KUL-TF-07-28 UGFT-221-07 CAFPE-91-07 A first-order formalism for timelike and spacelike brane solutions 8 0 0 2 Bert Janssen , Paul Smyth , Thomas Van Riet and Bert Vercnocke n † ∗ ‡ ‡ a J 4 2 Departamento de F´ısica Te´orica y del Cosmos † ] and Centro Andaluz de F´ısica de Part´ıculas Elementales h t Universidad de Granada, 18071 Granada, Spain - p [email protected] e h ∗ II. Institut fu¨r theoretische Physik der Universit¨at Hamburg [ Luruper Chaussee 149, 22761 Hamburg, Germany 2 v [email protected] 8 0 Institute for Theoretical Physics, K.U. Leuven, 8 ‡ Celestijnenlaan 200D, B-3001 Leuven, Belgium 2 . Bert.Vercnocke,[email protected] 2 1 7 0 : v i X r Abstract a We show that the construction of BPS-type equations for non-extremal black holes due to Miller et. al. can be extended to branes of arbitrary dimension and, more importantly, to time-dependent solutions. We call these first-order equations fake- or pseudo-BPS equations in light of the formalism that has been developed for domain wall and cosmological solutions of gravity coupled to scalar fields. We presentthefake/pseudo-BPSequationsforallstationarybranes(timelikebranes)and all time-dependent branes (spacelike branes) of an Einstein-dilaton-p-form system in arbitrary dimensions. Contents 1 Introduction 2 2 Four-dimensional Einstein–Maxwell theory 5 3 ( 1)-branes in D dimensions 9 − 4 p-branes in arbitrary dimensions 10 5 p-branes with type II deformations 11 6 Discussion 12 A Construction of the effective action 14 1 Introduction Many brane-type solutions have been constructed for simple truncations of supergravity theories to the form (see, for instance, [1]) S = dDx√ g 1 1(∂φ)2 1 eaφF2 , (1) − 2κ2DR− 2 − 2n! n Z (cid:16) (cid:17) where is the Ricci-scalar, κ2 is the D-dimensional gravitational coupling, φ is a scalar R D field (the dilaton) and F is the field strength of some (n 1)-form, F = dA if n > 0. n n n−1 − For the special case n = 0 we consider F2 to be a cosmological term (scalar potential). n The parameter a is fixed and is called the dilaton coupling. The equations of motion derived from this action admit electrically charged (n 2)- − branes and magnetically charged (D n 2)-branes. A brane solution can be stationary − − or time-dependent. The metric of a stationary p-brane is given by ds2 = e2A(r)η dxµdxν +e2B(r)dr2 +e2C(r)dΣ2, (2) D µν k whereη istheusualMinkowski metricinp+1dimensions, η = diag( ,+,...,+)anddΣ2 is − k the metric of a d-dimensional maximally symmetric space with unit curvature k = 1,0,1, − such that the Ricci scalar is given by = kd(d 1). When k = 1 the solutions possess a d R − rotational symmetry and can be asymptotically flat (in contrast to k = 1). For D = 10 − and specific values of a and n the solutions correspond to D-branes in string theory. The metric of the time-dependent branes is similar ds2 = e2A(t)δ dxµdxν e2B(t)dt2 +e2C(r)dΣ2, (3) D µν − k where δ is the usual flat Euclidean metric in p + 1 dimensions, δ = diag(+,+...+). µν µν In the k = 1 case the transverse space possesses a Lorentzian symmetry and can be − 2 asymptotically flat (in contrast to k = +1 solutions). These solutions are the spacelike branes (S-branes) introduced by Gutperle and Strominger [2], who conjectured that such branes correspond to specific time-dependent processes in string theory. From now on we shall call the stationary branes with spherical slicing (k = +1) timelike branes and the time-dependent branes with hyperbolic slicing (k = 1) spacelike branes. − All the other possible slicings are also covered here, but we choose to highlight only these two cases. It has been known for a long time that particular timelike p-brane solutions of super- gravity preserve some fraction of supersymmetry. Practically this means that the solutions fulfill some first-order differential equations that arise from demanding the supersymmetry transformations to be consistently satisfied for vanishing fermions. Such first-order equa- tions have become known as Bogomol’nyi or BPS equations, after Bogomol’nyi’s [3], and Prasad and Sommerfield’s [4] work on first-order equations and exact solutions for mag- netic monopoles