Accelerating Kerr-Newman black holes in (anti-)de Sitter space-time

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by  J. Podolsky

Accelerating Kerr–Newman black holes in (anti-)de Sitter space-time
J. Podolsky´1 and J. B. Griffiths2
1Institute of Theoretical Physics, Faculty of Mathematics and Physics,
Charles University in Prague, V Holeˇsoviˇck´ach 2, 18000 Prague 8, Czech Republic.
2Department of Mathematical Sciences, Loughborough University, Loughborough, Leics. LE11 3TU, U.K.
(Dated: February 7, 2008)
A class of exact solutions of the Einstein–Maxwell equations is presented which describes an accel-
erating and rotating charged black hole in an asymptotically de Sitter or anti-de Sitter universe.
The metric is presented in a new and convenient form in which the meaning of the parameters is
clearly identified,andfrom whichthephysicalpropertiesofthesolution canreadily beinterpreted.
I. INTRODUCTION theavailablefreedomismuchbetter usedtosimplify the
roots of these functions. In the rotating case, they ob-
6
tainedanewsolution[13]foranacceleratingandrotating
0 The Pleban´ski–Demian´ski [1] family of solutions is
black hole which differs from what is usually called the
0 known to include the case which describes an accelerat-
2 “spinning C-metric” [14]–[16] in which the linear terms
ingandrotatingblackhole. Significantly,theparameters
are set to zero. Surprisingly, it is the new solution of
n containedinthemetricincludeanarbitrarycosmological
HongandTeowhichrepresentstheNUT-freecase,while
a constantΛandelectricandmagneticchargeparameters.
J the“spinningC-metric”retainsNUT-likeproperties(i.e.
Thesolutionsmustthereforealsocontainparticularcases
0 which describe accelerating and rotating charged black part of the axis corresponds to a “torsion” singularity
3 which is surrounded by a region containing closed time-
holes in asymptotically de Sitter or anti-de Sitter uni-
like lines).
verses. However,intheoriginalformofthe metric,these
1
v special cases are not clearly identified. Moreover, that ForthecaseinwhichΛ=0,amoregeneralformofthe
0 form is not well suited for their physical interpretation. metricwaspresentedin[17]. Thishasconfirmedthatthe
3 As these solutions are now being used for various new “spinningC-metric”doesindeedpossessaneffectivenon-
1 purposes,itismostimportantthattheybebetterunder- zero NUT parameter. It also introduces new parameters
1 stood at the classical level. The purpose of the present which explicitly describe the acceleration and rotation
0 paper is to contribute to such an understanding, partic- of the sources. In general, it covers the complete family
6
ularly by expressingthe metric in a formthat is suitable ofexactsolutionswhichrepresentacceleratingandrotat-
0
/ for interpretation and in which the parameters involved ingblackholeswithpossibleelectromagneticchargesand
c have clear physical meanings. an arbitrary NUT parameter, and describes the internal
q
horizonandsingularitystructureasfarasthe associated
- For the non-rotating case, the Pleban´ski–Demian´ski
r acceleration horizon. The extension of this solution to
g family includes the C-metric whose analytic extension
boost-rotation-symmetric coordinates which cover both
: describes a causally separated pair of black holes which
v black holes is given in [18].
accelerate away from each other under the action of
i
X “strings” represented by conical singularities located It may also be mentioned that the Hong–Teo paper
r along appropriate sections of the axis of symmetry. [12], for the non-rotating case with Λ = 0, has already
a HawkingandRoss[2]haveusedthis solutionto describe been extremely useful in leading to a better understand-
the possible creation of a black hole pair by the break- ing of the higher-dimensional (rotating) black ring so-
ing of a cosmic string. The inclusion of a cosmological lution [19]. The improved factorizable structure intro-
constant could alternatively be considered to supply the duced in [12] was explicitly used in [20] and [21], and
potentialenergythatisnecessaryforsuchapaircreation this new version has now became commonly employed
process. Thishasbeenanalysedin[3]–[6],wherethecos- in more recent investigations of black rings such as [22]–
mological constant was included in the traditional way. [24]. To this form, we have here included both a Kerr-
However,asshownin[7],itwouldhavemademoresense likerotationandanarbitrarycosmologicalconstant. We
physically if Λ had been inserted in an alternative way. have also adopted more physically motivated (Boyer–
This has been investigated in [8, 9]. A particular case Lindquist-type) coordinates. It may be hoped that the
of the C-metric with Λ < 0 has also been used for the solution described here may enable further solutions in
costruction of a solution describing a black hole on the higher dimensions and different backgrounds to be ob-
brane [10, 11]. tained and analysed.
