Exact constraints on D$\leq 10$ Myers Perry black holes and the Wald Problem

0.65 MB·English

by  Jason Doukas

YITP-10-86
Exact constraints on D≤ 10 Myers-Perry black holes and the Wald Problem
Jason Doukas1,∗
1Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan
(Dated: February 1, 2011)
Exact relations on the existence of event horizons of Myers-Perry black holes are obtained in
D≤10dimensions. Itisfurthershownthatnakedsingularitiescannotbeproducedby“spinning-
up” these black holes by shooting particles into their (cid:98)D−1(cid:99) equatorial planes.
2
PACSnumbers: 04.70.Bw,04.20.Dw,04.50.Gh
I. INTRODUCTION secondly they have been shown in various guises to pro-
vide a novel solution to the hierarchy problem [6]. Since
thesetheoriestypicallyoperatewhengravitationaleffects
Singularities can be found in many physical theories.
1
are strong, higher dimensional black holes are expected
1 Generally speaking, rather than accepting the singular-
to play a key role in the experimental interrogation of
0 ities as legitimate physical quantities we usually inter-
thesetheories; insomecasesobservationalsignaturesare
2 pret them as non-physical solutions or a breakdown of
the theory itself. In black hole spacetimes various types evenpredictedattheLHC[7]. Inordertotestthesethe-
n
ories and aid experimental searches it is useful to have
a of singularities can exist; there are point-like structures
J found inside Schwarzschild solutions, ring singularities exact constraints on the black hole parameters.

1 found inside Kerr black holes, and even more unusual In higher dimensions uncharged black hole solutions
3 topologies can arise in higher dimensions [1, 2]. with angular momentum have been found by Myers and
When these singularities are concealed by event hori- Perry [1]. To date a systematic study of the horizon
]
h zonstheycannotcausallyinteractwithoutsideobservers constraints on these black holes has not been performed
t and their existence is of little consequence. However, a (although the qualitative features have been inferred in
-
p surprising feature of the Kerr solution [3] is that for cer- [8] and some results in the special case of n equal non-
e tainvaluesofmassandangularmomentumitdescribesa zero spins can be found in the recent work [9]). This
h nakedsingularity, thatisasingularityunshieldedbyany is possibly due to the difficulty in finding algebraically
[
horizon. Specifically, the uncharged solutions possess a closed expressions to the equation ∆ = 0. We report in

2 horizon when thisletterthatsuchconstraintscaninfactbedetermined
v exactlyinalldimensionslessthanorequaltotenwithout
8 ∆=r2+a2−2Mr =0, (1) having to solve this equation.
1
1 wherer istheradialcoordinate,aistheangularmomen- Furthermore,wewillshowinsectionIVthatthesecon-
6 tum per unit mass, and M is the mass of the black hole. straintsallowustoanalyzewhetherornotrotatingblack
.
9 Solving this quadratic equation leads to the conclusion holes can be spun up into naked singularities in higher
0 that naked singularities are present when M < a. Such dimensions. Recently an effort was made to repeat the
0 solutionsareusuallydismissedbyappealingtothecosmic Wald gedanken experiments for Myers-Perry black holes
1
censorship hypothesis [4], which posits that no singular- [10]. While their conclusions were that black holes could
:
v ity can be visible from future null infinity. Perhaps the not be spun up into naked singularities, such analysis
Xi bestevidencethatKerrblackholescannotbeturnedinto could only be performed for singly rotating black holes
naked singularities can be found in the intriguing results or those with all angular momenta equal. Even then, for
r
a of Wald [5]. He considers an extremal black hole and dimensionsgreaterthanfive,numericalmethodswerere-
tries to create a naked singularity by injecting particles quired. In this letter we go one step further and show
withenoughangularmomentumtoachievetheabovere- that none of the Myers-Perry black holes can be spun
lation. However,itisfoundthateithertheparticlemisses up into naked singularities. This is done by generalising
the black hole completely or is repelled by gravitational the setup of Wald to d particles with arbitrary angular
dipole-dipole type forces and so a naked singularity is momentum and energy where a single particle is taken
never formed. to fall into the black hole along each of the equatorial
In recent times the exploration of extra spatial dimen- planes.
sionshasbeenadominantthemeinhighenergyphysics.
