Table Of ContentON QUANTUM ANALOGUES OF P-BRANE BLACK HOLES
V.D. Ivashchuk1,† M. Kenmoku2‡ and V.N. Melnikov3†
† Center for Gravitation and Fundamental Metrology, VNIIMS, 3-1 M. Ulyanovoy Str., Moscow 117313, Russia
Institute of Gravitation and Cosmology, PFUR, 6 Miklukho-Maklaya St., Moscow 117198, Russia
‡ Department of Physics, Nara Women University, Nara 630, Japan
In a multidimensional model with several scalar fields and an m-form we deal with classical spherically symmetric
solutions with one (electric or magnetic) p-brane and Ricci-flat internal spaces and the corresponding solutions to
1 the Wheeler–DeWitt (WDW) equation. Classical black holes are considered and their quantum analogues (e.g. for
0 M2 and M5 extremal solutions in D = 11 supergravity, electric and magnetic charges in D = 4 gravity) are
0
suggested when the curvaturecoupling in the WDW equation is zero.
2
n
1. Introduction 2. The model
a
J
Consider the model governed by the action
1 Inthispaperwecontinueourinvestigationsof p-brane
1 solutions (see, e.g., [1, 2, 3] and references therein) 1
1 based on the sigma-model approach [4, 5, 6]. S = 2κ2 ZMdDz |g|{R[g]−hαβgMN∂Mϕα∂Nϕβ
v The model under consideration contains several 1 p
3 scalardilatonicfieldsandoneantisymmetricform. We exp[2λ(ϕ)]F2 +SGH (2.1)
− m! }
4 consider sphericallysymmetric solutions (see [15,14]),
0 when all functions depend on one radial variable and where g = gMNdzM dzN is the metric (M,N =
1 pay attention to black hole (BH) solutions with Ricci- 1,...,D), ϕ = (ϕα)⊗ Rl is a vector of dilatonic
0 ∈
01 flinat[1i6n,te1r7n,a1l8s])p.aces (see [15, 14] and special solutions (scla∈larNfi)e,ldFs, =(hαdβA) =is am1!nFoMn1-d...eMgmendezrMat1e∧l.×..l∧mdaztMrimx
/ ThecorrespondingsolutionstotheWDWequation is an m-form (m 1) on a D-dimensional mani-
c fold M and λ is a≥1-form on Rl: λ(ϕ) = λ ϕα,
q (in the spherically symmetric case) were considered in α
gr- [(1a2n,d1/4o]r. cHonerfeorwmealulys-ecotvhaericaonvta)riafonrtm“do’Af lethmebeWrtiDanW” αF2==1F,.M.1..,..lM.mIFnN(12....N1)mgwMe1dNe1n.o.t.egM|gm|N=m,|wdehte(rgeMSNG)|H,
: equation of Refs.[9, 10, 11]. We single out certain is the standard Gibbons-Hawking boundary term [8].
v
The signature of the metric is ( 1,+1,...,+1).
i classes of solutions to the WDW equations and sug- −
X The equations of motion corresponding to (2.1)
gest quantum analogues of BH solutions.
r have the following form:
a The plan of the paper is as follows. In Sec.2 we
consider the model and present the WDW equation. 1
R g R = T , (2.2)
Sec.3 is devoted to classical and quantum exact solu- MN − 2 MN MN
tions when spherical symmetry is assumed. In Sec.4 λα
[g]ϕα e2λ(ϕ)F2 = 0, (2.3)
we consider the classical black-hole solutions and for △ − m!
a∈∆
thecase a=0 (theterm aR[ ] intheWDWequation X
is responsiblefor the scalarcuGrvature ofminisuperme- ∇M1[g](e2λ(ϕ)FM1...Mm) = 0, (2.4)
tric ) we suggest quantum analogues of black-hole
G α = 1,...,l. In (2.3) λα = hαβλβ, where (hαβ) is
solutions. In the extremal case we consider severalex-
matrix inverse to (h ). In (2.2)
αβ
amples, e.g. for D = 4 Einstein-Maxwell theory and
D = 11 supergravity. It is shown that the brane part T =T [ϕ,g]+ e2λ(ϕ)T [F,g], (2.5)
MN MN MN
of the solution satisfying the outgoing-wave boundary
condition is regular for small brane quasi-volume (it where
lookslikeapseudo-Euclideanquantumwormhole). We T [ϕ,g]=h ∂ ϕα∂ ϕβ gMN∂ ϕα∂Pϕβ ,
MN αβ M N P
also compare our approach with that suggested by H. − 2
Lu¨,J.Maharana,S.MukherjiandC.N.Pope[20](with 1 (cid:16) 1 (cid:17)
T [F,g]= g F2
flat minisupermetric and classical fields of forms). MN m! −2 MN
(cid:20)
+mFa FM2...Mm .
