Randall-Sundrum black holes and strange stars

PHYSICAL REVIEW D 67, 124015 ~2003!

Randall-Sundrum black holes and strange stars

Malcolm Fairbairn* and Veronique Van Elewyck†

Service de Physique The´orique, CP225, Universite´ Libre de Bruxelles, B-1050 Brussels, Belgium~Received 31 August 2002; revised manuscript received 28 March 2003; published 11 June 2003!

It has recently been suggested that the existence of bare strange stars is incompatible with low scale gravityscenarios. It has been claimed that, in such models, high energy neutrinos incident on the surface of a barestrange star would lead to catastrophic black hole growth. We point out that, for the flat large extra dimensionalcase, the parts of parameter space which give rise to such growth are ruled out by other methods. We then goon to show in detail how black holes evolve in the Randall-Sundrum two brane scenario where the extradimensions are curved. We find that catastrophic black hole growth does not occur in this situation either. Wealso present some general expressions for the growth of five dimensional black holes in dense media.

DOI: 10.1103/PhysRevD.67.124015 PACS number~s!: 04.70.Dy, 04.50.1h

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I. INTRODUCTION

The idea that the geometry of extra dimensions mightresponsible for the hierarchy between the scale of etroweak physics and the Planck scale is extremely intering. In these models, the mass scale associated with grais around a TeV but appears to be much higher due tosmall overlap of the extra dimensional graviton wave funtion with our standard model brane@1,2#. In a gravity theorywith 41d space-time dimensions and a fundamental scMF , one expects black hole production at energy densihigher thanMF

41d , so there has been a great deal of interin the possibility of black hole production at the next geeration of super-TeV scale colliders@3–5,31#. The idea thatcolliders might produce small black holes is at first alarminbut these black holes are so small that they are expectedecay via Hawking evaporation before they are able to inact with their surroundings and grow.

A different situation would arise if the black hole waproduced in an extremely dense medium like the interior oneutron star, as in that case the black hole might interact wanother particle before it decays, so that the Hawking evaration would be balanced by the accretion of matter andblack hole might start to grow.

Production of the initial black hole requires that a nuclebelonging to the star be hit by an incident highly energeparticle such as a cosmic ray or a cosmic neutrino, withenergy of at least a few PeV to reach the threshold of blhole~BH! production,A2mNEi;MBH; a few TeV. Accord-ing to the hoop conjecture, the cross section for black hproduction can be taken to besBH5pr s

2 where r s is theSchwarzschild radius of the center of mass energy ofincident particle and the target. Cosmic neutrinos could bcandidate for black hole production sincesBH dominatesover all the standard model neutrino-nucleon interactionsneutrino energies above;100 PeV @6#. Ultrahigh energy~UHE! neutrinos are expected to exist~as well as the alreadyobserved UHE cosmic rays@7,8#!, although the current sen

*Email address: [email protected]†Email address: [email protected]

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sitivity of neutrino telescopes does not enable us to dethem @9#. The most straightforward mechanisms of prodution would be via the interaction of UHE cosmic rays withe cosmological microwave background@Greisen-Zatsepin-Kuzmin ~GZK! mechanism@10## and via collisions of accel-erated hadrons and photons inside astrophysical objectsas active galactic nuclei. Other, more exotic production pcesses involving ‘‘hidden sources’’ or decay of ultrahearelic particles have also been proposed, possibly givingto many neutrinos with energies as high as 1022–1023 eV@11#. We prefer however to retain a more conservative emate of the high energy neutrino flux, essentially basedthe assumption that neutrinos are produced by the samemologically distributed extra-galactic sources that wouldresponsible for the observed high energy cosmic rays:Waxman-Bahcall bound@12#. Using this bound, one can deduce the number of neutrinos of energyEmin,E,Emax fall-ing on a star of radiusR per unit time

N5800p2S R

1 kmD 2S 1 GeV

Emin2

1 GeV

EmaxD s21. ~1!

This rate would become comparable to the correspondexpression for cosmic rays as the energy increases. Fosurface of a star with a radius of 10 km this rate is aboutneutrinos per year with an energy between 1020 and 231020 eV, while the current measurements made on Eaalthough still quite imprecise, would imply approximate5–20 cosmic rays per year around 1020 eV.

A recent paper@13# has pointed out that such a black hoformed by high energy neutrinos on the outside of a neutstar will not in fact grow since the region in which the blachole first forms is not dense enough for the black holesinteract with more nucleons before it decays. The same paalso shows that the situation is fundamentally different incase of strange stars.

