k.a. bronnikov- block-orthogonal brane systems, black holes and wormholes

8/3/2019 K.A. Bronnikov- Block-Orthogonal Brane Systems, Black Holes and Wormholes

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Gravitation & Cosmology, Vol. 4 (1998), No. 1 (13), pp. 4956c 1998 Russian Gravitational Society

BLOCK-ORTHOGONAL BRANE SYSTEMS,

BLACK HOLES AND WORMHOLES

K.A. Bronnikov

Centre for Gravitation and Fundamental Metrology, VNIIMS, 3-1 M. Ulyanovoy St., Moscow 117313, Russia1

Received 27 February 1998

Multidimensional cosmological, static spherically symmetric and Euclidean configurations are described in a unifiedway for gravity interacting with several dilatonic fields and antisymmetric forms, associated with electric andmagnetic p -branes. Exact solutions are obtained when certain vectors, built from the input parameters of themodel, are either orthogonal in minisuperspace, or form mutually orthogonal subsystems. Some properties of blackhole solutions are indicated, in particular, a no-hair-type theorem and restrictions in models with multiple times.From the non-existence of Lorentzian wormholes, a universal restriction is obtained, applicable to orthogonal or

block-orthogonal subsystems of any p -brane system. Euclidean wormhole solutions are found, their action and radiiare explicitly calculated. Absence of Euclidean wormholes among the solutions of 11-dimensional supergravity isstated.

bLOK-ORTOGONALXNYE SISTEMY p -BRAN, ERNYE DYRY I KROTOWYE NORYk.a. bRONNIKOW

rASSMATRIWA@TSQ EDINYM OBRAZOM MNOGOMERNYE KOSMOLOGIESKIE, STATIESKIE SFERIESKI-SIMMETRINYE IEWKLIDOWY KONFIGURACII DLQ GRAWITACII, WZAIMODEJSTWU@]EJ S NESKOLXKIMI DILATONNYMI POLQMI I ANTI-SIMMETRINYMI FORMAMI, KOTORYE SWQZANY S LEKTRIESKIMI I MAGNITNYMI p -BRANAMI. tONYE RE[ENIQPOLUENY W PREDPOLOVENII, TO NEKOTORYE WEKTORY, POSTROENNYE IZ ISHODNYH PARAMETROW MODELI, LIBOWZAIMNO ORTOGONALXNY W MINISUPERPROSTRANSTWE, LIBO OBRAZU@T BLOK-ORTOGONALXNU@ SISTEMU. uKAZANY NEKO-TORYE SWOJSTWA RE[ENIJ, OPISYWA@]IH ERNYE DYRY, W ASTNOSTI, ANALOG IZWESTNYH TEOREM OB OTSUTSTWIIWOLOS I OGRANI ENIQ, WOZNIKA@]IE W MODELQH S NESKOLXKIMI WREMENAMI. iZ NESU]ESTWOWANIQ LORENCEWYHKROTOWYH NOR POLUENO UNIWERSALXNOE OGRANIENIE, SPRAWEDLIWOE DLQ ORTOGONALXNYH I BLOK-ORTOGONALXNYHPODSISTEM L@BYH SISTEM p -BRAN. nAJDENY RE[ENIQ, OPISYWA@]IE EWKLIDOWY KROTOWYE NORY, QWNO WYIS-LENY IH DEJSTWIE I RADIUS. uSTANOWLENO OTSUTSTWIE RE[ENIJ W WIDE EWKLIDOWYH KROTOWYH NOR W 11-MERNOJSUPERGRAWITACII.

1. Introduction. The model

Multiple self-gravitating dilatonic fields and antisym-metric forms, associated with p -branes, naturally eme-rge in bosonic sectors of supergravities [1], superstringand M-theory, their generalizations and modifications[26]. This paper continues the recent studies of such

models begun in Refs. [711].For fields depending on a single coordinate, we

present in Sec. 2 a general exact solution, assumingthat certain vectors Ys built from the input param-eters, form an orthogonal system (OS) in minisuper-space. This solution generalizes many previous ones([4, 12, 13], etc.) and was first found in its presentform in [10]. A new class of exact solutions, general-

izing the OS one, is built for systems where Ys arenot all orthogonal, but form a block-orthogonal sys-tem (BOS). All principal results of [10] are extendedto BOS: in Sec. 3, for black hole solutions in this gen-

eralized framework, a no-hair type theorem is proved

1e-mail: [email protected]

and it is shown that even in spaces with multipletimes a black hole may only exist with its unique, one-dimensional time. Other results are new for both OSand BOS. The absence of Lorentzian wormholes forfields with positive energy is used to obtain a univer-sal restriction on p -brane system parameters (Sec. 4).Unlike Lorentzian wormholes, Euclidean ones do ex-

ist and their actions and radii are explicitly calculatedin Sec.5. A few examples are given in Sec.6. It isshown, in particular, that Euclidean wormholes are ab-sent among our solutions for standard 11-dimensionalsupergravity, whereas for a D = 12 theory [6] they arefound.

We consider D -dimensional gravity interactingwith several antisymmetric ns -forms Fs and dilatonicscalar fields a , with the action

S =1

22

M

dDz

|g|

R[g] abgMNMaNb

sS

sns!

e2saa

F2s

, (1)


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50 K.A. Bronnikov

in a (pseudo-)Riemannian manifold M = R1(u) M0 . . . Mn with the metric

ds2 = gMNdsMdzN =

w e2(u)

du2

+

ni=0

e2i(u)

ds2i , w = 1, (2)

where u is a selected coordinate ranging in R1(u) R ; gi = ds2i are metrics on di -dimensional factorspaces Mi of arbitrary signatures i = sign gi ; |g| =| det gMN| and similarly for subspaces;

F2s = Fs, M1...MnsFM1...Mns

s ;

sa are coupling constants; s = 1 (to be specifiedlater); s S, a A, where S and A are finite sets.All Mi , i > 0 are assumed to be Ricci-flat, while M0is a space of constant curvature K0 = 0,

1.

We assume a = a(u) and use the indices s S to jointly describe u-dependent electric ( FeI) andmagnetic (FmI) F-forms, associated with subsets I ={i1, . . . , ik} (i1 < . . . < ik ) of the set of factor spacenumbers {i} = I0 = {0, . . . , n} : S = { eI} {mI} .Given a set of potential functions s(u), electric andmagnetic F-forms are defined for each I as follows:

FeI = d eI I,FmI = e

2mIaa [dmI I], (3)

where is the Hodge duality operator, I def= i1. . .ik , i are the volume forms of Mi , d = du and adot denotes d/du . By construction,

n eI = rank FeI = d(I) + 1,

nmI = rank FmI = D rank FeI = d(I), (4)

where Idef= I0 \ I and d(I) =

iI di , the dimension

of MI = Mi1 . . . Mik .Nonzero components of FeI carry coordinate in-

dices of the subspaces Mi, i I, those of FmI indices of Mi, i I. In p -brane studies [3] one of thecoordinates of MI is commonly identified with timeand a form (3) corresponds to an electric or magnetic

(d(I)1)-brane living in the remaining subspace ofMI.

Several, instead of one, electric and/or magneticforms might be attached to each I; this would changeactually nothing in the solution process but complicatethe notations.

Our problem setting covers various classes of mod-els: (A) isotropic and anisotropic cosmologies: u istimelike, w = 1; (B) static models with varioussymmetries (spherical, planar, etc.): u is spacelike,w = +1 and time is selected among Mi ; (C) Eu-clidean models with similar symmetries, or models

with a Euclidean external space-time, w = +1.In all Lorentzian models with the signature ( ++ . . . +), the energy density Ttt of the fields Fs is

non-negative if one chooses in (1), as usual, s = 1for all s . In models with arbitrary i , one obtainsTtt 0 if

eI =

(I)t(I), mI =

(I)t(I), (5)

where (I)def=

iI i ; the quantity t(I) = 1 if thetime (t) axis belongs to the factor space MI andt(I) = 1 otherwise. If t(I) = 1, this is a trueelectric or magnetic field; for t(I) = 1 the F-formbehaves as an effective scalar or pseudoscalar in theexternal subspace (such F-forms will be called quasis-calar). Solutions will be written for arbitrary s .

