Electric and Magnetic Black Holes in
Arbitrary Dimension
Pablo Díaz
[arXiv:0906.0489] Adil Belhaj, Pablo Díaz, Antonio Seguí
Contents
• Electric set-up
• Magnetic set-up
• Spherically symmetric solutions of both: electric
and magnetic black holes
• Existence of three horizons at the most
• Nonextreme black hole region
• Horizon coalescence and nonzero physical
distance
• Coalescence geometries
• Conclusions
Electric and Magnetic Black Holes in Arbitrary Dimension
Electric set-up
Perturbative solutions of Hilbert-Einstein-Maxwell action
abelian
Electric and Magnetic Black Holes in Arbitrary Dimension
Magnetic set-up (I)
Non-perturbative solutions of Hilbert-Einstein-YM action
nonabelian
Electric and Magnetic Black Holes in Arbitrary Dimension
Magnetic set-up (II)
The magnetic black holes we treat are Yang-type monopoles:
Static solutions of a YM theory on with
gauge group , , which fulfill:
1. Spherically symmetric configurations
2. Nontrivial topology
)
2
1
(
k
d
)
2
( k
SO
G
–
K=1
–
K=2
–
K=3,4,…
Yang monopoles are not Yang-type
)
1
(
)
2
( U
SO
G
3
d
Dirac monopole
2
)
2
(
)
2
(
)
4
(
Z
SU
SU
SO
G
5
d
Extended-Yang monopole
)
2
( k
SO
G
1
2k
d
)
2
( k
SO
-monopoles
Electric and Magnetic Black Holes in Arbitrary Dimension
Magnetic set-up (III)
Electric and Magnetic Black Holes in Arbitrary Dimension
Spherical symmetry (I)
1
d
Both magnetic and electric objects
are point-like in dimensions
Spherically
symmetric
Spherical ansatz:
Leads to…
Plugging into Einstein equations
Electric and Magnetic Black Holes in Arbitrary Dimension
Spherical symmetry (II)
Electric and Magnetic Black Holes in Arbitrary Dimension
Spherical symmetry (III)
For our cases and we will call
)
(
)
,
(
r
m
r
t
m
2
e
The electric configuration is consistent with the geometry
And the magnetic one with
)
2
1
(
k
d
Electric and Magnetic Black Holes in Arbitrary Dimension
Three horizons at the most
Roots of
)
(r