In addition, it was previously thought appropriate to Theimmediatepurposeofthepresentpaper,however,
use a coordinate freedom to remove the linear terms in istoclarifythephysicalinterpretationoftheclassofclas-
the quartic functions which characterise the Pleban´ski– sical solutions in 3+1-dimensions when the cosmological
Demian´ski family of solutions. This was thought to re- constant is non-zero. We will concentrate here on the
move the NUT parameter. However, Hong and Teo [12] physically most significant case in which the space-time
haverecentlyshownthat, atleastforsome specialcases, has no NUT-like properties.
2
II. ACCELERATING AND ROTATING relative to a natural null tetrad are
CHARGED BLACK HOLES WITH Λ6=0
1+αrcosθ
Ψ = m(1 iαa)+(e2+g2)
2
The Pleban´ski–Demian´skimetric coversa largefamily (cid:18)− − r iacosθ(cid:19)
−
3
of solutions which includes that of an accelerating and 1 αrcosθ
− ,
rotating charged black hole. Among the various sub- ×(cid:18)r+iacosθ(cid:19)
families identified in [25], we now present the following
1 (1 αrcosθ)4
mcaesteric as the most convenient form for this particular Φ11 = 2(e2+g2)(r2−+a2cos2θ)2, and Λ.
ds2 = 1 Q dt asin2θdφ 2 ρ2 dr2 Tityheasterin=di0ca,tθe=thπe.pTrehseenvcaenoisfhainKgeorfr-tlhikeecroinnfgorsminagluflaacr--
Ω2(cid:26)ρ2 − − Q 2
(cid:2) (cid:3) (1) tor Ω corresponds to conformal infinity. Thus, we may
ρ2dθ2 P sin2θ adt (r2+a2)dφ 2 , take the range of r as r (0,α−1secθ) if θ < π/2, and
−P − ρ2 − (cid:27) r (0, ) otherwise. Fo∈r θ (π,π], the r coordinate
(cid:2) (cid:3) ∈ ∞ ∈ 2
does not reach conformal infinity. In fact, an analytic
where extension through r = indicates [18] the presence of
∞
a second (mirror) region, as required for solutions ex-
Ω=1 αrcosθ, pressed in boost-rotation-symmetric coordinates [26].
−
As fully described in [17] for the case with Λ = 0,
ρ2 =r2+a2cos2θ,
conical singularities generally occur on the axis. How-
P =1 2αmcosθ ever, by specifying the range of φ appropriately, the
−
singularity on one half of the axis can be removed.
+ α2(a2+e2+g2)+ 1Λa2 cos2θ,
3 For example, that on θ = π is here removed by tak-
(cid:16) (cid:17)
ing φ 0,2π(1+ a a )−1 , where a = 2αm and
Q= (a2+e2+g2) 2mr+r2 (1 α2r2) ∈ 3 − 4 3
(cid:16) 1Λ(a2+r2−)r2. (cid:17) − a4 = −α2(cid:2)(a2+e2+g2)− 31Λa2(cid:1). In this case, the accel-
−3 eration of the “sources” would be achieved by “strings”
of deficit angle
This contains six arbitrary parameters m, e, g, a, α and
Λ which can each be varied independently. Besides the 8παm
δ = , (2)
cosmological constant Λ, these parameters have distinct 0 1+2αm+α2(a2+e2+g2)+ 1Λa2
physical interpretations: m is the mass of the black hole 3
(at least in the non-accelerating limit), e and g are its
connectingthemtoinfinity. Alternatively,thesingularity
electricandmagneticcharges,ameasuresits angularve-
onθ =0 could be removedby taking φ 0,2π(1 a
3
locity, and α is its acceleration. (The possible NUT pa- a )−1 , and the acceleration would then∈b(cid:2)e achiev−ed b−y
rameterlhasbeenputtozeroherealthough,asshownin 4
a “str(cid:1)ut” between them in which the excess angle is
[17],thePleban´ski–Demian´skiparameternmustthenbe
non-zero to avoid any NUT-like behaviour of the space-
8παm
time.) The metric is accompaniedby anelectromagnetic δ = . (3)
field F=dA, where the vector potential is − π 1−2αm+α2(a2+e2+g2)+ 31Λa2
er[dt asin2θdφ gcosθ[adt (r2+a2)dφ] Of course, the expressions (2) or (3) are closely related
A= − − − − . to the forces in the string or strut respectively and these
r2(cid:3)+a2cos2θ
shouldbe equal,at leastaccordingto Newtonian theory.