It is postulated that the energy per quark in normal~upand down! quark matter may be higher than that in stranquark matter@14# so it has been hypothesized that one psible end point of stellar evolution is a star entirely composed of up, down and strange quarks@15#. Since it isthought that some strange stars may be ‘‘bare’’ in as muchtheir density rises from zero to more than nuclear densitie

©2003 The American Physical Society15-1

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M. FAIRBAIRN AND V. VAN ELEWYCK PHYSICAL REVIEW D 67, 124015 ~2003!

a length scale of order;fm @15#, strange stars could provida medium in which TeV scale black holes could be creaand then subsequently grow. One would not expect a strastar to possess spectral lines and also for it to have a diffemass-radius relation and cooling rate to a neutron star. Thas recently been a lot of interest in a possible strangecandidate@16,17# although it is not clear yet if the identification of this source as a strange star is correct@18#.

The authors of@13# go on to say that the existence ofbare strange star would place constraints on the numbersize of extra dimensions. This is because for large enoextra dimensions and conservative estimates of high enneutrino fluxes, the growth of a neutrino-nucleon interactinduced black hole will continue until it consumes the whoof the star. The constraints obtained in that paper are mafor the case ofd<2 flat extra dimensions.

There are many other stronger constraints on the castwo extra flat dimensions from astrophysics and cosmol@19#. For the case of a single extra dimension which solthe hierarchy problem, it is not possible for the extra dimesion to be flat, since this would require physics to be efftively five dimensional at lengths up to solar system scaOne therefore requires a warped extra dimension as inmodel of Randall and Sundrum. Phenomenologically stheories are difficult to constrain since the graviton KaluKlein ~KK ! mode masses are of order of the fundamenscale;TeV @2#. This is fundamentally different from the flaextra dimension scenarios where the Kaluza-Klein masare far below a TeV and can therefore be excited at asphysical energies. Given the recent possible detectionstrange star candidates, it is interesting to find out ifexistence of strange stars would place any constraints usuch scenarios.

In this work we briefly review the 5D Randall-Sundrumodel and the black holes that can be formed in this theWe then write down some general equations describingevolution of 5D black holes in dense media. We show wTeV scale black holes created at the surface of neutron sdo not continue to grow and then describe the growthblack holes in strange stars assuming the existencesingle warped extra dimension.

II. TEV BLACK HOLES IN RANDALL-SUNDRUM

In the model of Randall and Sundrum with a compaextra dimension, a large apparent mass hierarchy betwgauge and gravitational mass scales is obtained via a warof the transverse space@2# ~the evolution of black holes inthe noncompact Randall Sundrum scenario is studied@20#!. In this study we will assume that there is only oextra dimension although the analysis could easily betended tod extra warped directions. The Schwarzschilddius of a black hole of massMB in a 41d dimensional flatspace-time with a gravitational scale ofMF is given by@21#1

1This definition of the Schwarzschild radius adoptsMFd12

5(2p)d/4pGd14.

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~2!

Since the cross section for accretion onto the black holtherefore proportional toMB

2/(11d) we will need a higher den-sity medium in order for the black hole to start to growd.1. Hence the situation with one extra warped directionmore likely to promote back hole growth.

Using the conventions of the original paper@2# we writethe five dimensional metric

ds25e22krcufuhmndxmdxn1r c2df2 ~3!

where f is the coordinate of the orbifold directio(0,ufu,p) and r c is the size of the compact space. In thRandall-Sundrum two brane scenario, more than in thelarge extra dimension compactifications, it is really notclear which mass scale (1019 GeV or 1 TeV! is fundamentaland which is derived from the geometry. We choose tonote the TeV scaleMF and the apparent four dimensionPlanck scaleM P . Then

MF5M Pe2pkrc ~4!

so that we needkrc;10 to solve the hierarchy problem. Thinverse curvature radius of the slice of AdS5 between thebranes as viewed from our brane is given by

m5ke2pkrc. ~5!

Black holes of sizeMF21<r<m21 see an effectively flat 5D

space-time@24# so they obey Eq.~2!:

r s50.651

MFS MB

MFD 1/2

. ~6!

Since we require that the mass of our black hole is grethan a few times the fundamental scale in order for semicsical assumptions about the formation and evaporationcess to be valid, we now have an expression telling uswhich black hole masses we can use the flat space equat

MF!MB&MF

3

m2. ~7!

In order to accommodate the hierarchy between the scalelectroweak and gravitational physics using this warpedometry, we simply need to ensure thatkrc;10, so it appearswe can reducem arbitrarily. However, the further belowMFwe takem to be, the less natural is the value ofr c required.Also there is a lower limit onm set by the lack of KK modeproduction at colliders@22#.

At mass scales such that the radius of the black holmuch larger than the AdS5 radius, full black hole solutionsare still out of reach and the behavior of black hole growthless clear~although see@23#!. It seems that there are twpossibilities for the subsequent behavior.

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RANDALL-SUNDRUM BLACK HOLES AND STRANGE STARS PHYSICAL REVIEW D67, 124015 ~2003!

The first is that the scattering cross section, and henceeffective size of the black hole, will increase logarithmica@24,25#. The horizon radius will therefore grow as

r s;1

mlnS m2MB

MF3 D ~8!

until it fills the space between the branes after which it wcontinue to grow as a 4D black hole.

Another possibility is that the black hole growth will bsuppressed in the radial direction, but will continue alongbrane according to the normal equation for a 5D black h~6!. If this occurs then once a black hole has reachedAdS5 radius it will rapidly become entropically favorable foit to split up into an ensemble of many smaller black hoeach of which obey the normal flat space relation@26#.

III. EVOLUTION OF 5D BLACK HOLES IN DENSE MEDIA

If the black hole comes within a distance equal to tSchwarzschild radius of the center of mass energy of aticle and the black hole itself, the black hole will accrete tparticle and continue with a correspondingly largSchwarzschild radius.

Although classically all of the matter approaching closthan 1.5 times the Schwarzschild radius will be absorbedthe black hole, in@5# it was pointed out that much of thenergy of a black hole formed with this enhanced cross stion will be ‘‘hair,’’ and would be radiated away in a timscale~for five dimensional black holes!

thair;S MF

MBD 3/2

tevaporation. ~9!

Since we only consider situations where the mass ofblack hole is at least several times larger than the fundamtal scale, we will assume this extra mass is lost on a tscale much shorter than the black hole lifetime, and we wuse the naive geometric cross sections5pr s

2 .2

A. Dense matter with T™m

Consider a black hole moving through a homogenemedium of particles of massm and number densityn at zerotemperature. The mean free path of the black holel before itaccretes another particle is given by

l51

ns5

1

npr s2

~10!

and the rate of increase of mass of the black hole is set byinverse of the time taken for the BH to cross one mean fpath (b5p/E, g5E/MB , E25p21MB

2)

2Here we neglect the interesting suppression mechanism ofloshin @27#. If we were to include this effect, the creation angrowth of black holes would be further suppressed.

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5bm

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Hawking radiation means that the black hole radiates wittemperature given by@28,5#

TH511d

4pr s~12!

and armed with this expression we can use Wien’s lawobtain the mass loss of the black hole@3#

dMB

dt Uevap

52p2ge f f

60ATH

41d52ge f f

960pr s2

~13!

where we uset to denote the time coordinate in the reframe of the BH andA for the horizon area which is given inthe Appendix. The parameterge f f is the number of effectiverelativistic degrees of freedom in the plasma which, sinceHawking temperature is typically below the mass of mostthe Kaluza-Klein modes, refers to the standard modelgrees of freedom on the brane. Combining the mass action and Hawking evaporation in the lab frame we endwith

dMB

dt5

dMB

dt Uacc

11

g

dMB

dt Uevap

5pr s2bmH n2

ge f f

960p2m

MB

prs4J . ~14!

This equation shows us straight away that for each blhole massMB and momentump there is a critical numberdensity of particles for which the rate of mass gained throuaccretion will be greater than the rate of mass lost throuevaporation.

One might expect that the solution of this differentiequation would lead to a full description of the behaviorthe black hole. If this were so a rapidly moving black howould accrete matter until it became stationary. This wooccur when its gamma factor goes down to close to 1 somass of the final black hole at rest would be of the saorder of its initial momentum. However, this is not the casince the momentump does not remain constant.

