Three dimensional black strings: instabilities and asymptotic charges
TThree dimensional black strings: instabilities andasymptotic charges
Ph. Spindel ∗ Service de Physique de l’Univers, Champs et Gravitation,Universit´e de Mons,Facult´e des Sciences,20, Place du Parc, B-7000 Mons, BelgiumService de Physique Th´eorique, Universit´e Libre de BruxellesBld du Triomphe CP225, 1050 Brussels, BelgiumJuly 25, 2019
Abstract
Three-dimensional Einstein gravity coupled to zero, one and twoforms is solved in terms of a polyhomogeneous asymptotic expansion,generalising stationary black string solutions. From first order termswe obtain, in closed form, a new solution evolving from the station-ary black string structure to a geometry developing a singularity inthe future. This new solution itself may be extended to more generalones. Taking into account subleading terms of the asymptotic expan-sion, both singularities in the past and the future occur. This demon-strates the unstable character of the stationary black string : tinyperturbations generated by terms breaking the rotational invarianceof the stationary black string configurations lead to cosmological-likesingularities. The symmetry algebra of the conserved charges also isdetermined: it is a finite dimensional one. In the general case the sur-face charge associated to energy is not integrable. However we identifya sub-class of solutions, admitting asymptotic symplectic symmetriesand as a consequence conserved charges that appear to be integrable.
Black holes are some of the most fascinating solutions of general relativity.They verify thermodynamic relations that provide a bridge between the ∗ E mail : [email protected] a r X i v : . [ h e p - t h ] J u l lassical theory of gravity and its quantized version. Three dimensionalblack strings appear naturally as target space geometries of gauged WZWmodels [1, 2, 3, 4] i.e. of marginal deformation of the SL (2 , R ) WZW model. They provide solutions of the field equations deduced from a lowenergy string Lagrangian describing the coupling of Einstein gravity to adilaton (a scalar resulting from quantum effects), a Maxwellian (abelian oneform) and a Kalb–Ramond (two form) field.Here we present a large class of solutions of 3 dimensional relativity in-spired from heterotic string theory. The interest of such solutions is thatthey unable us to understand some aspects of quantum gravity, at least inthe framework of low dimensional gravity. For instance they suggest inter-esting deformations of the underlying sigma-model, as the one presented insection of ref.[4], inspired from the new solutions presented hereafter(Eqs[71a–71h]). They also offer a framework to study asymptotic dynam-ics in a much more easier way that in four dimensions. Indeed it is wellknown that generic solutions of Einstein equations dont admit any localsymmetry (Killing vector field) that allows to define exact conserved quan-tities. However, since the seminal work of Arnowitt, Deser and Misner [5](see also refs[6, 7]), the existence and relevance of asymptotic symmetrieshas been put into evidence. In favorable case these asymptotic symmetrieslead to asymptotic conservation laws whose associated charges constitutethe generators of a symmetry algebra underlying the asymptotic dynamic.An important part of this work is devoted to the integration of Einsteinequations coupled to a dilaton (scalar), a Maxwellian (abelian one-form)and a Kalb-Ramond (two form) fields. This dynamical system offers, atleast, two distinct covariant phase space sectors exhibiting black holes. Thefirst one, where the rˆole of the cosmological constant is played by the Kalb-Ramond flux, leads to the well known btz black holes configurations [8].The second one, put into light by Horne and Horowitz [2], also providesblack string/brane configurations but where the (singular) behaviour of thedilaton field dictates the behaviour of the geometry. As we shall show, thelatter defines a covariant phase space where this (two degrees of freedom)dynamical problem appears to be exactly integrable in an asymptotic expan-sion scheme involving inverse powers and logarithms of a radial coordinate.These three dimensional configurations constitute toy models for study-ing several aspects of gravity. In particular, we derive closed forms of ex-act solutions exhibiting dynamical configurations. This was a totally unex-pected result. We have also obtained a large class of solutions, extendingthe stationary Horne and Horowitz black string solution, to much more gen-eral configurations, without any symmetry. However they all appear to be2nstable in the sense that under a tiny perturbation breaking the rotationalinvariance, the static black string configuration develops new cosmological-like curvature singularities. This result answers to a conjecture stated in ref.[2], conjecture based on the first order analysis developed in ref. [9].We have integrated the field equations similarly to the approach ofref. [10] in the three-dimensional framework of Maxwell-Einstein equations.However let us already mention that some non-trivial differences with thiswork will emerge along the thread of our analysis. Indeed the presence of adilaton, singular at spatial infinity, leads to a modification of the asymptoticbehaviour of other fields. It is the dilaton itself, multiplicatively coupled tothe abelian field, that will provide the expected logarithmic terms needed toobtain electric charges in three spacetime dimensions. They are extractedfrom sub-leading terms appearing during the expansion of the gauge fields.Of course the dilaton multiplying the abelian and the Kalb-Ramond can bereabsorbed in their definitions, but at the price of introducing (apparently)more complicated coupling terms.A second part of this work is devoted to constructing configurationsleading to finite asymptotic conserved charges. In particular we obtain thatsubdominant terms of the asymptotic expansion of the scalar fields occuras non integrable components in the expression of the charge associated toenergy [see eq. (114)]. Our considerations are based on a lemma [provedin appendix (C)] which guarantees the covariance of the expressions of thecharge density we consider. This allows us to perform the calculation in agauge where the expressions for the various field components are simpler andfrom them infer conditions that lead to finite charges in more general situ-ations, obtained by acting with so-called ”large diffeomorphisms” [a pointalso discussed in appendix (C)].
