Table Of ContentBlack Hole Evaporation and Compact Extra Dimensions
Roberto Casadioa and Benjamin Harmsb
aDipartimento di Fisica, Universit`a di Bologna, and I.N.F.N.,Sezione di Bologna, via Irnerio 46, I-40126 Bologna, Italy
bDepartment of Physics and Astronomy, The University of Alabama, Box 870324, Tuscaloosa, AL 35487-0324, USA
Westudytheevaporationofblackholesinspace-timeswithextradimensionsofsize L. Wefirst
obtainadescriptionwhichinterpolatesbetweentheexpectedbehaviorsofverylargeandverysmall
black holesandthenshowthat theluminosityis greatly dampedwhenthehorizon shrinkstowards
L from a larger value. Analogously, black holes born with an initial size smaller than L are almost
stable. This effect is due to the dependence of both the Hawking temperature and the grey-body
factor of a black hole on the dimensionality of space. Although the picture of what happens when
the horizon becomes of size L is still incomplete, we argue that there occurs a (first order) phase
transition, possibly signaled by an outburst of energy which leaves a quasi-stable remnant.
1
0
PACS: 04.50.+h, 04.70.Dy, 04.70.-s, 11.25.Mj
0
2
n
a I. INTRODUCTION
J
4
The current interest in the possibility that there exist large extra dimensions [1,2] beyond the four dimensions of
2
our every day experience is based on two attractive features of this proposal. First of all the hierarchy problem is
1 by-passedbecausetheradiativestabilityoftheweakscaleinthisproposalisduetotheidentificationoftheultraviolet
v cutoff of the theory with the electroweak energy scale ΛEW. Ancillary assumptions such as supersymmetry are not
4 required to achieve radiative stability. The other attraction of this proposal is that since the fundamental scale of
5 the theory is the electroweak energy scale, predictions drawn from the theory such as deviations from the 1/r2 law
1
of Newtonian gravity can be experimentally tested in the near future. In the extra-dimensions scenario all of the
1
interactions,gravityaswellgaugeinteractions,becomeunifiedatthe electroweakscale. Thismeansthatifthe model
0
is viable, particle accelerators such as the LHC, the VLHC and the NLC will be able to uncover the features of
1
0 quantum gravity as well as the mechanism of electroweak symmetry breaking.
/ The large-extra-dimension scenario also has significant implications for processes involving strong gravitational
h
fields, such as the bending of light near a massive object and the decay of black holes. The latter phenomenon has
t
- beendescribedwithinthe contextofthe microcanonicalensemble(seeRefs.[3,4])infourspace-timedimensions. Our
p
starting point is the idea that black holes are (excitations of) extended objects (p-branes), a gas of which satisfies
e
h the bootstrap condition. This yields a picture in which a black hole and the emitted particles are of the same nature
: andan improvedlaw ofblack hole decay whichis consistentwith unitarity (energyconservation). This idea has been
v
bolsteredbyrecentresultsinfundamentalstringtheory,whereitisnowacceptedthatextendedDp-branesareabasic
i
X ingredient [5]. States of such objects were constructed which represent black holes [6] and corroborate [7] the old
r idea that the area of the horizon is a measure of the quantum degeneracy of the black hole [8]. However, this latter
a
approachworksmostly for verytiny blackholes andsuffers fromthe same shortcomingthat the determinationof the
space-time geometry during the evaporation is missing.
Embedding a black hole in a space-time of higher dimensionality would seem, fromthe string theory point ofview,
to be a natural thing to do. In models with extra spatial dimensions the four dimensional spacetime is viewed as a
D3-braneembeddedinabulk spacetimeofdimension4+d. Inthe mostrecentmanifestationofstringtheory,matter
is described by open strings whose endpoints are Dirichlet branes. Therefore in the large-extra-dimension scenario
matter fields at low energy (in the present case this means at energies less than Λ ) are confined to live on the
EW
D3-brane. In string theory gravity is described by closed strings and hence it can propagate in the bulk. Black holes
in 4+d extra dimensions have been studied in both compact [9] and infinitely extended [10] extra dimensions (see
also [11] and references therein).
