Near-horizon limit of the global metric of a black hole
The near-horizon metric (NHM ) refers to the near-horizon limit of the global metric of a black hole . NHMs play an important role in studying the geometry and topology of black holes, but are only well defined for extremal black holes.[ 1] [ 2] [ 3]
r
{\displaystyle r}
[ edit ] The metric of extremal Reissner–Nordström black hole is
d
s
2
=
−
(
1
−
M
r
)
2
d
t
2
+
(
1
−
M
r
)
−
2
d
r
2
+
r
2
(
d
θ
2
+
sin
2
θ
d
ϕ
2
)
.
{\displaystyle ds^{2}\,=\,-{\Big (}1-{\frac {M}{r}}{\Big )}^{2}\,dt^{2}+{\Big (}1-{\frac {M}{r}}{\Big )}^{-2}dr^{2}+r^{2}\,{\big (}d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}{\big )}\,.}
Taking the near-horizon limit
t
↦
t
~
ϵ
,
r
↦
M
+
ϵ
r
~
,
ϵ
→
0
,
{\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \epsilon \to 0\,,}
and then omitting the tildes, one obtains the near-horizon metric
d
s
2
=
−
r
2
M
2
d
t
2
+
M
2
r
2
d
r
2
+
M
2
(
d
θ
2
+
sin
2
θ
d
ϕ
2
)
{\displaystyle ds^{2}=-{\frac {r^{2}}{M^{2}}}\,dt^{2}+{\frac {M^{2}}{r^{2}}}\,dr^{2}+M^{2}\,{\big (}d\theta ^{2}+\sin ^{2}\theta \,d\phi ^{2}{\big )}}
NHM of extremal Kerr black holes [ edit ] The metric of extremal Kerr black hole (
M
=
a
=
J
/
M
{\displaystyle M=a=J/M}
Boyer–Lindquist coordinates can be written in the following two enlightening forms,[ 4] [ 5]
d
s
2
=
−
ρ
K
2
Δ
K
Σ
2
d
t
2
+
ρ
K
2
Δ
K
d
r
2
+
ρ
K
2
d
θ
2
+
Σ
2
sin
2
θ
ρ
K
2
(
d
ϕ
−
ω
K
d
t
)
2
,
{\displaystyle ds^{2}\,=\,-{\frac {\rho _{K}^{2}\Delta _{K}}{\Sigma ^{2}}}\,dt^{2}+{\frac {\rho _{K}^{2}}{\Delta _{K}}}\,dr^{2}+\rho _{K}^{2}d\theta ^{2}+{\frac {\Sigma ^{2}\sin ^{2}\theta }{\rho _{K}^{2}}}{\big (}d\phi -\omega _{K}\,dt{\big )}^{2}\,,}
d
s
2
=
−
Δ
K
ρ
K
2
(
d
t
−
M
sin
2
θ
d
ϕ
)
2
+
ρ
K
2
Δ
K
d
r
2
+
ρ
K
2
d
θ
2
+
sin
2
θ
ρ
K
2
(
M
d
t
−
(
r
2
+
M
2
)
d
ϕ
)
2
,
{\displaystyle ds^{2}\,=\,-{\frac {\Delta _{K}}{\rho _{K}^{2}}}\,{\big (}dt-M\sin ^{2}\theta d\phi {\big )}^{2}+{\frac {\rho _{K}^{2}}{\Delta _{K}}}\,dr^{2}+\rho _{K}^{2}d\theta ^{2}+{\frac {\sin ^{2}\theta }{\rho _{K}^{2}}}{\Big (}Mdt-(r^{2}+M^{2})d\phi {\Big )}^{2}\,,}
where
ρ
K
2
:=
r
2
+
M
2
cos
2
θ
,
Δ
K
:=
(
r
−
M
)
2
,
Σ
2
:=
(
r
2
+
M
2
)
2
−
M
2
Δ
K
sin
2
θ
,
ω
K
:=
2
M
2
r
Σ
2
.
{\displaystyle \rho _{K}^{2}:=r^{2}+M^{2}\cos ^{2}\theta \,,\;\;\Delta _{K}:={\big (}r-M{\big )}^{2}\,,\;\;\Sigma ^{2}:={\big (}r^{2}+M^{2}{\big )}^{2}-M^{2}\Delta _{K}\sin ^{2}\theta \,,\;\;\omega _{K}:={\frac {2M^{2}r}{\Sigma ^{2}}}\,.}
Taking the near-horizon limit[ 6] [ 7]
t
↦
t
~
ϵ
,
r
↦
M
+
ϵ
r
~
,
ϕ
↦
ϕ
~
+
1
2
M
ϵ
t
~
,
ϵ
→
0
,
{\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \phi \mapsto {\tilde {\phi }}+{\frac {1}{2M\epsilon }}{\tilde {t}}\,,\quad \epsilon \to 0\,,}
and omitting the tildes, one obtains the near-horizon metric (this is also called extremal Kerr throat [ 6]
d
s
2
≃
1
+
cos
2
θ
2
(
−
r
2
2
M
2
d
t
2
+
2
M
2
r
2
d
r
2
+
2
M
2
d
θ
2
)
+
4
M
2
sin
2
θ
1
+
cos
2
θ
(
d
ϕ
+
r
d
t
2
M
2
)
2
.
