Bolting multicenter solutions

Journal of High Energy Physics, Jan 2017

We introduce a solvable system of equations that describes non-extremal multicenter solutions to six-dimensional ungauged supergravity coupled to tensor multiplets. The system involves a set of functions on a three-dimensional base metric. We obtain a family of non-extremal axisymmetric solutions that generalize the known multicenter extremal solutions, using a particular base metric that introduces a bolt. We analyze the conditions for regularity, and in doing so we show that this family does not include solutions that contain an extremal black hole and a smooth bolt. We determine the constraints that are necessary to obtain smooth horizonless solutions involving a bolt and an arbitrary number of Gibbons-Hawking centers.

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Bolting multicenter solutions

Received: November Published for SISSA by Springer Iosif Bena 1 Guillaume Bossard 0 Stefanos Katmadas 1 David Turton 1 Open Access c The Authors. 0 Centre de Physique Theorique, Ecole Polytechnique, CNRS, Universite Paris-Saclay 1 Institut de Physique Theorique, Universite Paris Saclay, CEA , CNRS We introduce a solvable system of equations that describes non-extremal multicenter solutions to six-dimensional ungauged supergravity coupled to tensor multiplets. The system involves a set of functions on a three-dimensional base metric. We obtain a family of non-extremal axisymmetric solutions that generalize the known multicenter extremal solutions, using a particular base metric that introduces a bolt. We analyze the conditions for regularity, and in doing so we show that this family does not include solutions that contain an extremal black hole and a smooth bolt. We determine the constraints that are necessary to obtain smooth horizonless solutions involving a bolt and an arbitrary number of Gibbons-Hawking centers. Black Holes; Black Holes in String Theory - multicenter solutions Contents 1 Introduction and discussion The supergravity ansatz The theory and the equations The solution Extremal limits Absence of black holes General properties of the solution Conditions for smooth solutions Local smooth geometry Absence of closed time-like curves A Relation to the Floating JMaRT system Vector elds Introduction and discussion providing quantum \hair" for the black hole. typical microstates are complicated quantum superpositions of such basis states. black holes has proven much more di cult.1 emission [35]. In a near-BPS limit, the solutions have a large AdS3 S3 region, with the constructions [39{41]. subalgebras [24, 44{46]. construction adds a Gibbons-Hawking center to the JMaRT solution, at a nite distance while the second angular momentum remained slightly over-rotating. cumbersome to solve in the form in which it was originally derived. in terms of new variables that simplify the di erential equations, and to nd a general axisymmetric BPS and almost-BPS multicenter solutions. control that fully-backreacted supergravity solutions o er. dimensional reduction, these con gurations become solutions to ve-dimensional supergroup SO(1; 1) SO(1; nT ). Our new variables also have the advantage of making this symmetry manifest. extremal limits. the present system contains similar solutions involving nite-size regular black objects in non-extremal solutions, as we will discuss momentarily.3 the more general solution of this paper. These geometries are supported by uxes on the independent, the corresponding uxes are not additive4 because the three-spheres that shrink are di erent at each Gibbons-Hawking center. equations restrict the positions of the various centers. However, a priori this does not rule out asymptotically ve-dimensional solutions. 