This paper is broken up into three sections; first we
One reason for this is that extra dimensions are neces-
describetheMPmetricthenwefindtheblackholemass
sary in quantum theories of gravity like string theory,
andangularmomentumconstraintsinsectionIII.Insec-
tion IV we repeat the Wald experiment in the higher
dimensional space and show that naked singularities can
not be created by classical processes once a MP black
∗ Email: [email protected] hole is formed.
2
II. THE MP METRIC where κ is the surface gravity:
∂ ∆(r )
SpinningblackholesinDspacetimedimensions,η = κ= r h . (9)
µν 4Mrσ
diag(−1,+1,··· ,+1), become somewhat more compli- h
cated compared with their 4 dimensional counterparts. However, we note that Π satisfies the differential equa-
Due to the symmetries of flat space an independent an- tion:
gularmomentumvariablea existsforeachofthe(cid:98)D−1(cid:99)
i 2 d
Casimirs of the little group of SO(D − 1). Since the (cid:88)
r∂ Π+ a ∂ Π=2dΠ, (10)
metricshaveslightlydifferentformsforoddandevendi- r i ai
mensions it will be convenient to define the variables d i=1
and σ: which can be used to write the area in the more conve-
nient form:
(cid:40) (cid:40)
D−2, if D is even 1, if D is even
d= D2−1, if D is odd. σ = 2 if D is odd. (2) A=2MAD−2rh. (11)
2
Since the location of the horizons are slightly different
Note that d is just the number of angular momentum for odd and even dimensions it will be convenient in the
parameters. Then the MP metric can be written: next section to analyze these cases separately.
d
(cid:88)
ds2 =−dt2+(2−σ)r2dα2+ (r2+a2)(dµ2+µ2dφ2)
i i i i III. MASS AND ANGULAR MOMENTUM
i=1 RELATIONS
(cid:32) d (cid:33)2
(Π−∆) (cid:88) ΠF
+ dt− a µ2dφ + dr2, (3)
ΠF i i i ∆ A. D-even
i=1
where[11], In D-even dimensions the condition for the location of
the horizon is:
(cid:88)d (cid:88)d a2µ2
1= µ2+(2−σ)α2, F ≡1− i i ,
i r2+a2 d
i=1 i=1 i ∆=Π−2Mr =0, Π=(cid:89)(r2+a2). (12)
d i
∆≡Π−2Mrσ, Π≡(cid:89)(r2+a2). (4) i=1
i
In their original work Myers and Perry pointed out that
i=1
equation (12) is a polynomial in r of order (D − 2),
From these metrics several other important properties
andthereforealgebraicallyclosedsolutionscouldonlybe
can be established [1] which we mention here for later
guaranteed for D = 4 and D = 6. They further noted
use. Firstly the mass and angular momenta of the black
thatthepolynomialisnotcompletelygenericsosolutions
hole are given by:
forlargerDmightbepossible,buttothepresentauthors
knowledge in the general case no such results have been
(D−2)A
M= D−2M, (5) reported.
8πG
The method we take, while different to the above,
2
Jxiyi = Ma , (6) makes use of many of the same arguments presented in
D−2 i
[1]. The necessary observations are that for all r > 0
∂ Πisastrictlyincreasingfunctionthatiszeroatr =0.
where A is the area of the (D−2)-sphere, A = r
D−2 D−2
2π(D−1)/2Γ((D−1)/2)−1, and G is Newton’s constant. Since M > 0 there is exactly one real and positive min-
ima, r˜, of ∆: ∂ ∆(r˜) = 0. One can further show that
SincethemassparameterM isproportionaltothemass, r
∆(r = 0) ≥ 0 (where equality occurs iff one or more of
M, we will often refer to M simply as the mass.