2. Solutions

Let us use, as in [14], the harmonic u coordinate:

(u) = d00 + 1 , 1def

=n

i=1 dii

(u).The Maxwell-like equations for the F-forms are

easily integrated giving

FuM1...Md(I)eI = Q eI e

22 eI M1...Md(I)/

|gI|,FmI, M1...Md(I) = QmI M1...Md(I)

|gI|, (6)

where |gI| =

iI |gi| , Qs = const are charges andbars over and denote summing in a . In whatfollows we restrict the set S = {s} to such s thatQs = 0.

Let us assume (as usual) that neither of our p-

branes involves the external subspace M0 , (that is,0 I if Q eI = 0 or QmI = 0). Then, if z belongsto M0 , for the total energy-momentum tensor (EMT)one has Tuu + T

zz = 0. The corresponding combination

of the Einstein equations RNM 12 NMR = TNM has theform of the Liouville equation giving

e0 = (d0 1)S(wK0, k , u), k = const, (7)

where one more integration constant (IC) is sup-pressed by choosing the origin of u and

S(1, h, t) =

h1 sinh ht, h > 0,t, h = 0,

h1 sin ht, h < 0;

S(1, h, t) = h1 cosh ht; h > 0;S(0, h, t) = eht, h R. (8)

With (7) the D -dimensional line element may be

written in the form ( ddef= d0 1 )

ds2 =e21/d

[dS(wK0, k , u)]2/d

w du2

[dS(wK0, k , u)]2+ ds20

+

ni=1

e2i

ds2i . (9)

For the remaining field equations let us use theso-called -model (minisuperspace) approach (for its


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Block-Orthogonal Brane Systems, Black Holes and Wormholes 51

more general form see [7]). Namely, let us treat theset of unknowns i(u), a(u) (i = 1, . . . , n) a s areal-valued vector function xA(u) i n a n (n + |A|)-dimensional vector space V. Our field equations for iand a can be derived from the Toda-like Lagrangian

L = GABxAxB VQ(y)

n

i=1

(i)2 +21d

+ abab VQ(y),

VQ(y) =

s

sQ2s e

2ys (10)

with the energy constraint

E = GABxAxB + VQ(y) =

d + 1

dK,

K = k2 sign k, wK 0 = 1;

k2, wK0 = 0, 1, (11)where the IC k has appeared in (7). The nondegener-ate symmetric matrix

(GAB) =

didj /d + diij 0

0 ab

(12)

defines a positive-definite metric in V; the functionsys(u) are defined as scalar products:

ys =iI

dii ss Ys,AxA,

(Ys,A) =

diiIs , ssa

, (13)

where iI = 1 if i I and iI = 0 otherwise; the signfactors s and s are

eI = eI(I), mI = wmI(I); eI = +1, mI = 1. (14)

The contravariant components and scalar products ofthe vectors Ys are found using the matrix GAB in-verse to GAB (D = D 2)):

(GAB

) = ij /di

1/D 0

0 ab

,

(YsA) =

iI d(I)

D, ssa

; (15)

Ys,AYsA Ys Ys

= d(Is Is) d(Is)d(Is)

D+ ss ss . (16)

2.1. Orthogonal systems (OS)

The field equations are entirely integrated if all Ys aremutually orthogonal in V, that is,

Ys Ys = ss

N2s ,

1

N2s = d(I)

1 d(I)/D+ s2 > 0. (17)

Then the functions ys(u) obey the decoupled Liouvilleequations ys = bs e2ys , whence

eys(u) =

|bs|S(s, hs, u + us), (18)

where bsdef

= sQ2s/N2s , hs and us are ICs and thefunction S(.,.,.) has been defined in (8). For thesought functions xA(u) = (i, a) and the conservedenergy (11) we then obtain:

xA(u) =

s

N2s YsAys(u) + c

Au + cA, (19)

E =

s

N2s h2s sign hs + c

2 =d0

d0 1 K (20)

where the vectors of ICs c and c are orthogonal to allYs : c

AYs,A = cAYs,A = 0, or

cidiiIs cassa = 0,cidiiIs cassa = 0. (21)

2.2. Block-orthogonal systems (BOS)

One can relax, at least partly, the orthogonality re-quirement (17), assuming that some of the functionsys (13) coincide. Suppose that the set S splits intoseveral non-intersecting subsets, S= S , |S| =m(), such that the vectors Y() (() S ) formmutually orthogonal subspaces V in V:

Y() Y() = 0,

= . (22)

Suppose, further, that for each the functions y , {1, . . . , m} = S , coincide up to additive con-stants, which may be then absorbed by re-defining

the charges Q , so that the expression y(u)def=

Y(),AxA does not depend on . This coincidence

condition overdetermines the set of equations. How-ever, the consistency relations may be written in termsof ICs. Indeed, let us fix and suppose that , , mcorrespond to this . Comparing the expressionsy = Y,AxA with different , found from the La-grange equations, we obtain a set of linear algebraic

equations for the charge factors q = Q

2

:

(Y Y ) Z = 0, Z def=

S

q Y, (23)

which must hold for each pair (, ). If there are l m linearly independent vectors Y , i.e. dim V = l m, their differences determine an (l 1)-dimensionalplane, and Z satisfying (23) must lie in its orthog-onal direction. But (23) must hold for all m vectorsY , i.e. their ends must all lie on the same (l 1)-plane. Therefore (23) always has a solution with onefree parameter (charge) if l = m and has, in general,

no solution if l < m. (All q must be nonzero; if (23)gives some q = 0, the consideration is repeated anewwithout this .)


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52 K.A. Bronnikov

Assuming that Eqs. (23) have been solved for each , the solution process is completed as in Subsec. 2.1.Define

q = S

q; b = Y() S

q Y;

Y =1

q

S

q Y; N2 =

Y2 =bq

, (24)

where b is -independent due to (23). (We assumeb = 0, otherwise y = 0, making this degree of free-dom trivial.) Then in the case m = 1 we recovera single member of the OS of Subsec. 2.1, with thecharge factor b = bs , the vector Y = Ys (orthog-onal to all others) and its norm N2s . Thus singlebranes and BOS subsystems are represented in a uni-fied way. Replacing s in (18)(20), but leaving(21) unchanged, one obtains a generalized solution forthe model (1), valid for any BOS. The OS solution ofSubsec. 2.1 is its special case ( m() = 1, ), there-fore in what follows we mostly deal with BOS solu-tions. (Note that the IC vectors c and c are, even in

a BOS, orthogonal to each individual Ys .)The metric has the form (9), where the function

1 is

1 = d0 1D 2

N2y(u)

S

qq

d(I)

+ un

i=1

ci +n

i=1

ci. (25)

For OS ( s) the sum in reduces to d(Is).The OS solution is general for a given set of di and

Ys and contains |S| independent charges. BOS givespecial solutions, with the number of charges equal to|{}| , the number of subsystems; however, we thusgain exact solutions for more general sets of input pa-rameters, e.g. a one-charge solution can be obtainedfor actually an arbitrary configuration of branes withlinearly independent vectors Y .

Other integrable cases of the model under studycan be found using the known methods of solving non-trivial Toda systems, see e.g. [11, 15].

3. Black holes

The positive energy condition (5), fixing the inputsigns s for Lorentzian models, in the notation (14)reads s = t(I); this essentially restricts the possi-ble solution behaviour. Table 1 shows the sign factorswK0 and s = sign qs (= sign bs for OS) for F-formswith positive energy density in various classes of mod-els.

In static, spherically symmetric models1 u is a ra-

dial coordinate (w = +1, M0 = Sd0 , K0 = +1) and1Cosmological solutions for OS are discussed e.g. in [8, 11].

time is defined as a factor space among Mi , say, M1 ,with d1=1, 1= 1. The range of u is (0, umax),where u = 0 corresponds to spatial infinity and umaxmay be finite or infinite. The general solution, com-bining hyperbolic, trigonometric and power functions

for various signs of k and h , shows a diversity ofbehaviours, but a generic solution has a naked singu-larity. Possible exceptions are black holes (BHs) andwormholes(WHs).

BH solutions are obtained in the case h > 0 ,umax = . The functions i (i = 0, 2, . . . , n) anda remain finite as u under the following con-straints on the ICs:

h = k, ;cA = k

N2YA kA1 (26)

where A = 1 corresponds to i = 1 (time). The con-straint (20) then holds automatically.

The family (26) exhausts all BH solutions underour assumptions, except the extreme case k = 0. Theregularity conditions hold only if 1Is = 1, s , givingrise to an analogue of no-hair theorems2:

Statement 1. In the model (1), with the time axis

M1 and the vectors Ys forming an OS or a BOS, inBH solutions all Ys possess the property 1Is = 1 .

This condition selects true electric and magneticF-forms, eliminating quasiscalar ones, even among in-

dividual members of BOS subsystems.Under the asymptotic conditions a 0, i 0as u 0, after the transformation

e2ku = 1 2Mrd

, ddef= d0 1, M def= kd (27)

the BH metric and the corresponding scalar fields ac-quire the form

ds2 =

HA

dt2

1 2M

rd

H2N2

+ dr2

1 2M/rd

+ r2d2d0+n

i=2

ds2i s

HAi ;

Adef=

2

b

S

qd(I)

D

OS= 2N2s

d(Is)

D;

Aidef= 2

b

S

qiIOS= 2N2s iIs ; (28)

a =

1

bln H

S

qa

OS=

s

N2s ssa ln Hs, (29)

2A no-hair theorem similar to the present one was obtainedfor OS in [10]; in [12] such a theorem in D -dimensional dilatongravity was proved even for cases when no solutions were found.


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Table 1: Sign factors for different kinds of model

Cosmology Static spaces Euclideanw = 1 w = +1 w = +1

wK0

K0 K0 K0electric none +1 none

smagnetic none +1 noneelectric quasiscalar 1 1 1magnetic quasiscalar 1 1 +1

whereOS= means equal for OS, with = s; d2d0

is the line element on Sd0 and H are harmonic func-tions in R+ Sd0 :

H(r) = 1 +p/rd, p

def=

M2 + b M. (30)The active gravitational mass M

gand the Hawking

temperature, calculated by standard methods, are

GNMg = M +

N2Y1

p;

TH =d

4kB(2M)1/d

2M

2M +p

N2, (31)

where GN is Newtons constant of gravity. The ex-treme case of minimum mass for given charges Qs isM 0 (k 0). TH is zero in the limit M 0 if theparameter

def=

N2 1/d > 0, is finite if = 0

and is infinite if < 0. In the latter case3 the horizon

turns into a singularity as M 0.The behaviour of TH as M 0 characterizes the

BH evaporation dynamics. As TH depends on N2 ,

which in turn depend on the p -brane setup, the latteris potentially observable via the Hawking effect.

Some recent unification models involve several timecoordinates (see [5, 16] and references therein). Oursolutions with arbitrary i include all such cases. Ifthere is another time direction, t , it is natural to as-sume that some branes evolve with t . If we try tofind a BH with two or more times on equal footing,such that for other times t there is also gtt = 0 at

the horizon, we have to consider d1 > 1. A calculationthen shows that the regularity conditions are at vari-ance with the constraint (20). We conclude that evenin a space-time with multiple times a BH can onlyexist with its unique preferred, physical time, whileother times are not distinguished from extra spatialcoordinates.

4. Lorentzian wormholes and a

universal restriction

Our solution describes a WH (a nonsingular configu-

ration with the radius e0

at both ends of3As is explicitly shown in [10], an infinite value of TH can

occur only at a curvature singularity.

the range of u) if k < 0 and, in addition, the closestpositive zero of S(+1, k , u) sin |k|u (i.e. u = /|k|)is smaller than that of sin[|h|(u + u)] for any h |h| for all h < 0. Fur-thermore, for k < 0 and h < 0 one needs wK0 = 1and b > 0 , respectively; the latter is only possiblewith F-forms having s = 1 . Table 1 shows thatthese conditions can hold in static or Euclidean modelswith spherical symmetry rather than pseudosphericalor planar one, and in static models true electric andmagnetic fields are necessary.

Suppose k < 0. Since in Eq.(20) c 2 0, therequirement |k| > |h| means that

{: h d0d0 1 . (32)

This inequality is not only necessary, but also suf-ficient for the existence of WHs with given input pa-rameters: di and the vectors Ys . Indeed, put c

A = 0and Q() = 0 in all subsystems with q < 0 (notethat, by (24), sign q = sign b .) Choose all h < 0to be equal, then due to (32) |h| < |k| . It is now easyto choose the ICs u so that sin[|h|(u + u)] > 0 onthe whole segment [0, /|k|] and this yields a WH.

