black hole spacetimes | vanishing angular momentum | positive angular momentum |
---|---|---|
vanishing charge | Schwarzschild spacetime | Kerr spacetime |
positive charge | Reissner-Nordstrom spacetime | Kerr-Newman spacetime |
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.
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
while “far” from the horizon means that
(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.
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 $X_{11-(p+2)}$;
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).
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
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.
The near/far horizon limits of the black M-branes:
The black M2-brane is given by the Riemannian metric
where $C(X_7)$ denotes the metric cone on a closed 7-dimensional Einstein manifold $X_7$ for cosmological constant $\Lambda = 5$, whence
and
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
where in the last step we set
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
and
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 is given by the Riemannian metric
where $C(X_4)$ denotes the metric cone on a closed 4-dimensional Einstein manifold $X_4$ for cosmological constant $\Lambda = 3$, whence
and
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
where in the last step we set
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
and
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 metric tensor of $N$ coincident KK-monopoles in 11-dimensional supergravity in the limit that $\ell_{th} \coloneqq N \ell_P \to 0$ is
subject to the identification
This is equation (47) in IMSY 98, where we used the condition $U/(\frac{N}{g^{2/3}_{YM}}) = U/(\frac{N}{(2\pi)^{4/3} \ell_P}) \gg 1$ from a few lines above, in the equation $y^2 = 2 N \ell^3_P U$ right above (47) to find $y^2 = 2(2\pi)^{-4/3} N^2 \ell_P^2 (U/(\frac{N}{ (2 \pi)^{4/3} \ell_P}))$, hence that $y$ is large in units of
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
on $\mathbb{R}^4 \simeq \mathbb{C}^2$, in terms of which (3) becomes
and which exhibit the identification (4) as indeed that of the A-type $\mathbb{Z}_N$-action (Asano 00, around (18)).
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
Gary Gibbons, Paul Townsend, Vacuum interpolation in supergravity via super p-branes, Phys. Rev. Lett. 71, (1993) 3754 (arXiv:hep-th/9307049)
Mike Duff, Gary Gibbons, Paul Townsend, Macroscopic superstrings as interpolating solitons, Phys. Lett. B. 332 (1994) 32 (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, Nucl. Phys. B207, (1982) 337;
Renata Kallosh, Amanda Peet, Phys. Rev. B46, (1992) 5223;
Sergio Ferrara, Gary Gibbons, Renata Kallosh, Nucl. Phys. B500, (1997) 75;
Ali Chamseddine, Sergio Ferrara, Gary Gibbons, Renata Kallosh, Phys. Rev. D55, (1997) 3647
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)
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.