H. Lu, C.N. Pope and K.-W. Xu- Black p-Branes and their Vertical Dimensional Reduction

8/3/2019 H. Lu, C.N. Pope and K.-W. Xu- Black p-Branes and their Vertical Dimensional Reduction

1/17

IC/96 /171CTP TAMU-46/96hep-th/9609126

United Nations Educational Scientific and Cultural OrganizationandInternational Atomic Energy AgencyINTERNATIONAL CENTRE FOR THEORETICAL PHYSICS

BLACK p-BRANESAND THEIR VERTICAL DIMENSIONAL REDUCTION

H. Lii t , C.N. Po pe tCenter for Theoretical Physics, Texas A&M University,College Station, Texas 77843, USAand

K.-W. XuInternational Centre for Theoretical Physics, Trieste, Italyand

Institute of Modern Physics, Nanchang University,Nanchang, People's Republic of China.

A B S T R A C TWe construct multi-center solutions for charged, dilatonic, non-extremal black holes in

D = 4. When an infinite array of such non-extremal black holes are aligned periodicallyalong an axis, the configuration becomes independent of this coordinate, which can there-fore be used for Kalu za-K lein com pactification. Th is generalises the vertical d imen sionalreduction procedure to include non-extremal black holes. We then extend the constructionto multi-center non-extremal (D 4)-branes in D dimensions, and discuss their verticaldimensional reduction.

MIRAMAR E - TR IESTENovember 1996

Research supported in part by DOE Grant DE-FG05-91-ER40633.


2/17

1 IntroductionThe supergravity theories that arise as the low-energy limits of string theory or M-theoryadmit a multitude of p-brane solutions. Ingeneral, these solutions are characterised bytheir mass per unit p volume, and the charge or charges carried by the field strengthsthat support thesolutions. Solutions can be extremal, in the case where thechargesand the mass per unit p-volume saturate a Bogomol 'nyi bound, or non-extremal if themass per unit p-volume exceeds the bound. There are two basic types of solution, namelyelementary p-branes, supported by field strengths of rank n= p + 2 , and solitonic p-branes,supported by field strengths of degree n = D p 2, where Dis the spacetime dimension.Typically, we are interested inconsidering solutions in a "fundamental" maximal theorysuch asD= 11 supergravity, which is the low-energy limit of M-theory, and its varioustoroidal dimensional reductions. A classification of extremal supersymmetric p-branes inM-theory compactified on a torus can be found in[?].

The various p-brane solutions inD < 11 can be represented as points on a "branescan" whose vertical and horizontal axes are the spacetime dimension D and the spatialdimension p of the p-brane world-volume. The same process of Kaluza-Klein dimensionalredu ction th at is used in order to construc t th e lower-dimensional toroidally-compactifiedsupergravities can also be used to perform dimensional reductions of the p-brane solutionsthemselves: Since theKaluza-Klein procedure corresponds to performing a consistentt runcat ion of the higher-dimensional theory, it is necessarily thecase that the lower-dimensional solutions will also besolutions of the higher-dimensional theory. There aretwo types of dimensional reduction that can be carried out on the p-brane solutions. Themore straightforward one involves a simultaneous reduction of the spacetime dimensionD and the spatial p-volume, from (D,p) to (D 1,p 1); this is known as "diagonaldimensional reduction" [?, ?]. It is achieved by choosing one of the spatial world-volumecoordinates as the compactification coordinate. The second type of dimensional reductioncorresponds to a vertical descent on the brane scan, from (D,p) to (D 1,p), implyingthat one of the directions in the space transverse to the p-brane world-volume is chosenas the compactification coordinate. This requires that one first construct anappropriateconfiguration of p-branes inD dimensions that has the necessary U(1) isometry alongthe chosen direction. It is not apriori obvious that this should bepossible, ingeneral.However, it is straightforward to construct such configurations in the case of extremalp-branes, since these satisfy a no-force condition which means that two or more p-branes


3/17

can sit in neutral equilibrium, and thus multi p-brane solutions exist [?]. By taking alimit correspond ing to an infinite continuum of p-bran es arrayed along a line, the requiredU(1)-invariant configuration can be constructed [?, ?, ?, ?].