On the other hand, using a (d0 + 2)-dimensionalformulation of the present model, one can prove (see

the Appendix) that under the positive energy condi-tion (5) Lorentzian WHs are absent.

The non-existence of spherical WHs is incompat-ible with (32), and the properly formulated oppositeinequality must hold. Thus, for OS

s

1IsN2s

d0d0 1 , or for sa = 0:

s

1Is

d(Is)

1 d(Is

D

1 d0

d0 1 , (33)

where the factor 1Is excludes quasiscalars. A moregeneral formulation of this conclusion is

Statement 2. Consider a vector space V with themetric (12), where di N , i = 0, . . . , n, d0 > 1 ,


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54 K.A. Bronnikov

d1 = 1 , D =n

i=0 di 1 , and a set of vectors Ysdefined in (13) (Is {1, . . . , n} , ssa R). Then,if all Ys are mutually orthogonal in V, the inequality(33) holds.

This formulation does not mention F-forms, time,etc., and is actually of purely combinatorial nature.No doubt there exists its combinatorial proof, but,surprisingly, it has been obtained here from physicallymotivated analytical considerations.

The corresponding result for BOS may be pre-sented as follows:

Statement 2a. In the model (1), under the condi-

tions of Statement 2, for vectors Ys forming a block-orthogonal system in V, the following inequality holds:

{: q>0}

N2

d0d0 1 (34)

where q and N2 are defined in (24) and, for all in-cluded qs , s = sign qs = 1 + 21Is .

The condition for s means that these sign factorsdepend on the inclusion or non-inclusion of the distin-guished one-dimensional factor space M1 (the timeaxis in Lorentzian models) into the world volumes ofBOS members. Evidently, even in Lorentzian modelsthere can be quasiscalar forms among such members.

In the OS case, (34) turns into (33).

Statements 2 and 2a are valid for any set of Fs -forms under the specified conditions, even if this set isonly a subsystem in a bigger model, for which maybeno solution is known.

5. Euclidean wormholes

As seen from Table 1, in Euclidean models with theaction (1) WH solutions can be built only with the aidof magnetic forms Fs . Just as in Lorentzian cosmol-ogy, all F-forms are quasiscalar since time is the ucoordinate, out of any I, but in cosmology no fields

are able to create hs < 0. The Wick rotation, pre-serving all s , changes w and hence mI, leaving thesame eI. This distinction is related to the propertyof the duality transformation to change the EMT signin Euclidean models [17].

Since all F-forms are now quasiscalar, we are nomore restricted to Is containing a distinguished index.The condition (32) is, as before, necessary and suffi-cient for WH existence, but there is a wider choice ofIs able to give hs < 0 or h < 0 and, as a result, tofulfil it.

Classical Euclidean WHs are used to describe

quantum tunnelling, where the finite action plays acrucial role. Let us find it for WHs appearing amongour solutions.

The Euclidean action SE corresponding to (1) is

SE =1

22

M

dDz

g

R[g] + abgMNMaNb

+sS

1ns!

e2saa

F2s

, (35)

where, for simplicity, we put s = 1 and all i =+1 (full Euclidean signature). Due to the trace ofthe Einstein equations, R[g] = TMM /(D 2), thecurvature cancels the scalar term in the action andthe remainder is expressed in terms of F2s .

WH solutions can contain both electric and mag-

netic forms and for both one has (1/ns!) e2+2 F2s =

Q2s e2ys (within each BOS subsystem all ys = y ). As

a result,

SE =1

22

M

dDz

n

i=1

gi

s

2ns2D 2 Q

2s e

2ys . (36)

Let us assume that (i) there are only BOS sub-systems with h < 0 (or, in an OS, only magneticforms Fs with hs < 0); (ii) the WH is symmetric

[i.e. y(v) = y(v) for all , v def= u /(2|k|)]and (iii) the boundary condition y(a)=y(a) = 0 isvalid, where 2a=/|k| (this is just normalization ofcharges, with no effect on generality). Then the solu-tion (18) for y can be written in the form e

y =

(b/|h|)cos |h|v , where b > 0.The extra dimensions, provided their volumes arefinite, can be integrated out in (36). The remainingintegrals in v are easily found. The final expression is

SE =1

16GN

2d/2+1

(d/2 + 1)

2

b

b h2

S

2n 2D 2 Q

2

. (37)

where, for correspondence, the factor 1/22 times theinternal space volumes (i = 1, . . . , n)) is identified

with (16GN)1

and the second factor in (37) is thevolume of Sd0 ( is Eulers gamma function).Another quantity of interest is the radius rth of

the WH throat, determining a length scale. For asymmetric WH it is just the value of e0 at v = 0 .Under the above assumptions,

rth =

|k|d

1/d

[y(0)]A/2,

y(0) =

b/|h|, (38)where A has been defined in (28). Simplifications for

OS are evident.Quantum versions of the present models can beobtained as described e.g. in [11].


