both must have opposite signs.

Some interesting conclusions can be derived. Assume that is spacelike in a region, then one of the is positive and the other is negative for all values of in that region. Letting (say), then at the maximum or the minimum of one can always show that

and thus the surface is untrapped. In fact, from the above reasoning follows the sufficiency that neither of the changes sign. Thus, there are no closed trapped surfaces at the hypersurfaces in any region where the are marginally trapped or absolutely non-trapped.

The last result has many applications too. As a simple but powerful one, consider once again the spherically symmetric line-element (15) and assume that is non-timelike everywhere. Then, the -spheres are either absolutely non-, or marginally, trapped everywhere. Besides, this implies that there cannot be any closed trapped surface in at all. For, due to the structure of the manifold, any closed must be osculating to some -sphere somewhere, and at this point their corresponding ’s of (14) coincide, proving that cannot be trapped there. In particular, globally static cases have non-timelike everywhere, hence they cannot contain closed trapped surfaces. These include flat spacetime, or Einstein’s and anti-de Sitter’s universes, for example. Of course, this is a known result, but the proof was rather indirect: if there were a closed trapped surface, the spacetime would be geodesically incomplete [1, 2, 3], which is not. It must be noted that any of the mentioned spacetimes, including flat one, certainly contains trapped surfaces (non-compact!, see [3] for examples), so that the previous result is not obvious in principle.

All in all, the significance and applicability of (13-14), which are wholly general, extremely simple, and easily computable, seems worth exploring in detail in the many different gravitational theories. Their potential applications seem to be very many.

Acknowledgement

I thank Roberto Emparan, Alex Feinstein, Marc Mars, and Raül Vera for their comments and suggestions.

References

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