thespinparametersiszero)and∂ ∆(r =0)<0. There-
Theangularvelocityofthehorizonintheplanedefined r
fore there are only three possibilities for the number of
by µ =1 is given by
i horizons, depicted in figure 1; either ∆ never crosses the
a axis leaving no solutions, just touches the axis in a de-
ω = i , (7)
i r2 +a2 generate solution or completely crosses the axis giving
h i
two distinct horizons.
where r is the radius of the outer horizon. The area of Our strategy to find the allowed black hole parameter
h
the horizon as originally presented [1] is: spaceproceedsasfollows. Letr˜betheuniqueminimaof
∆ for positive r. Then the black hole will have a horizon
A= AD−2M (cid:32)D−3−2(cid:88)d a2i (cid:33), (8) if:
κ r2 +a2 ∆(r˜)≤0, r˜>0. (13)
i=1 h i
3
(cid:68)
Weobservethatuponredefiningr˜ intermsofasquared
h
variable, the equation has algebraically closed solutions
for all even D ≤10.
This polynomial can also be used to find an upper
bound on the extremal radius, which may be useful for
example in larger than ten dimensions. Since P is less
2
than zero at the origin and equal to zero at the extremal
(cid:142) r
r horizonanupperboundonr˜h canbedeterminedbyfind-
ingavalueofrforwhichthepolynomialispositive. Since
FIG. 1: Thick line: ∆(r˜)>0 there is a naked it is only the constant term which is negative it is easy
singularity. Thin line: the minimum occurs at the to see,
r-intercept there is a degenerate horizon. Dashed line:
∆(r˜)<0 horizons occurs at r-intercepts. r˜h ≤min|ai|. (20)
i
As an example of how the polynomial can be used to
The reason for restricting, r˜, to positive values is that findanexactrelation,wecalculatetheconstraintinD =
when ∆(r˜ = 0) = 0, then the horizon at r˜ will have 6. In this case the polynomial reads:
zero area (see equation (11)) and so in this case a naked
singularitywouldbeexposed. Thereforestrictinequality 0=3r˜4 +(a2+a2)r˜2 −a2a2, (21)
h 1 2 h 1 2
must be assumed in this case (i.e., when one or more of
the spin parameters is zero). From here on this will be with solution given by
implied in our definition of the inequality symbol.
(cid:112)
Sincetheinequalityissaturatedwhen∆(r˜)=0,equal- −(a2+a2)+ (a2+a2)2+12a2a22
r˜2 = 1 2 1 2 1 , (22)
ity occurs iff r˜is also an r-intercept. Therefore, h 6
M ≥ Π(r˜h), (14) so the D =6 Myers-Perry black hole has a horizon iff
2r˜
h
(r˜2 +a2)(r˜2 +a2)
where r˜h is defined as a solution to the simultaneous set M ≥ h 1 h 2 . (23)
2r˜
of equations h
∆(r˜ )=0, ∂ ∆(r˜ )=0. (15) Itissatisfyingtonotethatifweputa =0theconstraint
h r h 2
becomes M > 0 reproducing the well known result that
The solution to these equations defines a surface in the
in even dimensions there is always a horizon if one or
parameter space for which the horizons are degener-
more of the spin parameters is zero [1].
ate. Since we have two constraint equations we can find
M˜({a }) and r˜ ({a }) solely in terms of the angular mo- In figures 3b and 2b we plot the allowed parameter
i h i space for black holes normalised with M = 1 for D =
mentum parameters, where M˜ and r˜ are the extremal
h 6 and D = 8 respectively. These figures nicely agree
mass and radius for a given set of {a }.
i with the qualitative features predicted by the technique
It is important to emphasize that finding the degen-
described in [8].