MM2...Mm N
i
In(2.3),(2.4) [g] and [g] aretheLaplace-Beltrami
1e-mail: [email protected] △ ▽
2e-mail: [email protected] andcovariantderivativeoperatorscorrespondingto g,
3e-mail: [email protected] respectively.
2 V.D. Ivashchuk, M. Kenmoku and V.N. Melnikov
Consider the manifold and I Ω. In (2.17) = [g] is the Hodge operator
s
∈ ∗ ∗
on (M,g). Theindices e and m correspondtoelectric
M =R (M =Sd0) (M =R) ... M (2.6)
0 1 n and magnetic p-branes, respectively.
× × × ×
Forthepotentialsin(2.16),(2.17)andthedilatonic
with the metric
scalar fields we put
n
g = e2γ(u)du du+ e2φi(u)gi, (2.7) Φs =Φs(u), ϕα =ϕα(u), (2.18)
⊗
i=0
X
s=e,m; α=1,...,l.
where u is a radial coordinate, gi = gi (y )dymi
dyni is a metric on M satisfying the emquinaitioin i ⊗ ¿From(2.16)and(2.17)weobtaintherelationsbe-
i i tween the dimensions of the p-brane worldsheets and
R [gi]=ξ gi , (2.8) the ranks of forms:
mini i mini
d(I )=m 1, d(I )=D m 1, (2.19)
m ,n = 1,...,d ; d = dimM , ξ = const, i = s s
i i i i i i − − −
0,...,n; n N. Thus (M ,gi) are Einstein spaces.
The function∈s γ,φi: (u ,ui) R are smooth. The for s=e,m, respectively.
− + → It follows from [5] that the equations of motion
metric g0 is a canonical metric on M = Sd0, ξ =
0 0 (2.2)–(2.4) and the Bianchi identities
d 1 and g1 = dt dt, ξ =0. Here (M ,g1) is a
0 1 1
− − ×
time manifold. dF =0, (2.20)
Each manifold M is assumed to be oriented and
i
connected, i = 0,...,n (for i = 0,1 this is satisfied forthefieldconfiguration(2.7),(2.15)–(2.18)areequiv-
automatically). Then the volume d -form alent to equations of motion for a σ-model with the
i
action
τ = gi(y ) dy1 ... dydi, (2.9)
i | i | i ∧ ∧ i S = θ du G φ˙iφ˙j +h ϕ˙αϕ˙β
and thepsignature parameter σ 2 N ij αβ
Z (cid:26)
ε(i)=signdet(gmi ini)=±1 (2.10) +εsexp[−2Us(φ,ϕ)](Φ˙s)2−2N−2V(φ) (2.21)
(cid:27)
arecorrectlydefinedforall i=0,...,n. Here ε(0)=1
where x˙ dx/du,
and ε(1)= 1. ≡
−
Let Ω be a set of all subsets of I 0,...,n : n
Ω0 = {∅0,{0},{1},...,{n},{0,1},...,0{0≡,1{,..., n}}} V =V(φ)= 12 ξidie−2φi+2γ0(φ) (2.22)
For any I = i ,...,i Ω , i < ... < i , we i=0
1 k 0 1 k X
{ } ∈
define the form
is the potential with
τ(I) τ ... τ , (2.11) n
≡ i1 ∧ ∧ ik γ (φ)= d φi; (2.23)
0 i
of rank
i=0
X
d(I) d , (2.12) furthermore,
i
≡
i∈I
X =exp(γ γ)>0 (2.24)
0
and the corresponding p-brane submanifold N −
is the lapse function,
M M ... M , (2.13)
I ≡ i1 × × ik
Us = Us(φ,ϕ)= χ λ(ϕ)+ d φi, (2.25)
s i
where p = d(I) 1, dimM = d(I). We also define −
the ε-symbol − I iX∈Is
ε = ( ε[g])(1−χs)/2ε(I )θ = 1 (2.26)
s s
− ±
ε(I) ε(i )...ε(i ). (2.14)
1 k
≡ for s = e,m, ε[g] = signdet(g ), χ = +1, 1, for
MN s
−
For I = we put τ( )=ε( )=1, d( )=0. s=e,m, respectively, and
∅ ∅ ∅ ∅
For the field of form we adopt the following 1-form
G =d δ d d (2.27)
ansatz ij i ij − i j
F = s, (2.15) are components of the “cosmological” minisupermet-
F ric, i,j =0,...,n [11].