In this paper, we shall investigate the dimensional-reduction procedures for non-extrem al p-bra nes. In fact the process of diagonal reduc tion is the sam e as in the ex-tremal case, since the non-extremal p-branes also have translational invariance in thespatial world-volume directions. The more interesting problem is to see whether one canalso describe an analogue of vertical dimensional reduction for non-extremal p-branes.There certainly exists an algorithm for constructing a non-extremal p-brane at the point(D 1,p) from one at (D,p) on the brane scan. It has been shown that there is a universalprescription for "blackening" any extremal p-brane, to give an associated non-extremalone [?]. Thus an algorithm, albeit inelegant, for performing the vertical reduction is tostart with the general non-extremal p-brane in D dimensions, and then take its extremallimit, from w hich an extrem al p-brane in D 1 dimensions can be obtained by the stand ardvertical-reduction procedure described above. Finally, one can then invoke the blackeningprescription to construct the non-extremal p-brane in D 1 dim ension s. However, unlikethe usual vertical dimensional reduction for extremal p-branes, this procedure does notgive any physical interpretation of the (D 1)-dimensional p-brane as a superposition ofD-dimensional solutions.

At first sight, one might think that there is no possibility of superposing non-extremalp-br anes , owing to the fact th at they do not satisfy a no-force condition. Indeed, it isclearly true that one cannot find well-behaved static solutions describing a finite numberof black p-branes located at different points in the transverse space. However, we do notrequire such general kinds of multi p-brane solutions for the purposes of constructing aconfiguration with a U(1) invariance in the transverse space. Ra ther , we require onlyth at there should exist static solutions corresponding to an infinite num ber of p-b rane s,periodically arrayed along a line. In such an array, the fact that there is a non-vanishingforce between any pair of p-branes is immaterial, since the net force on each p-brane willstill be zero. The configuration is in equilibrium, although of course it is highly unstable.For example, one can have an infinite static periodic array of D = 4 Schwarzschild blackholes aligned along an axis. In fact the instab ility problem is overcome in the K aluza-Klein reduction, since the extra coordinate z is compactified on a circle. Thus there is astable configuration in which the p-branes are separated by precisely the circumference ofthe compactified dimension. Viewed from distances for which the coordinates orthogonal


4/17

to z are large comp ared w ith this circumference, the fields will be effectively inde pen den tof z, andhence z can be used as the compactification coordinate for the Kaluza-Kleinreduction, giving rise to a non-extremal p-brane inD 1 dimensions.

In section 2, we obtain the equations of motion for axially symmetric p-branes in anarbitrary dimension D. We then construct multi-center non-extremal D = 4 black holesolutions in section 3, and show how they may beused forvertical dimensional reductionof non-extremal black holes. Insection 4, we generalise the construction to non-extremal(D 4)-branes inarbitrary dimension D.

2 Equations of motion for axially symmetric p-branesWe are interested in describing multi-center non-extremal p-branes inwhich thecenterslie along a single axis in the transverse space. Metrics with the required axial symmetrycan beparameterised in the following way:

ds2 = -e 2U dt2 + e2A dx idx i + e2V (d z2 + dr2) + e2B r2dQ 2 , (1)where (t,xi), i = 1,...,p, are the coordinates of thep-brane world-volume. The re-maining coordinates of th e D dimensional spacetime, i.e. those in the transverse space,are r, z and the coordinates on a d-dimensional unit sphere, whose metric is dfl2, withd = D p 3. The functions U, A, V and B depend on the coordinates r and z only.We find that the Ricci tensor for the metric (??) is given by