The geometries we consider in this work are solutions of the field equationsprovided by the diffeomorphism invariant action: S = π G (cid:90) √− g (cid:0) R − ∇ µ Φ ∇ µ Φ − H e − − k g F e − + δc e (cid:17) d x , (1)describing gravity coupled to a dilaton field Φ, an abelian (Maxwellian)2-form : F = dA , F = F µν F µν (2a)3nd a Kalb-Ramond 3-form : H = dB − k g A ∧ F , H = H µνρ H µνρ . (2b)This action is invariant with respect to diffeomorphisms and gauge trans-formations of the potentials. Denoting their infinitesimal generators by ξ µ ,Λ α and λ , the transformations of the various fields are given by : δg µν = L ξ g µν , (3a) δB αβ = L ξ B αβ + ∂ [ α Λ β ] − k g ∂ [ α λ ∧ A β ] , (3b) δA α = L ξ A α + ∂ α λ , (3c) δ Φ = L ξ Φ , (3d)where ξ is a vector field, Λ a one-form and λ a zero-form.Here after we make use of a natural length, that we denote by L , obtainedby rewriting the central charge as : δc =: 12 L . (4)Our interest for the problem at hand rests on the special solution that weobtained some time ago [3], in the framework of an analysis driven by defor-mations of the SL (2 , R ) wzw model. Written in term of a Boyer-Lindquistretarded time [11], it is given by : ds = Q (cid:16) − ρ + V (0)(0 , L ρ − k g L (cid:0) m (0)(0 , (cid:1) (cid:17) dτ − Q ρ dτ dρ + 2 L ω ρ dτ dθ + ρ dθ , (5a)Φ = − ln( ρ/L ) − ln Q , (5b) A = L m (0)(0 , ρ dτ + L c θ dθ , F = L m (0)(0 , ρ dτ ∧ dρ (5c) B = L Q ρ (cid:16) ω + k g Q m (0)(0 , c θ (cid:17) dτ ∧ dθ , H = ω L Q ρ dτ ∧ dρ ∧ dθ . (5d)in which we use coordinates τ, ρ and θ whose meaning will be clarified byeqs (49–50). Of course, if the range of the θ coordinate is the all real line, by The subscript (0 ,
0) and superscript (0) decorating the parameters V and m have beenintroduced in anticipation of more general solutions described in the following sections. Q = 1 and c θ = 0. However if, as from now we assume, θ is a cycliccoordinate varying between 0 and 2 π , both Q and c θ acquire a non trivialmeaning [12] .Geometrically the metric (5a) describes a black hole, displaying two Killinghorizons (an inner one ρ = r − and an outer one ρ = r + ) located at : r ± = L (cid:32) V (0)(0 , ± (cid:115)(cid:0) V (0)(0 , (cid:1) − k g (cid:0) m (0)(0 , (cid:1) − ω Q (cid:33) , (6)assuming (cid:0) V (0) (cid:1) ≥ (4 k g (cid:0) m (0) (cid:1) + 16 ω ) /Q . In terms of r + and r − theprevious field configurations, eqs (5a–5d), read : ds = − Q (cid:0) ρ − r + )( ρ − r − ) dτ + 2 ρ dρ (cid:1) dτ + (cid:0) ρ dθ + L ω dτ (cid:1) , (7a) F = 4 L (cid:112) k g ρ (cid:115) r + r − L − ω Q dτ ∧ dρ , H = ω L Q ρ dτ ∧ dρ ∧ dθ . (7b)Asymptotically ( i.e. for ρ/L → ∞ ) the terms involving the metric param-eters V (0) , m (0) and ω disappear and the metric reduces to : ds as = − Q ρ dτ − Q ρ dτ dρ + ρ dθ (8a)= L e √ Q x (cid:16) − dt + dx + dθ (cid:17) , (8b)where : τ = 12 Q (cid:0) t − x (cid:1) , ρ = L e Q x . (8c)It corresponds to the particular case presenting a naked singularity, ob-tained by setting V (0) = m (0) = ω = 0. An embedding of this geometry assubmanifold of a 4 dimensional Minkowski space is described in appendix(A).The curvature of this metric decreases as 1 /ρ . However the asymptoticstructure of the space is not the one of the usual 3-dimensional Minkowskispace. Asymptotically the domain of this coordinate chart is only geodesi-cally complete with respect to space-like and null geodesics. Time-likegeodesics always bounce at some maximal value (depending on the initialconditions) of the ρ coordinate, just as they do on a Rindler coordinatepatch. Also, in general, the angular variable θ along the space-like and nullgeodesics going to infinity performs an infinite number of turns with ρ (cid:55)→ ∞ : θ ∝ ± ln( ρ/L ), τ ∝ − ln( ρ/L ). 5 ield equations Field equations that determine the stationary configurations of the actioneq. (1) are : 16 π G δ ˙ Lδg µν = √− g ( T µν − G µν ) =: E µν = 0 , (9)where T µν = T µν Φ + T µνH + T µνF + T µνc with : T µν Φ = 4 (cid:18) ∇ µ Φ ∇ ν Φ − g µν ∇ α Φ ∇ α Φ (cid:19) , (10a) T µνH = 14 (cid:18) H µαβ H ναβ − g µν H αβγ H αβγ (cid:19) e − , (10b) T µνF = k g (cid:18) F µα F να − g µν F αβ F αβ (cid:19) e − , (10c) T µνc = 16 g µν δc e , (10d)and16 π G δ ˙ Lδ Φ = √− g (cid:18) H e − + k g F e − + 43 δc e + 8 (cid:3) Φ (cid:19) =: P = 0 , (11)16 π G δ ˙ LδA ν = k g √− g (cid:18) ∇ µ (cid:0) e − F µν − e − H µνρ A ρ (cid:1) + 14 e − H νµρ F µρ (cid:19) =: J ν = 0 , (12)16 π G δ ˙ LδB µν = √− g ∇ α (cid:0) e − H αµν (cid:1) =: K µν = 0 . (13)Before proceeding further note that : • From the last equation above, we see that the abelian field equationseq.(12) may be replaced by J ν = k g √− g ∇ µ (cid:0) e − F µν − e − H µνρ A ρ (cid:1) = 0 , (14) • As the space-time dimension is 3, the Kalb-Ramond field equation (13)is trivially solved in terms of a constant ω , the dilaton field and thevolume form η = √− g dx ∧ dx ∧ dx : H µ ν ρ = ω e L η µ ν ρ . (15)6o solve the remaining equations we start by expressing the fields in Bondigauge, i.e. by choosing coordinates (denoted now by u, r and φ ) obtainedby imposing the following gauge conditions :( g µν ) = g uu g u r g uφ g u r g uφ r , A r = 0 , B u r = B φ r = 0 . (16)To perform the integration of the field equations we follow mutatis mutandis the road exposed in ref.