However, it may be noticed that the deficit/excess an-
As advocated (for the case with Λ=0) by Hong and gles are the same fractions of the range of the periodic
Teo [13] and clarified in [17], a coordinate freedom has coordinate in each case. Thus, they do correspond to
been used in the derivation of (1) to simplify the roots identical expressions for the forces in the string or strut
of the Pleban´ski–Demian´ski quartic function P˜ in this as expected, at least in the linear approximation.
more general case. In particular, we can put P˜(p) = In view of the fact that the complete space-time con-
(1 p2)P(p). This enables us to put p = cosθ, so that tains two accelerating black holes while the above coor-
−
we can work with conventional spherical polar coordi- dinates only cover one of these, and also in view of the
nates (rather than complicated Jacobian elliptic func- necessary presence of conicalsingularities, it may be ob-
tions) with θ [0,π] and φ a periodic coordinate whose served that the space-time is not strictly asymptotic to
∈
precise period will be determined below. In accordance de Sitter or anti-de Sitter space in all directions, except
with this approach, the simpler function P(cosθ) has in the weak field limit.
been adopted above. Let us now note the following special cases in which
The onlynon-zerocomponentsofthe curvaturetensor one of the parameters α, Λ or a vanishes respectively.
3
A. The Kerr–Newman–(anti-)de Sitter solution with φ unchanged, A=α and
1
When α = 0, the metric (1) reduces to that for the G(y)= Q(r), G(x)= sin2θP(θ).
α2r4 −
Kerr–Newman–(anti-)de Sitter space-time. It can be
However, the Boyer–Lindquist-type coordinates em-
expressed in standard Boyer–Lindquist-type coordinates
ployedhere seem to be physically more natural than the
[27] using the simple rescaling
Pleban´ski–Demian´ski-type coordinates x and y. (The
t=t¯Ξ−1, φ=φ¯Ξ−1, (4) transformation in t′ is required for the existence of the
axis at θ =0,π.)
where Ξ=1+ 1Λa2. This puts the metric in the form
3
∆ 2 ρ2 ρ2 C. The charged C-metric with a cosmological
ds2 = r dt¯ asin2θdφ¯ dr2 dθ2 constant
Ξ2ρ2h − i − ∆r − ∆θ
(5)
∆θsin2θ adt¯ (r2+a2)dφ¯ 2, For the case in which a=0, the metric (1) reduces to
− Ξ2ρ2 h − i the simple diagonal form
where 1 Q r2
ds2 = dt2 dr2
(1 αrcosθ)2(cid:18)r2 − Q
ρ2 =r2+a2cos2θ, −
r2
∆ =(r2+a2)(1 1Λr2) 2mr+(e2+g2), (6) dθ2 P r2sin2θdφ2 ,
r − 3 − −P − (cid:19)
∆ =1+ 1Λa2cos2θ.
θ 3 where
Formally, there is no need to introduce the constant P =1 2αmcosθ+α2(e2+g2)cos2θ,
−
rescaling Ξ in t and φ. However, this is included (at
Q=(e2+g2 2mr+r2)(1 α2r2) 1Λr4.
leastfor φ) so that the metric has a well-behavedaxis at − − − 3
θ =0andθ =π withφ¯ [0,2π). Itshouldalsobenoted This may be considered as a generalized and modified
∈
that the metric (5) only retains Lorentzian signature for form for the charged C-metric that was introduced re-
all θ [0,π] provided 1Λa2 > 1. cently by Hong and Teo [12]. It describes a black hole
∈ 3 −
of mass m and electric and magnetic charges e and g
which accelerates along the axis of symmetry under the
B. Accelerating and rotating black holes in a action of forces representedby a topological(string-like)
Minkowski background
singularity,forwhichαis the acceleration,withanaddi-
tionalcosmologicalconstant. WhenΛ=0theblackhole
When Λ = 0, the metric (1) corresponds to that of horizons (7) and the accelerationhorizonat r =α−1 are
HongandTeo[13](anddescribedindetailin[17])which clearly displayed. However, when Λ = 0, the location of
6
representsanacceleratingandrotatingpairofblackholes all horizons is modified.