4The ux on a cycle linking points A and B is not the sum of the uxes on the cycles linking A and C and linking C and B with appropriate signs. from such smooth multicenter non-extremal solutions. one has much more control over their values [7, 12]. by a corresponding smooth scaling solution, with an arbitrarily long throat. Gibbons-Hawking centers. We believe that an exploration of this physics is of central black holes. in the system. We further present our general solution describing a non-extremal bolt given in the main text. The supergravity ansatz discussion of the extremal limits of the system in section 2.3. The theory and the equations being, we emphasize that we keep nT general. are asymptotically at in ve dimensions, and asymptotically R4;1 S1 in six dimensions. the following non-zero entries: 12 = 21 = 1; ab = for a; b = 4; : : : nT + 2 : 1 CIJK HI HJ HK = We also de ne the function jHj via jHj2 = corresponds to an angular momentum. The three-dimensional base space metric, ij , and the functions, V , V , are altogether a solution to the following nonlinear system of di erential equations: 2 V V = V = R( )ij = @(iV @j)V (1 + V V )2 the one computed using the metric ij . KI = LI = M = r 2 1 + V V 1 + V V 2 M rV speci ed by a solution to (2.4). next section, we write the metric as: ds2 = + w0)2 + ij dxidxj : The vectors A3 and k decompose as A3 = At3 (dt + !) + + w0) + w3 ; k = + w0) + ! ; and w0, ! are three scalars and two vector elds on the three-dimensional (V; V; KI ; LI ; M ) as follows: W = (1 + V ) M CIJK KI KJ KK 1 CIJK KI KJ KK M + 1 (1+V )CIJK LI LJ LK 1 CIJK KJ KK CILM LLLM ; HI = = (1 + V ) M 2 2 1 + V V KI M + 24 1 + V V 2 1 + V V 1 + V V (KJ LJ ) KI 2 CIJK LJ CKLP KLKP ; CIJK LI LJ LK CIJK KI KJ KK M + CIJK KJ KK CILM LLLM Similarly, the vector elds !, w0 and w3 are determined by the rst-order equations ?d! = dM ?dw0 = ?dwI = dLI 1 + V V (1 + V ) dM 24 1 + V V 4 1 + V V 2 M dV ; 1 1 2 V V 2 V 1 + V V 2 M dV + 1 CIJK KI KJ KK (1 + V V )2 4 (1 + V V )2 CIJK KJ KK V 2dV + dV ; discuss shortly. where the Hodge star in taken in the metric ij and we have given the wI in an SO(1; nT ) covariant form; the wa will appear in the matter sector, as we will The nT + 1 scalar elds, ta, are given by the expression physical scalar elds, namely the dilaton, , and the nT 1 real axions, &a, for a = 4 to nT + 2, as for a = 4; : : : nT + 2 leading to the expressions scalars as >>t1 = e ; t2 = e >:ta = e &a ta = e = &a = V = B@ 0 Mab = where 1 is the (nT which is given by 1)-dimensional identity matrix, so that V is a square (nT + 1)The inverse of M is M ab = ac bdMcd. The nT + 1 two-form potentials, Ca, give rise to one anti-self-dual and nT self-dual M ab ?6 Gb + abGb = 0 : We rst introduce the scalars Aa, a and t a with the latter identi ed as axions in the Finally, we de ne the two-forms in three dimensions, a, through d a = va ^ dw0 In terms of these quantities, we have a (dt + !) ^ (d ab wb ^ (dy + w3) + ba ^ (dt + !) + va ^ (d Note that the a ensure that the eld strengths, Ga, depend on the vectors wa, ba and va only through the gauge-invariant quantities dwa, dba and dva. The a vanish for axisyma to zero. The one-forms, va, ba in (2.17) are determined in terms of the functions appearing in the ansatz by solving the rst-order equations ?dbI = ?dvI = 1 + V V 1 + V V (1 + V V )2 (1 + V V )2 1) V dV + (1 + V )dV ; a = Similarly to the vectors wa, the electric potentials Ata and axions a in (2.17) are also extended by the scalars A3, 3 of (2.7) in the ve-dimensional reduction of the theory. For t in the following two equations) AtI = I = 2 (1 + V ) M V KI LI 1 + V V 1 V K1K2K3 1 + V V to obtain.5 CIJKHJ HK KLLL + LI CJKLKJ HKHL CIJLKJ CLP QLP CQRSHRHS : (2.23) transform linearly among themselves under the transformation de ned by 2 1 + V V 8 1 + V V CIJK kI KJ KK CIJK kI kJ KK + section to make use of the following invariant combinations 2 V 4 V 2 + V V K1K2K3 : and transforms: dbK ) + CIJK kI kJ (dvK where we have used the conserved currents ? d = (1 + V V )2 time coordinate t ! t The solution base to be the base space of the Euclidean Kerr solution: ij dxidxj = dr2 + (r2 c2 + a2 sin2 )d 2 + (r2 c2) sin2 d'2 ; (2.28) of the bolt, de ned by cos 1 respectively (we follow the terminology of [53]). Such to this bolt. the symmetry axis, i.e. at cos 1 and r > c. Similarly, all vectors on the 3D base are constrained to have a single component along ', for example ! = !'d' ; wI = w'Id' : the Euclidean Kerr-NUT solution: qI = kI Then we nd the solution for the KI hI + KeI = hI + kI V + X r + a cos + where hI , kI and nIA are integration constants and + was de ned in (2.30) above. Note structure is present in L I and M ; in order to parametrize this in what follows, we have introduced above the function KeI which asymptotes to zero. It turns out that a combination of the shift parameters, kI , and the asymptotic conquantities to use in the discussion of regularity in the next section. V = 1 + V = m (r + a cos ) + c2 is a (real-valued) constant of integration, and where we de ned the combination by the previous steps. The rst of (2.5) is homogeneous in the KI and allows for zero modes with simple denote by A the Euclidean distance A = With this notation, the solution for the LI takes the form LI = r + c cos X CIJK nJAnBK (r+a cos ) (RA a)(RB a) (r (RA +RB a) cos ) : pieces in the LI are not physically relevant. The same comments apply to the kI -dependent terms in M , for which we nd the CIJK KeI KeJ KeK + 1 + V V M = 2 V + 2 + V V c2 + a2 sin2 A; = r + a cos (2 + V + V V ) (RA + RB + RC (a + c) sin2 r + c cos (r + c cos )2 c2)(cos RA r) + a sin2 (r RA c) (r + c cos ) a m (r cos + c) + m a2 sin2 c2 + a2 sin2 c2 + a2 sin2 CIJK nIAnJBnCK V cos 2 a cos ) + (a2 1 + cos2 V + V V sin2 In the above, the term that contains (RA RB) in the denominator should be understood to PAI = CIJK nJA(pAK when considering vectors nIA of restricted rank, as we shall see later. Extremal limits setting either V or V to a constant. trivial only by holding m xed and non-zero and taking the limit a ! c, in which case V sending m to zero and a ! c, keeping the ratio (a2 c2)=m = p0 xed. In this case V with a single pole at r c cos . The KI become harmonic, as can be seen directly from Similarly, setting V to a constant simpli es in a di erent way the de ning equaintroducing the notation H , H vector of functions, one nds the following change of variables: V = M = KI = H1H2H3 : 2 H0 LI = H solution. In this limit, de ning dv0 to be the BPS limit of 12 kI , through a gauge transformation in H0 ! H0 that bI vanish identically in the BPS limit. General properties of the solution ? d! = ve dimension that amounts to the rede nition 2. We observe that this is consistent with the transformation (2.27), noting are interested in microstates of black holes in ve dimensions (and black strings in six dimensions) and so we are interested in solutions with R4;1 S1 asymptotics. horizons cannot be built using our ansatz, unless one takes an extremal limit. not impose any constraints on the parameters of the general solution. Speci cally, one can shift to zero the asymptotic values of the scalars a , a and Ata in (2.17) wa0 = wa + A va0 = va where primes denote rede ned quantities, we denote asymptotic values by , and we expressions, and likewise for the following two steps. Next, one may remove the asymptotic constants of At3 and y with t and at in nity, provided that one makes the rede nitions va0 = va + b0a = ba + A a0 = a + rede nitions. A nal rede nition we use is a di eomorphism mixing time with one of the compact directions, t = t0 + , and introducing the rede ned elds !0 = ! 0 = I 0 = vI0 = vI + where the value of will be determined by the asymptotic conditions below. We again immediately drop the primes on all the above expressions. norm vector of SO(1; nT ). To obtain our desired R4;1 S1 asymptotics, we impose the fall-o behaviour W = HI = = O It turns out that the obtained from (2.8) contains an asymptotic r 2 term that can be eliminated using the redundancy (3.3), for the speci c value We henceforth proceed with the solution obtained after (3.3) with as in (3.5) has been applied. In order to simplify the analysis, we take the same approach as in [47] and x the l0 = lI = hI = 1 ; kI = 1 + qI ; the following restrictions on the asymptotic constants l0, lI and hI : q0 = tions (3.4) are imposed by the function M in (2.35), as xing the parameter q0 that appears in the harmonic part of where we de ned the shorthand quantity CIJK qJ qK + X PAI + X CIJK nA J RA which will be useful in the following. duce an asymptotically R4;1 S1 solution. However, this solution does not yet possess the asymptotics of a single-center black hole in ve spacetime dimensions. The reason is constants governing the asymptotic fall-o of the scalars, EI , de ned as QI = 4 EI = 4 the conditions Q12 = E22 Q22 = E32 or in other words that all the components of the vector EI2 QI2 be equal. This only imposes two conditions on the various parameters. e 4 U = H1H2H3 S2 + JL2 sin2 where 16 2S > 0 is the horizon area. of the bolt. In order to have a black hole horizon at a given special point located at r = 0, the metric functions, W , and HI , be: wJL sin would-be horizon, and in particular that it has nite area. The horizon area of a vedimensional extremal black hole is controlled by the combination Centers away from the bolt. We start with the centers away from the bolt, so we set r = A where a denotes any such center. Near any of these centers, the base metric is but straightforward exercise to expand W , and HI for the solution given in section 2.2 imposing restrictions on the parameters of the solution. Considering rst the highest poles, and using the notation det nA n1An2An3A , we nd the behaviour6 W = 8 m2 JA(det nA) = (1 + )W 8 m JA(det nA) HI = 2 JA nIA RRA2A 6Recall that is the shift that imposes the correct asymptotics in , see below eq. (3.4). the result, we nd the following structure: W = m2 m2 (det nA)2 F1(RA; m ; a; c) cos2 A + FWA = m m2 (det nA)2 F2(RA; m ; a; c) cos2 A + F A W = m2 = (1 + )W + m n2APA2 controls all the unwanted poles and must vanish. the qI is added for later convenience) PAI = CIJK nJA(pAK nA. We thus henceforth adopt the rede nition (3.18), as anticipated in eq. (2.36). 2 At this stage the HI now have the desired behaviour described in (3.13), while both W and still contain 3 poles, which we now consider. In the interest of brevity we suppress in the following analysis the terms proportional to J and pI , anticipating our later result that J A 2 poles of the HI when the nIA are rank-2 vectors, then take the form W = 2 m a2) p~3A + (a2 HI = f n1Aq~A0 ; n2Aq~A0 ; n1A n2A (p~3A)2g 2 + O( A 1) ; where we used the shorthand de nitions 2 X sign(RA B6=A B6=A the rank of the double pole of HI is to set (RA2 a2) p~3A + (a2 c2)q~3A = 0. However, from the form of that H1H2H3 in (3.19), we see that cancelling the cubic pole in W automatically implies O( A 5), and therefore that the horizon area vanishes. This implies obtain a regular extremal black hole with nite horizon located at a nite distance from the non-extremal bolt. Poles of the bolt. We now turn to the poles of the bolt, where the metric behaves the we introduce coordinates centered on the North / South pole via 4c r cos 4c r cos r = with the function Then near the poles, the three-dimensional base metric ij behaves as $ ( )(dr 2 + r2 d 2) + r 2 sin2 Computing the expansions of W and , one obtains W = and therefore one must separately impose J these functions to vanish. We set J can examine the cubic poles of W and and the quadratic poles of the HI at r = 0, and pI implies that the sextic pole of H1H2H3 2 vanishes, as before. Therefore, we rule out Conditions for smooth solutions on speci c combinations of the parameters. construction of new explicit smooth horizonless solutions) for future work. we do not discuss their solution in this paper. Local smooth geometry the metric functions W , and HI behave as: W = HI = = O where W 2( ) and hI ( ) are strictly positive functions of . When the special points are away from the bolt, the function W 2( ) is a constant, and W = S1. This is a simple generalization of what is known as a Gibbons-Hawking center Hawking center. For more details, see the discussion in [47]. At the poles of the bolt, a factor of $ ( ) de ned in eq. (3.23) again enters, and W = preceding section. We must therefore impose = J A = 0 ; nds that PAI must be rank 1 in order for the quadratic poles in HI to vanish. This so we shall not consider it independently. and to impose CIJK nJAnAK = 0 ; qA0 = 4 RA2 symmetric Gibbons-Hawking center, with charges N at the poles as 7Since that solution was given in the context of a di erent parametrization for the system, a complete appendix A. (see for example [55, 56]). NA = 1 X nIApIA+1pI+2 RA