In this section we will introduce a general notion of a near-horizon geometry. This requires us to first introduce some preliminary constructions. Let be a smooth9 codimension-1 null hypersurface in a dimensional spacetime . In a neighbourhood of any such hypersurface there exists an adapted coordinate chart called Gaussian null coordinates that we now recall [179, 86].
Let be a vector field normal to whose integral curves are future-directed null geodesic generators of . In general these will be non-affinely parameterised so on we have for some function . Now let denote a smooth -dimensional spacelike submanifold of , such that each integral curve of crosses exactly once: we term a cross section of and assume such submanifolds exist. On choose arbitrary local coordinates , for , containing some point . Starting from , consider the point in a parameter value along the integral curve of . Now assign coordinates to such a point, i.e., we extend the functions into by keeping them constant along such a curve. This defines a set of coordinates in a tubular neighbourhood of the integral curves of through , such that . Since is normal to we have and on .
We now extend these coordinates into a neighbourhood of in as follows. For any point contained in the above coordinates , let be the unique past-directed null vector satisfying and . Now starting at , consider the point in an affine parameter value along the null geodesic with tangent vector . Define the coordinates of such a point in by , i.e., the functions are extended into by requiring them to be constant along such null geodesics. This provides coordinates defined in a neighbourhood of in , as required.
We extend the definitions of and into by and . By construction the integral curves of are null geodesics and hence everywhere in the neighbourhood of in in question. Furthermore, using the fact that and commute (they are coordinate vector fields), we have
These considerations show that, in a neighbourhood of in , the spacetime metric written in Gaussian null coordinates is of the form
The coordinates developed above are valid in the neighbourhood of any smooth null hypersurface . In this work we will in fact be concerned with smooth Killing horizons. These are null hypersurfaces that possess a normal that is a Killing field in . Hence we may set in the above construction. Since we deduce that in the neighbourhood of a Killing horizon , the metric can be written as Eq. (5) where the functions are all independent of the coordinate . Using the Killing property one can rewrite as on , where is now the usual surface gravity of a Killing horizon.
We may now introduce the main objects we will study in this work: degenerate Killing horizons. These are defined as Killing horizons such that the normal Killing field is tangent to affinely parameterised null geodesics on , i.e., . Therefore, , which implies that in Gaussian null coordinates . It follows that for some smooth function . Therefore, in the neighbourhood of any smooth degenerate Killing horizon the metric in Gaussian null coordinates reads
We are now ready to define the near-horizon geometry of a -dimensional spacetime containing such a degenerate horizon. Given any , consider the diffeomorphism defined by and . The metric in Gaussian null coordinates transforms where is given by Eq. (6) with the replacements , and . The near-horizon limit is then defined as the limit of . It is clear this limit always exists since all metric functions are smooth at . The resulting metric is called the near-horizon geometry and is given by
Intuitively, the near-horizon limit is a scaling limit that focuses on the spacetime near the horizon . We emphasise that the degenerate assumption is crucial for defining this limit and such a general notion of a near-horizon limit does not exist for a non-degenerate Killing horizon.
As we will see, geometric equations (such as Einstein’s equations) for a near-horizon geometry can be equivalently written as geometric equations defined purely on a -dimensional cross section manifold of the horizon. In this section we will write down general formulae relating the curvature of a near-horizon geometry to the curvature of the horizon . For convenience we will denote the dimension of by .
It is convenient to introduce a null-orthonormal frame for the near-horizon metric (7), denoted by , where , and10 The dual basis vectors are
It is worth noting that the following components of the Weyl tensor automatically vanish: and . This means that is a multiple Weyl aligned null direction and hence any near-horizon geometry is at least algebraically special of type II within the classification of . In fact, it can be checked that the null geodesic vector field has vanishing expansion, shear and twist and therefore any near-horizon geometry is a Kundt spacetime.11 Indeed, by inspection of Eq. (7) it is clear that near-horizon geometries are a subclass of the degenerate Kundt spacetimes,12 which are all algebraically special of at least type II .
Henceforth, we will drop the “hats” on all horizon quantities, so and refer to the Ricci tensor and Levi-Civita connection of on .
