# nLab near-horizon geometry

### Context

#### Gravity

gravity, supergravity

## Spacetimes

black hole spacetimesvanishing angular momentumpositive angular momentum
vanishing chargeSchwarzschild spacetimeKerr spacetime
positive chargeReissner-Nordstrom spacetimeKerr-Newman spacetime

# Contents

## Idea

The spacetime geometry of a black hole or black brane close to the horizon is called its near-horizon geometry. Accordingly, the geometry far from the horizon might be called its far-horizon geometry. One commonly says that the black hole/brane solution interpolates between its near and far horizon geometry.

## Properties

In the following “small” and “large” radius is in units of Planck length $\ell_P$ times a root of the integer charge (“number” $N$) of the branes, i.e. being “near” to the horizon means that

$r/\ell_P N^{1/k} \ll 1$

while “far” from the horizon means that

$r/\ell_P N^{1/k} \gg 1$

This means that the near/far horizon limit may also be thought of as corresponding to large/small $N$, respectively.

Since the Planck length “is tiny” and due to the higher roots of $N$ appearing here, this means that $N$ must be “huge” for the near horizon limit to be visible at macroscopic scale, while, conversely, any “moderate” value of $N$ means implies that every macroscopic radius is “far” from the horizon.

### Near-horizon geometry

For black M-branes (black branes in 11-dimensional supergravity) of dimension $p+1$ that preserve some supersymmetry, the near horizon geometry is always a Cartesian product of an anti-de Sitter spacetime $AdS_{p+2}$ with a compact Einstein manifold $X_{11-(p+2)}$;

$Ads_{p+2} \times X_{d-(p+2)} \,;$

while for black D-branes and NS5-branes in type II supergravity the near horizon geometry is conformal to a geometry of this form $AdS_{p+2} \times X_{d-(p+2)}$ (see AFFHS 98, section 2).

### Far-horizon geometry

In contrast, the “far-horizon geometry” of all those black branes whose near horizon geometry is $AdS_{p+2} \times X_{d-(p+2)}$ (i.e. actual anti-de Sitter spacetime without conformal factor) is of the form

$\mathbb{R}^{p,1} \times C\left(X_{d-(p+2)}\right) \,,$

where $\mathbb{R}^{p,1}$ is Minkowski spacetime (the brane worldvolume) and $C(X_{d-(p+2)})$ is the metric cone over $X_{d-(p+2)}$ (see AFFHS 98, section 3). Since this “far-horizon limit” is still a solution to the supergravity equations of motion away from the tip of the cone, it may in itself be regarded as a “cone brane”-solution (see AFFHS 98, section 3.1).

If $X_{d-(p+2)}$ is a smooth quotient space by the action of a finite subgroup of SU(2), then the corresponding cone brane is a brane “at an ADE-singularity”.

Examples and applications of such cone branes, in the context of M-theory on G2-manifolds, are discussed in Atiyah-Witten 01.

## References

• Hari K. Kunduri, James Lucietti, Classification of Near-Horizon Geometries of Extremal Black Holes (web)

That the near horizon geometry of the extremal Reissner-Nordström black hole in $\mathcal{N}=2$ 4d supergravity is $AdS_2 \times S^2$ was observed in

That the near horizon geometry of black branes in 11-dimensional supergravity is (conformal to) anti de Sitter spacetime times some compact space is apparently due to

The observation that the resulting isometry group is the bosonic body of one of the orthosymplectic super groups is due to

A decent account is in

reviewed in

Examples and applications of cone branes in the context of M-theory on G2-manifolds are discussed in

Conical D-branes are discussed in