where In the electric case for finite “internal space” vol-
umes V the action (2.21) (the time manifold should
i
s = dΦs τ(Is), s=e, (2.16) be S1) coincides with the action (2.1) if θ = 1/κ2,
Fs = e−2λ∧(ϕ) (dΦs τ(I)), s=m (2.17) κ2 =κ2V ...V . − 0
F ∗ ∧ 0 0 n
On quantum analogues of p-brane black hole 3
The action (2.21) may be also written in the form 2.1. Wheeler–DeWitt equation
S = θ du (X)X˙AˆX˙Bˆ 2 −2V (2.28) Here we fix the gauge as follows:
σ 2 N GAˆBˆ − N
Z n o γ γ =f(X), = ef, (2.42)
where X = (XAˆ) = (φi,ϕα,Φs) RN, and the min- 0− N
∈
isupermetric = (X)dXAˆ dXBˆ on minisuper- where f: R is a smooth function. Then we
space = RGN, GNAˆBˆ= n + 2 +⊗l, is defined by the obtain the LMagr→ange system with the Lagrangian
M
relation
θ
Gij 0 0 Lf = 2efGAˆBˆ(X)X˙AˆX˙Bˆ −θe−fV (2.43)
( (X))= 0 h 0 . (2.29)
GAˆBˆ αβ and the energy constraint
0 0 ε e−2Us(X)
s
θ
E = ef (X)X˙AˆX˙Bˆ +θe−fV =0. (2.44)
The minisuperspace metric may be written as fol- f 2 GAˆBˆ
lows:
Thestandardprescriptionsofcovariantandconfor-
=G¯+εse−2Us(x)dΦs dΦs (2.30) mallycovariantquantization(see,e.g.,[11,9,10])lead
G ⊗
where x=(xA)=(φi,ϕα), G¯ =G¯ dxA dxB, to the Wheeler-DeWitt (WDW) equation
AB
⊗
(G¯ )= Gij 0 , (2.31) HˆfΨf ( 1 ∆ ef + aR ef + e−fθV)Ψf =0
AB 0 h ≡ −2θ G θ G
αβ
(cid:18) (cid:19)
(cid:2) (cid:3) (cid:2) (cid:3) (2.45)
Us(x)=UsxA is defined in (2.25) and
A
(Us)=(d δ , χ λ ). (2.32) where
A i iIs − s α
N 2
Here δiI is an indicator of i belonging to I: δiI = 1, a=ac(N)= − . (2.46)
8(N 1)
i I and δ =0, i / I.
iI −
∈ ∈
The potential (2.22) reads
Here Ψf =Ψf(X) is the wave function corresponding
n ξ to the f-gauge (2.42) and satisfying the relation
V = jd e2Uj(x), (2.33)
j
2
j=0 Ψf = ebfΨf=0, b=(2 N)/2, (2.47)
X
−
where
∆[ ] and R[ ] denote the Laplace-Beltrami oper-
1 1
Uj(x) = UjxA = φj +γ (φ), (2.34) atoGr and the Gscalar curvature corresponding to .