[10]. We assume u and φ to be dimensionless timeand angular coordinates and r a radial coordinate (of dimension L ) andadopt their parametrisation of the metric : g uu := r U ( u, r, φ ) + L e β ( u,r,φ ) V ( u, r, φ ) , g u r := − L e β ( u,r,φ ) , g uφ := r U ( u, r, φ ) ,g ur := − e − β ( u,r,φ ) /L , g rr := − e − β ( u,r,φ ) V ( u, r, φ ) , g rφ := e − β ( u,r,φ ) U ( u, r, φ ) /L . (17)Moreover, we rewrite the dilaton as :Φ =: − ln( r/L ) + f ( u, r, φ ) . (18)We proceed to integrate the field equations in two steps. First we showthat all the metric components and the temporal component of the Maxwellfield ( A u ) can be expressed in terms of radial integrals involving only thedilatonic field f ( u, r, φ ), the abelian potential component A φ ( u, r, φ ) andfive, a priori , non-trivial functions of the u and φ . At this stage the fieldequations E r r = 0, J u = 0, E r φ = 0, E r u = 0 are satisfied. If moreoverthe dilaton and the φ component Maxwell equation are verified ( P = 0 and J φ = 0), thanks to the Bianchi identities we only have to require that theremaining equations J r = 0, E u φ = 0, and E u u = 0 are satisfied for a singlevalue of the radial coordinate, while the remaining equation E φ φ = 0 is thenautomatically satisfied.Thus we shall first obtain the expressions of the various functions interms of the dilaton and the relevant component of the Maxwell field. Thenwe introduce an asymptotic expansion scheme and start to integrate thehierarchy of equations it generates. In the very first steps of the procedurewe shall see that the remaining arbitrariness that the Bondi gauge con-ditions left are just what will allows to drastically simplify the remainingequations. Going back and forth between the equations, we shall obtain aninfinite triangular system of inhomogeneous equations, involving a “massive7eat differential operator” on the circle. At this stage the complete inte-gration is reduced to a (cumbersome but elementary) algebraic problem. Itis the occurence of this heat operator that will made the evolution of thegeneric solution blowing up exponentially in time (both in the past and inthe future), rendering the black string solution unstable.Of course, the difficult task of proving the convergence of the all proce-dure will not even be evoked in what follows (see for instance ref. [13] for adiscussion of the convergence of polyhomogeneous expansions of zero-rest-mass fields in asymptotically flat spacetimes). However a truncate closedsubset of the solutions so constructed will be exhibited, showing that theprocedure could made sense. Elementary integrations solving : E r r = 0 , J u = 0 , E r φ = 0 , E r u = 0 All functions depend on the coordinates u, r, φ (however we didn’t alwayswritten them explicitly in the expressions that follow). Derivatives withrespect to φ are denoted by a prime ( F (cid:48) := ∂ φ F ), those with respect to u by a dot ( ˙ F := ∂ u F ). The r integrals that follows are indefinite. Thus,arbitrary functions of u and φ appear as integration constants; they will bewritten explicitly. • From equation E r r = 0 we obtain : β = (cid:90) (cid:16) k g L r e − f (cid:0) ∂ r A φ (cid:1) + 1 r (cid:0) − r ∂ r f (cid:1) (cid:17) dr + β (0 , ( u, φ )(19) • From J u = 0, and setting F µν := √ g (cid:0) e − F µν − ωL η µ ν ρ A ρ (cid:1) weobtain : m := F r u = e − (4 f + β ) r L (cid:0) U ∂ r A φ − ∂ r A u (cid:1) − ωL A φ , (20a)= − (cid:90) rL (cid:0) e − f ∂ r A φ (cid:1) (cid:48) dr + m (0 , ( u, φ ) (20b). A u = (cid:90) (cid:16) U ∂ r A φ − L r e f + β ( m + ωL A φ ) (cid:17) dr + L a u (0 , ( u, φ ) . (21)8 From E r φ = 0, we obtain : n := r e − β ∂ r U (22a)= 1 r (cid:90) (cid:16) L (cid:0) − r ∂ r f ) f (cid:48) + L (cid:0) β (cid:48) − r ∂ r β (cid:48) (cid:1) + k g (cid:0) L m + ω A φ (cid:1) ∂ r A φ (cid:17) dr + L r n (0 , ( u, φ ) , (22b) U = (cid:90) e β r n dr + U (0 , ( u, φ ) . (23) • Finally from E u r = 0, and using g u r = − e β /L , (along with the factthat g r r = 0 and g r φ = 0 implies, using the gravitational field equa-tions, that R r r = T r r and R r φ = T r φ ) we obtain : V = (cid:90) (cid:16) e f + β ω L r + e f + β (cid:2) − r + k g r ( L m + ω A φ ) (cid:3) + e β r (cid:2) f (cid:48) ) + 12 ( β (cid:48) ) + β (cid:48)(cid:48) (cid:3) + e β L r n + 1 L (cid:0) U + 1 r e β n (cid:1) (cid:48) (cid:17) dr + V (0 , ( u, φ ) (24) • The calculation of the asymptotic charges requires the knowledge ofthe Kalb-Ramond field. In the Bondi gauge that we adopt, its onlynon-zero component is given by : B φ u = (cid:90) (cid:16) e f + β ω L r + k g (cid:0) A u F r φ + A φ F u r (cid:1)(cid:17) dr + L b φ u (0 , ( u, φ ) , (25) Global electric and Kalb-Ramond charges and asymptotic ex-pansions
The first assumption that we will impose on the asymptotic behaviour of thefields is the requirement that the black string solution eqs (5a–5d) belongsto the phase space we are looking for. Accordingly we assume as asymptoticfall-off conditions on the geometry : U ( u, r, φ ) = O (1) , V ( u, r, φ ) = O ( r ) , β ( u, r, φ ) = ln( r/L ) + O (1) . (26)and on the dilaton field : f ( u, r, φ ) = O (1) . (27)9s a consequence, we obtain that asymptotically √− g = O ( r ).From eq. (14), we infer that, on shell, an electric charge may be definedas the integral on a circle u = u (cid:63) , r = r (cid:63) : Q M = 116 π G (cid:73) C [ u (cid:63) , r (cid:63) ] k g F αβ (cid:15) α β γ dx γ = 116 π G (cid:90) π k g F u r [ u (cid:63) , r (cid:63) , φ ] dφ . (28)Equation (20b) shows that this indeed make sense (is independent of r (cid:63) );we obtain : Q M = − π G (cid:90) π k g m (0 , ( u (cid:63) , φ ) dφ (29)while Stokes theorem implies that the coefficient of the zero mode in theFourier expansion of − k g / (8 G ) m (0 , ( u, φ ) must be a constant, see eq. (63),equal to the so defined conserved electric charge.Moreover a (non a priori conserved) “topological charge” (specific