We review recent results about the modelling of gravitational collapse to black holes in higher dimensions. The models are constructed through the junction of two exact solutions of the Einstein field equations: an interior collapsing fluid solution and a vacuum exterior solution. The vacuum exterior solutions are either static or containing gravitational waves. We then review the global geometrical properties of the matched solutions which, besides black holes, may include the existence of naked singularities and wormholes. In the case of radiating exteriors, we show that the data at the boundary can be chosen to be, in some sense, arbitrarily close to the data for the Schwarzschild-Tangherlini solution.

Higher dimensional black holes have recently been the subject of an increasing number of research works (see [

Mathematically, black hole formation can be analysed by constructing appropriate matched spacetimes which settle down through gravitational collapse to black hole solutions. The junction (or matching) of two spacetimes requires the equality of the respective first and second fundamental forms at some matching boundary hypersurface. These conditions amount to a set of differential equations that the spacetime metric functions need to satisfy at that matching boundary. This is often not an easy problem as the metric functions are also required to satisfy the respective Einstein field equations (EFEs) on both sides of the matching surface. In what follows, we review some results about models of black hole formation with emphasis on models resulting from the spacetime matching with an exterior which is either vacuum or a cosmological background.

For a zero cosmological constant

The collapse to nonspherical black holes has been less studied. Smith and Mann [

There is also a vast literature about the formation of black holes in cosmological models. There are, essentially, three types of models which are geometrically different, namely, models with (i) junctions with no shells where two metrics are involved, for example, swiss-cheese-type models (see, e.g., [

In the context of primordial black hole (PBH) formation, the gravitational collapse due to first-order density perturbations in FLRW models of the early universe radiative phase was first studied by Zeldovich and Novikov [

In this paper, we will be mostly interested in cases which involve spacetime junctions with no shells and, in particular, we will focus on nonspherical spacetimes containing a non-zero

We also review the interesting case of a model of radiating gravitational collapse. In particular, we take the anisotropic Bizoń-Chmaj-Schmidt (BCS) solution in

Let

Given the basis

We now define a 1-form

The second fundamental forms are given by

The theory is also fully developed for the case where the matching boundary

We are interested in the particular cases for which the matching surface inherits a certain symmetry of the two space times

We note that if the exterior is a vacuum spacetime and the interior contains a fluid then the normal pressure of the fluid has to vanish at the boundary

An

Let

It is easy to see that the above metric generalises the Kottler metric to arbitrary dimensions. Nevertheless, black holes with the above metric do not immediately integrate into the usual intuition of a black hole in four dimensions. For instance, since the metric on the sections of null infinity

The metrics (

Therefore, by comparison with the four-dimensional case [

With

With

With

The solutions in the previous class with

We will fill in the solutions of the previous section with (interior) collapsing dust solutions, so that the resulting global spacetime is a model of gravitational collapse with a black hole as the end state. In order to do that, we take the following classes of generalised FLRW spacetimes.

The

We consider the matching of an interior metric (

The metric (

The matching boundary is comoving with the collapsing fluid whose dynamics is given by (

Penrose diagram for

Penrose diagram for

Penrose diagram for

If

Interestingly, [

If

The case

For

If

If

If

We consider now higher dimensional versions of the inhomogeneous Lemaître-Tolman-Bondi (LTB) solutions, generalising those of [

The

This metric has three free functions of

The metric (

Note that the matching boundary is again comoving with the collapsing inhomogeneous fluid and its dynamics is now given by (

In this section, we consider models of gravitational collapse with a gravitational wave exterior, so that the exterior metrics will be time-dependent generalisations of (

We will give data

As interior metrics, we will consider three classes of FLRW-like solutions based on Riemannian Bianchi-IX spatial metrics which are, respectively, the Eguchi-Hanson metric (with

We summarize now our main result and leave the details of the proof to the next three sections (see also [

In each case, the interior metric gives consistent data for the metric (

Eguchi and Hanson found a class of self-dual solutions to the Euclidean Einstein equations with metric given by [

The

We consider the Riemannian Taub-NUT metric with a cosmological constant (

The work summarised in this paper was done in collaboration with José Natário and Paul Tod. The author is supported by FCT Projects PTDC/MAT/108921/2008, CERN/FP/116377/2010, and Est-C/MAT/UI0013/2011 and by CMAT, Universidade do Minho, through FCT plurianual FEDER Funds “Programa Operacional Factores de Competitividade COMPETE.”