A − 0 G1
(Uj) = ( δj +d ,0). (2.35) We note that parameter a may be arbitrary if we do
A − i i not care about the conformal covariance of the WDW
TheintegrabilityoftheLagrangesystem(2.28)de- equation.
pendsonthescalarproductsofco-vectors Uj, Us cor- For the scalar curvature of minisupermetric (2.30)
responding to G¯: we get (see (2.29) in [5]):
(U,U′)=G¯ABU U′ , (2.36)
A B R[ ]= 2(Us,Us). (2.48)
G −
where
For the Laplace operator we obtain
Gij 0
(G¯AB)= (2.37)
0 hαβ ∂ ∂
(cid:18) (cid:19) ∆[ ]= eUs(x) G¯ABe−Us(x)
is the matrix inverse to (2.31). Here (as in [11]) G ∂xA ∂xB
(cid:18) (cid:19)
2
Gij = δij + 1 , (2.38) +εse2Us(x) ∂∂Φs . (2.49)
di 2 D (cid:18) (cid:19)
−
i,j = 0,...,n. These products have the following The WDW equation (2.45) for f = 0 (in the har-
form: monic time gauge)
δ
(Ui,Uj) = ij 1, (2.39) 1 a
dj − HˆΨ= ∆[ ]+ R[ ]+θV Ψ=0, (2.50)
−2θ G θ G
(d(I ))2 (cid:18) (cid:19)
(Us,Us) = d(I )+ s +λ λ hαβ, (2.40)
s α β
2 D may be rewritten, using Eqs.(2.48), (2.49) and
−
(Us,Ui) = δ , (2.41)
− iIs d(I )
Usi =GijUs =δ s , Usα = χ λα, (2.51)
s=e,m. j iIs − D 2 − s
−
4 V.D. Ivashchuk, M. Kenmoku and V.N. Melnikov
as follows: (φi,ϕα), with Eqs.(3.7) substituted, are equivalent to
the Lagrange equations for the Lagrangian
∂ ∂ ∂ ∂
2θHˆΨ= Gij hαβ
(cid:26)− ∂φi∂φj − ∂ϕα∂ϕβ LQ = 1G¯ABx˙Ax˙B VQ (3.8)
2 −
2
∂
−εse2Us(φ,ϕ) ∂Φs where
(cid:18) (cid:19) 1
+ ∂ d(Is) n ∂ χ λα ∂ VQ =V + 2εsQ2sexp[2Us(x)], (3.9)
∂φi − D 2 ∂φj − s as∂ϕα
(cid:20)iX∈Is − Xj=0 (cid:21) (G¯AB) and V aredefinedin(2.31)and(2.22),respec-
tively. The zero-energy constraint (2.44) reads
+2aR[ ]+2θ2V Ψ=0. (2.52)
G 1
(cid:27) E = G¯ x˙Ax˙B +V =0. (3.10)
Q AB Q
2
Here Hˆ Hˆf=0 and Ψ Ψf=0.
≡ ≡ When the conditions (i)–(iii) are satisfied, exact
solutions for the Lagrangian (3.8) with the potential
3. Exact solutions
(3.9) and V ¿from (3.4) have the following form [14]:
Hereweusethefollowingrestrictionontheparameters xA(u)= U0A ln f (u u )
of the model: −(U0,U0) | 0 − 0 |
UsA
(i) ξ =d 1, ξ =...=ξ =0, (3.1) ln f (u u ) +cAu+c¯A (3.11)
0 0− 1 n − (Us,Us) | s − s |
onespace M =Sd0 isaunitsphereandall M (i>1)
0 i where u , u are constants,
0 s
are Ricci-flat;
(d 1)
0
f (τ)= − sinh( C τ), (3.12)
(ii) 0∈/ Is, (3.2) 0 √C0 0
p
and
i.e. the“brane”submanifold M (see(2.13))doesnot
Is
contain M0, and f (τ)= |Qs| sinh( C τ), ε <0; (3.13)
s s s
ν √C
(iii) (Us,Us)>0. (3.3) Qs s p
s
| | cosh( C τ), C >0, ε >0(;3.14)
s s s
The latter is satisfied in most of examples of interest. ν √C
s s
¿From (i), (ii) we get for the potential (2.33): p
C and C are constants. Here sinh(√Cx)/√C = x
0 s
V = 21ξ0d0e2U0(x), (3.4) forTCh=e c0o.ntravariantcomponents U0A =G¯ABUB0 are
where (see (2.39)) U0i = δ0i, U0α =0. (3.15)
−d
0
(U0,U0)=1/d 1<0. (3.5)
0− (For UsA,s S, see (2.51)).
∈
¿From (iii) and (2.41) we get The vectors c=(cA) and c¯=(c¯A) satisfy the lin-
ear constraint relations (due the configuration space
(U0,Us)=0. (3.6) splitting into a sum of three mutually orthogonalsub-
spaces, see [14])
3.1. Classical solutions
n
U0(c)=c0+ d cj =0, U0(c¯)=0, (3.16)
Consider a solution to the Lagrange equations corre- j
sponding to the Lagrangian (2.43) with the energy- Xj=0
constraint(2.44)undertherestrictions(3.1)–(3.3). We Us(c)= d ci χ λ cα =0, Us(c¯)=0. (3.17)
i − s asα
put f =0, i.e., use the harmonic time gauge.
iX∈Is
Integrating the Maxwell equations (for s=e) and
The zero-energy constraint E = E +E +(1/2)
the Bianchi identities (for s=m), we get G¯ cAcB = 0, with C = 2E (U00,U0)s, C =
AB 0 0 s
×
d exp( 2Us)Φ˙s =0 Φ˙s =Q exp(2Us), 2Es(Us,Us) may be written as
s
du − ⇐⇒ n
d
(cid:16) (cid:17) (3.7) C 0 =C ν2+h cαcβ + d (ci)2
0d 1 s s αβ i
0
− i=1
X
where Qs =const. We put Qs 6=0. 1 n 2
For fixed Qs, the Lagrange equations for the La- + d ci . (3.18)
i
grangian (2.43) with f = 0 corresponding to (xA) = d0 1 !
− i=1
X
On quantum analogues of p-brane black hole 5
The followingexpressionsfor the metric andscalar As usual,weseek asolutionto the WDWequation
fields follows from (2.51 (3.15)) and (3.15): (2.50) by separation of variables, i.e., we put
g =[fs2(u−us)]d(Is)νs2/(D−2)× Ψ∗(z)=Ψ0(z0)Ψs(zs)eiPsΦseipaza. (3.27)
× [f02(u−u0)]d0/(1−d0)e2c0u+2c¯0[du⊗du+f02(u−u0)g0] It follows from (3.26) that Ψ∗(z) satisfies the WDW
(cid:26) equation (2.50) if
ϕ+αXi=6=0ν[f2s2χ(uλ−α ulns)f]−νs2+δicIsαeu2+ciuc¯+α2.c¯igi(cid:27), ((33..2109)) 2Hˆ0Ψ0 ≡((cid:18)∂∂z0(cid:19)2+θ2ξ0d0e2q0z0)Ψ0
s s as | s|
=2 Ψ ; (3.28)
0 0
¿From the relation exp(2Us) = f−2 (following from E
s ∂ ∂
(3.6), (3.11) (3.17)) we get for the forms: 2Hˆ Ψ eqszs e−qszs
s s ≡ − ∂zs ∂zs
(cid:26) (cid:18) (cid:19)
s = Q f−2du τ(I ), (3.21)
F s s ∧ s +ε P2e2qszs Ψ =2 Ψ , (3.29)
s = e−2λ(ϕ) Q f−2du τ(I ) =Q¯ τ(I¯) (3.22) s s s Es s
F ∗ s s ∧ s s s (cid:27)
for s = e,m, res(cid:2)pectively, where(cid:3)Q¯ = Q ε(I )µ(I ) and
s s s s
and µ(I) = 1 is defined by the relation µ(I)du
τ(I )=τ(I¯) ±du τ(I) [14]. ∧ 2 0+ηabpapb+2 s+2aR[ ]=0, (3.30)
0 E E G
∧ ∧
Thus we obtain exact spherically symmetric so-
with a and R[ ] from (2.46) and (2.48), respectively.
lutions with internal Ricci-flat spaces (M ,gi), i = G
i Linearly independent solutions to Eqs.(3.28) and
2,...,n in the presence of several scalar fields and
(3.29) have the following form:
one form. The solution is presented by the relations
(3.20), (3.21)–(3.19)with the functions f0, fs defined eq0z0
ionn(t3h.1e2s)–o(lu3t.1io4n) apnadratmheetreerlsatcioAn,sc¯(A3.1(6A)–(=3.1i7,α),)(,3.C18), Ψ0(z0) = Bω00 −θ2(d0−1)d0 q0 !, (3.31)
0 p
Cs, νs. Ψ (zs) = eqszs/2Bs ε P2 eqszs , (3.32)
This solution describes a charged p-brane (electric s ωs s s q
(cid:18) s (cid:19)
or magnetic) “living” on the submanifold MIs (2.13), p
where the set I does not contain 0, i.e. the p-brane where
s
lives only in the “internal” Ricci-flat spaces.
1
Inthe non-compositecasewithseveralintersecting ω = 2 /q , ω = 2 ν2, (3.33)
0 E0 0 s 4 − Es s
p-branes,solutionsof this type wereconsideredin [12, r
p
13] (the electric case) and [15] (the electro-magnetic B0,Bs =I ,K are the modified Bessel function.
ω ω ω ω
case). For the composite case see [14]. The general solution of the WDW equation (2.50)
is a superposition of the “separated” solutions (3.27):
3.2. Quantum solutions
The truncated minisuperspace metric (2.31) may be Ψ(z)= dpdPd C(p,P, ,B)Ψ∗(z p,P, ,B),
E E | E
diagonalized by the linear transformation XB Z
(3.34)
zA =SA xB, (zA)=(z0,za,zs) (3.23)
B
where p = (p ), P = (P ), = ( ), B = (B0,Bs),
a s s
as follows: B0,Bs = I,K, and Ψ = ΨE(z p,EP, ,B) is given by
∗ ∗
| E
G¯ = dz0 dz0+ηsdzs dzs+dza dzbηab, (3.24) therelations(3.27),(3.31)–(3.33)with E0 ¿from(3.30).
− ⊗ ⊗ ⊗ Here C(p,P, ,B) are smooth enough functions. For
E
where a,b=1,...,n; η =η δ ;η = 1, and several intersecting p-branes (non-composite electric
ab aa ab aa
±
and composite electro-magnetic) see [12] and [14], re-
q z0 =U0(x), q zs =Us(x), (3.25)
0 s spectively.
with q = (U0,U0)1/2 = [1 1/d ]1/2 > 0, q =
0 0 s
| | −
ν−1 = (Us,Us)1/2. 4. Black hole (BH) solutions
s | |
¿From (2.49), (3.23), (3.24) we get
4.1. Classical BH solutions
∂ 2 ∂ ∂
∆[ ]= +ηab Letussingleoutsolutionswithahorizon(withrespect
G − ∂z0 ∂za∂zb
(cid:18) (cid:19) to time t). We put
2
∂ ∂ ∂
+ eqszs e−qszs +ε e2qszs . (3.26)
∂zs ∂zs s ∂Φs 1 Is, (4.1)
(cid:18) (cid:19) (cid:18) (cid:19) ∈
6 V.D. Ivashchuk, M. Kenmoku and V.N. Melnikov
i.e., the p-brane contains the time manifold. Let Recall that ν2 =(Us,Us)−1.
s
ε = 1 (4.2) Extremal case. Inthe extremalcase µ +0 we get
s − for the metric (4.7) →
This is a physical restriction satisfied when a pseudo-
Euclidean brane in pseudo-Euclidean space is consid- g =H2d(Is)νs2/(D−2) dR dR+R2dΩ2
ered. s ⊗ d0
(cid:26)
We single outthe solutionwith a horizon: for inte- n
gration constants we put c¯A =0, Hs−2νs2dt dt+ Hs−2νs2δiIsgi , (4.15)
− ⊗
i=2 (cid:27)
UrA X
cA = µ¯ µ¯δA, (4.3) where H =H′ ( ′ = ) in (4.10), s=e,m.
(Ur,Ur) − 1 s s Ps Ps
r=0,s
X Remark. This solutionhas a regular horizonat R
C = C =µ¯2, (4.4) →
0 s +0 (with a finite limit of the Riemann tensor squared
for R +0) if ν2d(I )d¯ D 2 (see [19]).
where µ¯ > 0. Here A = (iA,αA) and A = 1 means → s s ≥ −
i =1. Itmaybe verifiedthatthe restrictions(3.16)–
A
4.2. Quantum analogues of BH solutions
(3.17) and (3.18) are satisfied identically.