R00 = e2U-2V([J"+ U + U'2 + U2+p(U'A' + uA) + d(U'B' + UB) + dv rRrr = - ( [ / " - U'V+U12 + UV + V+ V" +p{A" -V A1 + A I2 + VA)

+d(B" - V'B' --V' + VB+ B'2 + -B'r rRzz = - (U - UV + U 2 + U'V+V+ V" +

+d(B -VB + V'B' +B2 + 1Vr+d(-B' +VB' + -V--B- BB' + V'B)r r

Rab = -e 2B ~2V (B" + B + V'B' + UB + -U' + p(A'B' + AB + -A')+d(B'2 + B2 + -B')) r2gab + (d - 1)(1 - e2i?-2y)^, ,

i? tJ = -e2A ~2V (A" + A + U'A' + UA+ p{A'2 + A2) +d{A'B' + AB + -A'


5/17

where the primes and dots denote derivatives with respect to r and z respectively, gat, ist h e metric on the unit d- sphere, and the components are referred to a coordinate frame.

Let us consider axially- symmetric solutions to th e theory described by the Lagrangian^ - ^ F r a 2 , (3)

where Fn is an n- rank field strength. The constant a in th e dilaton prefactor can beparameterised as

(a A | ( 4 )

where the constant A is preserved under dimensional reduction [?]. (For supersymmetricp- branes in M- theory compactified on a torus, th e values of A are 4/N where N is aninteger 1 < N < Nc, and Nc depends on D an dp [?]. Non- supersymmetric (D 3)-braneswith A = 24/(N(N + 1)(N + 2)) involving N 1-form field strengths were constructed in[?]. Their equations of motion reduce to the SL(N + 1, R) Toda equations. Further non-supersymmetric p- branes with other values of A constructed in [?] however cannot beembedded into M- theory owing to the complications of the Chern- Simons modificationst o th e field strengths.) We shall concentrate on the case where Fn carries an electriccharge, and thus th e solutions will describe elementary p- branes with p = n 2. (Thegeneralisation to solitonic p- branes th at carry magnetic charges is straightforward.) Thepotential for Fn takes the form AOil...ip = je^...^, where is a function of r and z. Thust h e equations of motion will be

0 = - \aS 2 , RUN = \dM- 2Ua n d d = p + 1.

3 D = 4 black holes and th eir dimensional reduc tion3.1 Single- cen ter black ho lesLet us first consider black hole solutions in D = 4, whose charge is carried by a 2-form fieldstrength. By appropriate choice of coordinates, and by making use of the field equations,


6/17

we may set B = U. Defining also V = K U, we find that equations ofmotion (??)for the metric

d s 2 = _eW d t 2 + e2K-2U( d r 2 + d z 2 ) + e-2U f2 ^ 2 ( 7 )

c a n be reduced toV2U = |e- ^ - 2 f / (7 2 + Y2) , V2K - \K' = | e - ^ - 2 f / 7 / 2 - 2U'2 - \f ,

l \ + \ 4 x t > ' , (8)V2 7 = (a^' + 2U')i + (a

where V2 = d2 2+1d+d 22 isth e Laplacian for axially- symmetric functions in cylindricalObi t Obi OZ

polar coordinates.We shall now discuss th ree cases, with increasing generality, beginning with the pure

Einstein equation, with = 0 and = 0 . The equations (??) then reduce toV2U = 0 , K' = r{U2 - U2) , K = 2rU'U , (9)

thus giving a Ricci- flat axially- symmetric metric for any harmonic function U. The so-lution forK then follows byquadratures. Asingle Schwarzschild black hole isgiven bytaking Uto be the Newtonian potential for a rod ofmass M and length 2M [?], i. e.

f f lwhere = r2 + (z M)2 an d a = r2 + (z + M)2. The solution for KisK = (^ i2MK^ i 2M)

4(7(7Now we shall show that this isrelated t o the standard Schwarzschild solution in isotropiccoordinates, i.e.

