# nLab near-horizon geometry

Contents

### Context

#### Gravity

gravity, supergravity

# 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 $X_{d-(p+2)}$, hence an orbifold (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.

## Examples

The near/far horizon limits of the black M-branes:

### The black M2-brane

The black M2-brane is given by the Riemannian metric

(1)$g_{M2} \;\coloneqq\; H^{- 2/3} g_{(\mathbb{R}^{2,1})} + H^{1/3} g_{C(X_7)}$

and the C-field strength

$F_{M2} \;\coloneqq\; dvol_{\mathbb{R}^{2,1}} \wedge d H^{-1}$

where $C(X_7)$ denotes the metric cone on a closed 7-dimensional Einstein manifold $X_7$ for cosmological constant $\Lambda = 5$, whence

$g_{C(X)} \;\coloneqq\; (d r)^2 + r^2 g_{X_7} \,,$

and

$H \;\coloneqq\; 1 + \frac{\ell_{th}^6}{r^6} \:, \phantom{AAA} \ell_{th} \;\coloneqq\; 2^{5/6} \pi^{2/6} N^{1/6} \ell_P$

with $N$ the number of M2-branes and with $\ell_P$ the Planck length in 11 dimensions.

In the near-horizon/large $N$-limit $\ell_{th} \to \infty$ this becomes

\begin{aligned} g_{M2} \overset{\ell_{th} \gg 1}{\longrightarrow} \;\;\; & \left( r/\ell_{th} \right)^{4} g_{(\mathbb{R}^{2,1})} + \left( r/\ell_{th} \right)^{-2} g_{C(X_7)} \\ = & \left( r/\ell_{th} \right)^{4} g_{(\mathbb{R}^{2,1})} + \left( r/\ell_{th} \right)^{-2} (d r)^2 + \ell_{th}^{2} g_{X_7} \\ = & \underset{ = g_{AdS} }{\underbrace{ \frac{1}{z^2} \left( 2^4 g_{(\mathbb{R}^{2,1})} + (d z)^2 \right) }} + \ell_{th}^{2} g_{X_7} \end{aligned} \,,

where in the last step we set

$r \;\coloneqq\; 2 \ell_{th} \frac{1}{\sqrt{z}}$

This reveals the first summand as being the metric tensor of anti-de Sitter spacetime of AdS radius $\ell_{th}$ in horospheric coordinates, and the second summand as that of $X_{7}$ rescaled to radius $\ell_{th}$.

In contrast, in the far-horizon/small $N$-limit $\ell_{th} \to 0$ (1) becomes

$g_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; g_{\mathbb{R}^{2,1}} + g_{C(X_7)}$

and

$F_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; 0$

which is the metric on a Cartesian product of flat Minkowski spacetime worldvolume of an M2-brane with the metric cone on $X_7$.

### The black M5-brane

The black M5-brane is given by the Riemannian metric

(2)$g_{M5} \;\coloneqq\; H^{- 1/3} g_{(\mathbb{R}^{5,1})} + H^{2/3} g_{C(X_4)}$

and the C-field strength

$F_{M2} \;\coloneqq\; \pm 3 \star_5 \wedge d H$

where $C(X_4)$ denotes the metric cone on a closed 4-dimensional Einstein manifold $X_4$ for cosmological constant $\Lambda = 3$, whence

$g_{C(X)} \;\coloneqq\; (d r)^2 + r^2 g_{X_4} \,,$

and

$H \;\coloneqq\; 1 + \frac{\ell_{th}^3}{r^3} \:, \phantom{AAA} \ell_{th} \;\coloneqq\; \pi^{1/3} N^{1/3} \ell_P$

with $N$ the number of M5-branes and with $\ell_P$ the Planck length in 11 dimensions.

In the near-horizon/large $N$-limit $\ell_{th} \to \infty$ this becomes

\begin{aligned} g_{M5} \overset{\ell_{th} \gg 1}{\longrightarrow} \;\;\; & \left( r/\ell_{th} \right)^{4} g_{(\mathbb{R}^{2,1})} + \left( r/\ell_{th} \right)^{-2} g_{C(X_7)} \\ = & r/\ell_{th} g_{(\mathbb{R}^{2,1})} + \left( r/\ell_{th} \right)^{-2} (d r)^2 + \ell_{th}^{2} g_{X_7} \\ = & \underset{ = g_{AdS} }{\underbrace{ \frac{1}{z^2} \left( 2 g_{(\mathbb{R}^{2,1})} + (d z)^2 \right) }} + \ell_{th}^{2} g_{X_4} \end{aligned} \,,

where in the last step we set

$r \;\coloneqq\; \ell_{th}\tfrac{1}{2} \frac{1}{z^2}$

This reveals the first summand as being the metric tensor of anti-de Sitter spacetime of AdS radius $\ell_{th}$ in horospheric coordinates, and the second summand as that of $X_{4}$ rescaled to radius $\ell_{th}$.

In contrast, in the far-horizon/small $N$-limit $\ell_{th} \to 0$ (2) becomes

$g_{M5} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; g_{\mathbb{R}^{5,1}} + g_{C(X_4)}$

and

$F_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; 0$

which is the metric on a Cartesian product of flat Minkowski spacetime worldvolume of an M5-brane with the metric cone on $X_4$.