We will consider spacetimes that are solutions to Einstein’s equations:
An important fact is that if a spacetime containing a degenerate horizon satisfies Einstein’s equations then so does its near-horizon geometry. This is easy to see as follows. If the metric in Eq. (6) satisfies Einstein’s equations, then so will the 1-parameter family of diffeomorphic metrics for any . Hence the limiting metric , which by definition is the near-horizon geometry, must also satisfy the Einstein equations.
The near-horizon limit of the energy momentum tensor thus must also exist and takes the form8), it is then straightforward to verify that the and components of the Einstein equations for the near-horizon geometry give the following equations on the cross section : 17), (18) and the matter field equations, as follows.
The matter field equations must imply the spacetime conservation equation . This is equivalent to the following equations on :12). The first equation in (21) and the identity (13) imply that the equation is satisfied as a consequence of the equation. Finally, substituting Eqs. (17) and (18) into the identity (14), and using the second equation in (21), implies the equation. Alternatively, a tedious calculation shows that the equation follows from Eqs. (17) and (18) using the contracted Bianchi identity for Eq. (17), together with the second equation in (21).
Although the energy momentum tensor must have a near-horizon limit, it is not obvious that the matter fields themselves must. Thus, consider the full spacetime before taking the near-horizon limit. Recall that for any Killing horizon and therefore . This imposes a constraint on the matter fields. We will illustrate this for Einstein–Maxwell theory whose energy-momentum tensor is6.
It is worth remarking that the above naturally generalises to -form electrodynamics, with , for which the energy momentum tensor is
The Einstein equations for a near-horizon geometry can also be interpreted as geometrical equations arising from the restriction of the Einstein equations for the full spacetime to a degenerate horizon, without taking the near-horizon limit, as follows. The near-horizon limit can be thought of as the limit of the “boost” transformation . This implies that restricting the boost-invariant components of the Einstein equations for the full spacetime to a degenerate horizon is equivalent to the boost invariant components of the Einstein equations for the near-horizon geometry. The boost-invariant components are and and hence we see that Eqs. (17) and (18) are also valid for the full spacetime quantities restricted to the horizon. We deduce that the restriction of these components of the Einstein equations depends only on data intrinsic to : this special feature only arises for degenerate horizons.13 It is worth noting that the horizon equations (17) and (18) remain valid in the more general context of extremal isolated horizons [163, 209, 28] and Kundt metrics .
The positivity of and can be related to standard energy conditions. For a near-horizon geometry . Since is timelike on the horizon, the strong energy condition implies . Hence, noting that we deduce that the strong energy condition implies28) and (29). Observe that Einstein–Maxwell theory with satisfies both of these conditions.
In this review, we describe the current understanding of the space of solutions to the basic horizon equation (17), together with the appropriate horizon matter field equations, in a variety of dimensions and theories.
So far we have considered near-horizon geometries independently of any extremal–black-hole solutions. In this section we will assume that the near-horizon geometry arises from a near-horizon limit of an extremal black hole. This limit discards the asymptotic data of the parent–black-hole solution. As a result, only a subset of the physical properties of a black hole can be calculated from the near-horizon geometry alone. In particular, information about the asymptotic stationary Killing vector field is lost and hence one cannot compute the mass from a Komar integral, nor can one compute the angular velocity of the horizon with respect to infinity. Below we discuss physical properties that can be computed purely from the near-horizon geometry [123, 79, 154].
Area. The area of cross sections of the horizon is defined by
For definiteness we now assume the parent black hole is asymptotically flat.
Angular momentum. The conserved angular momentum associated with a rotational symmetry, generated by a Killing vector , is given by a Komar integral on a sphere at spacelike infinity :1423) one can show that, in the gauge ,
In five spacetime dimensions it is natural to couple Einstein–Maxwell theory to a Chern–Simons (CS) term. While the Einstein equations are unchanged, the Maxwell equation now becomes
Gauge charges. For Einstein–Maxwell theories there are also electric, and possibly magnetic, charges. For example, in pure Einstein–Maxwell theory in any dimension, the electric charge is written as an integral over spatial infinity: