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\begin{document}

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\section*{near-horizon geometry}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{gravity}{}\paragraph*{{Gravity}}\label{gravity}

[[!include gravity contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak
\noindent\hyperlink{NearHorizonGeometry}{Near-horizon geometry}\dotfill \pageref*{NearHorizonGeometry} \linebreak
\noindent\hyperlink{FarHorizonGeometry}{Far-horizon geometry}\dotfill \pageref*{FarHorizonGeometry} \linebreak
\noindent\hyperlink{Examples}{Examples}\dotfill \pageref*{Examples} \linebreak
\noindent\hyperlink{BlackM2}{The black M2-brane}\dotfill \pageref*{BlackM2} \linebreak
\noindent\hyperlink{BlackM5Brane}{The black M5-brane}\dotfill \pageref*{BlackM5Brane} \linebreak
\noindent\hyperlink{MK6Brane}{The MK6-brane}\dotfill \pageref*{MK6Brane} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak
\noindent\hyperlink{general}{General}\dotfill \pageref*{general} \linebreak
\noindent\hyperlink{for_extremal_black_holes}{For extremal black holes}\dotfill \pageref*{for_extremal_black_holes} \linebreak
\noindent\hyperlink{for_black_branes}{For black branes}\dotfill \pageref*{for_black_branes} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

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

\hypertarget{properties}{}\subsection*{{Properties}}\label{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

\begin{displaymath}
r/\ell_P N^{1/k} \ll 1
\end{displaymath}
while ``far'' from the horizon means that

\begin{displaymath}
r/\ell_P N^{1/k} \gg 1
\end{displaymath}
(e.g. \hyperlink{AFFHS98}{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.

\hypertarget{NearHorizonGeometry}{}\subsubsection*{{Near-horizon geometry}}\label{NearHorizonGeometry}

For [[black brane|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 topological space|compact]] [[Einstein manifold]] $X_{11-(p+2)}$;

\begin{displaymath}
Ads_{p+2} \times X_{d-(p+2)}
  \,;
\end{displaymath}
while for [[black brane|black]] [[D-branes]] and [[NS5-branes]] in [[type II supergravity]] the near horizon geometry is [[conformal transformation|conformal]] to a geometry of this form $AdS_{p+2} \times X_{d-(p+2)}$ (see \hyperlink{AFFHS98}{AFFHS 98, section 2}).

\hypertarget{FarHorizonGeometry}{}\subsubsection*{{Far-horizon geometry}}\label{FarHorizonGeometry}

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

\begin{displaymath}
\mathbb{R}^{p,1} \times C\left(X_{d-(p+2)}\right)
  \,,
\end{displaymath}
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 \hyperlink{AFFHS98}{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 \hyperlink{AFFHS98}{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 \hyperlink{AtiyahWitten01}{Atiyah-Witten 01}.

\hypertarget{Examples}{}\subsection*{{Examples}}\label{Examples}

The near/far horizon limits of the [[black brane|black]] [[M-branes]]:

\begin{itemize}%
\item \emph{\hyperlink{BlackM2}{The black M2-brane}}


\item \emph{\hyperlink{BlackM5Brane}{The black M5-brane}}


\item \emph{\hyperlink{MK6Brane}{The MK6-brane}}



\end{itemize}
\hypertarget{BlackM2}{}\subsubsection*{{The black M2-brane}}\label{BlackM2}

The [[black brane|black]] [[M2-brane]] is given by the [[Riemannian metric]]

\begin{equation}
g_{M2} \;\coloneqq\;
  H^{- 2/3} g_{(\mathbb{R}^{2,1})}
  +
  H^{1/3} g_{C(X_7)}
\label{BlackM2BraneMetric}\end{equation}
and the [[C-field]] [[field strength|strength]]

\begin{displaymath}
F_{M2} \;\coloneqq\; dvol_{\mathbb{R}^{2,1}} \wedge d H^{-1}
\end{displaymath}
where $C(X_7)$ denotes the [[metric cone]] on a [[closed manifold|closed]] 7-[[dimension|dimensional]] [[Einstein manifold]] $X_7$ for [[cosmological constant]] $\Lambda = 5$, whence

\begin{displaymath}
g_{C(X)}
  \;\coloneqq\;
  (d r)^2 + r^2 g_{X_7}
  \,,
\end{displaymath}
and

\begin{displaymath}
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
\end{displaymath}
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{displaymath}
\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}
  \,,
\end{displaymath}
where in the last step we set

\begin{displaymath}
r \;\coloneqq\; 2 \ell_{th} \frac{1}{\sqrt{z}}
\end{displaymath}
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 $\ell_{th} \to 0$ \eqref{BlackM2BraneMetric} becomes

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

\begin{displaymath}
F_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; 0
\end{displaymath}
which is the metric on a [[Cartesian product]] of flat [[Minkowski spacetime]] [[worldvolume]] of an M2-brane with the [[metric cone]] on $X_7$.

\hypertarget{BlackM5Brane}{}\subsubsection*{{The black M5-brane}}\label{BlackM5Brane}

The [[black brane|black]] [[M5-brane]] is given by the [[Riemannian metric]]

\begin{equation}
g_{M5} \;\coloneqq\;
  H^{- 1/3} g_{(\mathbb{R}^{5,1})}
  +
  H^{2/3} g_{C(X_4)}
\label{BlackM5BraneMetric}\end{equation}
and the [[C-field]] [[field strength|strength]]

\begin{displaymath}
F_{M2} \;\coloneqq\; \pm 3 \star_5 \wedge d H
\end{displaymath}
where $C(X_4)$ denotes the [[metric cone]] on a [[closed manifold|closed]] 4-[[dimension|dimensional]] [[Einstein manifold]] $X_4$ for [[cosmological constant]] $\Lambda = 3$, whence

\begin{displaymath}
g_{C(X)}
  \;\coloneqq\;
  (d r)^2 + r^2 g_{X_4}
  \,,
\end{displaymath}
and

\begin{displaymath}
H \;\coloneqq\; 1 + \frac{\ell_{th}^3}{r^3}
  \:,
  \phantom{AAA}
  \ell_{th} \;\coloneqq\; \pi^{1/3} N^{1/3} \ell_P
\end{displaymath}
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{displaymath}
\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}
  \,,
\end{displaymath}
where in the last step we set

\begin{displaymath}
r \;\coloneqq\; \ell_{th}\tfrac{1}{2} \frac{1}{z^2}
\end{displaymath}
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 $\ell_{th} \to 0$ \eqref{BlackM5BraneMetric} becomes

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

\begin{displaymath}
F_{M2} \;\overset{\ell_{th} \to 0}{\longrightarrow}\; 0
\end{displaymath}
which is the metric on a [[Cartesian product]] of flat [[Minkowski spacetime]] [[worldvolume]] of an M5-brane with the [[metric cone]] on $X_4$.

\hypertarget{MK6Brane}{}\subsubsection*{{The MK6-brane}}\label{MK6Brane}

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

\begin{equation}
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)
\label{SmallNMK6}\end{equation}
subject to the identification

\begin{equation}
(\varphi, \phi)
  \;\sim\;
  (\varphi, \phi) + (2\pi/N ,2\pi/N)
  \,.
\label{OrbifoldIdentificationForKKMonopole}\end{equation}
This is equation (47) in \hyperlink{IMSY98}{IMSY 98}, which applies subject to the condition

\begin{displaymath}
U/\left(\frac{N}{g^{2/3}_{YM}}\right) 
  \;=\; 
  U/\left(\frac{N}{(2\pi)^{4/3} \ell_P}\right) 
  \;\gg\; 
  1
\end{displaymath}
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{displaymath}
\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}
\end{displaymath}
hence that the distance $y$ from the locus of the MK6-brane is large in units of

\begin{displaymath}
\ell_{th} \;=\; \sqrt{2} (2\pi)^{-2/3} N \ell_P
  \,.
\end{displaymath}
The identification \eqref{OrbifoldIdentificationForKKMonopole} means that this is the [[orbifold]] [[metric cone]] $\mathbb{R}^{6,1} \times \left( \mathbb{R}^4/(\mathbb{Z}_N)\right)$, hence an [[ADE classification| A-type]] [[ADE-singularity]]. To make this more explicit, introduce the complex coordinates

\begin{displaymath}
v \;\coloneqq\; y \, e^{i \varphi} \sin \theta
  \;\;\;
  w \;\coloneqq\; y \, e^{i \phi} \cos \theta
\end{displaymath}
on $\mathbb{R}^4 \simeq \mathbb{C}^2$, in terms of which \eqref{SmallNMK6} becomes

\begin{displaymath}
g_{MK6} 
  \;\coloneqq\;
  d v d \overline v
  + 
  d w d \overline w
\end{displaymath}
and which exhibit the identification \eqref{OrbifoldIdentificationForKKMonopole} as indeed that of the [[ADE classification|A-type]] $\mathbb{Z}_N$-action (\hyperlink{Asano00}{Asano 00, around (18)}).

[[!include KK-monopole geometries -- table]]

\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[piecewise flat spacetime]]


\item [[anti-de Sitter spacetime]]


\item [[asymptotic boundary]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\hypertarget{general}{}\subsubsection*{{General}}\label{general}

\begin{itemize}%
\item Hari K. Kunduri, James Lucietti, \emph{Classification of Near-Horizon Geometries of Extremal Black Holes} (\href{http://relativity.livingreviews.org/Articles/lrr-2013-8/title.html}{web})

\end{itemize}
\hypertarget{for_extremal_black_holes}{}\subsubsection*{{For extremal black holes}}\label{for_extremal_black_holes}

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

\begin{itemize}%
\item [[Gary Gibbons]], in F. del Aguila, [[J. de Azcarraga]], [[Luis Ibanez]] (eds.) \emph{Supersymmetry, Supergravity and Related Topics}

\end{itemize}
Description of the near-horizon geometry of near-extremal black holes by [[Jackiw-Teitelboim gravity]]:

\begin{itemize}%
\item Pranjal Nayak, Ashish Shukla, Ronak M Soni, [[Sandip Trivedi]], V. Vishal, \emph{On the Dynamics of Near-Extremal Black Holes} (\href{https://arxiv.org/abs/1802.09547}{arXiv:1802.09547})


\item Upamanyu Moitra, [[Sandip Trivedi]], V. Vishal, \emph{Near-Extremal Near-Horizons} (\href{https://arxiv.org/abs/1808.08239}{arXiv:1808.08239})



\end{itemize}
\hypertarget{for_black_branes}{}\subsubsection*{{For black branes}}\label{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

\begin{itemize}%
\item [[Gary Gibbons]], [[Paul Townsend]], \emph{Vacuum interpolation in supergravity via super p-branes}, Phys. Rev. Lett. 71, (1993) 3754 (\href{https://arxiv.org/abs/hep-th/9307049}{arXiv:hep-th/9307049})


\item [[Mike Duff]], [[Gary Gibbons]], [[Paul Townsend]], \emph{Macroscopic superstrings as interpolating solitons}, Phys. Lett. B. 332 (1994) 32 (\href{https://arxiv.org/abs/hep-th/9405124}{arXiv:hep-th/9405124})


\item [[Gary Gibbons]], [[Gary Horowitz]], [[Paul Townsend]], \emph{Higher-dimensional resolution of dilatonic black hole singularities}, Class. Quant. Grav. 12 (1995) 297 (\href{https://arxiv.org/abs/hep-th/9410073}{arXiv:hep-th/9410073})


\item [[Gary Gibbons]], Nucl. Phys. B207, (1982) 337;


\item [[Renata Kallosh]], [[Amanda Peet]], Phys. Rev. B46, (1992) 5223;


\item [[Sergio Ferrara]], [[Gary Gibbons]], [[Renata Kallosh]], Nucl. Phys. B500, (1997) 75;


\item [[Ali Chamseddine]], [[Sergio Ferrara]], [[Gary Gibbons]], [[Renata Kallosh]], Phys. Rev. D55, (1997) 3647



\end{itemize}
The observation that the resulting [[isometry group]] is the bosonic body of one of the [[orthosymplectic super groups]] is due to

\begin{itemize}%
\item P. Claus, [[Renata Kallosh]], J. Kumar, [[Paul Townsend]], [[Antoine Van Proeyen]], \emph{Conformal Theory of M2, D3, M5 and D1+D5 Branes}, JHEP 9806 (1998) 004 (\href{https://arxiv.org/abs/hep-th/9801206}{arXiv:hep-th/9801206})