As the black hole accretes matterp is indeed conservedbut each time the black hole evaporates a particle of mmout the black hole will lose a fraction of its momentumpsuch thatDp/p52mout /MB . This is associated with thefact that the wavelengths of the quanta emitted via Hawkradiation are greater than the radius of the black hole sothe process is effectivelys-wave emission in the black holrest frame. We therefore have an additional equation whgoverns the evolution ofp

dp

dt5

p

MB

1

g

dMB

dt Uevap

~15!o-

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FIG. 1. Evolution of black hole mass vs timwhen the black hole is much smaller thanm21

for different neutrino energies forT51 GeV,MF51 TeV andMBinitial55 TeV. This growththreshold is not valid for black holes of sizer s

.m21.

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which we have to solve simultaneously with Eq.~14! in or-der to obtain the evolution of the black hole.

B. Application to outer neutron star crust

The outside;300 m of a neutron star consists of a dgenerate electron gas with the nucleons becoming incringly neutron rich as one moves inwards to higher densi@29#. At a nucleon number density of about 231024 fm23,it is energetically favorable for the neutrons to occupy cotinuum states and they begin to drip out of the nuclei.

In the outer parts of the neutron star crust, the pressenergy density is much less than the mass energy denand the total mass of the crust is only 1% of the neutronmass. The Tolman-Oppenheimer-Volkoff equation@30# is thegeneral relativistic equation for hydrostatic equilibrium

dP

dr52

~r1P!G@m~r !14pr 3P#L~r !

r 2~16!

whereL(r )5@122Gm(r )/r #21. In the crust of the neutronstar r(r d);231030 eV4 and P(r d);831027 eV4 so P/r,1% @29#. Also, in the crustm(r ) varies from the total masof the neutron starM* by at most a few percent, so we cawrite

dP

dr.2

GM* L~R* !

R*2

r. ~17!

HereR* is the radius of the neutron star which we take to10 km and we setM* 51.4M ( . We denote the radius awhich neutron drip occurs asr d and assume thatr5mnnwhere n is the number density of nucleons. The electrdensityne is relativistic so that3 P5(p/2)(3/8p)1/3ne

4/3 andif we make the simplifying assumption thatne5np5n wecan write

n~r d!1/32n~r !1/35GM* L~R* !

4R*2

mn~r 2r d! ~18!

with n(r d);1022 eV3. An incident neutrino with momentum1019 eV will collide with a neutron to create a black hole

3This is in reasonable agreement with the values obtained inmore detailed analysis of@29#.

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massMB;1014 eV. Using the expression for the mean frepath~10! one finds that such a neutrino moving through thouter crustal region will have a mean free path ofl;(eV21) and so will not penetrate any deeper into the sbefore becoming a black hole. The critical density for tgrowth of a black hole with such a mass and momentumbe calculated from Eq.~14! and isn;1029 eV3 so the blackhole will not grow in this outer region of the star where itcreated.

The medium will therefore become optically thick duethe neutrino-nucleon black hole production cross sectihowever, this will happen at a depth much smaller than twhere the surrounding density of matter is high enoughthe resulting black hole to accrete more matter than it wevaporate. The black hole will therefore decay thermally instandard model particles which will join the surrounding stFor very large incident neutrino energies, some of the deproducts may be able to produce secondary black holes,these black holes will also decay in a time scale musmaller than the time required for them to travel into tdepths of the neutron star where they may be able to grOur conclusion for neutron stars is therefore the same asauthors of@13#.

C. Radiation with Tšm

Now if we consider the situation where the black holemoving through a medium of relativistic particles at temperatureTbath , it becomes more convenient to work in threst frame of the black hole. Following@13#, the effectivetemperature of the medium in the black hole frame becom

Te f f5TbathAgS 11b2

3 D 1/4

5TbathS 114

3

p2

MB2 D 1/4

.

~19!

If one assumes that all the effectively light degrees of frdom are constrained to lie on the brane, the mass of ablack hole will evolve in its own rest frame as

dMB

dt5

p3ge f f

15r s

2S Te f f4 2

1

4~2p!4r s4D ~20!

and the evolution of momentum will still be given by Eq~15!. In Figs. 1 and 2 we show the evolution of the mass amomentum of black holes formed by neutrinos with varioe

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FIG. 2. Evolution of black hole momentum vtime for r s,m21 for different neutrino energiesT51 GeV, MF51 TeV andMBinitial55 TeV.

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momenta propagating in a medium of temperatureT51 GeV. These figures show the evolution in the regwherer s,m21 so that Eq.~2! gives the cross section. As wwill see in the next section, whenr s.m21 it becomes muchmore difficult to obtain growth. Put another way, black hothat are able to grow during the flat regime often cancontinue to grow once they become larger than the curvaof the compact space.