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6. Examples

1. The simplest and even degenerate example of Fs isthe Kalb-Ramond field FMN P in 4 dimensions, wherethe indices do not contain u . It is a magnetic form

with I = , the coupling = 0, so that its vector Y =0, and the solution containing only FMN P is trivial.The Euclidean WH solution obtained in this case, withits SE and rth , coincides with that of Refs. [17] afteradjusting the notations.

2. In 11-dimensional supergravity [3], with a = s =0 , the orthogonality conditions (17) are satisfied by2-branes, d(Is) = 3, and 5-branes, d(Is) = 6, if theintersection rules hold: d(3 3) = 1 , d(3 6) = 2,d(6 6) = 4 (the notations are evident). In particular,with d0 = 2 or d0 = 3 and other di = 1, there canbe seven mutually orthogonal 2-branes, but no more

than three of them can have 1Is = 1, i.e., describetrue electric or magnetic forms in a static space-time[8, 10]. Since for all 2- and 5-branes N2s = 1/2, the BHtemperature (31) in the extreme limit tends to infinityif there is one such brane, to a finite limit if there aretwo and to zero if there are three branes. LorentzianWHs are absent since (32) requires

s N

2s > 2 for

d0 = 2 and > 3/2 for d0 = 3 . In the Euclidean casewe can have as many as 7 magnetic 2-branes, eachwith N2s = 1/2, and WHs are easily found. Though,the latter is true if one considers FmI of rank 7. If oneremains restricted, as usual, to Fs of rank 4 [3], then

for magnetic forms d(I) = 6, and Euclidean WHs donot exist. One can show that a consideration of BOSdoes not change this conclusion.

3. Example of a BOS in the same model: let thedigits 1, . . . , 7 label 1-dimensional factor spaces, andconsider a set of 6 branes with the following Is :

a : 123456,b : 123,

c : 234,d : 147,

e : 257,f : 367.

The F-forms associated with a,b,c , with wrongintersections (3- and 2-dimensional), form a BOS sub-

system; d,e,f are separate forms with Ys orthogo-nal to all others. In this example our scheme gives a(probably) new solution with 4 charges.

4. Consider the 12-dimensional field model [6], admit-ting the bosonic sector of 11-dimensional supergravityas a truncation. The model contains one scalar , twoF-forms of ranks 4 and 5, and the coupling constants21 = 1/10, 2 = 1 , and admits d(Is) = 3, 4(electric 2- and 3-branes) and d(Is) = 6, 7 (magnetic6- and 5-branes); for all branes N2s = 1/2. Orthogonalintersections obey the rules

d(3, 3) = d(3, 4) = 1;

d(4, 4) = d(3, 6) = d(4, 6) = d(3, 7) = 2;

d(4, 7) = 3; d(6, 6) = d(6, 7) = 4;

d(7, 7) = 5. (39)

The parameters 1,2 are pure imaginary, hence theICs cA in our solutions are imaginary as well, makingit easy to obtain both Lorentzian and Euclidean WHs(see footnote 5). It is of interest, however, to look forWHs under the old conditions, which restore if one

puts cA = 0. Such an example will follow.Since the orthogonal intersection rules of item 2

enter into (39) as a part, all previous examples forD = 11 are valid for this model; now, however, notonly the choice of branes is wider, leading to numer-ous OS and BOS, but one more dimension widens theopportunities even for d(Is) = 3, 6. Let us here give just one example of an OS of four magnetic 5-branes(d(Is) = 6), using the notation of item 3, but with 8world-volume dimensions:

a : 123456,b : 123478,

c : 125678,d : 345678.

