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Pair Production of Topological anti de Sitter Black Holes

Pair Production of Topological anti de Sitter Black Holes

R.B.Mann1

Dept.of Physics,University of Waterloo Waterloo,ONT N2L3G1,Canada

PACS numbers:04.70.Dy,04.40.Nr,04.60.-m

July30,1996

Abstract

The pair creation of black holes with event horizons of non-trivial topology is described.The space-times are all limiting cases of the cosmological C metric.They are generalizations of the(2+1)dimen-sional black hole and have asymptotically anti de Sitter behaviour.Domain walls instantons can mediate their pair creation for a wide range of mass and charge.

1email:mann@avatar.uwaterloo.ca

Pair creation of black holes continues to a?ord us interesting insights into quantum gravity and the relationship between entropy and the number of quantum states of a black hole.It is a tunnelling process in which the mass-energy of the created pair of black holes is balanced by their negative potential energy in some background ?eld,such as that of an electromagnetic ?eld [1],a positive cosmological constant [2],a cosmic string [3]or a domain wall [4].The amplitude for the process is approximated by e ?I i ,where I i is the action of the relevant instanton i.e.a Euclidean solution to the ?eld equations which interpolates between the states before and after a pair of black holes is produced.

The C metric solution of the Einstein-Maxwell equations may be interpreted as describing two oppositely-charged black holes undergoing uniform acceleration [5].It contains conical singularities,which in general cannot be eliminated at both poles.These singularities are interpreted as representing “rods”or “strings”which provide the force necessary to accelerate the black holes.Removal of these singularities is generally obtained by adding additional forms of stress-energy which generate the background ?eld required to provide the necessary accelerating force.

The purpose of this paper is to point out that the cosmological C -metrics [6]contain a rich array of Euclidean instantons that mediate pair production of black holes whose topology is of arbitrary genus.The genus zero solution corresponds to the set of Reissner-Nordstr¨o m de Sitter instantons studied previously in the context of cosmological black hole pair production [2].The higher genus solutions are asymptotically anti-de Sitter,and correspond to instantons that are 4dimensional generalizations of the 3dimensional black hole [7].Pair production of these black holes can take place in the presence of domain walls whose topology is the same as that of the produced black hole pairs.

The cosmological charged C -metric solution is [6]

ds 2=1A 2(x ?y )2

H (y )dt 2?H ?1(y )dy 2+G ?1(x )dx 2+G (x )d?2 ,(1)where H (z )=a ?bz 2?2mAz 3?q 2A 2z 4=G (z )?kl 2

A

(2)Λ=3k/l 2being the cosmological constant and A the acceleration parameter and k =±1.The gauge ?eld is

F M =?qdx ∧d?,F E =?qdt ∧dy,(3)

in the magnetic and electric cases respectively.Under the coordinate transformation y =x ?1/Ar ,t =Au ? y dz/H (z ),the metric (1)may also be written as

ds 2=H (x ?1Ar )A 2r 2du 2?2dudr ?2Ar 2dudx +r 2 G ?1(x )dx 2+G (x )d?2

.(4)

The electric ?eld becomes F E =?qdu ∧(Adx +dr

r 2)and the magnetic ?eld is unchanged.

The inner,outer and acceleration horizons are given by the ?rst three roots of H in ascending order respectively.The coordinate r ∈(0,∞),whereas x lies between either the smallest (x 1,x 2)or largest (x 3,x 4)pair of roots of G so that the metric (4)has proper signature.

Removal of conical singularities in the (x,?)sector implies that

G (x 3)=?G (x 4),(5)

with ?periodically identi?ed with period ??=4π/|G (x 4)|.This condition can only be satis?ed if x 3=x 4.A similar analysis for the smallest pair of roots implies that conical singularities can be removed if x 1=x 2.Conical singularities may also be avoided if x 2=x 3,in which case the point x =x 3is an in?nite proper distance from any allowed value of x .