nIA+1nIB+2 I; B6=A jRAj jRA qI+1)(pIA+2 RA The same analysis applies in the vicinity of the poles of the bolt. One nds that in function. The only consistent solution is therefore to set We therefore concentrate on option (ii), which sets the nIA to be of rank-1 at each center: I = 0 : where we introduced the integer x = such that N+ + N A NA = 1, because the bolt is only regular in six dimensions for that x must be an integer with the same parity as a +cc (1 + P qI+1)(pIA+2 Hawking centers are given by8 I6=J I; B6=A 8Note that we use the rank 1 condition (4.5) of nIA to simplify these formulae. Absence of closed time-like curves of solutions, a rst strong requirement is that the vector !, describing the time the symmetry axis. potential discontinuity on the bolt. In order to write the conditions for the the vector eld ! to be continuous, let us introduce the following shorthand quantity, that will also be useful below: as well as the sign, "AB, depending on the position of centers A;I6=J nIApJA + 2 m qI+1)(pIA+2 to vanish on hold, one obtains: c !' B = X NA + 1 + RA where x is the integer de ned in (4.11). imposing the BPS limit a ! c; m ! 0 with a2m c2 kept xed as in section 2.3, one centers de ned by the harmonic functions H , H of (2.37) with restricted poles according South pole of the bolt turn out to vanish identically (in particular N = 0 in (4.10)). It is also important to compute the value of the vector w0 de ning the bration over on the bolt, since the regularity conditions at the bolt imply that = m the poles of the bolt are automatically integers: N+ = = n quotient of R4 FaA = 4 2 the parameters qI ; a; m at the bolt are understood to parametrize the two integers m and n characterizing its topology, the S1 parametrized by two integers NA and MA (for the action on the additional circle), and its presence introduces one new 3-cycle that supports nT + 1 Ga [47]. We thus see that each Gibbons-Hawking center is parametrized by derived in [47]. Acknowledgments by a CEA Enhanced Eurotalents Fellowship. Relation to the Floating JMaRT system solution, denoted by E and KI (old) for the functions appearing in [43]. identi cations E+ = + = 2 + K3 + 2 V 1 + K3 + 2 V 2 + K3 + 2 V 2 V 1 ; V 1 for the Ernst potentials, where is a free parameter, set to = m =e when comparing tions we use are such that E not independent. are those of an extremal solution and are therefore fH1; H2; H3g ! 16 2 f2 H1; 2 H2; H3g ; fw0; w1; w2; w3g ! 4 2 f2 w0; w1; w2; 2 w3g ; (A.5) sign of the two gauge elds A1 and A2. The nal two functions in the Floating JMaRT system are identi ed as 2 2 + K3 + 2 V 2 + K3 + 2 V Ka(old) = 2 (V + 1) 2 1 + V V 1) (V + 1) (K1 + L2) (K2 + L1) 1 + V V 2 + K3 + 2 V (KaLa + K1K2) + 1 (K1 + L2) (K2 + L1) 1 + V V 2 + K3 + 2 V 8 1 + V V K1K2K3 2 1 + V V (KaLa + K1K2) which we nd La(old) = K3(old) = L3(old) = in this paper: 2 dt + 2 d and we rescaled all elds appropriately in order to remove rst de ne some useful functions cos + a S(r; ) (r a cos + m ) ; a S(r; ) (r a cos ) ; which we use for brevity. Additionally, we use the shorthand c cos for the remainder of this appendix. from (2.18) the '-components (vI )' = c2) c + (r + a c ) (RA vector elds wI , w0 (wI )' = 2 W m CIJK hJ (hK + 2 qK ) W+ X CIJK (qJ +hJ ) nAK nIA+1nIB+2 +S(r; ) (r2 c2) 2 (r + a c ) (RA + RB 2 a) + 4 (a2 (w0)' = ((k1k2k3 +2q1q2q3) W0 (k1k2k3 +q1q2q3) W )+ (q1q2q3 k1k2k3) W+ a) A A; = 1 (RA X CIJK nIAnJB (pBK = 1 A; = 1 (q1 + h1) (q2 + h2) (q3 + h3) r + a c + m (1 +(RA + a) RA (J+ + J ) A; = 1 (RA n1An2B n3C Finally, the vector eld ! is also determined from (2.9) as !' = hI hJ ) vK + RB (a + c) RA + 2 S(r; ) (r2 ((2 r + (a (RA a)W+ +a(a2 c2)S(r; )c CIJK hI qJ (hK + qK )W+ 1 I r c + a + +2 S(r; ) m (r2 c2) (r + a c ) a) W+ + a (a2 A;B;C (RA a) (RB a) (RC a) where WABC is given by 2 m S(r; ) RB RARBRC W+ + c2 sin2 c (RA + RB + RC a S(r; ) (a r a S(r; ) (a r + c2 c ) RARB + RARC + RBRC + r c ) (RA + RB + RC) + c2 Open Access. 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Iosif Bena, Guillaume Bossard, Stefanos Katmadas. Bolting multicenter solutions, Journal of High Energy Physics, 2017, 127, DOI: 10.1007/JHEP01(2017)127