This system gives rise to a Euclidean WH solutionfor a closed isotropic cosmology of correct physical di-mension, d0 = 3.

Appendix. Non-existence of Lorentzian

wormholes

In general relativity the neighbourhoods of WH throatsare known to necessarily violate the standard en-ergy requirements to matter [18]. Let us show thatsuch a conclusion extends to the model (1), namely,Lorentzian spherically symmetric WHs are impossibleif the positive energy requirement (5) is valid.

Let us first consider an arbitrary static, sphericallysymmetric (d0 + 2) -dimensional metric

ds2 = e2(u)dt2 + e2(u)du2 + e2(u)d2d0 . (A.1)Taking the harmonic u coordinate in (A.1), = +d0, we can write for the Ricci tensor components:

Rtt Ruu = d0 e2[ 2 (d0 1)2] (A.2)(a dot is d/du). A wormhole neck is the value of uwhere = 0 while4 > 0. Therefore the difference

(A.2) is positive as well, implying, via the Einsteinequations, the inequality

Ttt Tuu > 0 (A.3)for the EMT components, that manifests a violation ofthe so-called null energy condition. Indeed, a generalform of this condition is Tmnmn 0 for mm = 0,m, n = 0, 1, . . . , d0+2. For the metric (A.1) with

m =( e2 , e2, 0, . . . , 0) it takes the form Ttt Tuu 0.

On the other hand, integrating out in (1) all co-ordinates of the internal spaces Mi , i = 2, . . . , n

4It can happen that = 0 , then the first nonvanishing

derivative is even and positive. In this case our reasoning isslightly modified but valid. For a more detailed discussion ofthe properties of wormhole throats see Ref. [18]).


8/8

56 K.A. Bronnikov

and using the well-known conformal mapping5 gmn =e22/d0gmn in order to pass to the Einstein con-formal frame, one arrives at the following (d0 + 2)-dimensional action (up to a constant factor and a di-vergence):

Sd0+2 =

dd0+2z

|g|

R 1d0

(2)2

n

i=2

di (i)2 aA

(a)2 + . . .

(A.4)

where (f)2 def= gmnf,mf,n ; m, n label the phys-ical space-time coordinates, bars mark the metricand quantities obtained from it in the Einstein frame, 2 =

ni=2 di

i is the logarithm of the extra-dimensions volume factor, and dots replace the term

with F-forms in (1) where a proper dimensional re-duction is made (its sign is unchanged and its explicitform is now irrelevant).

As is evident from (A.4), the additional kineticterms due to dimensional reduction have the samesign as that for the scalar fields a . Their (d0 + 2)-dimensional EMT has the correct sign and their EMTpossesses the property Ttt Tuu = 2Ttt < 0. The samerelation holds for quasiscalar F-forms. For true elec-tric and magnetic ones we have Ttt Tuu 0. Com-bined, these properties make impossible the validity of(A.3).

Thus in the present class of models, just as in 4-

dimensional theory, wormhole solutions appear onlyat the expense of explicitly violating the conventionalenergy requirements6.

Acknowledgement

I would like to thank V. Ivashchuk, V. Melnikov and J.Fabris for many helpful discussions and the colleaguesfrom DF-UFES, Vitoria, Brazil, for kind hospitality. Igratefully acknowledge partial financial support fromCAPES (Brazil), Russian State Committee for Scienceand Technology and Russian Basic Research Founda-

tion, grant 98-02-16414.

References

[1] A. Salam and E. Sezgin, eds., Supergravities in Di-verse Dimensions, 2 vols., World Scientific, 1989.

[2] M.B. Green, J.H. Schwarz, and E. Witten, Super-string Theory, 2 vols., Cambridge UP, 1987.

5In the case of interest, the conformal factor is regular atu = 0 and u = /|k| , consequently, a WH solution in the initialmetric maps to a WH solution in the Einstein conformal frame,and vice versa.

6Indeed, if at least one of a is pure imaginary, its ca is also

pure imaginary, and, as is clear from (20), wormhole solutionsare readily obtained (cf. [14, 19]). Cancelling the positive energyrequirement for quasiscalar F-forms leads to the same result.

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[18] D. Hochberg and M. Visser, gr-qc/9704082.

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k.a. bronnikov- block-orthogonal brane systems, black holes and wormholes

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