Let us introduce the new radial variable R=R(u) Let us put a = 0 in the WDW equation. In this case
by the relations there exist a map that puts into correspondence some
2µ quantum solution to WDW equation to any classical
e−2µ¯u =1 , µ=µ¯d¯>0, d¯=d 1 (4.5)
− Rd¯ 0− BH solution. We also put
and put u0 =0, us <0, E0 =E0, Es =Es, (4.16)
Q i.e., the classical energies of subsystems coincide with
s
|µ¯νs|sinhβs =1, βs ≡µ¯|us|, (4.6) the eigenvalues of the Hamiltonians Hˆ0 and Hˆs, re-
spectively, and
s=e,m.
Then the solutions for the metric and the scalar ′νs
P =Q = , (4.17)
fields (see (3.20), (3.19)) are: s s −Ps d¯
i.e., the classicalcharges coincide with the eigenvalues
g =H2d(Is)νs2/(D−2) dR⊗dR +R2dΩ2 of the momentum operators Pˆ = ∂/∂Φs.
s 1 2µ/Rd¯ d0 s −
(cid:26) − Then it may be shown that
n
H−2νs2 1 2µ dt dt+ H−2νs2δiIsgi ,(4.7) p za =c xA. (4.18)
− s − Rd¯ ⊗ s a A
(cid:18) (cid:19) i=2 (cid:27)
X Using these relations, we get solutions to the WDW
ϕα =ν2χ λα lnH , (4.8)
s s as s equation(with a=0)ofspecialtype. These solutions
where correspond to classical BH solutions and have an am-
biguity in the choice of the Bessel functions. We note
Q d¯
H =1+ Ps, | s| e−βs. (4.9) that the quantum energy constraint (3.30) is satisfied
s Rd¯ Ps ≡ νs identically due to our choice a=0.
The form field is given by (2.16), (2.17) with Extremal case. In the extremal case we have
ν
Φs = s , (4.10) = =p za =0. (4.19)
H′ E0 Es a
s
′ Hence the wave function (4.20) reads
H′ =1+ Ps , (4.11)
s Rd¯+ ′ Ψ =Ψ Ψ exp(iQ ), (4.20)
Ps−Ps ∗ 0 s s
′ Qsd¯ where
. (4.12)
Ps ≡− ν
s Ψ = B0 iθ r v0 , (4.21)
s=e,m. It follows from (4.6), (4.9) and (4.12) that 0 0 | | 0
′ = µ = eβs = ( +2µ). (4.13) Ψs = vs1/2(cid:0)B1s/2(iQ(cid:1)svs), (4.22)
|Ps| sinhβ Ps Ps Ps r = d (d 1) and
s 0 0 0
−
p
The Hawking “temperature” corresponding to the p n
solution is (see also [15, 18]) v0 = eq0z0 =exp( φ0+ diφi), (4.23)
−
i=0
d¯ 2µ νs2 v = eqszs =exp( χ λ(ϕX)+ d φi), (4.24)
T = . (4.14) s s i
H 4π(2µ)1/d¯(cid:18)2µ+Ps(cid:19) − iX∈Is
On quantum analogues of p-brane black hole 7
are “quasivolumes”, s = e,m. The gravitational part The (extremal) solutions read
of the wave function, i.e. Ψ , coinsides with that of
0
Refs.[22, 23], see also[21, 24]. For small quasivolumes Ψ =B0 iθ √2v0 v1/2
v 0 we get ∗ 0 | | s ×
s → Bs (cid:16)( i vs)(cid:17)exp( i ν Φs), (4.32)
× 1/2 − Ps − Ps s
Ψ vs 2iQ /π, B =I, (4.25)
s s
∼ with v = exp(φ1), s = e,m, and v = exp(φ0 +
Ψs ∼ pπ/2iQs, B =K. (4.26) φ1). Wse see that here electric and mag0netic solutions
For apbig brane quasivolume v we get coincide. (This fact may be concidered as a simple
s
→∞ manifestationofelectro-magneticdualityataquantum
exp(iQsvs) level).