erate subspace turns out to be significantly easier than
solvingequation(12). Furthermore,oncer˜ isfounditis
h
arelativelystraightforwardtasktoinputitintoequation
B. D-odd
(14), and thereby obtain the constraint. Eliminating M
from equations (15) results in the single equation:
Next we consider the D-odd case. In this case, it is
0=r˜h∂rΠ(r˜h)−Π(r˜h). (16) more convenient to use l =r2. The location of the hori-
zon is found by the equation:
Using the product expansion:
j j d
(cid:89)(r2+a2)=(cid:88)r2iAj−i, (17) ∆=Π(l)−2Ml=0, Π=(cid:89)(l+a2). (24)
i j i
i=1 i=0 i=1
with the coefficients conveniently defined as
As pointed out by Myers and Perry, by Galois’ theorem
Ak = (cid:88) a2 a2 ...a2 , (18) this equation has closed solutions for D = 5,7,9 (and
n ν1 ν2 νk even a closed solution in terms of elliptic functions for
ν1<ν2<...νk D = 11). Our method unfortunately does not get any
where vk are summed over the range [1,n], 0 ≤ k ≤ n further in the number of dimensions. However, it does
and A0k ≡1, equation (16) can be written as allow the constraint to be easily determined.
Starting from equation (24), as in the even case ∂ Π is
d l
P2(r˜h)=(cid:88)(2i−1)r˜h2iAdd−i =0. (19) strictly increasing for l>0, however now ∂lΠ(l=0) can
be greater than zero. This implies ∆ will have a unique
i=0
4
minimum for l > 0 iff 2M > ∂ Π(l = 0). First consider
l
thesolutionsforwhich2M ≤∂ Π(l=0). Because∆(l=
l
0) ≥ 0 and ∆(l → ∞) = +∞ and because there are no
turningpointsforl>0theonlypositionahorizoncould
occur at is l = 0, but this would be a zero area horizon
and would therefore expose a singularity. Therefore, we
must have 2M > ∂ Π(l = 0). Since ∆(l = 0) ≥ 0 we
l
find that we have the same possibilities for the number
of horizons as in the even case. Performing the same
analysis again we find that:
(a)D=7 (b)D=8
Π(˜l )
˜l ∂ Π(˜l )−Π(˜l )=0, M ≥ h . (25) FIG. 2: The solid region indicates parameters for which
h l h h 2˜l black holes have a horizon. M =1. (left) D=7, the
h
prongs extended out towards infinity, in the a =0
i
Where again the inequality in equation (25) must be re-
plane the remaining two spins are bounded by
placed by a strict inequality when ˜l = 0. The polyno-
h hyperbolas (31). (right) D=8, in this case the a =0
i
mial for the odd case is:
planes are always part of the allowed solutions.
d
P (r˜ )=(cid:88)(i−1)˜liAd−i =0. (26)
1 h h d
i=0 C. Other MP varieties
Using a similar argument to the one used in the even
case we can use this polynomial to find an upper bound Since solutions with the full set of non-zero angular
on the extremal radius: momentum parameters can be difficult to examine, sev-
eralvarietiesoftheseblackholesarecommonlyexplored.
√
r˜h ≤min{ aiaj}. (27) The most important of these being those with all spins
i(cid:54)=j
equal, these are also known as cohomogeneity-1 black
It is easy to verify that for D = 5 the polynomial holes. We find for these that:
becomes ˜l2 =a2a2 and we recover the known relation:
h 1 2 (cid:40)
(2d)d|a|(2d−1)(2d−1)1/2−d, if D is even
2M ≥2|a a |+a2+a2. (28) 2M ≥
1 2 1 2 a2d−2dd(d−1)1−d, if D is odd,
When D =7 we have: (32)
(cid:18) (cid:19) inagreementwithwhathasbeenpresentedpreviouslyin
1 1
˜l = −(a2+a2+a2)+ (a2+a2+a2)2+p , the literature [12].
h 6 1 2 3 p 1 2 3
√ Itisinterestingtoconsiderthelargedimensionlimitof
p3 =−(a2+a2+a2)3+54a2a2a2+6 3 (29) thesesolutions. Whenthenumberofangularmomentum
1 2 3 1 2 3
(cid:113) parameters, d→∞, the solutions will have a horizon for
× −(a2+a2+a2)3a2a2a2+27(a4a4a4),
1 2 3 1 2 3 1 2 3 a < 1, regardless of the value of M > 0. In the same
limit, for a ≥ 1 the mass would need to be infinitely
and
large to avert naked singularities.