In Ref. [12] we studied the evaporation of black holes in space-times with extra dimensions of size L. We first
obtained a description which interpolated between the expected behaviors of very large and very small black holes
andthenshowedthata(firstorder)phasetransition,possiblysignaledbyanoutburstofenergy,occursinthesystem
whenthehorizonshrinksbelowLfromalargervalue. Thephasetransitionisrelatedtobothachangeinthetopology
ofthe horizonandthe breakingoftranslationalsymmetryalongextradimensions. Inthe presentworkweextendthe
results of [12] by investigating several factors which affect the decay rate of a black hole.
In SectionII we reviewand improveonthe effective potentialdiscussedin Ref. [12]. In SectionIII we then analyze
the evaporationofablackhole in4+ddimensionsbothfromthe canonicalandthe microcanonicalpointofviewand
1
showthat the luminosity ofsmallblackholes,suchas primordialblackholes (PBHs)whichmight be sourcesofγ-ray
bursts, strongly depends on the number of space-time dimensions. Such a dependence is further enhanced by the
analysis of wave propagation in Section IIIC where we argue that the heighth of the potential barrier which affects
the grey-body factor depends on d. The phase transition and the accompanying topology change which a black hole
undergoes as its horizon radius becomes of the order of the size of the extra dimensions (R L) are discussed in
H
∼
Section IV. Finally, in Section V we summarize and discuss our results. We use units with c=h¯ =1.
II. THE EFFECTIVE POTENTIAL
If large extra spatial dimensions exist in nature, deviations from Newton’s law will be detected at the scale of
the extra dimensions. Assuming that all of the matter described by the standard model lives on a four-dimensional
D3-brane, the form of Newton’s law can be obtained for a point-like mass by means of Gauss’ law [1]. Denoting by
r the radial distance in 4+d dimensions and by r the radial distance as measured on the D3-brane, we find for
b
distances r much greater than the typical size of the extra dimension L a potential of the form
M
V = G , (II.1)
(4) − N r
b
where G =m−2 is Newton’s constant in four dimensions. On the other hand for r L the potential becomes
N p ≪
M
V = G , (II.2)
(4+d) − (4+d) r1+d
withG =M−2−d =LdG . ThisimpliesthatthehugePlanckmassm2 =M2+d Ld and,forsufficientlylargeL
(4+d) (4+d) N p (4+d)
and d, the bulk mass scale M (eventually identified with the fundamental string scale)can be as small as 1TeV.
(4+d)
Since
L 1TeV/M(4+d) 1+d2 103d1−16mm , (II.3)
∼
(cid:2) (cid:3)
requiring that Newton’s law not be violated for distances largerthan 1mm restricts d 2 [1,13]. Further bounds are
≥
obtained by estimating the production of KK-gravitonsand support higher values of d [14].
Since our purpose is to study the decay of black holes, we would like to be able to describe the metric tensor
elements of a space surrounding a black hole for all values of the horizon R of the black hole, including R L.
H H
∼
Black holes with both very large horizon radii (R L) and very small radii (R L) have been extensively
H H
≫ ≪
investigated. In the former case, the compact extra dimensions can be unwrapped and the real singularity can be
regarded as spread along a (black) d-brane [15] of uniform density M/Ld, thus obtaining the Schwarzschild metric
on the orthogonal D3-brane, in agreement with the weak field limit V , and an approximate “cylindrical” horizon
(4)
topology S2 IRd. In the latter case a solution is known [10], for one infinite extra dimension [2], which still has
×
the form of a black string extending all the way through the bulk AdS . However, this solution is unstable [16] and
5
believedto further collapse into one point-like singularity[10]. This canbe also arguedfromthe observationthat the
Euclidean action of a black hole is proportional to its horizon area and is thus minimized by the spherical topology
S2+d. Hence, small black holes are expected to correspond to a generalization of the Schwarzschild metric to 4+d
dimensions [17] and should be colder and (possibly much) longer lived [9].
As a black hole evaporates the topology of the horizon changes from S2 Rd to S2+d when R decreases from
H
×
R > L to R < L. For small black holes a complete description would provide an explicit matching between the
H H
cylindrical metric (for r L) and the spherical metric (for r L). This is not a trivial detail, since the ADM mass
≫ ≪
ofaspherical4+ddimensionalblackholeiszeroasseenfromtheD3-branebecausethereisno1/r terminthe large
b
r expansion of the time-time component of the metric tensor [17]. Thus, one concludes that the four-dimensional
ADM mass M of a small black hole can be determined as a function of the horizon radius R (which governs the
H
evaporation) only after such a matching is provided explicitly. One can further guess that M emerges either as a
boundary effect (i.e., due to the size of the extra-dimensionbeing bounded) or as a consequence ofa non-zerotension
(energy density) on the D3-brane (this latter possibility will not be consideredhere). Evenless is known about black
holesofsizeR L,andacompletedescriptionislikelytobeachievedonlybysolvingtheentiresetoffieldequations
H
∼
for an evaporating black hole in 4+d dimensions.
Rather than attempting the formidable problem of obtaining a complete description of an evaporating black hole
from the field equations, we approximate the (time and radial components of the) metric in 4+d dimensions as
2
g 1 2V(r)
tt
≃− −
(II.4)
g g−1 ,
rr ≃− tt
where
M d C LnG MΘ(L r )
n N b
V(r)= G Θ(r L) − , (II.5)
− N r b− − (1+n)/2
b nX=1 hrb2+DnΘ(L−rb) di=1 yi2i
P
where Θ is the Heaviside function (or a smooth approximationof it), the y s are cartesian coordinates in the d extra
i
dimensions and C , D are numerical coefficients. This yields M as the ADM mass (on the D3-brane) of the black
n n
hole and the radius of the horizon is determined by
g−1 =0 . (II.6)
rr
Theaboveansatz doesnotprovideanexactsolutionofthe vacuumEinsteinequations,sincesomeofthe components
of the corresponding Einstein tensor in 4+d dimensions G = 8πG T = 0. However, the coefficients C and
ij (4+d) ij n
6
D can be chosen in such a way that the “effective matter contribution” T from the region outside the black hole
n ij
horizon is small. In particular, one can require the contribution to the ADM mass to be negligible,
∞
m d4+dxTt
≡Z t
RH
∞
1
= d4+dxGt ( C ) M . (II.7)
8πG Z t { n} ≪
(4+d) RH
In this sense one can render the above metric a good approximation to a true black hole in 4+d dimensions. For
instance, for d=1, we obtain
M
G r >aL L
N
− rb ≫
V(r)= −GN Mrb −C Lrb2G+NyM2 L<bL<r<aL (II.8)
−C Lrb2G+NyM2 r <bL ,
where the various coefficients must satisfy
m a b
=C − 1 , (II.9)
M ab ≪
which determines the size of the region in which both 1/r and 1/r2 terms are turned on [18]. Correspondingly one
finds
2G M M M
N c
≫
RH(y =0)= GNM (cid:18)1+r1+ G2NCML (cid:19) M ∼Mc (II.10)
√2LGNM M ≪Mc ,
whereM isthevalueofM forwhichtheusualfour-dimensionalhorizonradius2G M =L[thevalueofM forwhich
c N
R = L can be obtained only after solving Eq. (II.6), which is an algebraic equation of order d+2]. For M M
H c
∼
the radius of the horizonalso depends on the coordinatey and is neither cylindrically nor spherically symmetric (see
Fig. 1). Analogous estimates can be obtained for d>1.
3
r
b
2.75
2.5
2.25
y
0.2 0.4 0.6 0.8 1
1.75
1.5
1.25
FIG. 1. Plot of the horizon for d = 1 and M = Mc, 0.8Mc, 0.6Mc, 0.4Mc (in units with G = L = 1). For d = 0 the
corresponding horizons would be at rb =2, 1.6, 1.2, 0.8.
Finally, we note that, althoughthe ansatz in Eq.(II.4) is not a true vacuum solution, sucha solution wouldnot be
very useful for R L in any event, because once Hawking radiation is included, its backreaction on the metric at
H
≪
small r is likely to be significant. Therefore, it is sufficient to impose the condition that m be comparable with the
energy density of Hawking’s radiation integrated from r =R to r = , i.e. the amount of mass lost into Hawking
H
∞
particles. This quantity of course depends on the initial mass of the black hole.
III. BLACK HOLE EVAPORATION
There are several factors affecting the rate of black hole evaporation due to Hawking radiation. We begin with a
review of the expressionfor the decay as obtained from the canonicalensemble and then discuss the modifications to
this expressionwhich arisewhen the more basic microcanonicalensemble is used. Finally, we estimate the grey-body
factor by studying the Klein-Gordan equation.
A. The canonical picture
When the size of the black hole is large compared to the extra dimensions (R L), r r and the metric (II.4)
H b
≫ ≃
is approximately cylindrically symmetric (along the extra dimensions). Further, when L is set to 0, m = 0, and
Eq. (II.6) coincides with the usual four dimensional Schwarzschildradius
M
R 2ℓ . (III.1)
H p
≃ m
p
Correspondingly the inverse Hawking temperature (β =T−1) and Euclidean action are [19,8]
H H
M
β> 8πℓ (III.2)
H ≃ p m
p
2
M
S> 4π A(4) , (III.3)
E ≃ (cid:18)m (cid:19) ≃ 4ℓ2
p p
where is the area of the horizon in D space-time dimensions and the condition R L translates into
(D) H
A ≫
L L
M m M 1027g . (III.4)
p c
≫ ℓ ≡ ∼(cid:18)1mm(cid:19)
p
The fact that the extra dimensions do not play any significantrole at this stage is further confirmedby T> 2π/L.
H ≪
Since 2π/L m(1) is the mass of the lightest Kaluza-Klein(KK) mode, no KK particles can be produced.
≡
The energy loss per unit time for an evaporating black hole is given by
4
dM
= F> F> , (III.5)
dt − (4+d) ≃− (4)
where F is the total luminosity as measured in D space-time dimensions. Since microcanonical corrections have
(D)
been shown to become significant only for M m (in four space-time dimensions [4]), one can further approximate
p
∼
the luminosity by employing the (simpler) canonical expression [19]
∞ Γ (ω)ωD−1
s
F dω
(D) ≃A(D) Xs Z0 eβHω∓1
= N (T )D , (III.6)
(D) (D) H
A
where Γ is the grey-body factor and N is a coefficient which depends upon the number of available particle species
s with energy smaller than T and is also affected by the value of Γ. For D = 4 and β = β> one obtains the well
H H H
known result [19]
N m 2
F> (4) p . (III.7)
(4) ∼ ℓ2p (cid:16)M (cid:17)
For R L the parameters in the potential (II.5) can be chosen to minimize the contribution of m to the ADM
H
mass and t≪his is generally obtained by switching off all terms except that going as 1/r1+d for r <L. Eq. (II.6) then
leads to
R 2LdG M 1/1+d , (III.8)
H N
≃
(cid:0) (cid:1)
and the consistency conditions ℓ R L hold for
p H
≪ ≪
d
ℓp 2+d
m M M . (III.9)
p c
(cid:18)L(cid:19) ≪ ≪
Since we have assumed that the spherical symmetry extends to 4+d dimensions, one obtains [9]
1
M 1+d
β< L (III.10)
H ∼ (cid:18)M (cid:19)
c
S< L 2 M 12++dd A(4+d) , (III.11)
E ∼(cid:18)ℓ (cid:19) (cid:18)M (cid:19) ∼ ℓ2Ld
p c p
which reduce back to (III.2) and (III.3) if one pushes down L ℓ (M m ). For M M , the temperature
p c p c
T< is sufficient to excite KK modes, although it is lower than t→hat of a fo→ur-dimensional b≪lack hole of equal mass.
H
Correspondingly, the Euclidean action S<(M) S>(M), yielding a smaller tunnelling probability [19,3]
E ≥ E
P exp( S ) , (III.12)
E
∼ −
that is, a smaller probability for the Hawking particles to be created in the 4+d dimensional scenario.
Although ordinary matter is confined on the D3-brane, a black hole can emit particles via Hawking’s process into
all of the 3+d spatial directions of the bulk. For R L the evaporation of a black hole can be obtained from
H
Eq. (III.6) with D =4+d and T =T<, ≪
H H
2
dM = F< N(4+d) Mc 1+d . (III.13)
dt − (4+d) ∼ L2 (cid:18)M (cid:19)
The luminosity (III.6) can be written as
N
(D)
F = . (III.14)
(D) R2
H
Thus, a comparison of the radius of the (apparent) horizon (III.8) with the analogous quantity in Eq. (III.1) shows
thatthe luminosity ofablackhole ofgivenADM massM <M ismuchsmallerin4+ddimensionsthanitwouldbe
c
5
with no extra dimensions. This assertion might have to be qualified if the N term in the luminosity were greater
(D)
in 4+d dimensions than the one in four dimensions. We show below that this is not the case.
Conservation of energy in 4+d dimensions requires that
F =F +F (III.15)
(4+d) (4) KK
and since F N [Eq. (III.6)] and the number of degrees of freedom of KK gravitons is much less than the
(D) (D)
∼
number of standard model particles (N N ), the energy emitted into the KK modes must be a small fraction
KK (4)
≪
of the total luminosity (similar results were obtained in Ref. [11])
F N
KK KK
1 . (III.16)
F ∼ N +N ≪
(4+d) (4) KK
StandardmodelparticlesontheD3-branewithhighenoughenergy(e.g.,largerthanΛ )mightbeabletoovercome
EW
the confining mechanism. In this case the bulk standard model fields should be included among the KK modes and
N shouldbeconsideredasanincreasingfunctionofthetemperature. InthisscenariotheratioF /F would
KK KK (4+d)
eventually reach unity when T< Λ .
H ≫ EW
B. The Microcanonical Picture
The canonical ensemble description used above is in principle incorrect because a black hole in asymptotically flat
space-timecannotbeinthermalequilibriumwithitsHawkingradiation. Howeverforlargeblackholesthisdescription
is averygoodapproximationto the truepicture. The correctstatisticalmechanicaldescriptionofblackholesutilizes
the microcanonical ensemble [3,4].
For R L the topology of the horizon is “cylindrical”, and the Euclidean action is
H
≫
2
M
S> 4π . (III.17)
E ≃ (cid:18)m (cid:19)
p
The number density in the microcanonical enesemble for this case is
[[M/ω]] exp 4π(M lω)2/m2
n>(ω)= − p , (III.18)
e(cid:2)xp(4πM2/m2) (cid:3)
Xl=1 p
where[[X]]denotestheintegerpartofX. InthelimitM ,n> isequaltothecanonicalensemblenumberdensity
→∞
used in Eq. (III.6). The decay rate for a black hole in 4+d dimensions is given by
∞
dM
= n(ω)Γ(ω)ωd+3dω . (III.19)
dt A(4+d) Z
0
Making the substitutions n = n>, x = M lω and assuming that Γ 1, the decay rate in four dimensions can be
− ≃
written as
∞
dM 1 π4
= I = I , (III.20)
dt A(4) l4 90 A(4)
Xl=1
where
M (M x)3 exp 4πx2/m2 dx
I = − p . (III.21)
Z exp 4π(cid:0)M2/m2 (cid:1)
0 p
(cid:0) (cid:1)
A plot of the decay rate as a function of the mass is shown in Fig. 2. However, we remark that, in the presence of
extra dimensions, this description holds only for M > M (R > L), therefore the plot in Fig. 2 is probably not
c H
accurateforsmallmasses. Itshouldinfactbe consideredonlyinthe regionM/m >L/ℓ wherethereis noessential
p p
difference between the microcanonical and the canonical luminosities (III.7).
6
dM
dt 0.012
0.01
0.008
0.006
0.004
0.002
0.5 1 1.5 2
M/m
p
FIG.2. Decay rate for a large (RH ≫L) black hole.
For R <L the Euclidean action is
H
2 (d+2)/(d+1)
L M
S< =4π , (III.22)
E (cid:18)ℓ (cid:19) (cid:18)M (cid:19)
p c
where M =m L/ℓ . The number density for horizon radii of this size is
c p p
d+2 2
[[M/ω]] exp 4π M−lω d+1 L
n<(ω)= (cid:20) (cid:16) Mc (cid:17) (cid:16)ℓp(cid:17) (cid:21) . (III.23)
d+2 2
Xl=1 exp 4π M d+1 L
(cid:20) (cid:16)Mc(cid:17) (cid:16)ℓp(cid:17) (cid:21)
The decay rate corresponding to this number density as calculated from Eq. (III.19) is exhibited for d = 0,...,4 in
Fig. 3. In this plot, we have matched the microcanonical decay rates for M < M with the canonical rate given in
c
Eq. (III.7) for M >M by multiplying the former by a suitable constant in order to show that there is a peak in the
c
luminosity aroundM . In fact, the curves show clearly that the decay rates for all cases with d>1 have a maximum
c
at M M and black holes with masses less than M thus have a significantly slower evaporation rate which results
c c
≃
inalongerlifetime (seeFig.4). Thecased=1appearsslightlydifferent,butthis valueofdmustbe rejectedbecause
it leads to contradictions with experimental tests of Newton’s law [1,13].
dM
dt
8
6
4
2
0.2 0.4 0.6 0.8 1 1.2 1.4
M/M
c
FIG.3. Decay rate for a small black hole in increasing number of extra dimensions (d = 0 uppermost curve, d = 4 lowest
curve). Vertical unitsare arbitrary.
7
M
Mc 1
0.8
0.6
0.4
0.2
0.5 1 1.5 2
t
FIG. 4. History of a small [M(0) ∼ Mc] black hole in increasing number of extra dimensions (d = 0 lowest curve, d = 4
uppermost curve). Horizontal unitsare arbitrary.
We wish to make two remarks about the above estimates: firstly, we expect the precise shapes of the decay rates
in Fig. 3 at M M depend on the details of the phase transition which we are able to describe only qualitatively
c
∼
(see Section IV). As a consequence, the curves in Fig. 4 are not totally reliable around t = 0 (M M ), but can
c
∼
be used to describe black holes which are produced with M(0) significantly less than M ; secondly, we have always
c
set Γ 1, but the grey-body factor actually depends strongly on the number of extra dimensions d causing a further
∼
suppression of the decay rate, as we show in Section IIIC.
A reduced decay rate for black holes in large extra dimensions has severalinteresting implications for cosmological
phenomenainvolvingPBHs. Ford=2andM 1TeV,thesizeoftheextradimensionisL 1mm[fromEq.(II.3)]
(6)
∼ ∼
and M 1027g [from Eq. (III.4)]. Since the initial mass of an astrophysicalblack hole is known to exceed at least a
c
few solar∼masses (M > 107M ) [23], its T> is cooler than the cosmic background radiation and it cannot decay to
0 c H
M in any reasonableamountof time. However,PBHs (see, e.g., Refs. [24]) couldhave been producedwith aninitial
c
mass M M , so that R L from the very beginning and, according to Fig. (3) the lifetimes of such black holes
0 c H
≪ ≪
wouldbe many ordersof magnitude longerthan those living in four spacetime dimensions. For instance, a PBH with
M 1015g, which has a life-time comparable with the age of the Universe according to Eq. (III.7) (the canonical
0
ense∼mbleisagoodapproximationinthis massrangeford=0),inthis newscenariowoulddecayinτ 1016 times
(6)
∼
theageoftheUniverse. Ontheotherhand,ifthePBHhadbeenproducedwithR >L,wecanspeculatethat,after
H
it shrank down to near L according to Eq. (III.7), either the evaporation continued and its mass eventually entered
into the regime of very slow decay, or the evaporation stopped (because of the backreaction) at M M . The latter
c
∼
option is even more dramatic, since it would leave remnants of mass as large as M (to be compared with possible
c
remnants of Planck size [24] as follows from microcanonical estimates in four dimensions [4]).
One can therefore highlight a list of topics involving PBHs upon which models with extra dimensions might have
a bearing [9]:
1. PBHs could be the source of γ-ray bursts and cosmic rays only if their initial mass was such that M 105g
≪
[25] or M M at the time of the emission; the allowed density of PBHs which fits into bounds obtained from
c
∼
current observations could then be significantly changed, with consequences on
2. models of the early Universe,
3. dark matter (in the form of KK particles [1]) and
4. density and size of microlensing sources [26].
C. Angular Momentum Barrier to Decay
A scalar wave in 4+d dimensions satisfies the equation
2Φ=0 , (III.24)
8
where the D’Alembertian is given by
1
2= ∂ √ ggµν∂ , (III.25)
µ ν
√ g −
− (cid:0) (cid:1)
with (in spherical coordinates for which θ =θ and θ =φ)
1 2
d+2
√ g =r2+d (sinθ )d+2−i (III.26)
i
−
iY=1
For L R <r we can neglect the extra dimensions and simply take the standard Schwarzschildline element on
H
≪
the brane,
dr2
ds2 ∆ dt2+ +r2 dθ2+sin2θdφ2 , (III.27)
≃− (4) ∆
(4) (cid:0) (cid:1)
where
R D−3
H
∆ =1 . (III.28)
(D) −(cid:18) r (cid:19)
ForR <r Lonecananalogouslyconsiderasphericallysymmetricblackholein4+ddimensions[17]. However,
H
≪
in order to take into account the fact that d spatial dimensions have size L, we shall instead use the following form
dr2 d
ds2 ∆ dt2+ +r2 dθ2+sin2θdφ2 +r2 dφ2 , (III.29)
≃− (4+d) ∆ i
(4+d) (cid:0) (cid:1) Xi=1
where R < r < L is the 4+d dimensional areal coordinate and dxi rdφ are cartesian coordinates in the extra
H i
≃
dimensions.