(\ M_\2 = W dt2

2 R )

(\ M_\ Mds2 = - jr- Wr2 dt2 + (1 + 3 W + dy2 + 2d 2) , (12)

where R = ' 2 + y2. To do this, wenote that (??) is of the general form (??), butwith B U. Asdiscussed in [?], setting B =U depends firstly upon having a fieldequation for which the R00 and R components of the Ricci tensor are proportional, andsecondly upon performing a holomorphic coordinate transformation from = r + iz tonew variables = + iy. Comparing the coefficients ofd 2 in (??) and (??), we seet h a t we must have 3?(r/) = 3ft()(l ~~ m2/ (4C ))> a n dhence we deduce that the requiredholomorphic transformation isgiven by

oT fl


7/17

I t is now straightforward to verify that this indeed transforms the metric (??), with Ua n d K given by (??) and (??), into the standard isotropic Schwarzschild form (??).

N o w let us consider the pure Einstein- Maxwell case, where is non- zero, but a = 0a n d hence = 0. We find that t he equat ions of motion (??) can be solved by making theansatz

IT U on ^ on /1 0 ou\~^ /- , i\e = e u - c2eu , = 2ce 2U ( l - c2e2U j , (14)where c is an arbitrary constant and U satisfies V2f7 = 0. Substituting into (??), we findt h a t all th e equation s are th en satisfied if

K' = r{U'2- U ) , K = 2rUU (15)

( O u r solutions in th is case are in agreement with [?], after correcting some coefficients andexponents.) The solution for a single Reissner-Nordstrm black hole is given by takingthe harmonic function U to be the Newtonian potential for a rod of mass \k and lengthk, implying that U and K are given by

~ 1 + a-k11 = \ lo a + - + f c

1( + a k)( + a + k)K = 1lo g( k ) . (16)4 ( 7 ( 7where = r2 + (z k/2)2 and a = r2 + (z + k / 2 ) 2. The metric can be re- expressed int e r m s of the standard isotropic coordinates (i,p,y,9) by performing the transformations

where = + iy an d k = (1 c2)k, giving

= - ( 1

^ ^ Sinh ^) (1 +

where c = tanh/ / , an d again R = 2 + y2. (It is necessary to rescale th e time coordinate ,as in (??), because th e function e~u given in (??) tends to (1c2) rath er th an 1 at infinity.)E q u a t i o n (??) is the standard Reissner-Nordstrm metric in isotropic coordinates, withmass M and charge Q given in terms of the parameters k and JJ L by

M = ksmh2fi + 12k , Q = \ksmh2fi . (19)


8/17

The extremal limit is obtained by taking k > 0 at th e same time as sending // > ,while keeping Q finite, implying that Q = 21M. This corresponds to setting c 1 f in( ? ? ) . The description in the form (??) becomes degenerate in this limit, since the lengthand mass of th e Newtonian rod become zero. However, the rescalings (??) also becomesingular, and the net result is th at the metric (??) remains well- behaved in the extremallimit. The previous pu re Einstein case is recovered if \i is instead sent to zero, implyingt h a t Q= 0 and c = 0.

Finally, let us consider the case of Einstein- Maxwell- Dilaton black holes. We find thatt h e equations of motion (??) can be solved by making the ansatze

0 = 2a(U- U), e~ AU = (e~U - c2 eU)e- a2U ,7 = 2ce2U(l- c2e2Uy1 , (20)

where, as in th e pure Einstein- Maxwell case, c is an arbitrary constant and U satisfiesV2f7 = 0. Substituting the ansatze into (??), we find that all the equations are satisfiedprovided that th e function K satisfies (??). The solution for a single dilatonic blackhole for generic coupling a is given by again taking the harmonic function U to be th eNewtonian potential for a rod of mass \k and length k . After performing th e coordin atetransformations