M ≥ (˜lh+a21)(˜lh+a22)(˜lh+a23). (30) While equating all the spin parameters greatly sim-
2˜l plifies the solutions much of the interesting behaviour is
h
lostinthisprocess. However,onecangeneralisethisidea
Equation (29) actually defines three solutions; the cube to higher cohomogeneity so that some of the features of
root must be taken so that ˜l is non-negative. When multiple parameters can be probed while simultaneously
h
we set one spin to zero, we get the expected exceptional keeping a lot of symmetry. Cohomogeneity-2 black holes
behaviour ˜l = 0 and the bound is replaced by a strict are constructed by taking the whole set of non-zero spin
h
inequality. In fact, in any odd dimension for a =0: parameters and setting each one of them to either the
1
value a or the value b. Plots of the angular momenta
d
1(cid:89) constraints for cohomogeneity-2 black holes can be seen
M > a2. (31)
2 i in figure 3. Likewise the process can be generalised to
i=2 cohomogeneity-3 black holes where three sets of param-
Furthermore, setting one more spin to zero gives the eters a,b and c are used, we show the cohomogeneity-3
usual unbounded behaviour [1]. black holes in dimensions D =9 and D =10 in figure 4.
In figures 3a and 2a we plot the black hole parameter Next we will use the constraints obtained to show, us-
space (M = 1) for D = 5 and D = 7 dimensions re- ing arguments similar to that of Wald, that in higher
spectively. This again validates the qualitative features dimensions MP black holes can not be spun-up past the
predicted by the technique described in [8]. extremal solution.
5
(a)D=5;a=a1,b=a2 (b)D=6;a=a1,b=a2 (a)D=9;a=a1=a4,b=a2, (b)D=10;a=a1=a4,
c=a3 b=a2,c=a3
FIG. 4: Cohomogeneity-3 black hole solids M =1.
bythrowinginparticleswithorbitalangularmomentum.
We consider d particles falling along each of the d-
equatorialplanes[13]. Eachparticleisgivenanindepen-
dent energy E and angular momentum L .
i i
From our previous results the extremal solution was
given by:
Π(r˜ ,a )
(c)D=7;a=a1=a3,b=a2 (d)D=8;a=a1=a3,b=a2 M˜ = h i . (33)
2r˜σ
h
Usingequations(16)or(25)itcanbeshownthat∂ M˜ =
r˜h
0 therefore, as we expect, the extremal mass is only a
function of the spin parameters and we can write:
dM˜ =(cid:88)d ∂M˜ da = 1 (cid:88)d a ∂Πdai, (34)
∂a i 2r˜σ i∂a a
i=1 i h i=1 i i
Thespinparameterisafunctionofboththeangularmo-
mentuminthei-thdirectionandthemass(5),therefore,
the spin parameter will change when there is an increase
in the mass of the black hole even if there is no angular
(e)Db==9;aa2==aa14=a3, (f)D=b=10a;2a==aa41=a3, momentum absorbed in that direction. Since dM˜ = ET
where E is defined as the sum of all the energies E we
T i
FIG. 3: Cohomogeneity-2 black hole regions M =1. obtain:
Plots (a) and (b) are the usual D =5 and D =6
(cid:18) (cid:19)
da 1 (D−2)L
dimensional black hole regions with a unique parameter i = i −E . (35)
on each axis. In the other cases some of the parameters ai M˜ 2 ai T
have been equated as shown in the subtext.