We assume the scalar field Φ can be factorized according to
Φ=eiωtR(r)S(θ)eimφei niφi , (III.30)
P
with n positive integers, so that Φ satisfies periodic boundary conditions at the edges of the bulk (y = L/2). The
i i
±
radial equation obtained from Eq. (III.24) for the metric (III.29) then becomes (∆ ∆ )
(4+d)
≡
∆ d d ∆
r2+d∆ +ω2 A R=0 (III.31)
(cid:20)−r2+d dr (cid:18) dr(cid:19) − r2 (cid:21)
and the angular equation is
1 ∂ ∂ m2
sinθ S =λS , (III.32)
(cid:20)sinθ ∂θ (cid:18) ∂θ(cid:19)− sin2θ(cid:21)
where A=λ+ n2, the separation constant λ=l(l+1) and S =Ym is a standard spherical harmonic.
i l
The radial eqPuation can be further simplified by defining a tortoise coordinate
dr
dr∗ , (III.33)
≡ ∆
and introducing a rescaled radial function,
W r1+d/2R , (III.34)
≡
which then satisfies a Schr¨odinger-like equation
d2
+V W =ω2W , (III.35)
(cid:20)−dr∗2 (cid:21)
where the potential V =V +V is given by
1 2
9
d ∆
V = (1+d) 1+ +A
1 (cid:20) (cid:18) 2(cid:19) (cid:21) r2
(III.36)
d 2 ∆2
V = 1+ .
2 − (cid:18) 2(cid:19) r2
The contributionV vanishes sufficiently fast near R and is negligible there. The potential V generates a barrier
2 H 1
located outside the horizon which suppresses the grey-body factor for all modes of the scalar field including those
with zero angular momentum [27]. The effect for d=0 is mild, however for d>0 the barrier increases, significantly
reducingtheblackholedecayrate. FromtheplotofthepotentialinFig.5onecanestimatethe(frequencydependent)
suppression factors with respect to the purely four-dimensional case (for which Γ 1) by making use of the W.K.B.
∼
approximation for the transmission probability
Γ(ω) exp 2 dr V ω , (III.37)
∼ (cid:18)− Z | − |(cid:19)
p
where the integralmust be performed between the two values of r at which V =ω (for ω smaller than the maximum
of V). For the typical frequency ω T< ( 1 in the units of Fig. 5) one obtains
∼ H ∼
Γ 1
d=1
∼
Γ 0.73
d=2
∼
Γ 0.22
d=3
∼
Γ 0.05 . (III.38)
d=4
∼
One then observes that, for frequency ω <T< the transmission probability is smaller, while for quanta of frequency
H
ω T< theprobabilityofcreationisactuallynegligible. WeremarkthatthecurvesshowninFig.5wereobtainedby
≫ H
setting A=0 (i.e., l =n =0) and, since the n s cannot be zero, the curves represent lower bounds on the heighths
i i
ofthecurvesandthe valuesinEq.(III.38)representupper boundsonthegrey-bodyfactors. Thisbarriereffecttaken
together with the decay rate reduction for d>1 (again, we recallthat the case d=1 is ruled out) as shown in Fig. 3
wouldseemtosuggestthatblackholeswithhorizonradiiontheorderofthesizeoftheextradimensionsorlessdecay
very little if at all.
V
4
3
2
1
1.5 2 2.5 3 3.5 4
r/R
H
FIG.5. Potential V +V for different numbersof extra dimensions (d=4 uppermost curve,d=0 lowest curve).
1 2
IV. HORIZON TOPOLOGY CHANGE AND PHASE TRANSITION
In the foregoing analysis horizon radii on the order of the size of the extra dimensions (R L) have not been
H
∼
considered in great detail. For example, the expressions for β and S in this region are discontinuous because they
H E
contain numerical coefficients which depend on the topology and number of the extra dimensions. For instance,
10