V=(l- c2) -^^- ^), t=(l- c2)it, (21)where k = ( 1 c2)l^Ak, and writing c = t&nh/i, we find t ha t the metric takes the standardisotropic form

ds2 = - M + sinhV) ( M di2V ( R+ \ k )2 J KR+\k }

^ 4 ( d 2 + d y 2 + 2 d 2 ) . (22)J 4R

The mass M and charge Q are given by4 V U ^ u . (23)A l 4VA

Again, the extremal limit is obtained by taking k > 0, \ i - ^ , while keeping Q finite,implying that Q =


9/17

extende d object. In order to carry out the reduction , it is necessary tha t the higher-dimensional solution be indepe nden t of the chosen compactification co ordin ate. In thecase of extremal p-branes, this can be achieved by exploiting the fact that there is azero-force condition between such objects, allowing arbitrary multi-center solutions to beconstru cted. M athem atically, this can be done because the equations of mo tion reduceto a Laplace equation in the transverse space, whose harmonic-function solutions can besuperposed . Thu s one can choose a configuration with an infinite line of p-bran es alongan axis, which implies in the continuum limit that the the solution is independent of thecoordinate along the axis.

As we saw in the previous section, the equations of motion for an axially-symmetricnon-extremal black-hole configuration can also be cast in a form where the solutions aregiven in terms of an arb itrary solution of Laplace's equa tion. Thu s again we can superposesolution s, to describe multi-black-hole c onfiguration s. We shall discuss the general caseof dilatonic black holes, since the a = 0 black holes and the uncharged black holes arem erely special cases. Specifically, a solu tion in wh ich U is taken to be the Newtonianpotential for a set of rods of mass 21kn and length kn aligned along the z axis will describea line of charged, dilatonic black holes:

_ i4m,n=1

WmOn + (Z-Zm- \km){z - zn + 21kn) + r2][(7m (Jn T \Z Zm 2n'rn)\Z Zn 2""n) T ' J

[ffmffn + (z ~ zm + 21k m)(z ~ Zn~ \k n) + r4 m,n=1

where o\ = r2 + (z zn 12 kn)2 and a% = r2 + (z zn + 12 kn)2. (Multi-center Schwarzschildsolutions were obtained in [?], correspon ding to (??) and (??) w ith kn = 2M n, and U equalto U rath er th an th e expression given in (??).) This describes a system of N non-extremalblack holes, which remain in equilibrium because of the occurrence of conical singularitiesalong the z axis. These singularities correspon d to the existence of (unphysical) "stru ts"that hold the black holes in place [?, ?]. If, however, we take all the constants kn to beequal, and take an infinite sum over equally-spaced black holes lying at points zn = nbalong the entire z axis, the conical singularities disapp ear [?]. In the limit when theseparation goes to zero, the resulting solution (??) becomes independent of z. For smallk = kn, we have U \k{r2 + (z - nb )2)~ 1/ 2 + O(k 3/r 3), and thus in the limit of small


10/17

b, the sum giving U in (??) can be replaced by an integral:(26)

+ z'2in the limit L > . Subtractin g out th e divergent constan t k(log2 + log L)/b, thisgives the z- independent result [?]

U=^logr. (27)Similarly, one finds that K is given by

K = kb22logr. (28)One can of course directly verify that these expressions for U and K satisfy the equationsof motion (??). Since the associated metric and fields are all z- independent, we can nowperform a dimensional reduction with z as the compactification coordinate, giving rise toa solution in D = 3 of the dimensionally reduced theory, which is obtained from (??),with D = 4 and n = 2, by th e stan dard Kaluza- Klein reduction procedure. A detaileddiscussion of this procedure may be found, for example, in [?]. From the formulae givent h e r e , we find that the relevant part of the D = 3 Lagrangian, namely the part involvingt h e fields that participate in our solution, is given by

e~ lC = R- \{d(t>)2 - \{d^)2 - \e-*-* F2 , (29)where

), occurring in the exponential prefactor of thefield strength F2 th at supports the solution, is non- zero. In other words, the orthogonalcombination should vanish, i.e. a


11/17

t h e D = 4 multi- black- hole solution becomes a single black hole with k = L > oo inthis case. If r is large compared with z, th e solution is effectively independent of z, an dthus one can reduce to D = 3 with z as th e Kaluza- Klein compactification coordinate.(This is rat her different from the situation in the extremal limit; in t h a t case, the lengthsa n d masses of the individual rods are zero, and the sum over an infinite array does n otdegenerate to a single rod of infinite length.)