in the Yang–Mills–Higgs theory. It was then later shown that this limit is intimately linked to the preserved supersymmetry of solitons in supersymmetric theories by Witten and Olive [5]. The term BPS equation is now generically used for equations of motion that are inferred by rewriting the action as a sum of squares. Supersymmetric solutions, in general, belong to this class. Stationary non-extremal and time-dependent solutions cannot preserve supersymmetry in ordinary supergravity theories. Naively one therefore expects that such solutions cannot be found from BPS equations, but rather by solving the full second-order equations of motion. To our knowledge there are three instances in the literature where it has been shown that this view is too pessimistic: (I) Not all extremal black hole solutions of supergravity have to be supersymmetric. It turnsout that manynon-supersymmetric but extremal solutions fulfillfirst-order equations in a given supergravity theory (see for instance [6–8]). More surprisingly, Miller et. al have shown that the non-extremal Reissner–Nordstr¨om black hole solution of Einstein–Maxwell theory can be found from first-order equations [9], by a clever rewriting of the action as a sum of squares `a la Bogomol’nyi. The method of [9] is the main tool for the present paper. (II) Many stationary domain wall solutions that do not preserve any supersymmetry have been shown to allow for first order-equations by the construction of a fake superpo- tential [10–12]. The domain walls in question are solutions to the following Lagrangian = √ g 1G (Φ)gµν∂ Φi∂ Φj V(Φ) (4) L − R− 2 ij µ ν − (cid:16) (cid:17) where Φi are scalars fields, G (Φ) the metric on the target space and V(Φ) the scalar ij potential. The metric Ansatz for a flat domain wall is1 ds2 = e2B(z)dz2 +e2A(z)η dxadxb, (5) ab where η is diag( ,+,...,+). The high degree of symmetry of this Ansatz is only con- ab − sistent when the fields that support the solution depend solely on the z-coordinate i.e. 1 For simplicity we only discuss flat cosmologies and flat domain walls. 3 Φi = Φi(z). We then suppose that a scalar function W(Φ) exists such that V = 1Gij∂ W∂ W D−1 W2, (6) 2 i j − 4(D−2) which allows the action to be written as a sum of squares (neglecting boundary terms) [13] 2 S = dze(D−1)A+B (D−1) W 2(D 2)e−BA′ 2 1 e−B(Φi)′ +Gij∂ W , (7) 4(D−2) − − − 2 j Z n (cid:12)(cid:12) (cid:12)(cid:12) o (cid:2) (cid:3) (cid:12)(cid:12) (cid:12)(cid:12) where a prime denotes a derivative with respect to z. Sol(cid:12)(cid:12)utions are obtained(cid:12)(cid:12)when each square in the action is zero. If W is a superpotential of some supersymmetric theory, these first-order equations are the standard BPS equations for domain walls that would arise by demanding that the supersymmetry variations are satisfied for vanishing fermions. However, for every W that obeys (6) we can find a corresponding DW-solution. If W is not related to the quantity appearing in the supersymmetry transformations the resulting solutions are called fake supersymmetric. (III) FLRW-cosmologies are very similar to domain walls [14–16], the difference in metrics being given by a few signs ds2 = e2B(t)dt2 +e2A(t)δ dxadxb. (8) ab − When the relation (6) is changed by an overall minus-sign V = 1Gij∂ W∂ W + D−1 W2, (9) −2 i j 4(D−2) the same first-order equations for domain walls exist for cosmologies, where now the primes indicate derivatives with respect to time. These relations have become known as pseudo- BPS conditions [15,16] (see also [17,18] for the first-order framework in cosmology). As for domain walls one readily checks that these first-order equations arise from the fact that the action can be written as a sum of squares [19]. The structure underlying the existence of these first-order equations can be understood from Hamilton–Jacobi theory [20–22].2 The examples given above (I-III) are only a subset of the different p-branes that exist, namely timelike 0-branes in D = 4 (the RN black holes) and (D 2)-branes (domain − walls and FLRW-cosmologies). It is the aim of this paper to understand in general when stationary and time-dependent p-branes in