Ψ , B =I, (4.27)
s
∼ √2πiQ
s
exp( iQ vs)
Ψ − s , B =K, (4.28) 4.2.3. WDW equation with fixed charges
s
∼ 2iQ /π
s
There exists another quantization scheme, where the
Thus,forappositivecharge s or,equivalently, Qs <0 fields of forms are considered to be classical. This
P
(see(4.17)),thebranepartofthesolutionwith B =K scheme is based on the zero-energy constraint relation
satisfies the outgoing-wave boundary condition, used (3.10), see [20]. The corresponding WDW equation in
first in quantum cosmology in [25], and is regular for the harmonic gauge reads
small brane quasi-volumes.
1 ∂ ∂
Hˆ Ψ G¯AB +θV Ψ=0 (4.33)
4.2.1. Example: D =11 supergravity Q ≡ −2θ ∂xA∂xB Q
(cid:18) (cid:19)
Consider D = 11 supergravity [7] with the truncated where the potential V is defined in (3.9). This equa-
Q
bosonic action (without a Chern-Simons term) tion describes quantum cosmology with classicalfields
offormsandquantumscalefactorsanddilatonicfields.
1
S = d11z g R[g] F2 , (4.29) The basis of solutions is given by the following re-
11tr
ZM | |(cid:26) − 4! (cid:27) placements in (3.27), (3.30), (3.32) and (3.33): Ps
p Q , 2aR[ ] 0, ω 2 ν2. and Ψ (zs) 7→
where F is a 4-form. Here we have two types of so- s G 7→ s 7→ − Es s s 7→
lutions: an electric 2-brane with d(Is) = 3 (s = e) Bωss εsQ2seqszs/qs . p
and a magnetic 5-brane with d(Is) = 6 (s = m). In In(cid:16)pthis approach(cid:17)there is no problem with the a
both cases (Us,Us) = 2 = νs−2, (s = e,m. We put parameter when quantum analogues of black-hole so-
ε(1)= 1 and ε(k)=1, k >1. lutions are constructed (since in the harmonic gauge
−
In the extremal case we get for an M2-brane (s= R(G¯)=0).
e) and an M5-brane (s=m): Letuscomparequantumblack-holesolutionsinthe
two approaches. The function Ψ is the same in both
Ψ =B0 iθ r v0 v1/2 0
∗ 0 | | 0 s × approachesbutforthebranepartofthewavefunction
Bs (cid:0) i 1 v(cid:1)s exp i νsΦs , (4.30) we get
× 1/2 − Ps2d¯ − Ps d¯
(cid:18) (cid:19) (cid:18) (cid:19)
Ψ =Bs(iQ vs). (4.34)
s 0 s
with v = exp( d φi), s = e,m, and v defined
s i∈Is i 0
in (4.23). For small quasivolumes v 0 we get
s
P →
Ψ 1, B =I, (4.35)
4.2.2. Example: D =4 Einstein-Maxwellgrav- s ∼
ity Ψs ln(iQsvs/2), B =K. (4.36)
∼−
Consider D =4 gravity:
For a big brane quasivolume, v , we get
s
→∞
S4 =ZMd4z |g|(cid:26)R[g]− 21!F2(cid:27), (4.31) Ψs ∼ e√xp2π(iiQQssvvss), B =I, (4.37)
p
where F is a 2-form. Here we have two types of exp( iQsvs)
Ψ − , B =K, (4.38)
s
solutions: an electric 0-brane (electric charge) with ∼ 2iQ vs/π
s
d(I ) = 1 (s = e) and a magnetic 0-brane (mag-
s
p
netic charge) with d(I ) = 1 (s = m). In both cases Thusfor Q <0 thebranepartofthesolutionwith
s s
(Us,Us)=1/2=ν−2 (s=e,m). We put ε(1)= 1. B =K satisfiestheoutgoing-waveboundarycondition
s −
Here n=1, d =2 and d =1. [25]butitisnotregularforasmallbranequasi-volume.
0 1
8 V.D. Ivashchuk, M. Kenmoku and V.N. Melnikov
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