If k and b are not equal, th e dimensional reduct ion of the D = 4 array of black holeswill of course still yield a 3- dimensional solution of the equations following from (??), butnow with the orthogonal combination a


12/17

feature t h a t the combination of scalar fields orthogonal to (f>i in dimensions vanishes, i.e.2oii 4>t+\ + at+\ fe = 0. This ensures t h a t a single- scalar solution in D dimensions remainsa single- scalar solution in all th e reduct ion steps. Thus we have the following recursiverelations

a4V -&)-


13/17

discussed in section 3, it will describe a single- scalar solution if k = b. In this case, we haveK = U = logr, and hence we find t h a t the (D 1)- dimensional metric ds2D_l , obtainedby taking z as the compactification coordinate, so t h a t ds2D = e2aifds2D_1 + e~2(- D~^aifdz2,is given by

ds2D_! = - e2fj dt2 + dxidxi + e^~m dr2 + r2e2fj- iU d 2 ,= - r 2 dt2 + dxidxi + (1 - c2r2)^ (dr2 + d 2) . (36)

Although th e condition t h a t the length k of the rods and their spacing b be equal isdesirable from the point of view t h a t it gives rise to a single- scalar solution in the lowerdimension, it is clearly undesirable in the sense t h a t the individual single p- brane solutionsare being placed so close together t h a t their horizons are touching. This reflects itself int h e fact t h a t th e sum over the single- rod potent ials is just yielding the potential for onerod, of infinite length and infinite mass, and accordingly, th e higher- dimensional solutionjust describes a single infinitely- massive p- brane. The corresponding vertically- reducedsolution (??), which one might have expected to describe a black ((D 1) 3)- branein (D 1) dimensions,1 thus does not have an extremal limit. This can be understoodfrom another point of view: A vertically- reduced extremal solution is in fact a line ofuniformly distribu ted extremal p- branes in one dimension higher. In order to obtain ablack ((D 1) 3)- brane in (D 1)- dimension t h a t has an extremal limit, we should beable to take a limit in the higher dimension in which the configuration becomes a line ofextremal p- branes. Thus a more appropriat e superposition of black p- branes in th e higherdimension would be one where the spacing b between t he rods was significantly largert h a n the lengths of th e rods. In particu lar, we should be able to pass to the extremallimit, where the lengths k tend to zero, while keeping the spacing b fixed. In this case, thefunctions U an d K will take the form (??) and (??) with k < b. Defining = k/b, we t h e nfind t h a t th e lower- dimensional met ric, after compactifying the z coordinate, becomes

213(13- 1) 2/3(/ 3- l) 4

This can be interpreted as a black ((D 1) 3)- brane in (D 1) dimensions. (In otherwords, what is normally called a (D 3)- brane in D dimensions.) The extremal limit isobtained by sending k = b to zero and \i to infinity, keeping b and the charge parameterQ = (fcsinh2/ i)/(4VA) finite. At t he same time, we must rescale th e r coord inate so t h a t

1The somewhat clumsy notation is forced upon us by the lack of a generic D-independent name for a (D 3)- brane in D dimensions.13


14/17

r > r(cosh/ x)4'A, leading to the extremal metricds2 = - dt2 + dxidxi + (l - ^ ^ log r ) A (dr2 + r2d 2) . (38)

Thus the solution (??) seems to be the nat ural non - extremal generalisation of th e extremal((D 1) 3)- brane (??). N o t e t h a t the black solutions (??) involve two scalar fields, aswe discussed previously, although in the extremal limit the additional scalar decouples.