arbitrary dimensions can be found from BPS equations. One of the main subtleties that arises in this generalisation is that there exist two kinds of black deformations of timelike p-branes which coincide for the special case of black holes 2 InordinarysupergravitytheoriesthepseudoBPSrelationscannotberelatedtosupersymmetrypreser- vation. However, in the case of supergravity theories with ‘wrong sign’ kinetic terms the pseudo-BPS relationsare relatedto true supersymmetry[23–27]. In this paper we shallconsider ordinarysupergravity theories and therefore pseudo-BPS conditions are not related to supersymmetry. Practically this means that we have first-order equations which can be understood to originate from a Bogomol’nyi rewriting of the action. 4 infourdimensions. ForthisreasontheexampletreatedbyMilleret.al.[9]isnotcompletely representative. Secondly, for time-dependent solutions, it has yet to be understood if the concept of pseudo-supersymmetry could be extended beyond cosmologies (time-dependent (D 2)-branes) to general time-dependent p-branes (see [25] for initial progress in this − direction). The rest of the paper is organized as follows. In section 2 we consider Einstein– Maxwell theory and repeat the construction of the first-order equations for the non- extremal Reissner-Nordstr¨om black hole. We immediately show that the same technique allows one to rederive the S0-brane solution of Einstein–Maxwell theory [2]. In section 3 we discuss the special case of ( 1)-branes in arbitrary dimensions. In section 4 we explain − how the BPS equations for the ( 1)-branes imply the BPS equations for general p-branes − in arbitrary dimensions via an uplifting procedure. We then discuss the issue of different black deformations in section 5 and finish with conclusions in section 6. 2 Four-dimensional Einstein–Maxwell theory Einstein-Maxwell theory in four dimensions is described by the action S = d4x√ g 1 1F2 , (10) − 2κ2R− 4 Z (cid:16) (cid:17) and has electric and magnetic 0-branes solutions. Following [9] we shall choose a particular Ansatz for the 0-brane metric which turns out to be useful ds2 = ǫe2A(u)dz2 +e−2A(u)+2B(u) ǫe2C(u)du2 +dΣ2 . (11) − k (cid:16) (cid:17) If ǫ = +1 then z is time z = t and the metric is static. For spherical slicings (k = +1) this is the appropriate Ansatz for a black hole, where u is then some function of the familiar radial coordinate r. When ǫ = 1 the metric is time-dependent and for hyperbolic slicings − (k = 1)thisistheappropriateAnsatzforaS0-brane[2]withaone-dimensionalEuclidean − worldvolume labelled by z, and u is some function of the time-coordinate τ used in the Milne patch of Minkowski spacetime. The general Ricci scalar is given by = 2ǫe2(A−B−C) A¨ A˙2 +A˙B˙ A˙C˙ 2B¨ +2B˙C˙ B˙2 +2ke2(A−B), (12) R − − − − (cid:16) (cid:17) where a dot indicates a derivative with respect to u. For electrical solutions, the Maxwell and Bianchi equations are solved by F = Qe2A−B+C . (13) uz Plugging the Ans¨atze (11) and (13) into the Einstein field equations derived from (10), one can ask whether the resulting second-order equations of motion in the one variable u can be interpreted as field equations for A,B and C derived from a one-dimensional 5 effective action. It is straightforward to see that the equations of motion can be obtained by varying the following action S = du eB−C 2B˙2 2A˙2 +2ǫke2C ǫκ2Q2e2(A−B+C) . (14) − − Z (cid:16) (cid:17) This action cannot be obtained from direct substitution of the Ans¨atze in the four-dimen- sional action as the sign of the resulting Q2-term would be wrong. This sign discrepancy does not appear for purely magnetic solutions, for which the Ans¨atze can be plugged into the action consistently. We discuss this point in detail in appendix A and refer to [28] for a careful derivation of the black hole effective action in a more general setting. The field C does not appear with a derivative in the action and is therefore not a propagating degree of freedom. This was to be expected since C is related to the re- parametrization freedom of u. The field C acts as a Lagrange multiplier enforcing the following constraint 2B˙2 2A˙2 2ǫke2C +ǫκ2Q2e2(A−B+C) = 0. (15) − − As long as this contraint is satisfied we are free to pick a gauge choice for C as we like. In the following we choose the gauge B = C. It turns out that it is easy to generalize the Bogomol’nyi bound found in [9] to include both stationary and time-dependent configurations with arbitrary slicing of the transverse space k = 0, 1. The action (14) is, up to total derivatives, equivalent to ± 2 2 S = du 2 B˙ + ǫke2B +β2 2 A˙ + ǫκ2Q2e2A +β2 , (16) 1 − 2 2 Z (cid:16) q (cid:17) (cid:16) q (cid:17) where β and β are constants. The BPS equations are 1 2 B˙ = ǫke2B +β2, A˙ = ǫκ2Q2e2A +β2. (17) − 1 − 2 2 q q The constraint (15) implies that β2 = β2 β2. Note that for time-dependent solutions 1 2 ≡ with charge (ǫ = 1, Q = 0) the limit of β 0 does not exist, while for Q = 0 the limit − 6 → only exists for k = 1. The BPS equations are all of the form − D˙ = β2 K2e2D± , (18) ± − ± p where K is a constant, depending on the case under consideration. The solutions to these equations are given by K K e−D+ = sinh(βu+c ), e−D− = cosh(βu+c ), (19) + − β β where c are constants of integration. In the extremal limit β 0 the solution becomes ± → e−D+ = e−D− = Ku+c. 6 Rediscovering Reissner–Nordstr¨om black holes For the black hole Ansatz (ǫ = +1,k = +1) it was shown in [9] that solving the BPS equa- tion described above leads to the non-extremal Reissner–Nordstr¨om solutions. We shall now quickly review this for completeness, and draw attention to some further subtleties. The solutions of the first-order equations (17) are Qκ 1 e−A = sinh(βu+c ), e−B = sinhβu, (20) a √2β β where we put the integration constant in the solution for B to zero by shifting the origin of the u-axis, leading to the following metric 2β2 β2κ2Q2sinh2(βu+c ) κ2Q2sinh2(βu+c ) ds2 = dt2 + a du2 + a dΩ2. −Q2κ2sinh2(βu+c ) 2sinh4βu 2sinh2βu 2 a (21) We can identify the radial coordinate r2 as the function in front of dΩ2. In order to obtain 2 the standard form of the Reissner-Nordstr¨om solution, one has to perform the following coordinate transformation: κQsinh(βu+c ) √2β a r = , τ = t, (22) √2sinhβu κQsinhc a such that the solution takes the form Q ds2 = H(r)dτ2 +H(r)−1dr2 +r2dΩ2, F = , (23) − 2 τr −r2 where 2κQcoshc κ2Q2 a H(r) = 1 + . (24) − √2r 2r2 It is then clear that the ADM mass corresponds to M = κQcoshc /√2. Note that for a coshc = 1,theabovesolutionreducestotheextremeReissner-Nordstr¨ommetric, implying a that coshc is related to the non-extremality parameter β. Indeed, from (20) we have that a 2β2e−2A(0) coshc = 1+ , (25) a s κ2Q2 such that the limit coshc = 1 corresponds to β = 0, as one expects from the action (16). a Rediscovering spacelike 0-branes For spacelike branes (ǫ = k = 1) we find − κQ 1 e−A = cosh(βu+c ), e−B = sinh(βu). (26) a √2β β 7 Once again, shifting the origin of the u-axis, the integration constant in the equation for B has been put to zero. Using the coordinate transformation κQcosh(βu+c ) √2β a τ = , x = z, (27) √2sinhβu κQcoshc a the solution then takes the following form Q ds2 = G(τ)dx2 G(τ)−1dτ2 +τ2dH2, F = , (28) − 2 τx τ2 with sinh(c )κQ Q2κ2 a G(τ) = 1 2 , (29) − √2τ − 2τ2 where we introduced the metric for the hyperboloid dH2 = dΣ2 . Again, this solution is 2 −1 asymptotically flat. Moreover, we see that this reduces to the metric for the S0-brane of [2] after a constant rescaling of x and τ. Taking the limit β 0, the metric is easily seen to → describe flat space in Milne coordinates. Addition of a dilaton Before we proceed to the case of p-branes in arbitrary dimensions let us first consider the coupling of the vector field to a dilaton, as this is the generic situation in supergravity theories. The action describing four-dimensional Einstein-Maxwell-dilaton theory is S = dx4√ g 1 1(∂φ)2 1eaφF2 . (30) − 2κ2R− 2 − 4 Z (cid:16) (cid:17) The Ansatz for electrical solutions is now given by F = Qe2A−B+C−aφ. In the gauge uz B = C the effective action becomes S = du 2B˙2 2A˙2 κ2φ˙2 +2ǫke2B ǫκ2Q2e2A−aφ. (31) − − − Z It turns out to be convenient to define new variables A and φ 1 1 a a A = A φ, φ = A+φ. (32) 1 − 2 1 κ2 With these new variables the Bogomol’nyi form is obvious and similar to the previous case without a dilaton. Writing the action as a sum of squares, we now introduce three constants β ,β and β 1 2 3 2 2 2 κ2 S = du 2 B˙ + ǫke2B +β2 A˙ + ǫ∆κ2Q2e2A1 +β2 (φ˙ β )2, (33) 1 − ∆ 1 2 2 − ∆ 1 − 3 Z (cid:16) q (cid:17) (cid:16) q (cid:17) where ∆ = 1+(a2/2κ2). 