to the3-dimensional context we are considering) may also be defined from theMaxwell field : Q W ( u (cid:63) ) = 116 π G (cid:90) C [ u (cid:63) , ∞ ] A φ dφ (30)where (cid:82) C [ u (cid:63) , ∞ ] denotes the integral on a circle at infinity, the limit for r (cid:63) →∞ of the same integral evaluated on the circles C [ u (cid:63) , r (cid:63) ]. As this only makesense if this limit exists, we assume henceforth that there exists a gauge suchthat the asymptotic behaviour of the φ component of the abelian potentialreduces to : A φ = O (1) . (31)Finally, the constant ω in eq. (15) may also be seen as a conserved charge.It is obtained by integrating eq. (13) contracted with the component of aclosed, but non exact 1-form, i.e. withΛ = Λ µ dx µ = dF ( u, r, φ ) + Λ (0) φ dφ , Λ (0) φ = Cte , (32)which leads to ∇ α (cid:0) e − H α µ β Λ β (cid:1) = 0 , (33) More generally, on any closed curve of winding number 1. p -forms in arbitrary dimen-sions), by integration on a circle :Λ (0) φ Q KR := 116 π G (cid:73) C [ u (cid:63) , r (cid:63) ] e − H α β µ Λ µ η αβγ dx γ = 116 π G (cid:90) π √− g e − H u r φ Λ (0) φ dx φ = − ω G L Λ (0) φ . (34)All these considerations lead us to adopt the following asymptotic be-haviour for the two main functions that drive the all dynamic : f ( u, r, φ ) = N (cid:88) p =0 L p p (cid:88) q =0 f ( q,p ) ( u, φ ) ln q [ r/L ] r p + O (cid:16) L N +1 ln N +1 [ r/L ] r N +1 (cid:17) , (35) A φ ( u, r, φ ) = L N (cid:88) p =0 L p p (cid:88) q =0 a φ ( q,p ) ( u, φ ) ln q [ r/L ] r p + O (cid:16) L N +1 ln N +1 [ r/L ] r N +1 (cid:17) . (36) Two fundamental simplifying relations
Inserting these ansatz into eqs (19–23) we obtain as leading terms of theasymptotic expansions of β , A u and U : β = ln( r/L ) + β (0 , − Lr (cid:0) ln( r/L ) f (1 , + f (0 , (cid:1) + O (cid:16) L ln ( r/L ) /r (cid:17) , (37) A u = L a u (0 , + L r (cid:16) ln ( r/L ) (cid:0) ∂ φ ( e β (0 , a φ (1 , ) + e β (0 , a φ (1 , ∂ φ (4 f (0 , + β (0 , ) (cid:1) + ln( r/L ) (cid:0) U (0 , a φ (1 , + ∂ φ ( e β (0 , a φ (1 , ) + e β (0 , a φ (1 , ∂ φ (4 f (0 , + β (0 , ) (cid:1) + U (0 , a (0 , + e f (0 , + β (0 , ( m (0 , + ω a (0 , )+ ∂ φ ( e β (0 , a φ (1 , ) + e β (0 , a φ (1 , ∂ φ (4 f (0 , + β (0 , ) (cid:1)(cid:17) + O (cid:16) L ln ( r/L ) /r (cid:17) , (38) U = e β (0 , (cid:0) f (cid:48) (0 , + β (cid:48) (0 , (cid:1) ln( r/L ) + U (0 , + O (cid:16) L ln ( r/L ) /r (cid:17) . (39)Accordingly, in order to satisfy the first of the asymptotic conditions eq.(26) we have to impose that : β (0 , ( u, φ ) = − f (0 , ( u, φ ) + b (0 , ( u ) . (40)11n the other hand, solving the dilaton equation : P = 0, eq. (11), atorder 1 (the leading order for this equation) we obtain the condition : U (cid:48) (0 , ( u, φ ) = − U (0 , ( u, φ ) f (cid:48) (0 , ( u, φ ) + 2 ˙ f (0 , ( u, φ )+ e b (0 , ( u ) − f (0 , ( u,φ ) (cid:0) f (cid:48)(cid:48) (0 , ( u, φ ) − f (cid:48) (0 , ( u, φ ) (cid:1) (cid:1) . (41)The general solution of this equation reads : U (0 , ( u, φ ) = 2 e b (0 , ( u ) − f (0 , ( u,φ ) f (cid:48) (0 , ( u, φ ) + e − f (0 , ( u,φ ) ∂ u F ( u, φ ) (42)with F ( u, φ ) = (cid:90) φ e f (0 , ( u,ξ ) dξ + h ( u ) . (43)But in order for U (0 , ( u, φ ) to be periodic, i.e. to have U (0 , ( u,
0) = U (0 , ( u, π ), we require that : ∂ u (cid:90) π e f (0 , ( u,φ ) dφ = 0 , (44)or equivalently that : (cid:90) π e f (0 , ( u,ξ ) dξ = 2 π/Q , (45)where Q is a (positive) constant.Here comes a crucial simplification for the continuation of our discus-sion on the asymptotic solutions. Since the Bondi gauge conditions do notcompletely fix the coordinates, we may still perform two kinds of diffeomor-phisms : τ = T ( u ) and ρ = r/∂ φ H ( u, φ ) , θ = H ( u, φ ) . (46)Of course to be well defined we have to impose the usual conditions (assum-ing a choice of the orientations of the time and angular coordinates) : ∂ u T ( u ) > , ∂ φ H ( u, φ ) > , H ( u, π ) − H ( u,
0) = 2 π ;(47)we remind that both φ and θ are assumed to be cyclic variables of period2 π .These conditions are exactly fulfilled by choosing T ( u ) and H ( u, φ ) as : T ( u ) = (cid:90) u e b ( υ ) dυ and H ( u, φ ) = Q F ( u, φ ) . (48)12hus by redefining the coordinates as : τ = (cid:90) u e b (0 , ( υ ) dυ , ρ = e − f (0 , ( u,φ ) r/Q , θ = Q F ( u, φ ) , (49)we see that we may always assume that we are working in a coordinatesystem { τ, ρ, θ } such that : f (0 , ( τ, θ ) = − ln( Q ) , b (0 , ( τ ) = 0 i.e. β (0 , ( τ, θ ) = 2 ln( Q ) , U (0 , ( τ, θ ) = 0 . (50)Here Q is a constant and the new angular coordinate, denoted by θ , stillvarys between 0 and 2 π . It is on such a chart that actually the black stringgeometry is described by eqs (5a–5d). A glance on the formulae that governthe transformation of the asymptotic expansions of the various unknownfunctions that have to be evaluated convinces rapidly on the usefulness towork on a { τ, ρ, θ } chart, where eqs (50) are satisfyed. We could evenassume that Q = 1 and restoring it at the end of the integration by asimple rescaling of the cyclic variable and an appropriate redefinition on thesummation index in the Fourier expansion of the various functions.Note that all these diffeomorphisms are usually considered as being largediffeomorphisms. Indeed the expressions of the asymptotic Killing vectorswe shall consider later, when discussing the asymptotic charges, are not leftinvariant under their action. Thus, in principle, the values of the variouscharges they define change when such diffeomorphisms act on a physical con-figuration : large diffeomorphisms transform into distinct dynamical statesof the