I n fact th e above proposal for the non- extremal generalisation of (D 3)-branes inD dimensions receives support from a general analysis of non- extremal p- brane solutions.The usual prescription for construct ing black p- branes, involving a single scalar field, asdescribed for example in [?], breaks down in the case of (D 3)-branes in D dimensions,owing to the fact t h a t the transverse space has dimension 2, and hence d = 0. Specifically,o n e can show in general t h a t there is a universal procedure for "blackening" t he extrem alsingle- scalar p- brane ds2 = e2A(- dt2 + 0 and \i > , th e metr ic becomesds2 = - dt2 + dxidxi + (1 + QR)^(dR 2 + d 2) , (42)

where R = logr. Unlike th e situat ion for non - zero values of d, where the analogouslimit of th e black p- branes gives a normal isotropic extremal p- brane, in th is d = 0 caset h e extremal limit describes a line of (D 3)- branes in D dimensions, lying along the9 direction, rather t h a n a single (D 3)- brane. (In fact th is line of (D 3)- branes canbe further reduced, by compactifying the coordinate, to give a domain- wall solution ino n e lower dimension [?, ?].) Thus it seems t h a t there is no appropriate single- scalar non-extremal generalisation of an extremal (D 3)- brane in D dimensions, and the two- scalarsolution (??) t h a t we obt ained by vertical reduct ion of a black (D 3)- brane in one higherdimension is the na tural non- extremal generalisation.

14


15/17

5 ConclusionsIn this paper, we raised thequestion as to whether one can generalise theprocedure ofvertical dimensional reduction to the case of non-extremal p-branes. It is of interest todo this, since, combined with the more straightforward procedure ofdiagonal dimensionalreduction, it would provide a powerful way of relating the mult i tude of black p-branesolutions of toroidally-compactified M-theory, analogous to the already well-establishedprocedures for extremal p-branes. Vertical dimensional reduction involves compactifyingone of the directions transverse to the p-brane world-volume. Inorder toachieve the nec-essary translational invariance along this direction, one needs to construct multi-centerp-brane solutions in the higher dimension, which allow a periodic array of single-center so-lutions tobe superposed. This is straightforward for extremal p-branes, since the no-forcecondit ion pe rmits the construction of arbitrary multi-center configurations that remain inneutral equilibrium. No analogous well-behaved multi-center solutions exist ingeneral inthe non-extremal case, since there will be net forces between the various p-branes. How-ever, an infinite periodic array along a line will still be in equilibrium, albeit an unstableone. This is sufficient for the purposes of vertical dimensional reduction.

The equations of motion for general axially-symmetric p-brane configurations arerather complicated, and in this paper we concentrated on the simpler case where thetransverse space is 3-dimen sional. Th is leads to simplifications in the equations of m o-tion, and we were able to obtain the general axially-symmetric solutions for chargeddilatonic non-extremal (D 4)-branes inD dimensions. These solutions are determinedby a single function U that satisfies a linear equation, namely the Laplace equation on aflat cylindrically-symmetric 3-space, and thus multi-center solutions can be constructedas superpositions ofbasic single-center solutions. The single-center p-brane solutions cor-respond to the case where Uis theNewtonian potential for a rod of mass k and length

The rather special features that allowed us to construct general multi-center blacksolutions when the transverse space is 3-dimensional also have a counterpart in specialfeatures of the lower-dimensional solutions that we could obtain from them by verticaldimensional reduction. The reduced solutions are expected to describe non-extremal(D 3)-branes inD dimensions. Although a general prescription for constructing single-scalar black p-branes from extremal ones for arbitrary p and Dwas given in [?], we foundt h a t an exceptional case arises when p = D3. In this case thegeneral analysis in