D. Application to strange star interiors

The surface of a strange star is expected to have ashell of electrons with thickness of the order of a few hudred fm@15#. The Coulombic repulsion of this shell will nobe able to stop matter falling freely onto the star from infiity, however, if the accretion onto the star is in the form ofluid the incoming matter may lose energy via the normaccretion processes. A crust would then build up on theterior of the strange star which might create the same bato black hole growth as the outer crust of a neutron sHowever, if strange stars exist, one would expect at lesome of them to exist outside binary systems and therepossess surface density profiles very close to step functiIt therefore seems possible that such stars would indeedvide a suitable medium for the growth of TeV scale blaholes. The interior of a strange star consists of quarks witemperature of aboutTbath;1 GeV @15#. Since we haveshown that neutrinos of sufficiently high energy will creablack holes that grow in a medium ofT51 GeV, we need toconsider what will happen once the size of the black hreachesm21.

1. Black holes with r.µ21

As discussed earlier, once such a black hole becomelarge as the AdS5 curvature radiusm21, there are two pos-sibilities as to its future evolution.

The first is that presented in@26# where the black holesplits up into many smaller black holes once it grows outthe AdS5 radius. In this scenario, there is a minimum evapration temperature for 5D black holes, since if they grolarger thanm21 they will decay into smaller holes withhigher temperatures. As the holes grow and split up, twill also gain mass while losing momentum via radiation aultimately the following criterion applies to such a systemblack holes: If they are to have any chance of growth,rest frame temperature of the medium must be higher t

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the minimum black hole temperature (;m). Using Eq.~20!to obtain a more precise estimate we find we require a vaof m,20 GeV in order for a 5D Randall-Sundrum blachole to have any chance of growing in this medium. Thregion of parameter space has already been ruled out byfact that Kaluza-Klein modes have not lead to loop corrtions of the oblique parameters in electroweak interacti@22#.

The second scenario for growth of black holes is ourtrapolation of the scenario described in@25# where the size ofthe black hole will only increase logarithmically untenough mass has been added so that the black hole fillsbulk. Once the black hole fills the bulk, its subsequent elution will obey the normal 4D mass radius relationMB

;M P2 r s . This will typically occur at a mass more than 3

orders of magnitude higher than the mass at which the rareachesm21. The fact that the radius of the hole now onincreases slowly with mass means that the temperaturemains high as the mass increases. This would suggestonce black holes are big enough to feel the curvature ofcompact space, their growth will be suppressed sinceradius fails to increase as rapidly with mass after that poThis is exactly what we find in our numerical analysis.

2. Distance traveled by black hole

We also need to check whether or not the black holetually stays inside the star, as black holes that have sufficmomentum to grow may quickly traverse the star and eThe distance traveled by the black holes is easily calculavia

distance5E bdt5E p~ t !

E~ t !dt5E p~t!

E~t!g~t!dt

5E p~t!

MB~t!dt. ~21!

We have assumed a black hole travelling more than 10will exit the star, but we will see that the exact figure is ncritical.

If a black hole should exit the star, it is then necessarycalculate the velocity of the black hole at that time and copare it with the escape velocity of the star given by

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bescape;A2GM*R*

50.94 ~22!

where we have assumed the star has a massM* equal to thesun ;1057 GeV and a radiusR* of 10 km. Again, we willsee that the exact values are not critical. Black holes whexit the star with velocitybexit.bescapewill simply escape toinfinity. For the case of black holes which exit the star wbexit,bescape, we need to check if they will evaporate awacompletely before they fall back onto the surface of tstrange star and start to accrete again. Therefore it is nesary to compute the decay time:

tdecay5960p

ge f f

1

mE0

MBln2S m2M

MF3 D dM

5960p

ge f f

MB

m2 F ln2S m2MB

MF3 D 22 lnS m2MB

MF3 D 12G

~23!

which uses the expression for evaporation~13! and the equa-tion for the radius~8! since the black holes in our simulationhave r @m21. We must then compare this with the timtaken for the black hole to fall back onto the surface ofstar

t return52ER*

r turn 1

b~r !dr

52ER*

r turnFbexit2 12GM* S 1

r2

1

R*D G21/2

dr

52bexitR*

2

GM*~24!

where the maximum distance of the black hole from the sis given by

r turn5S bexit2

2GM*1

1

R*D 21

. ~25!