\end{itemize}
A decent account is in

\begin{itemize}%
\item [[Bobby Acharya]], [[Jose Figueroa-O'Farrill]], [[Chris Hull]], B. Spence, \emph{Branes at conical singularities and holography}, Adv. Theor.Math. Phys.2:1249-1286, 1999 (\href{https://arxiv.org/abs/hep-th/9808014}{arXiv:hep-th/9808014})

\end{itemize}
reviewed in

\begin{itemize}%
\item [[Jose Figueroa-O'Farrill]], \emph{Near-horizon geometries of supersymmetric branes}, talk at \href{http://inspirehep.net/record/971430/}{SUSY 98} (\href{https://arxiv.org/abs/hep-th/9807149}{arXiv:hep-th/9807149}, \href{http://www.maths.ed.ac.uk/~jmf/CV/Seminars/SUSY98.pdf}{talk slides})

\end{itemize}
The near horizon geometry of coincident [[KK-monopoles]] in [[11-dimensional supergravity]] is discussed in

\begin{itemize}%
\item Nissan Itzhaki, [[Juan Maldacena]], Jacob Sonnenschein, Shimon Yankielowicz, section 9 of \emph{Supergravity and The Large $N$ Limit of Theories With Sixteen Supercharges}, Phys. Rev. D 58, 046004 1998 (\href{https://arxiv.org/abs/hep-th/9802042}{arXiv:hep-th/9802042})


\item Nissan Itzhaki, [[Arkady Tseytlin]], S. Yankielowicz, \emph{Supergravity Solutions for Branes Localized Within Branes}, Phys.Lett.B432:298-304, 1998 (\href{https://arxiv.org/abs/hep-th/9803103}{arXiv:hep-th/9803103})


\item Akikazu Hashimoto, \emph{Supergravity Solutions for Localized Intersections of Branes}, JHEP 9901 (1999) 018 (\href{https://arxiv.org/abs/hep-th/9812159}{arXiv:hep-th/9812159})


\item Masako Asano, section 3 of \emph{Compactification and Identification of Branes in the Kaluza-Klein monopole backgrounds} (\href{https://arxiv.org/abs/hep-th/0003241}{arXiv:hep-th/0003241})



\end{itemize}
Examples and applications of cone branes in the context of [[M-theory on G2-manifolds]] are discussed in

\begin{itemize}%
\item [[Michael Atiyah]], [[Edward Witten]] \emph{$M$-Theory dynamics on a manifold of $G_2$-holonomy}, Adv. Theor. Math. Phys. 6 (2001) (\href{http://arxiv.org/abs/hep-th/0107177}{arXiv:hep-th/0107177})

\end{itemize}
Conical [[D-branes]] are discussed in

\begin{itemize}%
\item [[Koji Hashimoto]], Shunichiro Kinoshita, Keiju Murata, \emph{Conic D-branes}, Progress of Theoretical and Experimental Physics, Volume 2015, Issue 8, August 2015 (\href{https://arxiv.org/abs/1505.04506}{arXiv:1505.04506}, \href{http://www2.yukawa.kyoto-u.ac.jp/~qft.web/2015/slides/kinoshita.pdf}{slides pdf})

\end{itemize}
See also

\begin{itemize}%
\item Wikipedia, \emph{\href{https://en.wikipedia.org/wiki/Near-horizon_metric}{Near-horizon metric}}

\end{itemize}
[[!redirects near-horizon geometries]]

[[!redirects near horizon geometry]] [[!redirects near horizon geometries]]

[[!redirects far-horizon geometry]] [[!redirects far-horizon geometries]]

[[!redirects far horizon geometry]] [[!redirects far horizon geometries]]

[[!redirects near-horizon limit]] [[!redirects near-horizon limits]]

[[!redirects near horizon limit]] [[!redirects near horizon limits]]

[[!redirects near-horizon metric]] [[!redirects near-horizon metrices]]

[[!redirects far-horizon metric]] [[!redirects far-horizon metrices]]

[[!redirects far-horizon limit]] [[!redirects far-horizon limits]]

[[!redirects far horizon limit]] [[!redirects far horizon limits]]

[[!redirects cone brane]] [[!redirects cone branes]]



\end{document}