First we assume that none of the spins are zero. At the
extremalradiusequation(10)reducestotherelationship:
IV.MCYLEARSSS-PICEARLRLYYBSLPAINCNKINHGOLUEP A (cid:88)d ai∂Π∂(ar˜h) =(D−3)Π(r˜h), (36)
i
i=1
Fromouranalysisofthehorizonstructurewewereable Usingthiswefindthatfortheparticlestocreateanaked
todeterminetheconditiononM andai underwhichthe singularity the total energy absorbed by the black hole
black hole was an extremal solution. Having these re- must be less than:
lations are equivalent to having in the 4D case the well
d d
known equation M2 = a2+e2. It is then natural to see E <(cid:88) aiLi =(cid:88)ω L . (37)
whether the Wald analysis, for which the extremal con- T r˜2 +a2 i i
straintequationwascritical,canbecarriedoutinhigher i=1 h i i=1
dimensions. In effect, starting with an extremal solu- If one or more of the spin parameters are zero then r˜ =
h
tion, we want to see whether we can destroy the horizon 0. In this case there is no extremal solution since the
6
horizon area vanishes. Nevertheless, our approach would Given that:
still be valid for arbitrarily small but non-zero values of
∆r2
one or more spin parameters. g2 −g g = ≡ð (44)
Wenowinvestigatehowmuchenergytheinfallingpar- tφ φφ tt ΠF
ticles can have at the horizon. Since all d equatorial
it is relatively straight-forward to invert the metric and
planes are symmetrical, we only need to consider the
find that along the trajectory E satisfies the equation:
geodesic of a single particle in the metric defined along
the µ1 = 1 equatorial plane [12]; the following analysis 0=gφφE2+2LEgtφ+gttL2+(δ−grrr˙2)ð (45)
will be identical for each of the other infalling particles.
In this plane the metric (3) becomes: Solving this quadratic equation, keeping the solution for
which the energy at infinity is positive, we obtain:
(cid:18) (Π−∆)(r2+a2)(cid:19) Πr2
ds2 =− 1− 1 dt2+ dr2
1 (cid:18) a2(ΠΠr−2 ∆)(r2+a2)(cid:19)∆(r2+a21) Ei ≥−Lgiφgφtφ. (46)
+ r2+a2+ 1 1 dφ2 The relevant metric components at the horizon are:
1 Πr2 1
− 2a1(Π−Π∆r)2(r2+a21)dtdφ1. (38) gtφ =−a1(r˜h2r˜h2+a21), gφφ = (r˜h2 +r˜h2a21)2. (47)
We are then able to deduce that the sum of the energies
The geodesic equation [5, 14] is given by
of all the d particles at the horizon indeed satisfies:
D2xµ d2xµ dxρdxσ
= +Γµ =0, (39) d
Ds2 ds2 ρσ ds ds E ≥(cid:88)L ω , (48)
T i i
i=1
which can be derived from the Lagrangian
which is precisely the inequality necessary to prevent
1 dxµdxν
L= g (x) , these processes from destroying the horizon. Therefore
2 µν ds ds nakedsingularitiescannotbeformedbyspinningupMP
= 1(g t˙2+g φ˙2+2g t˙φ˙+g r˙2). (40) black holes.
2 tt φφ tφ rr Itisworthmentioningthatevenifablackholesatisfies
the constraints presented it may not be stable; classical
Since the Lagrangian is independent of t and φ there
instabilities [15] are known to arise in the ultra spinning
are two conserved quantities along the particle worldline
regimes. InarrivingatourmainresultinsectionIVclas-
which are associated with the energy and angular mo-
sical particles obeying well-defined geodesic trajectories
mentum at infinity respectively:
were used, it would be interesting to investigate whether
the quantum nature of these particles alters our conclu-
∂L
−E =pt = ∂t˙ =gttt˙+gtφφ˙, (41) sions [16].
The author thanks W. Naylor for encouraging him to
∂L
L=p = =g φ˙+g t˙. (42) look for these constraints and H. T. Cho & A. S. Cor-
φ ∂φ˙ φφ tφ nell for discussions. The author also thanks D.Page who
at the APCTP focus program on the frontiers of black
We consider both time-like (δ = −1) and null (δ = 0 )
holephysics,Pohang,noticedaninconsistencyassociated
trajectories. Then from the particle 4-velocity we have:
with an error in an earlier version of equation (35). This
workwassupportedbytheJapanSocietyforthePromo-
dxµdxν
δ =g =2L. (43) tion of Science (JSPS), under fellowship no. P09749.
µν ds ds
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