{\displaystyle ds^{2}\simeq {\frac {1+\cos ^{2}\theta }{2}}\,{\Big (}-{\frac {r^{2}}{2M^{2}}}\,dt^{2}+{\frac {2M^{2}}{r^{2}}}\,dr^{2}+2M^{2}d\theta ^{2}{\Big )}+{\frac {4M^{2}\sin ^{2}\theta }{1+\cos ^{2}\theta }}\,{\Big (}d\phi +{\frac {rdt}{2M^{2}}}{\Big )}^{2}\,.}
[ edit ] Extremal Kerr–Newman black holes (
r
+
2
=
M
2
+
Q
2
{\displaystyle r_{+}^{2}=M^{2}+Q^{2}}
[ 4] [ 5]
d
s
2
=
−
(
1
−
2
M
r
−
Q
2
ρ
K
N
)
d
t
2
−
2
a
sin
2
θ
(
2
M
r
−
Q
2
)
ρ
K
N
d
t
d
ϕ
+
ρ
K
N
(
d
r
2
Δ
K
N
+
d
θ
2
)
+
Σ
2
ρ
K
N
d
ϕ
2
,
{\displaystyle ds^{2}=-{\Big (}1-{\frac {2Mr-Q^{2}}{\rho _{KN}}}\!{\Big )}dt^{2}-{\frac {2a\sin ^{2}\!\theta \,(2Mr-Q^{2})}{\rho _{KN}}}dtd\phi +\rho _{KN}{\Big (}{\frac {dr^{2}}{\Delta _{KN}}}+d\theta ^{2}{\Big )}+{\frac {\Sigma ^{2}}{\rho _{KN}}}d\phi ^{2},}
where
Δ
K
N
:=
r
2
−
2
M
r
+
a
2
+
Q
2
,
ρ
K
N
:=
r
2
+
a
2
cos
2
θ
,
Σ
2
:=
(
r
2
+
a
2
)
2
−
Δ
K
N
a
2
sin
2
θ
.
{\displaystyle \Delta _{KN}\,:=\,r^{2}-2Mr+a^{2}+Q^{2}\,,\;\;\rho _{KN}\,:=\,r^{2}+a^{2}\cos ^{2}\!\theta \,,\;\;\Sigma ^{2}\,:=\,(r^{2}+a^{2})^{2}-\Delta _{KN}a^{2}\sin ^{2}\theta \,.}
Taking the near-horizon transformation
t
↦
t
~
ϵ
,
r
↦
M
+
ϵ
r
~
,
ϕ
↦
ϕ
~
+
a
r
0
2
ϵ
t
~
,
ϵ
→
0
,
(
r
0
2
:=
M
2
+
a
2
)
{\displaystyle t\mapsto {\frac {\tilde {t}}{\epsilon }}\,,\quad r\mapsto M+\epsilon \,{\tilde {r}}\,,\quad \phi \mapsto {\tilde {\phi }}+{\frac {a}{r_{0}^{2}\epsilon }}{\tilde {t}}\,,\quad \epsilon \to 0\,,\quad {\Big (}r_{0}^{2}\,:=\,M^{2}+a^{2}{\Big )}}
and omitting the tildes, one obtains the NHM[ 7]
d
s
2
≃
(
1
−
a
2
r
0
2
sin
2
θ
)
(
−
r
2
r
0
2
d
t
2
+
r
0
2
r
2
d
r
2
+
r
0
2
d
θ
2
)
+
r
0
2
sin
2
θ
(
1
−
a
2
r
0
2
sin
2
θ
)
−
1
(
d
ϕ
+
2
a
r
M
r
0
4
d
t
)
2
.