15


16/17

[?] degenerates, and the single-scalar black solutions take the form (??), rather than thenaive d > 0 limit of (??) and (??) where onewould simply replace r~d by l ogr . Theextremal limit of (??) in fact fails to give the expected extremal (D 3)-brane, but insteadgives the solution (??), which describes a line of (D 3)-branes. Interestingly enough,we found that the vertical reduction of the non-extremal p-branes obtained in this papergives a class of (D 3)-branes which are much more natural non-extremal generalisationsof extremal (D 3)-branes. In particular, their non-extremal limits do reduce to thestandard ext remal (D 3)-branes. Theprice that onepays for this, however, is t h a tthe non-extremal solutions involve two scalar fields (i.e. the original dilaton of the higherdimension andalso theKaluza-Klein scalar), rather than just one linear combination ofthem . Thus we see tha t anumber of special features arise in the cases we have considered.It would be interesting to see what happens in the more generic situation when d > 0.

AcknowledgementWe are grateful toGary Gibbons for useful discussions, and toTuan Tran for drawing ourattention to some errors in an earlier version of the paper. H.L. andC.N.P. are gratefulto SISSA for hospitality in theearly stages of this work. K.-W.X. is grateful to TAMUfor hospitality in the late stages of this work.

References[1] H. Lii and C.N. Pope, p-brane solitons inmaximal super gravities, Nucl. Phys. B 4 6 5

(1996) 127.[2] M.J. Duff, P.S. Howe, T. Inami and K.S. Stelle, Superstrings in D= 10 from super-

membranes in D= 11, Phys. Lett . B 1 9 1 (1987) 70.[3] H. Lii, C.N. Pope, E. Sezgin andK.S. Stelle, Stainless super p-branes, Nucl. Phys.

B 4 5 6 (1995) 669.[4] Multi-center extremal static solutions have been considered in:

D. Majumdar, Phys. Rev 72 (1947) 390; A. Papapetrou, Proc . R. Irish Acad.A51(1947) 191; G.W. G ibbons a nd S.W. Haw king, Phys. Lett. 78B(1978) 430.

[5] R. Khuri, Nucl. Phys. 387(1992) 315.[6] J.P. Gauntlett, J.A. Harvey andJ.T. Liu, Nucl. Phys. 3 87 (1993) 363.

16


17/17

[7] C.M. Hull and P.K. Townsend, Unity of superstring dualities, Nucl. Phys. B 2 9 4(1995) 196.

[8] H. Lii, C.N. Pope and K.S. Stelle, Vertical versus diagonal dimensional reduction forp-branes, hep-th/9605082, to appear in Nucl. Phys. B.

[9] M.J. Duff, H. Lii and C.N. Pope, The black branes of M-theory, Phys. Lett . B 3 8 2(1996) 73.

[10] R.C. Myers, Higher-dimensional black holes in compactified spacetime, Phys. Rev.D 3 5 (1987) 455.

[11] H. Lii and C.N. Pope, SL(N +1, R) Toda solitons in super gravities, hep-th/9607027.[12] W. Israel and K.A. Khan, Collinear particles a nd Bondi d ipoles in general relativity,

Nuovo Cimento 33 (1964) 331.[13] J.L. Synge, Relativity: the general theory, Chap. VIII (Amsterdam, 1960).[14] G.W. Gibbons, Non-existence of equilibrium configurations of charged black holes,

Proc. R. Soc. Lond. A 3 7 2 (1980) 535.[15] G.W. Gibbons, Commun. Math. Phys. 35 (1974) 13.[16] H. Miiller zum Hagen and H.J. Seifert, Int. J. Theor. Phys. 8 (1973) 443.[17] P.M. Cowdall, H. Lii, C.N. Pope, K.S. Stelle and P.K. Townsend, Domain walls in

massive supergravities, hep-th/9608173.[18] E. Bergshoeff, M. de Ro o, M .B. Gree n, G. Pap ado po ulo s an d P.K. Townsen d, Duality

of Type II 7-branes and 8-branes, hep-th/9601150.

17

Documents

H. Lu, C.N. Pope and K.-W. Xu- Black p-Branes and their Vertical Dimensional Reduction