### The MK6-brane

The metric tensor of $N$ coincident KK-monopoles in 11-dimensional supergravity in the limit that $\ell_{th} \coloneqq N \ell_P \to 0$ is

(3)$g_{MK6} \;=\; g_{\mathbb{R}^{6,1}} + (d y)^2 + y^2 \big( (d \theta)^2 + (\sin \theta)^2 (d \varphi)^2 + (\cos \theta)^2 (d \phi)^2 \big)$

subject to the identification

(4)$(\varphi, \phi) \;\sim\; (\varphi, \phi) + (2\pi/N ,2\pi/N) \,.$

This is equation (47) in IMSY 98, which applies subject to the condition

$U/\left(\frac{N}{g^{2/3}_{YM}}\right) \;=\; U/\left(\frac{N}{(2\pi)^{4/3} \ell_P}\right) \;\gg\; 1$

from a few lines above. Inserting this condition into the definition $y^2 \coloneqq 2 N \ell^3_P U$ right above (47) shows that

\begin{aligned} y^2 & = 2 N \ell^3_P U \\ & = 2(2\pi)^{-4/3} N^2 \ell_P^2 \; \underset{ \gg 1 }{ \underbrace{ \left(U/\left(\frac{N}{ (2 \pi)^{4/3} \ell_P}\right)\right) }} \end{aligned}

hence that the distance $y$ from the locus of the MK6-brane is large in units of

$\ell_{th} \;=\; \sqrt{2} (2\pi)^{-2/3} N \ell_P \,.$

The identification (4) means that this is the orbifold metric cone $\mathbb{R}^{6,1} \times \left( \mathbb{R}^4/(\mathbb{Z}_N)\right)$, hence an A-type ADE-singularity. To make this more explicit, introduce the complex coordinates

$v \;\coloneqq\; y \, e^{i \varphi} \sin \theta \;\;\; w \;\coloneqq\; y \, e^{i \phi} \cos \theta$

on $\mathbb{R}^4 \simeq \mathbb{C}^2$, in terms of which (3) becomes

$g_{MK6} \;\coloneqq\; d v d \overline v + d w d \overline w$

and which exhibit the identification (4) as indeed that of the A-type $\mathbb{Z}_N$-action (Asano 00, around (18)).

geometry transverse to KK-monopolesRiemannian metricremarks
Taub-NUT space:
geometry transverse to
$N+1$ distinct KK-monopoles
at $\vec r_i \in \mathbb{R}^3 \;\; i \in \{1, \cdots, N+1\}$
$\array{d s^2_{TaubNUT} \coloneqq U^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U \coloneqq 1 + \underoverset{i = 1}{N+1}{\sum} U_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}$(e.g. Sen 97b, Sect. 2)
ALE space
Taub-NUT close to $N$ close-by KK-monopoles
e.g. close to $\vec r = 0$: $\frac{{\vert \vec r_i\vert}}{R/2}, \frac{{\vert \vec r\vert}}{R/2} \ll 1$
$\array{d s^2_{ALE} \coloneqq U'^{-1}(d x^4 + \vec \omega \cdot d \vec r)^2 + U' (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) \\ U' \coloneqq \underoverset{i = 1}{N+1}{\sum} U'_i\,, \phantom{AA} \vec \omega \coloneqq \underoverset{i = 1}{N+1}{\sum} \vec \omega_i \\ U'_i \coloneqq \frac{R/2}{ {\vert \vec r - \vec r_i\vert} }\,, \phantom{AA} \vec \nabla \times \vec \omega= \vec \nabla U_i}$e.g. via Euler angles: $\vec \omega = (N+1)R/2(\cos(\theta)-1) d\psi$
(e.g. Asano 00, Sect. 2)
$A_N$-type ADE singularity:
ALE space in the limit
where all $N+1$ KK-monopoles coincide at $vec r_i = 0$
$\array{d s^2_{A_N Sing} \coloneqq \frac{\vert\vec r\vert }{(N+1)R/2}(d x^4 + \vec \omega \cdot d \vec r)^2 + \frac{ (N+1)R/2}{\vert \vec r\vert} (d \vec r)^2 \,, \\ \vec r \in \mathbb{R}^3,\, x^4 \in \mathbb{R}/(2 \pi R\mathbb{Z}) }$(e.g. Asano 00, Sect. 3)

## References

### General

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

### For extremal black holes

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

Description of the near-horizon geometry of near-extremal black holes by Jackiw-Teitelboim gravity:

### For black branes

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

The near horizon geometry of coincident KK-monopoles in 11-dimensional supergravity is discussed in

• Nissan Itzhaki, Juan Maldacena, Jacob Sonnenschein, Shimon Yankielowicz, section 9 of Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges, Phys. Rev. D 58, 046004 1998 (arXiv:hep-th/9802042)

• Nissan Itzhaki, Arkady Tseytlin, S. Yankielowicz, Supergravity Solutions for Branes Localized Within Branes, Phys.Lett.B432:298-304, 1998 (arXiv:hep-th/9803103)

• Akikazu Hashimoto, Supergravity Solutions for Localized Intersections of Branes, JHEP 9901 (1999) 018 (arXiv:hep-th/9812159)

• Masako Asano, section 3 of Compactification and Identification of Branes in the Kaluza-Klein monopole backgrounds (arXiv:hep-th/0003241)

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

Conical D-branes are discussed in