without any NUT-like behaviourand in whichthe accel- FurtherpropertiesofthechargedC-metricinadeSit-
erationisidentifiedasα. Inthiscase,ifm2 a2+e2+g2, teroranti-deSitterbackgroundhavebeenanalysedin[7]
≥
the expression for Q factorises as and[28]–[32],usinghoweverdifferentandlessconvenient
formsofthemetrictothatpresentedabove. Whentrans-
Q=(r− r)(r+ r)(1 α2r2), formedtoboost-rotation-symmetriccoordinates,thenew
− − −
form above has a particularly simple structure, at least
where
when Λ=0, as given in [18].
r± =m m2 a2 e2 g2. (7)
± − − −
p
III. ACCELERATING TEST PARTICLES AND
The expressions for r± are identical to those for the
THE NATURE OF THE NEW COORDINATES
locations of the outer and inner horizons of the non-
accelerating Kerr–Newman black hole. However, in the
present case, there is another horizon at r = α−1 which To elucidate the nature of the new coordinates intro-
duced in the metric (1), we now consider the weak field
is already familiar in the context of the C-metric as an
limit in which m, a, e and g are reduced to zero while α
acceleration horizon.
and Λ remain arbitrary. The resulting metric is
In this case with Λ = 0, the metric is equivalent to
that given in equations (11)–(13) of Hong and Teo [13], 1
iunsewdhhicehrethbeyirthcoeotrrdainnsaftoersm(at′t,ixon,y,φ)arerelatedtothose ds2 = (1−αrcosθ)2h(cid:0)1−(α2+ 13Λ)r2(cid:1)dt2 (8)
dr2
r2(dθ2+sin2θdφ2) ,
t′ =−α(t−aφ), x=cosθ, y =1/(αr), −1−(α2+ 13Λ)r2 − i
4
which reduces to the standard form of the Minkowski or Z = Z = 0, Z = const. This actually represents the
2 3 4
(anti-)de Sitter metric in static coordinates when α=0. trajectoriesofapairofuniformlyacceleratedparticlesin
Let us first observethat for Λ=0, the transformation a de Sitter or anti-de Sitter space-time (see e.g. [7, 28,
29, 33]).
√α−2 r2
In the alternative case of a test particle with
T = − sinh(αt),
1 αrcosθ small acceleration in an anti-de Sitter universe, for
−
which α2+ 1Λ<0, the metric (8) corresponds to the
√α−2 r2 3
Z = − cosh(αt), (9) parametrization
±1 αrcosθ
−
rsinθ r2 (α2+ 1Λ)−1
R= 1−αrcosθ , Z0 = q 1− αrcos3θ sin(q−(α2+ 13Λ)t),
−
leadstothestandardformoftheMinkowskilineelement
r2 (α2+ 1Λ)−1
ds2 =dT2 dZ2 dR2 R2dφ2, Z = q − 3 cos( (α2+ 1Λ)t), (12)
− − − 4 1 αrcosθ q− 3
−
confirming that all points with constant values of r, θ α (α2+ 1Λ)rcosθ
and φ are in uniform acceleration in the positive or Z = − 3 ,
1
negative Z-direction relative to the Minkowski back- 1 Λ (α2+ 1Λ)(1 αrcosθ)
ground. Inparticular,atestparticlelocatedattheorigin q3| |q− 3 −
r = 0 of the new coordinates has acceleration given ex- with Z ,Z as in (10). In this case, the trajectory r =0
2 3
actly by α as it moves along either of the trajectories represents the motion of a single uniformly accelerated
T =α−1sinh(αt), Z = α−1cosh(αt), R=0. test particle in an anti-de Sitter universe [28, 32, 34].