Hence removal of conical singularities in the cosmological C metric implies that G (x )must have a double root.This apparently implies that the (x,?)sector shrinks to a point,but this is just a poor choice of coordinates.The proper distance between any adjacent pair (x i ,x i +1)of roots of G is actually ?nite,as can be seen by setting ?=φ/ and and x = f (λ),where the roots x i = f (λi )and x i +1= f (λi +1)coincide as →0.The parameters of G must then be chosen so that it has a double root at x =x i =x i +1as →0.This constrains the acceleration parameter A in terms of m ,q and l .

1

For all possible pairs of degenerate roots the metric(4)becomes,as →0,

ds2=?V(r)dT2+

dr2

V(r)

+r2

dλ2+s2(λ)dφ2

,(6)

where a coordinate transformation on u has been performed,

V(r)=?k

l2

r2+b?

2m

r

+

q2

r2

(7)

andφhas period2π.

If the largest two roots of H are degenerate then b>0,and the class of metrics obtained are of the Reissner-Nordstr¨o m(anti)-de Sitter(RN(a)dS)type,of mass m and charge q.However there is a surprise in that the parameter b is completely arbitrary.A simple rescaling of parameters and coordinates allows b to be set to unit magnitude without loss of generality if it is nonzero.If the middle two roots of H are degenerate,then b=?1,and if the largest three roots are degenerate then b=0.There are therefore three possible forms for the function s(λ):

b=1,k=±1s(λ)=sin(λ)(8)

b=0,k=?1s(λ)=1(9)

b=?1,k=?1s(λ)=sinh(λ)(10) It is easily checked that all of these spacetimes satisfy the Einstein-Maxwell equations with cosmological constant.The gauge?eld becomes

F M=qs(λ)dλ∧dφF E=?q

r

dT∧dr(11)

in the magnetic and electric cases respectively.

The regularity requirements for the C metric in the b=1case for positiveΛhave been discussed previously[2].The other spacetimes,however,have been overlooked in previous studies.The b=1,Λ<0 case is simply Reissner-Nordstr¨o m anti de Sitter spacetime.

The two remaining spacetimes all haveΛ<0and b<0.Their q=m=0versions were studied recently by Aminneborg et.al.[8]who showed that they can be understood as four-dimensional analogues of the three-dimensional black hole[7],by compactifying the(λ,φ)sector.For the metrics derived here this construction can also be carried out.For b=1the topology of this sector is the2-sphere.For b=0,the coordinateλmay be identi?ed,and the(λ,φ)sector is a torus,whose unit area shall be chosen to be4πby identifying the lambda-coordinate with period2.For b=?1,the identi?cations may be carried out by mapping the(λ,φ)sector to the Poincar′e disk under the transformationρ=tanh(λ/2),yielding

dλ2+s2(λ)dφ2=

1

(1?ρ2)2

dρ2+ρ2dφ2

(12)

where0≤ρ<1.The Poincar′e disk has an isomorphism group SO(2,1).Identifying points on the disk under any discrete subgroup of SO(2,1)yields a compact two-dimensional space of negative curvature,which necessarily has genus g≥2.The unit area of such surfaces is4π(g?1).These spaces may be constructed by symmetrically placing a polygon of4g sides at the center of the Poincare disk and identifying opposite sides.The edges of the polygon are geodesics of the Poincar′e disk;these are circles whose extensions are orthogonal to the disk boundary.The simplest case is the octagon with g=2.

The b≤0constructions hold for all values of r and T in(6).An analysis of the behaviour of V(r)in(7) indicates that in this case it has at most two roots,corresponding to an inner and outer horizon,as with the usual RNadS metric.For b=0,there will be two horizons,provided

27l2m4≥16Q6(13) with the extremal case saturating the inequality.For nonzero b there will be event horizons provided

m2≤l2

27

16?24e2b?16b

1?e2b e2+6b2e4+16

1?e2b

e6

(14)

2

where e =2√2q .