8 In this case the equivalent of the constraint (15) implies that only two of the three integration constants are independent 2 κ2 2β2 β2 β2 = 0. (34) 1 − ∆ 2 − ∆ 3 ˙ The BPS equations are the same as before apart from the extra equation φ = β . 1 3 When the solutions for A,B and φ are plugged into the Ansatz one reproduces the familiar dilatonic black hole solution [29]. One then also notices that the two indepen- dent β-parameters appear in a fixed combined way as to effectively form one deformation parameter. 3 ( 1)-branes in D dimensions − A ( 1)-brane couples electrically to a 0-form gauge potential, χ, known as the axion. The − worldvolume is zero-dimensional and, in the case of a timelike ( 1)-brane, this implies − that the whole space is Euclidean since it is entirely transverse. The action is S = dxD√ g 1(∂φ)2 +ǫ1ebφ(∂χ)2 . (35) − R− 2 2 Z (cid:16) (cid:17) Notethe‘wrongsign’kinetictermfortheaxionwhenǫ = +1,whichisnormalforEuclidean theories. The ( 1)-brane Ansatz is − ds2 = ǫe2C(z)dz2 +e2A(z)dΣ2, φ = φ(z), χ = χ(z). (36) D k If we consider the axion equation of motion, ∂ (√ ggµνebφ∂ χ) = 0, then the solution is µ ν − of the form χ˙ = Q e−bφ. (37) The one-dimensional effective action that reproduces the equations of motion for A and φ is S = dz (D−1)(D−2) A˙2e(D−1)A−C+ǫke(D−3)A+C e(D−1)A−Cφ˙2 ǫeC−(D−1)A−bφQ2. (38) κ2 − − Z (cid:16) (cid:17) As we discussed before this form differs from that obtained by direct substitution of the Ansatz into the original action (appendix A). The field C is not propagating and we can choose it at will; the gauge C = (D 1)A is obviously useful. As before we must consider − the constraint that arises from varying the action with respect to C. In this gauge, the BPS form of the action is then equal to 2 2 S = dz (D−1)(D−2) A˙ + ǫke2(D−2)A +β2 φ˙ ǫQ2e−bφ +β2 , (39) κ2 1 − − 2 Z (cid:16) q (cid:17) (cid:16) q (cid:17) supplemented with the constraint (D−1)(D−2) A˙2 ǫke−2(D−2)A φ˙2 +ǫe−bφQ2 = 0. (40) κ2 − − (cid:16) (cid:17) 9 The constraint equation tells us that there is only one effective deformation parameter since β2 = (D−1)(D−2)β2. (41) 2 κ2 1 We now first solve the BPS equations with vanishing deformation parameters for k = ǫ = 1. If we define the coordinate ρ via dρ = e(D−1)Adz, then the BPS equation, A˙ = e(D−1)A, implies that ρ = eA +c. Shifting ρ su−ch that c = 0 we find that the metric − describes the Euclidean plane in spherical coordinates ds2 = dρ2 +ρ2dΩ2 . (42) D D−1 The solutions for the scalar fields are b b b b e2φ = Qb ρ−D+2 +e2φ∞ , χ = 2|Q| (e−2φ e−2φ∞)+χ . (43) −2(D−2) − bQ − ∞ This is indeed the extremal instanton solution, see for instance [30,31]. For non-zero β the solution becomes (in the frame C = (D 1)A) − e(2−D)A = 1 sinh[(D 2)β z +c ], (44) β1 − 1 1 b e−2φ(z) = Q sinh(β2b z +c ), χ(z) = 2 Q2e−bφ +β2 +c , (45) β2 2 2 −bQ 2 3 q where c ,c and c are arbitrary constants of integration. These solutions correspond to 1 2 3 the super-extremal instantons that were constructed in [31,32]. Finally, the time-dependent S( 1) brane solution (with k = ǫ = 1) that was first − − constructed in [33] can be rederived (again in the frame C = (D 1)A) − e(2−D)A = 1 sinh[(D 2)β t+c ], (46) β1 − 1 1 b e−2φ(t) = Q cosh(β2b t+c ), χ(t) = 2 Q2e−bφ β2 +c . (47) β2 2 2 −bQ − 2 3 q 4 p-branes in arbitrary dimensions Let us now consider the following theory in d = D +p+1 dimensions S = ddx√ g 1 1(∂φ)2 1 eaφF2 . (48) − 2κ2R− 2 − 2(p+2)! p+2 Z n o The corresponding p-brane solutions can all be reduced to ( 1)-brane solutions in D − dimensions via reduction over their flat worldvolumes. Therefore we should be able to reproduce the BPS bounds and the BPS solutions using the ( 1)-brane calculation of the − previous section. A typical p-brane Ansatz takes the form ds2 = e2αϕ(z)ds2 +e2βϕ(z) ηǫ dymdyn, φ = φ(z), D mn A (z) = χ(z) dy1 dy2 ... dyp+1, (49) p+1 ∧ ∧ ∧ 10