±
Similarly, when Λ = 0, the metric (8) just describes In fact, any world-line xµ(τ)=(t(τ),r ,θ ,φ ) in the
6 0 0 0
a de Sitter or anti-de Sitter universe but expressed in space-time (8), where r ,θ ,φ are constants and τ is
0 0 0
new accelerating coordinates. These space-times can be the proper time, represents the motion of a uniformly
represented as the four-dimensional hyperboloid acceleratedtestparticle. Its4-velocityisUµ =(t˙,0,0,0),
Z 2 Z 2 Z 2 Z 2 ǫZ 2 = 3/Λ, t˙ = 1/ gtt(r0,θ0) = const., and the 4-acceleration has
0 1 2 3 4
− − − − − constanpt components
in the flat five-dimensional space
g g
ds2 =dZ02−dZ12−dZ22−dZ32−ǫdZ42, Aµ ≡Uµ;νUν =(cid:16)0,2−gtttgt,rrr,2−gtttgt,θθθ,0(cid:17).
where ǫ=signΛ. SinceAµU =0,itisaspatialvectorintheinstantaneous
µ
Providedα2+1Λ>0,themetric(8)canbeexpressed rest frame orthogonal to the 4-velocity, and its constant
3
in this notation by the parametrization magnitude is
(α2+ 1Λ)−1 r2 (1 αr cosθ )2
Z0 = q 1−α3rcosθ− sinh(qα2+ 13Λt), A2 ≡−AµAµ =(α2+31Λ)1−−(α20+ 31Λ)0r02 −13Λ. (13)
(α2+ 1Λ)−1 r2 In particular, the uniform acceleration of a test particle
Z1 =±q 1−α3rcosθ− cosh(qα2+ 13Λt), dloecpaetneddeanttltyheofoθr0ig,iφn0ro=r Λ0.isFogrivtehnisexreaacstolyn,bwyeAm=ayαc,oinn--
rsinθsinφ (10) clude that the new form of the line element (1) may be
Z = ,
2 1 αrcosθ interpreted as using most convenient accelerated coordi-
−
nates in a Minkowski or (anti-)de Sitter background(8).
rsinθcosφ
Z = , (Ofcourse,theaccelerationisonlythatofarealphysical
3
1 αrcosθ
− particle when m is non-zero and there exists a physical
α (α2+ 1Λ)rcosθ cause that can be modelled by (2) or (3).)
Z = − 3 .
4
1 Λ α2+ 1Λ(1 αrcosθ)
q3| |q 3 −
IV. CONCLUSION
(Notice that, in this case, a possibility exists to perform
either of the limits α 0 or Λ 0.) The trajectory of
→ → The metric (1) is presented here as the most conve-
a test particle located at r =0 is given by
nient form with which to analyse the properties of a ro-
Z = (α2+ 1Λ)−1/2 sinh( α2+ 1Λt), tating and accelerating, possibly charged, black hole in
0 3 q 3 an asymptotically de Sitter or anti-de Sitter space-time.
(11)
Inparticular,theparametersemployedallpossessanex-
Z1 =±(α2+ 31Λ)−1/2 cosh(qα2+ 13Λt), plicit physical interpretation.
5
This form of the metric nicely represents the hori- both when α = 0 and when Λ = 0, so that it is actu-
zon and singularity structure of the solution. It cov- ally a better representation of an accelerating charged
ers the space-time from the singularity, through the in- blackholeintheabovebackgroundsthanthatgivenpre-
ner and outer black hole horizons, through the exterior viously in [7] (to which it is related by the rescaling
region, and even through the acceleration horizon. It t t 1+ 3α2, r r 1+ 3α2). In addition, the
also nicely describes the conical singularity that is re- → q Λ →− q Λ
metric functions depend on (powers of) r and cosθ only.
quired to produce the acceleration. However, it it does
More importantly, a non-vanishing Kerr-like rotation is
not represent the complete analytical extension of the
now also included.
space-time,either throughthe blackhole horizonsorbe-
yondtheaccelerationhorizon. Forsuchextensions,either
a Kruskal–Szekeres-like transformation or a transforma-
tion to boost-rotation-symmetric coordinates is required Acknowledgements
respectively. Such extensions would reveal multiple pos-
sible sources inside the black hole horizon and mirror, Theauthorsaregratefultotherefereesforsomehelpful
causally separated sources beyond the acceleration hori- comments on an earlier draft of this work, which was
zon. partly supported by grants from the EPSRC and (JP)
Let us finally note that the metric (1) has clear limits by GACR 202/06/0041.
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