3. Numerical analysis

The longest known strange star candidate was obseonly about 10 years ago@16# and using the Waxman-Bahcabound ~1! we can estimate the maximum energy neutrthat may have been incident on that object since its discery. The answer is 231023 eV which is very high comparedto observed cosmic ray showers. However, since the groof a black hole is more likely for higher energy incominparticles, adoption of this energy makes our constraint strger.

The results of our investigations are summarized in TaI. We find thatno black holes formed at these energies grquickly enough to remain inside the star. For very high vues ofm (m>202 GeV, see Fig. 3! black holes are not ableto continue their initial growth and start to lose mass fasthan they accrete. These black holes exit the star and es

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to infinity ~although they will of course decay rather quickonce they are outside the star!. For lower values ofm, theblack holes are able to grow but exit the star long before tengulf it. These black holes also escape with a velochigher than the escape velocity. For the lowest values omthe black holes grow more quickly due to the fact that thspend a longer period of time in the flat regime where thcross sections grow rapidly. They therefore exit the star wlower velocities, but still decay before they are able to fback onto the star.

To see this, we remember that the lowest value ofm thatis not ruled out at collider experiments ism;20 GeV. Forthis value ofm, a black hole created by a neutrino with thinitial momentum described earlier leaves the star withvelocity of bexit50.6 which corresponds to a return timet return5631019 GeV21. The decay time for this black hole itdecay5131016 GeV21. In order to find a black hole thacould fall back onto the star before it decays we would thefore have to consider values ofm less than the permittedexperimental lower limit. Thus black holes formed from higenergy neutrinos will not grow to engulf strange stars.

IV. CONCLUSION

In this work we investigated the possibility that thRandall-Sundrum 2-brane model of TeV scale gravity iscompatible with the existence of bare strange stars due togrowth of black holes seeded by high energy neutrinos.pointed out that the growth of such black holes is suppresin these models when they reach the radius correspondinthe curvature radius of the AdS5 in between the branes. Wperformed detailed simulations to see if black holes wogrow and engulf the star. We saturated the Waxman-Bahbound to find the highest energy neutrino that one coexpect to have hit the oldest known strange star candid@16# in the time since it has been discovered. In doing so,found that the regions of parameter space where growthsuch holes may be permitted has already been ruled ouaccelerator experiments.

In @13# it was shown that catastrophic black hole growin strange stars could only work if there were 1 or 2 larextra dimensions. The case of 2 large flat extra dimensihas already been tightly constrained by astrophysical

TABLE I. Summary of evolution of black holes created byneutrino of energy 231023 eV for different values of the curvatureparameterm. No such black holes remain in the star. We assumstrange star of radius 10 km and temperature 1 GeV, and an inmass for the black hole of 5 TeV.

202 GeV<m black hole does not growbexit.bescape

50 GeV<m<201 GeV black hole growsbexit.bescape

m<49 GeV black hole growsbexit,bescape

t return@tdecay

5-6

h

e


FIG. 3. Evolution of black hole form5201and 202 GeV showing that forMF51 TeV,MBinitial55 TeV andEn5231023 eV this valueof m is the critical one for growth. Note that suca black hole will leave the star after a timet583109 GeV21 with a mass much smaller than thtotal mass of the star (.1057 GeV).

ll-rs

5Di

si

t,efn

cosmological constraints@19#. In this paper we have ruledout the possibility of black hole growth in the RandaSundrum model. We therefore do not expect strange stacollapse due to TeV gravity black hole seeding.

We have also presented formalisms for the growth ofblack holes in dense media which might be of interestother areas such as cosmology and hadronic collider phy

ACKNOWLEDGMENTS

We would like to thank Nicolas Borghini, Robert BrouDominic Clancy, Jean-Marie Frere, Steve Giddings, JamGray, Raf Guedens, Scott Thomas, and Toby Wisemaninteresting discussions. M.F. is funded by an IISN grant athe IUAP program of the Belgian Federal Government.

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APPENDIX: AREA OF 2 ¿d SPHERES

The surface area of a sphere in 31d space dimensions isgiven by the expression

A5~31d!p (31d)/2

@~31d!/2#!r s

21d ~A1!

where for half integer factorials we use

~1/21n!! 5Ap~2n12!!

~n11!!4n11. ~A2!

y

ci.

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Randall-Sundrum black holes and strange stars