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near-horizon geometry

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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 P\ell_P times a root of the integer charge (“number” NN) of the branes, i.e. being “near” to the horizon means that

r/ PN 1/k1 r/\ell_P N^{1/k} \ll 1

while “far” from the horizon means that

r/ PN 1/k1 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 NN, respectively.

Since the Planck length “is tiny” and due to the higher roots of NN appearing here, this means that NN must be “huge” for the near horizon limit to be visible at macroscopic scale, while, conversely, any “moderate” value of NN 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+1p+1 that preserve some supersymmetry, the near horizon geometry is always a Cartesian product of an anti-de Sitter spacetime AdS p+2AdS_{p+2} with a compact Einstein manifold X 11(p+2)X_{11-(p+2)};

Ads p+2×X d(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×X d(p+2)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×X d(p+2)AdS_{p+2} \times X_{d-(p+2)} (i.e. actual anti-de Sitter spacetime without conformal factor) is of the form

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

where p,1\mathbb{R}^{p,1} is Minkowski spacetime (the brane worldvolume) and C(X d(p+2))C(X_{d-(p+2)}) is the metric cone over X d(p+2)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)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 M2H 2/3g ( 2,1)+H 1/3g C(X 7) g_{M2} \;\coloneqq\; H^{- 2/3} g_{(\mathbb{R}^{2,1})} + H^{1/3} g_{C(X_7)}

and the C-field strength

F M2dvol 2,1dH 1 F_{M2} \;\coloneqq\; dvol_{\mathbb{R}^{2,1}} \wedge d H^{-1}

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

g C(X)(dr) 2+r 2g X 7, g_{C(X)} \;\coloneqq\; (d r)^2 + r^2 g_{X_7} \,,

and

H1+ th 6r 6,AAA th2 5/6π 2/6N 1/6 P 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 NN the number of M2-branes and with P\ell_P the Planck length in 11 dimensions.

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

g M2 th1 (r/ th) 4g ( 2,1)+(r/ th) 2g C(X 7) = (r/ th) 4g ( 2,1)+(r/ th) 2(dr) 2+ th 2g X 7 = 1z 2(2 4g ( 2,1)+(dz) 2)=g AdS+ th 2g X 7, \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

r2 th1z 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 th\ell_{th} in horospheric coordinates, and the second summand as that of X 7X_{7} rescaled to radius th\ell_{th}.

In contrast, in the far-horizon/small NN-limit th0\ell_{th} \to 0 (1) becomes

g M2 th0g 2,1+g C(X 7) g_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; g_{\mathbb{R}^{2,1}} + g_{C(X_7)}

and

F M2 th00 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 7X_7.

The black M5-brane

The black M5-brane is given by the Riemannian metric

(2)g M5H 1/3g ( 5,1)+H 2/3g C(X 4) 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±3 5dH F_{M2} \;\coloneqq\; \pm 3 \star_5 \wedge d H

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

g C(X)(dr) 2+r 2g X 4, g_{C(X)} \;\coloneqq\; (d r)^2 + r^2 g_{X_4} \,,

and

H1+ th 3r 3,AAA thπ 1/3N 1/3 P H \;\coloneqq\; 1 + \frac{\ell_{th}^3}{r^3} \:, \phantom{AAA} \ell_{th} \;\coloneqq\; \pi^{1/3} N^{1/3} \ell_P

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

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

g M5 th1 (r/ th) 4g ( 2,1)+(r/ th) 2g C(X 7) = r/ thg ( 2,1)+(r/ th) 2(dr) 2+ th 2g X 7 = 1z 2(2g ( 2,1)+(dz) 2)=g AdS+ th 2g X 4, \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 th121z 2 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 th\ell_{th} in horospheric coordinates, and the second summand as that of X 4X_{4} rescaled to radius th\ell_{th}.

In contrast, in the far-horizon/small NN-limit th0\ell_{th} \to 0 (2) becomes

g M5 th0g 5,1+g C(X 4) g_{M5} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; g_{\mathbb{R}^{5,1}} + g_{C(X_4)}

and

F M2 th00 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 4X_4.

The MK6-brane

The metric tensor of NN coincident KK-monopoles in 11-dimensional supergravity in the limit that thN P0\ell_{th} \coloneqq N \ell_P \to 0 is

(3)g MK6=g 6,1+(dy) 2+y 2((dθ) 2+(sinθ) 2(dφ) 2+(cosθ) 2(dϕ) 2) 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)(φ,ϕ)(φ,ϕ)+(2π/N,2π/N). (\varphi, \phi) \;\sim\; (\varphi, \phi) + (2\pi/N ,2\pi/N) \,.

This is equation (47) in IMSY 98, where we used the condition U/(Ng YM 2/3)=U/(N(2π) 4/3 P)1U/(\frac{N}{g^{2/3}_{YM}}) = U/(\frac{N}{(2\pi)^{4/3} \ell_P}) \gg 1 from a few lines above, in the equation y 2=2N P 3Uy^2 = 2 N \ell^3_P U right above (47) to find y 2=2(2π) 4/3N 2 P 2(U/(N(2π) 4/3 P))y^2 = 2(2\pi)^{-4/3} N^2 \ell_P^2 (U/(\frac{N}{ (2 \pi)^{4/3} \ell_P})), hence that yy is large in units of

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

The identification (4) means that this is the orbifold metric cone 6,1×( 4/( N))\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

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

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

g MK6dvdv¯+dwdw¯ 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 N\mathbb{Z}_N-action (Asano 00, around (18)).

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 𝒩=2\mathcal{N}=2 4d supergravity is AdS 2×S 2AdS_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

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 NN Limit of Theories With Sixteen Supercharges, Phys. Rev. D 58, 046004 1998 (arXiv:hep-th/9802042)

  • 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

See also

Last revised on October 22, 2018 at 03:11:26. See the history of this page for a list of all contributions to it.