physical system. But this conclusion presuppose that after a largediffeomorphism charges associated to the reducibility parameters given bytheir original expressions are still defined. For the problem we consider hereit will be shown that this is not the case. In this section we describe the path we have followed to integrate order byorder the remaining field equations. We almost completely fix the gaugeby imposing the choices (50). The remaining freedom only consists into thechoice of the lower bound of the integrals defining T ( u ) and F ( u, φ ), corre-sponding to translations of the τ and θ variables.The equations we have to consider consist of two standard evolution equa-tions J θ = 0 and P = 0 governing the evolution of f and A θ and three13omplementary equations that will fix the integration functions m (0 , ( u, φ ), n (0 , ( u, φ ) and V (0 , ( u, φ ), introduced in eqs (20b, 22b, 24).We start with the expressions of β (0 , ( τ, θ ) = − Q ) and U (0 , ( τ, θ ) = 0obtained in eqs (40, 50), that solve the equation P = 0 at order one. Fromnow we shall proceed order by order, taking into account at each order theresults established at the previous ones and represent τ -derivatives by dotsand θ -derivatives by primes.At order (1), the equation J θ = 0 implies that : a τ (0 , ( τ, θ ) = ˙ λ ( τ, θ ) , a θ (0 , ( τ, θ ) = λ (cid:48) ( τ, θ ) + c θ , (51)where c θ is a constant. In other words these components reflect the expectedresidual gauge freedom that preserves the Bondi gauge but also encode thetopological charge eq.(30) that, as a consequence of eq. (31), defines aconserved charge.At order ln( ρ/L ) /ρ and 1 /ρ , respectively, we obtain from the dilatonequation P = 0 (11):˙ f (1 , ( τ, θ ) = − Q f (cid:48)(cid:48) (1 , ( τ, θ ) , (52a)˙ f (0 , ( τ, θ ) = − Q f (cid:48)(cid:48) (0 , ( τ, θ ) + 2 ˙ f (1 , ( τ, θ ) + 4 f (1 , ( τ, θ ) , (52b)and from the Maxwell equation J θ = 0 (12):˙ a θ (1 , ( τ, θ ) = − Q a (cid:48)(cid:48) θ (1 , ( τ, θ ) , (52c)˙ a θ (0 , ( τ, θ ) = − Q a (cid:48)(cid:48) θ (0 , ( τ, θ ) + 2 ˙ a θ (1 , ( τ, θ ) + 4 a θ (1 , ( τ, θ ) . (52d)At order ρ − (in the field equations), we obtain nine equations : • From P = 0 at order ln ( ρ/L ) /ρ :3 (cid:0) f (cid:48) (1 , ( τ, θ ) (cid:1) − f (1 , ( τ, θ ) f (cid:48)(cid:48) (1 , ( τ, θ ) = 0 , (53a)whose only periodic solution compatible with eq. (52a) is f (cid:48) (1 , ( τ, θ ) = 0 i.e. f (1 , ( τ, θ ) = f (0)(1 , , (53b)with f (0)(1 , an arbitrary constant. • As a direct consequence of eq. (53b), equation J θ = 0 :3 a (cid:48) θ (1 , ( τ, θ ) f (cid:48) (1 , ( τ, θ ) − a θ (1 , ( τ, θ ) f ” (1 , ( τ, θ ) = 0 (53c)is satisfied at order ln ( ρ/L ) /ρ .14 From J θ = 0 at order ln ( ρ/L ) /ρ : a θ (1 , ( τ, θ ) f (cid:48)(cid:48) (0 , ( τ, θ ) − a (cid:48) θ (1 , ( τ, θ ) f (cid:48) (0 , ( τ, θ ) = f (0)(1 , a (cid:48)(cid:48) θ (1 , ( τ, θ ) . (54a) • From P = 0 at order ln ( ρ/L ) /ρ : (cid:0) a (cid:48) θ (1 , ( τ, θ ) (cid:1) − a θ (1 , ( τ, θ ) a (cid:48)(cid:48) θ (1 , ( τ, θ ) = 32 k g Q f (0)(1 , f (cid:48)(cid:48) (0 , ( τ, θ ) . (54b)Assuming k g > : a (cid:48) (1 , ( τ, θ ) = 0 i.e. a θ (1 , ( τ, θ ) = a (0) θ (1 , (55a)with a (0) θ (1 , a constant, see eq. (52c), and : f (0 , ( τ, θ ) = j (0 , ( τ ) or f (0)(1 , = 0 . (55b)To proceed further let us restrict ourselves to the dynamical solution sectorand choose the solution f (0)(1 , = 0.Then, the solutions of eqs (52d,52b) are given by the superpositions ofmodes : f (0 , ( τ, θ ) = (cid:88) n f ( n )(0 , e i n θ + Q n τ , (56a) a θ (0 , ( τ, θ ) = (cid:88) n a ( n ) θ (0 , e i n θ + Q n τ + 4 τ a (0) θ (1 , . (56b)Of course we have to impose the reality conditions (always understood inthe following and that we shall not repeat explicitly) : f ( n )(0 , = (cid:0) f ( − n )(0 , (cid:1) (cid:63) , a ( n ) θ (0 , = (cid:0) a ( − n ) θ (0 , (cid:1) (cid:63) . (57)These modes are obviously unstable : they blow up exponentially when τ increases.But this is not the end of the story. It is interesting to pursue thediscussion to next order. Set κ = (cid:112) k g Q/ k = j (1 , κ , f (0 , = κ f , ζ = a θ (1 , + i f , ξ = ( k + i a θ (1 , ).Equations (54a, 54b) imply that ζ (cid:48) ξ (cid:48) = ξ ζ (cid:48)(cid:48) , from which we conclude. Note that if k g < From the equation P = 0, at order ln ( ρ/L ) /ρ , we obtain :˙ f (2 , ( τ, θ ) = − Q f (cid:48)(cid:48) (2 , ( τ, θ ) − f (2 , ( τ, θ ) + 23 Q (cid:16) f (0 , ( τ, θ ) f (cid:48)(cid:48) (0 , ( τ, θ ) − (cid:0) f (cid:48) (0 , ( τ, θ ) (cid:1) (cid:17) + 124 k g Q (cid:16) Q a (0)(1 , a (cid:48)(cid:48) θ (0 , ( τ, θ ) − (cid:0) a (0)(1 , (cid:1) (cid:17) . (58a) • From the equation J θ = 0, at order ln ( ρ/L ) /ρ :˙ a θ (2 , ( τ, θ ) = − Q a (cid:48)(cid:48) θ (2 , ( τ, θ ) − a θ (2 , ( τ, θ )+ 23 Q (cid:16) ( a θ (0 , ( τ, θ ) − a (0)(1 , ) f (cid:48)(cid:48) (0 , ( τ, θ ) − a (cid:48) θ (0 , ( τ, θ ) f (cid:48) (0 , ( τ, θ ) − a (0)(1 , n (cid:48) (0 , ( τ, θ ) (cid:17) . (58b) • The equation P = 0, at order ln( ρ/L ) /ρ , leads to a similar equationbut involving the not yet fixed functions m (0 , ( τ, θ ) and n (0 , ( τ, θ ) :˙ f (1 , ( τ, θ ) = − Q f (cid:48)(cid:48) (1 , ( τ, θ ) − f (1 , ( τ, θ )+ Q (cid:16) k g Q (cid:0) ( a θ (0 , ( τ, θ ) + 176 a (0)(1 , ) a θ (0 , ( τ, θ ) (cid:48)(cid:48) − ( a θ (0 , ( τ, θ ) (cid:48) ) (cid:1) − k g a (0)(1 , a θ (0 , ( τ, θ ) (cid:17) + 4 Q (cid:16) f (0 , ( τ, θ ) f (cid:48)(cid:48) (0 , ( τ, θ ) − (cid:0) f (cid:48) (0 , ( τ, θ ) (cid:1) − f (cid:48)(cid:48) (2 , ( τ, θ ) (cid:17) + Q (cid:0) n (0 , ( τ, θ ) f (cid:48) (0 , ( τ, θ ) − n (cid:48) (0 , ( τ, θ ) f (0 , ( τ, θ ) (cid:1) + k g Q a (0)(1 , (cid:0) m (cid:48) (0 , ( τ, θ ) + ω λ (cid:48)(cid:48) ( τ, θ ) − a (0)(1 , (cid:1) + 409 f (2 , ( τ, θ ) . (59a) • Equation J θ = 0, at the same order ln( ρ/L ) /ρ , implies that :˙ a θ (1 , ( τ, θ ) = − Q a (cid:48)(cid:48) θ (1 , ( τ, θ ) − a θ (1 , ( τ, θ )+ Q (cid:16)(cid:0) a θ (0 , ( τ, θ ) − a (0)(1 , (cid:1) f (cid:48)(cid:48) (0 , ( τ, θ ) − a (cid:48)(cid:48) θ (2 , ( τ, θ ) − f (0 , ( τ, θ ) a (cid:48)(cid:48) θ (0 , ( τ, θ ) + 29 f (cid:48) (0 , ( τ, θ ) a (cid:48) θ (0 , ( τ, θ )+ 13 (cid:0) n (0 , ( τ, θ ) a (cid:48) θ (0 , ( τ, θ ) − n (cid:48) (0 , ( τ, θ )( a θ (0 , ( τ, θ ) + 76 a (0)(1 , ) (cid:1)(cid:17)
