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New static black holes withnonspherical horizon topology
(work done together with B. Kleihaus and J. Kunz)
Eugen Radu
Institut fur Physik, Universitat Oldenburg, Germany
Motivation The solutions
Further remarks
based on:
B. Kleihaus, J. Kunz and E. Radu:
d 5 static black holes with S2Sd4 event horizon topology, e-Print: arXiv:0904.272(Phys. Lett. B: 301, 2009)
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Motivation:
gravity in higher dimensions a lot of interest inthe last decade
the d = 5 back ring solution (Emparan and Reall2001)
d > 5 solutions with nonspherical topology of the horizon?
the phase structure becomes increasingly intricate and diverse
no closed form useful solutions (apart from the Myers-Perry black holes)
approximate solutions: the method of asymptotic expansions (Emparan et. al.); however, limited
valability...
numerical (-nonperturbative) approach?
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our proposal:
metric ansatz (d 5):
ds2 = f0(r, z)dt2 + f1(r, z)(dr2 + dz2)+f2(r, z)d
2 + f3(r, z)d2d4
(generalized Weyl coordinates+rod structure)
the Einstein equations:
2f0 12f0
(f0)2 + 12f2
(f0) (f2)
+(d 4)
2f3(f0) (f3) = 0,
2f1 + . . . = 0,2f2 + . . . = 0,2f3 + . . . = 0.
no obvious structure except for d = 5
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Known solutions
the d = 5 static black ring:
the metric functions are known in closed form (Emparan and Reall 2001):
f0 =R2 + 2
R1 + 1,
f1 =(R1 + 1 + R2 2)((1 c)R1 + ( 1 + c)R2 + 2cR3)
8(1 + c)R1R2R3,
f2 =(R2 2)(R3 + 3)
R1 1, f3 = R3 3 ,
where
i = z
zi, Ri = r2 + 2i and z1 =
a, z2 = a, z3 = b,
a and b positive constants, c = a/b < 1.
M(5), A(5)H , T(5) : functions of (a, b),
conical singularity!
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d 5: the Schwarzschild-Tangerlini solution: Sd2 topology of the event horizon
f0 =
v(d3)/2 1v(d3)/2 + 1
2, f1 = c
(v(d3)/2 + 1)4/(d3)
4v(z2 (v21)2
(v2+1)2+ r2 (v
2+1)2
(v21)2 ),
f2 = . . . , f 3 = . . . , with v = v(r, z)
complicated expressions (see arXiv:0904.2723)
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New solutions:
Black holes with S2 Sd4 topology of the event horizonrod structure:
d=5: static black ring!
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numerical solutions only!
computation done with the new metric functions Fi
f0i the static black ring solution: impose the rod structure
fi = f0i Fi
boundary conditions:
rFi|r=0 = 0, for < z b,rF0|r=0 = 0, rF1|r=0 = 0, rF2|r=0 = 0, F1|r=0 = F3|r=0
and Fi = 1 as r
or z
.
solve the elliptic partial differential equations for Fi
main idea : increase the value of d (dimension of spacetime)
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d = 6, 7 solutions with good accuracy
d > 7?
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properties of the solutions:
The horizon metric:
d2 = f1(0, z)dz2 + f2(0, z)d
2 + f3(0, z)d2d4,
The event horizon area:
AH = Vd4aa
dz
f1f2fd43
= 2Vd
42(d4)/2a
a + b a
adz (b
z)(d5)/2F1F2Fd43
The Hawking temperature:
T =1
4a
a + b
2
F0
F1,
Mass: (from the asymptotics of f0)
f0 1 16GM(d 2)Vd2(r2 + z2)(d3)/4
+ . . . .
the Smarr law: (test of numerics!)
M =
d
2
4G(d 3)T AH .
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However, all solutions have a conical singularity!
(the same pathology of the d = 5 static black ring)
the value of the conical excess of for a < z < b is
= 2
1
b + a
b a
F2
F1
.
the relative angular excess (1 0):
=/(2)
1 /(2),
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(left): The relative angular excess is shown as a function of the ratio between the two length scales a/b.
(right): The scale free ratio AH/M(d2)/(d3) is shown as a function of the relative angular excess . The value of the
parameter cd there is cd = (d2)/(16)(d2)/(d3)V1/(d3)d2 and has been choosen such that the point (1,1) on the plot
corresponds to the Schwarzschild-Tangerlini black hole.
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Further remarks/Conclusions
Q: Charged static solutions?
A: Yes! (however, the same pathology...)
electrically carged solutions generated from vacuum via an Harrison transformation
I =1
16G
ddxg
R 1
2g 1
4e2aF2
a second Harrison transformation = regular solutions in a magnetic Melvin Universe (details in
e-Print: arXiv:0904.2723)
situation similar to the d = 5 static black rings...
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Q: rotating solutions? d > 5 black rings? (within this approach)
A: very likely. (however, difficult numerical problem (3D eqs., unclear metric ansatz etc) not
impossible...)
Q: asymptotically (A)dS solutions with this method? (e.g. d = 5 AdS black rings)
A: unlikely
generalizations (currently under study):
Schwarzschild-Tangerlini + S2 Sd4 solution
new solutions with bubbles