If b =0(the genus 1case)then the range of e is from 0to 1.Analysis of (14)in this case indicates that event horizons can (but need not)exist provided q

The topology of the outer event horizon is H 2g ,where H 2g is a two-dimensional surface of genus g .The

entire spacetime has topology R 2×H 2g .An analysis of the quasilocal mass [11]and charge contained within

a surface of topology H g at a ?xed value of the coordinate r centered about the origin indicates,in the limit of large r ,that q (|g ?1|+δg,1)and m (|g ?1|+δg,1)are the conserved charge and mass parameters of the black hole.

Pair production of these black holes may be achieved using the domain wall construction of ref.[4].The

topology of the Riemannian section is R 2×H 2g ,where the R 2factor is like a bell.Two copies of this manifold

may be matched together at a radius r determined by the matching condition [9,10] =2πσ(15)

where σis the energy per unit area of the domain wall,whose topology is S 1×H 2g ,and the overdot refers

to the derivative with respect to Euclidean proper time.The Riemannian section is two bells glued together

along their open ends at a ridge;it has topology S 2×H 2g and corresponds to static a domain wall con?guration

with two surfaces at which the Killing ?eld d dτvanishes,where τis the Euclidean time parameter.

Equation (15)may be interpreted as the equation describing the motion of a ?ctitious particle in a potential v =V ?(2πσr )2.Static solutions,which have energy zero,may be obtained by solving (15)under the condition ?v/?r =0.Non-static solutions are obtained by matching the period of the RNadS black hole with an integer multiple of

βw = rmax rmin dτ= rmax rmin dr V (V ?(2πσr ))(16)which is the period of the domain wall evolution between the extrema rmin and rmax where ˙r vanishes.For simplicity,I shall consider static solutions;details including non-static solutions shall appear in a forthcoming paper [12].These occur when

r s =3mb +√22

(17)if b =0and at r s =2q 2

3m if b =0.Note that for b <0there is no solution with zero charge.The squared

mass of the created black holes is m 2=154 36q 2+b 4π2σ2l 2?1 l 3+ (l 2+12q 2(1?4π2σ2l 2)3 (18)for nonzero b and by

m 2=4q 33√ 4π2σ2l 2?1(19)

if b =0.

The Euclidean action for these instantons is I = d 4x √g ?R 16π+F 216π+L c +L d (20)where L c is the cosmological Lagrangian and L d the domain-wall Lagrangian.The former may be taken to be that of the squared ?eld strength of a 3-form or simply the constant 3k 8πl 2.The domain wall Lagrangian

can be that of a membrane current coupling to the 3-form [10]or that of a scalar ?eld Φwhose potential V (Φ)is everywhere positive [4](and so its Euclidean action is always negative).There are no boundary terms because the instantons considered here are compact and without boundary.

Using the Einstein ?eld equations (20)becomes

I = d 4x √g ?3k 8πl 2+F 216π

?V (Φ) (21)3

which yields in,say,the g=2case

P=exp

2πσr2s

s

?

q2

r s r+

(r s?r+)β+

k(r3+?r3s)β

l

?1

8πσ

(22)

for the probability of pair creation of the black holes in the magnetic case;the electric case is similar but entails the incorporation of an additional surface term that vanishes in the magnetic case[2].Here r+is the location of the outer horizon andβthe instanton period for the RNadS black hole.The above expression includes the case k=1(RNdS);if theσ-dependent terms are omitted,and r s is taken to be the location of the cosmological horizon,then the results of ref.[2]are recovered.The probability(22)is relative to that for creation of a domain wall with no black holes or relativisitic3-form.

To summarize,the C-metric has been shown to reduce to a set of metrics whose Euclidean sections can be interpreted as instantons corresponding to the pair creation of black holes with event horizons of arbitrary topology.The Lorentzian sections are spacetimes that are asymptotically anti de Sitter and which can be interpreted as generalizations of the(2+1)dimensional black hole.A more detailed study of these black holes and their pair creation will appear in a forthcoming paper[12].

Acknowledgements

This work was supported by the Natural Sciences and Engineering Research Council of Canada.I am grateful to J.Creighton for interesting discussions on this subject.

References

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