16 43 (cid:0) a (0)(1 , f (0 , ( τ, θ ) − ( m (0 , ( τ, θ ) + ω λ (cid:48) ( τ, θ )) f (cid:48) (0 , ( τ, θ ) (cid:1) + 13 V (0 , ( τ, θ ) a (0)(1 , + 13 ω a (cid:48) θ (0 , ( τ, θ ) − a θ (2 , ( τ, θ ) . (59b)The occurence, on the righthand sides, of a term proportional to whatis an arbitrary gauge parameter : λ (cid:48) ( τ, θ ) in eq. (59b), λ (cid:48)(cid:48) ( τ, θ ) ineq. (59a) is not surprising. Indeed, as we may see from eq. (20b)it is the combination m ( τ, θ ) + ω A θ ( τ, θ ) /L and more specifically, inthe asymptotic expansion : m (0 , ( τ, θ ) + ω a θ (0 , (see eq. (38)), thatis gauge invariant with respect to the residual gauge transformationthat preserves the Bondi gauge. That this gauge invariant term and itsderivatives occur in equations governing subdominant terms of order1 /ρ in the expansions of f and A θ results, as eq. (38) shows, fromthe fact that “ m ( τ, θ ) + ω A θ ( τ, θ ) /L ” contributes to the order 1 /ρ ofthe expansion of A u : it is like an “ a u (0 , ” term.17 From the equation P = 0, expanded at order 1 /ρ , we obtain :˙ f (0 , ( τ, θ ) = − Q f (cid:48)(cid:48) (0 , ( τ, θ ) − f (0 , ( τ, θ ) − Q (cid:16) k g a (0)(1 , a θ (0 , ( τ, θ )+ 124 k g a (cid:48) θ (0 , ( τ, θ ) (cid:0) m , ( τ, θ ) + ω a θ (0 , ( τ, θ ) (cid:1) − k g a θ (0 , ( τ, θ ) (cid:0) m (cid:48) (0 , ( τ, θ ) + ω a (cid:48) θ (0 , ( τ, θ ) (cid:1) + 16 k g a θ (0 , ( τ, θ ) − f (cid:48) (0 , ( τ, θ ) n (0 , ( τ, θ ) − f (0 , ( τ, θ ) n (cid:48) (0 , ( τ, θ ) − (cid:0) f (cid:48) (0 , ( τ, θ ) (cid:1) − f (0 , ( τ, θ ) f (cid:48)(cid:48) (0 , ( τ, θ ) + 29 f , , ( τ, θ )+ 827 f (cid:48)(cid:48) (0 , ( τ, θ ) + 5144 k g a (0)(1 , (cid:0) m (cid:48) (0 , ( τ, θ ) + ω a (cid:48)(cid:48) θ (0 , ( τ, θ ) (cid:1) − k g a (0)(1 , + 112 n , ( τ, θ ) (cid:17) − k g Q a (0)(1 , a (cid:48)(cid:48) , ( τ, θ ) − k g Q (cid:0) a (cid:48) , ( τ, θ ) (cid:1) + 1172 k g Q a θ (0 , ( τ, θ ) a (cid:48)(cid:48) , ( τ, θ )+ 13 f (0 , ( τ, θ ) V (0 , ( τ, θ ) + 209 f (1 , ( τ, θ ) + 827 f (2 , ( τ, θ ) + ω Q . (60a) • Similarly, at order 1 /ρ , the equation J θ = 0 results in :18 a θ (0 , ( τ, θ ) = − Q a (cid:48)(cid:48) θ (0 , ( τ, θ ) − a θ (0 , ( τ, θ ) − Q (cid:16) a (cid:48)(cid:48) θ (2 , ( τ, θ ) + 29 a (cid:48)(cid:48) θ (1 , ( τ, θ )+ 5227 a (0)(1 , f (0 , , ( τ, θ ) − a (0)(1 , n (cid:48) (0 , ( τ, θ )+ 269 a (cid:48)(cid:48) θ (0 , ( τ, θ ) f (0 , ( τ, θ ) − a (cid:48) θ (0 , ( τ, θ ) f (cid:48) (0 , ( τ, θ ) + 518 a (cid:48) θ (0 , ( τ, θ ) n (0 , ( τ, θ )+ 1727 a θ (0 , ( τ, θ ) f (cid:48)(cid:48) (0 , ( τ, θ ) + 118 a θ (0 , ( τ, θ ) n (cid:48) (0 , ( τ, θ ) (cid:17) + 827 a , ( τ, θ ) + 209 a , ( τ, θ ) + 329 a (0)(1 , f (0 , ( τ, θ )+ 718 ω a (cid:48) θ (0 , ( τ, θ ) + 163 a θ (0 , ( τ, θ ) f (0 , ( τ, θ ) − (cid:0) m (cid:48) (0 , ( τ, θ ) + ω a θ (0 , (cid:48)(cid:48) ( τ, θ ) (cid:1) f (0 , ( τ, θ )+ (cid:0) m (0 , ( u, θ ) + ω a (cid:48) θ (0 , ( τ, θ ) (cid:1)(cid:0) ω Q + 13 n (0 , ( τ, θ ) − f (0 , , ( τ, θ ) (cid:1) + (cid:0) a θ (0 , ( τ, θ ) − a (0)(1 , (cid:1) V (0 , ( τ, θ ) . (60b)Before discussing the general structure of the hierarchy of equations, let usturn to the complementary equations.Solving J ρ = 0. • The order ln( ρ/L ) term of the expansion of J ρ is proportional to :˙ a (cid:48) θ (0 , ( τ, θ ) = U ( τ ) a (cid:48)(cid:48) θ (0 , ( τ, θ ) − Q a (cid:48)(cid:48)(cid:48) θ (0 , ( τ, θ ) , (61)and thus, as a consequence of eqs (52d) and (55a), the correspondingequation is satisfied at this order. • At order 1, we obtain the equation : (cid:0) m (0 , ( τ, θ ) + ω a θ (0 , ( τ, θ ) (cid:1) ˙ = − Q (cid:0) m (0 , ( τ, θ ) + ω a θ (0 , ( τ, θ ) (cid:1) (cid:48)(cid:48) + 4 Q a (cid:48) θ (0 , − Q a (cid:48)(cid:48)(cid:48) θ (0 , , (62)19hose general solution is given by a mode superposition similar tothose displayed in eqs (56a, 56b) : m (0 , ( τ, θ ) + ω a θ (0 , ( τ, θ ) = (cid:88) n m ( n )(0 , e i n θ + Q n τ + τ (cid:0) Q a (cid:48) θ (0 , − Q a (cid:48)(cid:48)(cid:48) θ (0 , (cid:1) . (63)So J ρ vanishes at infinity and thus, as a consequence of the Bianchiidentities, it vanishes everywhere.Solving ρ E uφ (cid:39) ρ E uφ (the term of order ln( ρ/L ) vanishes as a consequence of eqs (52b) and (53b). • At order 1, this equation leads to the equation :˙ n (0 , ( τ, θ ) = − Q n (cid:48)(cid:48) (0 , ( τ, θ ) − f (cid:48) (0 , ( τ, θ ) + 8 Q f (cid:48)(cid:48)(cid:48) (0 , ( τ, θ ) , (64)whose general solution reads n (0 , ( τ, θ ) = (cid:88) p n ( p )(0 , e i p θ + Q p τ − τ (cid:0) f (cid:48) (0 , + 8 Q f (cid:48)(cid:48)(cid:48) (0 , (cid:1) . (65)The equation ρ E uu = 0 also has to be only satisfied for a single value of ρ . • Taking into account all previous equations, only the term of order 1in the expansion of ρ E uu , provides a new condition, that fixes the lastunknown function V (0 , . It has to satisfy the equation :˙ V (0 , ( τ, θ ) = − Q V (cid:48)(cid:48) (0 , ( τ, θ ) + Q (cid:0) f (cid:48)(cid:48) (0 , ( τ, θ ) − n (cid:48) (0 , ( τ, θ ) (cid:1) + Q (cid:0) n (cid:48)(cid:48)(cid:48) (0 , ( τ, θ ) − f (cid:48)(cid:48)(cid:48)(cid:48) (0 , ( τ, θ ) (cid:1) , (66)whose solution is : V (0 , ( τ, θ ) = (cid:88) n V ( n )(0 , e i n θ + Q n τ + τ Q (cid:0) f (cid:48)(cid:48) (0 , ( τ, θ ) − n (cid:48) (0 , ( τ, θ ) (cid:1) + τ Q (cid:0) n (cid:48)(cid:48)(cid:48) (0 , ( τ, θ ) − f (cid:48)(cid:48)(cid:48)(cid:48) (0 , ( τ, θ ) (cid:1) . (67)20he general pattern for the complete integration of the field equationsemerges from these first steps. It is the following : at each order in 1 /ρ p ,the equations driving, both f ( q,p ) and a θ ( q,p ) (here after generically denotedby X ( q,p ) ) have the following general structure : D p X ( q,p ) = S ( q,p ) , (68)where D p is the parabolic differential operator (a massive heat flow operator): D p := ∂ τ + Q (2 p − ∂ θ + 4 p ( p − p − . (69)The general solution of the homogeneous equation D p X ( q,p ) = 0 is given bythe mode superposition : X hom ( q,p ) = (cid:88) n x ( n )( q,p ) e i n θ + ( Q n − p ( p − p − τ . (70)Except for at most one of them, all the remaining modes are exponentiallyunstable. Those with n > (cid:112) p ( p − /Q blow up for τ going to + ∞ ; thosewith n < (cid:112) p ( p − /Q diverge in the limit where τ → −∞ .The right-hand side of the equation (68) is given by a sum of products ofpower τ k of time variable τ and modes solving the homogeneous equation forlower (or equal) values of the p index. As product of such modes constitutein general eigenmodes of the operator D p but with a non zero eigenvalue, it isimmediate that the special solution of the non-homogeneous equation withthe source term S ( q,p ) will be given by a sum of products of the same eigen-modes multiplied by a polynomial of the τ variable. This polynomial maybe chosen to be equal to τ k +1 / ( k + 1) if the eigenmode of D p is of zero eigen-value ( for instance, see eqs (56b, 63, 65, 67) or equal to e − s τ ∂ kσ (cid:0) e σ τ /σ (cid:1) | σ = s ,if the eigenmode is of non-zero eigenvalue s . But actually this procedure isformal as it requires in general to manage products of series. A special solution
As we have shown in the preceding sections, all the asymptotic solutions maybe obtained following a procedure involving solutions of the heat equationon a circle. There is a particular subset of solutions that can be written inclosed form. Let us assume that for p ≥ f ( q,p ) ( τ, θ ) and a φ ( q,p ) ( τ, θ ) are vanishing. The hierarchy of equations is almost all trivially21atisfied. The only remaining non-zero functions are : f ( τ, ρ, θ ) = f (0 , ( τ, θ ) = −
12 ln( Q ) , (71a) β ( τ, ρ, θ ) = ln( ρ/L ) + β (0 , ( τ, θ ) = ln( ρ/L ) + 2 ln( Q ) , (71b) U ( τ, ρ, θ ) = L ωρ , (71c) V ( τ, ρ, θ ) = − ρL + (cid:88) n V ( n )(0 , e i n θ + Q n τ − k g Lρ (cid:16) (cid:88) n m ( n )(0 , e i n θ + Q n τ (cid:17) − L ω Q ρ , (71d) A τ ( τ, ρ, θ ) = L ρ (cid:88) n m ( n )(0 , e i n θ + Q n τ + L ∂ τ λ ( τ, θ ) , (71e) A θ ( τ, ρ, θ ) = L c θ + L ∂ θ λ ( τ, θ ) , (71f) B τθ ( τ, ρ, θ ) = ω L Q ρ + k g A τ ( τ, ρ, θ ) A θ ( τ, ρ, θ ) + L b τθ (0 , ( τ, θ ) , (71g) H τρθ ( τ, ρ, θ ) = ω L Q ρ . (71h)Actually this solution was obtained from a direct integration of the fieldequations, under the working assumption f ( q,p> ( τ, θ ) = 0, a θ ( q,p> ( τ, θ ) =0 [see appendix (B)], but it could be guessed from the previous equationsby noticing that eq. (60b) implies n (0 , = − ω/Q which is (fortunately)compatible with eq. (60a).The Gauss curvature of the corresponding geometry (see eqs (17)) is givenby R = − Q ρ − L (cid:80) n V ( n )(0 , e i n θ + Q n τ Q ρ + 5 L ω Q ρ + k g L (cid:16) (cid:80) n m ( n )(0 , e i n θ + Q n τ (cid:17) Q ρ . (72)Solution (71a-71h) coincides with the black string (5a–5d) when τ goes to −∞ . It diverges on the timelike singularity ρ = 0 but also, when dynamical,on the surface τ = + ∞ , developing a ”cosmological singularity”. Let usremind that the more general solutions, discussed previously, present suchsingularities both on τ = + ∞ and on τ = −∞ [See eq. (70)]. These surfacescorrespond to Cauchy horizons of the stationary solution (5a–5d). These Guessed, because we have limited the hierarchy of equations at order p = 2. ± ) coinciding withthem asymptotically in the past. The evolution of these surfaces is obtainedby solving the partial differential equation : ∂ τ R = − k g L R (cid:16) (cid:88) n m ( n )(0 , e i n θ + Q n τ (cid:17) − (cid:0) Q ∂ θ R − ω L (cid:1) Q R− R + L (cid:88) n V ( n )(0 , e i n θ + Q n τ (73)with the asymptotic condition [see eq.(6)] : lim τ →−∞ R = r ± .A large degree of arbitrariness characterises the geometry defined by theconfiguration [71a–71h]. It is encoded in the function V ( τ, ρ, θ ) (Eq.[71d])that appears in the g τ τ component of the metric. This function V ( τ, ρ, θ )involves two infinite series of arbitrary coefficients : V ( n )0 , and m ( n )0 , . Thisarbitrariness make it difficult (if not impossible) to discuss the general struc-ture of these null surfaces for τ >