{\displaystyle ds^{2}\simeq {\Big (}1-{\frac {a^{2}}{r_{0}^{2}}}\sin ^{2}\!\theta {\Big )}\left(-{\frac {r^{2}}{r_{0}^{2}}}dt^{2}+{\frac {r_{0}^{2}}{r^{2}}}dr^{2}+r_{0}^{2}d\theta ^{2}\right)+r_{0}^{2}\sin ^{2}\!\theta \,{\Big (}1-{\frac {a^{2}}{r_{0}^{2}}}\sin ^{2}\!\theta {\Big )}^{-1}\left(d\phi +{\frac {2arM}{r_{0}^{4}}}dt\right)^{2}\,.}
NHMs of generic black holes [ edit ] In addition to the NHMs of extremal Kerr–Newman family metrics discussed above, all stationary NHMs could be written in the form[ 1] [ 2] [ 3] [ 8]
d
s
2
=
(
h
^
A
B
G
A
G
B
−
F
)
r
2
d
v
2
+
2
d
v
d
r
−
h
^
A
B
G
B
r
d
v
d
y
A
−
h
^
A
B
G
A
r
d
v
d
y
B
+
h
^
A
B
d
y
A
d
y
B
{\displaystyle ds^{2}=({\hat {h}}_{AB}G^{A}G^{B}-F)r^{2}dv^{2}+2dvdr-{\hat {h}}_{AB}G^{B}rdvdy^{A}-{\hat {h}}_{AB}G^{A}rdvdy^{B}+{\hat {h}}_{AB}dy^{A}dy^{B}}
=
−
F
r
2
d
v
2
+
2
d
v
d
r
+
h
^
A
B
(
d
y
A
−
G
A
r
d
v
)
(
d
y
B
−
G
B
r
d
v
)
,
{\displaystyle =-F\,r^{2}dv^{2}+2dvdr+{\hat {h}}_{AB}{\big (}dy^{A}-G^{A}\,rdv{\big )}{\big (}dy^{B}-G^{B}\,rdv{\big )}\,,}
where the metric functions
{
F
,
G
A
}
{\displaystyle \{F,G^{A}\}}
h
^
A
B
{\displaystyle {\hat {h}}_{AB}}
intrinsic metric of the horizon, and
y
A
{\displaystyle y^{A}}
isothermal coordinates on the horizon.
Remark: In Gaussian null coordinates, the black hole horizon corresponds to
r
=
0
{\displaystyle r=0}
^ a b Kunduri, Hari K.; Lucietti, James (2009). "A classification of near-horizon geometries of extremal vacuum black holes". Journal of Mathematical Physics . 50 (8): 082502. arXiv :0806.2051 Bibcode :2009JMP....50h2502K . doi :10.1063/1.3190480 . ISSN 0022-2488 . S2CID 15173886 . ^ a b Kunduri, Hari K; Lucietti, James (2009-11-25). "Static near-horizon geometries in five dimensions". Classical and Quantum Gravity . 26 (24). IOP Publishing: 245010. arXiv :0907.0410 Bibcode :2009CQGra..26x5010K . doi :10.1088/0264-9381/26/24/245010 . ISSN 0264-9381 . S2CID 55272059 . ^ a b Kunduri, Hari K (2011-05-20). "Electrovacuum near-horizon geometries in four and five dimensions". Classical and Quantum Gravity . 28 (11): 114010. arXiv :1104.5072 Bibcode :2011CQGra..28k4010K . doi :10.1088/0264-9381/28/11/114010 . ISSN 0264-9381 . S2CID 118609264 . ^ a b Hobson, Michael Paul; Efstathiou, George; Lasenby., Anthony N (2006). General relativity : an introduction for physicists . Cambridge, UK New York: Cambridge University Press. ISBN 978-0-521-82951-9 OCLC 61757089 . ^ a b Frolov, Valeri P; Novikov, Igor D (1998). Black hole physics : basic concepts and new developments . Dordrecht Boston: Kluwer. ISBN 978-0-7923-5145-0 OCLC 39189783 . ^ a b Bardeen, James; Horowitz, Gary T. (1999-10-26). "Extreme Kerr throat geometry: A vacuum analog of AdS2 ×S2 ". Physical Review D . 60 (10): 104030. arXiv :hep-th/9905099 Bibcode :1999PhRvD..60j4030B . doi :10.1103/physrevd.60.104030 . ISSN 0556-2821 . S2CID 17389870 . ^ a b Amsel, Aaron J.; Horowitz, Gary T.; Marolf, Donald; Roberts, Matthew M. (2010-01-22). "Uniqueness of extremal Kerr and Kerr-Newman black holes". Physical Review D . 81 (2): 024033. arXiv :0906.2367 Bibcode :2010PhRvD..81b4033A . doi :10.1103/physrevd.81.024033 . ISSN 1550-7998 . S2CID 15540019 . ^ Compère, Geoffrey (2012-10-22). "The Kerr/CFT Correspondence and its Extensions" . Living Reviews in Relativity . 15 (1). Springer Science and Business Media LLC: 11. arXiv :1203.3561 Bibcode :2012LRR....15...11C . doi :10.12942/lrr-2012-11 ISSN 2367-3613 . PMC 5255558 PMID 28179839 .
