nLab near-horizon geometry

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

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

while “far” from the horizon means that

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

(e.g. AFFHS 98, (2), (7) and below (11)).

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.

See also at flat space holography.

Examples

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

• The black M2-brane

• The black M5-brane

• The MK6-brane

The black M2-brane

The black M2-brane is given (we follow AFFHS98, §2.1.1) 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 (we follow AFFHS98, §2.1.2) 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

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

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-monopoles	Riemannian metric	remarks
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)

Related concepts

• piecewise flat spacetime

• anti-de Sitter spacetime

• asymptotic boundary

References

General

See also

• Wikipedia, Near-horizon metric

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

• Gary Gibbons, in F. del Aguila, J. de Azcarraga, Luis Ibanez (eds.) Supersymmetry, Supergravity and Related Topics

Review:

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

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

• Pranjal Nayak, Ashish Shukla, Ronak M Soni, Sandip Trivedi, V. Vishal, On the Dynamics of Near-Extremal Black Holes, J. High Energ. Phys. 2018 48 (2018) [doi:10.1007/JHEP09(2018)048, arXiv:1802.09547]

• Upamanyu Moitra, Sandip Trivedi, V. Vishal, Near-Extremal Near-Horizons (arXiv:1808.08239)

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

• Gary Gibbons, Paul Townsend, Vacuum interpolation in supergravity via super p-branes, Phys. Rev. Lett. 71 (1993) 3754-3757 [arXiv:hep-th/9307049, doi:10.1103/PhysRevLett.71.3754]

• Mike Duff, Gary Gibbons, Paul Townsend, Macroscopic superstrings as interpolating solitons, Phys. Lett. B 332 (1994) 32 [doi:10.1016/0370-2693(94)91260-2, arXiv:hep-th/9405124]

• Gary Gibbons, Gary Horowitz, Paul Townsend, Higher-dimensional resolution of dilatonic black hole singularities, Class. Quant. Grav. 12 (1995) 297 [arXiv:hep-th/9410073]

• Gary Gibbons, Antigravitating black hole solitons with scalar hair in $N=4$ supergravity, Nucl. Phys. B 207 2 (1982) 337-349 [doi:10.1016/0550-3213(82)90170-5]

• Renata Kallosh, Amanda Peet, Dilaton black holes near the horizon, Phys. Rev. B 46, (1992) R5223(R) [doi:10.1103/PhysRevD.46.R5223]

• Sergio Ferrara, Gary Gibbons, Renata Kallosh, Black Holes and Critical Points in Moduli Space, Nucl. Phys. B 500 (1997) 75-93 [doi:10.1016/S0550-3213(97)00324-6arXiv:hep-th/9702103]

The observation that the resulting isometry group is the bosonic body of an orthosymplectic super group:

• Piet Claus, Renata Kallosh, J. Kumar, Paul Townsend, Antoine Van Proeyen, Conformal Theory of M2, D3, M5 and D1+D5 Branes, JHEP 9806 (1998) 004 [arXiv:hep-th/9801206]

A decent account is in:

• Bobby Acharya, Jose Figueroa-O'Farrill, Chris Hull, B. Spence, Branes at conical singularities and holography, Adv. Theor. Math. Phys. 2 (1999) 1249-1286 [arXiv:hep-th/9808014, spire:474212, doi:10.4310/ATMP.1998.v2.n6.a2]

reviewed in

• Jose Figueroa-O'Farrill, Near-horizon geometries of supersymmetric branes, talk at SUSY 98 (arXiv:hep-th/9807149, talk slides)

Other review:

• Adémólá Adéìféoba: Brane Solutions in Supergravity and the Near-Horizon Geometries, seminar talk at Holography and large-$N$ dualities, Heidelberg (2018) [pdf, pdf]

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, Shimon 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

• Michael Atiyah, Edward Witten $M$-Theory dynamics on a manifold of $G_2$-holonomy, Adv. Theor. Math. Phys. 6 (2001) (arXiv:hep-th/0107177)

Conical D-branes are discussed in

• Koji Hashimoto, Shunichiro Kinoshita, Keiju Murata, Conic D-branes, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 8, August 2015 (arXiv:1505.04506, slides pdf)

Near horizon geometries of black branes which KK-compactify to black holes:

• Mirjam Cvetič, Paulo J. Porfírio, Alejandro Satz, Gaussian null coordinates, near-horizon geometry and conserved charges on the horizon of extremal non-dilatonic black p-branes (arXiv:2003.09304)

Further discussion for M2-branes:

• Harvendra Singh, M2-branes on a resolved $\mathbb{C}^4/\mathbb{Z}_4$, JHEP 0809:071, 2008 (arXiv:0807.5016)

• Chethan Krishnan, Carlo Maccaferri, Harvendra Singh, M2-brane Flows and the Chern-Simons Level, JHEP 0905:114, 2009 (arXiv:0902.0290)

Discussion for heterotic supergravity:

• Karl-Philip Gemmer, Alexander S. Haupt, Olaf Lechtenfeld, Christoph Nölle, Alexander D. Popov, Heterotic string plus five-brane systems with asymptotic $AdS_3$, Adv. Theor. Math. Phys. 17 (2013) 771-827 (arXiv:1202.5046)