0. Of course, for τ → −∞ only V (0)0 , = 4 ( r + + r − ) L and m (0)0 , = 4 (cid:112) k g (cid:115) r + r − L − ω Q (74)contribute and we recover the “Kerr-like horizon structure” discussed in ref.[3].Preliminary numerical integrations of eq. (73), for particular choices of the V ( n )(0 , and m ( n )(0 , arbitrary coefficients, illustrate some aspects of the causalstructure of the dynamical geometries. In particular they show that some ofthe Σ ± generators may hit the singularity ρ = 0, while others run to infinity.We illustrate this behaviour by considering the special configuration where ω = 0 and where the equation : ∂ θ V ( τ, ρ, θ ) = 0 (75)admits constant solutions θ = θ (cid:63) , i.e. zeros independent of τ and ρ . Insuch case the curves { θ = θ (cid:63) , ρ = ρ (cid:63) ( τ ) where ρ (cid:63) ( τ ) is the solution of the Only subjected to reality conditions : V ( n )0 , = ( V ( − n )0 , ) (cid:63) and m ( n )0 , = ( m ( − n )0 , ) (cid:63) ) For instance that will be the case if, for n (cid:54) = 0, the arbitrary coefficients V ( n )(0 , and m ( n )(0 , are all real ( resp. purely imaginary). In this case the series occurring in the dρdτ = L V ( τ, ρ, θ (cid:63) ) , (76a)= − (cid:16) ρ − (cid:0) r + + r − + L/ (cid:88) n (cid:54) =0 V ( n )(0 , e i n θ (cid:63) + Q n τ (cid:1) + k g ρ (cid:0)(cid:113) r + r − /k g + L/ (cid:88) n (cid:54) =0 m ( n )(0 , e i n θ (cid:63) + Q n τ (cid:1) (cid:17) , (76b) ρ (cid:63) ( −∞ ) = r + ( resp. ρ (cid:63) ( −∞ ) = r − ) , (76c)are generators of the external ( resp. internal) horizon.To go further let us specialise some more the arbitrariness of the geometrywe examine by fixing, for n (cid:54) = 0, all the m ( n )(0 , coefficients to zero. Equation(76b) reduces to : dρdτ = − ρ − r + )( ρ − r − ) ρ + L (cid:88) n (cid:54) =0 V ( n )(0 , e i n θ (cid:63) + Q n τ . (77)For large positive value of τ , it is the values of κ (cid:63) ( u ) = (cid:88) n (cid:54) =0 V ( n )(0 , e i n θ (cid:63) + Q n τ (78)that drive the behaviour of the function ρ (cid:63) ( τ ). If κ (cid:63) ( τ ) grows to + ∞ likeexp( k θ ) ( k ∈ N ), ρ (cid:63) ( τ ) also grows as exp( k θ ). If on the other hand κ (cid:63) ( τ ) decreases exponentially to −∞ , ρ (cid:63) ( τ ) also will decrease and reachzero for a finite value τ (cid:63) of τ . For illustrative purposes let us choose (cid:80) n (cid:54) =0 V ( n )(0 , e i n θ + Q n τ = cos(3 θ ) exp(9 τ ), r + = 4 L , r − = 2 L , ω = 0 and Q = 1. A numerical integration, whose results are displayed in Fig.(1) showsthe evolution of the external horizon from its asymptotically past circularstructure of radius r + to a trefoil-like one. Three of the generators evolvingat fix values of θ : those with θ (cid:63) = π/ , π/ , π/ ρ = 0, whilethose with θ (cid:63) = 0 , π/ , π/ ∞ . expression of V ( τ, ρ, θ ) only involves functions cos( n θ ) ( resp. sin( n θ )) and we obtain θ (cid:63) = 0 and π ( resp. θ (cid:63) = ± π/ n (cid:54) = 0, the coefficients V ( n )(0 , and m ( n )(0 , are non-zero only for one value of n than Eq.(75) will admit 2 n solutions, equally spacedby π/n . =0 λ −∞ = 4 . λ = 4 . λ τ (cid:63) / = 4 . λ . τ (cid:63) = 3 . λ τ (cid:63) = 3 . r + = 4 L , r − = 2 L , ω = 0, Q = 1, m ( n> , = 0, V ( n> , = 0 excepted V (3)(0 , = V ( − , = 1 /
2. Thesections are considered at times : τ = −∞ , 0, τ (cid:63) /
2, and τ . (cid:63) up to themoment τ (cid:63) = 0 .
46 where it hits the singularity, that becomes naked. Thelengths λ τ (in units 2 π L ) of the horizon sections also are indicated.The main purpose of this illustrative discussion is to show that the struc-ture of the horizon of the black string is geometrically rich, far from beingcompletely understood, and deserves extra studies. Before stopping it, letus show another sequence of horizon sections and their respective lengths[Fig. (2) ] for a more general configuration, where ω (cid:54) = 0 and which in-volves both sine and cosine functions : r + = 4 L , r − = 2 L , Q = 1, ω = 5, (cid:80) n (cid:54) =0 V ( n )(0 , e i n θ + Q n τ = sin( θ ) exp( τ ) + cos(2 θ ) exp(4 τ ). Here again wesee that some horizon generators hit the singularity (despite the absence ofsymmetry) while others (apparently) go to infinity.To make an end to these numerical considerations, let us mention that theyalso indicate that the length of the horizon diminishes when the time vari-able τ increases. This is not in contradiction with the usual theorem onblack holes as here the horizons are past horizons. A simple way to recon-cile the behaviour of the horizon with the thermodynamic law of entropyincrease is to change τ into − τ . But then the τ variable has to be inter-preted as a advanced time labelling J − and not as a retarded time on J + .The corresponding solutions describe configurations where initially diver-25 −∞ = 4 . λ = 4 . λ τ (cid:63) / = 3 . λ . τ (cid:63) = 3 . λ τ (cid:63) = 3 . r + = 4 L , r − = 2 L , Q = 1, ω = 5, (cid:80) n (cid:54) =0 V ( n )(0 , e i n θ + Q n τ = sin( θ ) exp( τ ) + cos(2 θ ) exp(4 τ ). . The sectionsare considered at times : τ = −∞ , 0, τ (cid:63) /
2, and τ . (cid:63) up to the moment τ (cid:63) = 0 .
86 where it hits the singularity (represented by black dots) that be-comes naked. The lengths λ τ (in units 2 π L ) of the horizon sections alsoare indicated.gent field modes vanish and futur horizons appear, masking an initial nakedsingularity. Solutions of gauge invariant theories that admit gauge transformations leav-ing the field configuration invariant, i.e. a set of reducibility parameters { ζ } , allow to define elementary charges, and, if integrable, finite charges(see for example refs [14, 15, 16] and references therein). The elementarycharges are given by integration on closed surfaces of co-dimension 2, hereon closed curves C , of the elementary surface charge density (154), here : k { ζ } ρ := k µν { ζ } η µνρ . Those that we consider hereafter, are specifyed by choos-ing the solutions of the linearised field equations, the definition of k ζ requires,given by arbitrary variations of the constants occurring in the expressionsof the background solutions. Let us notice that if { ξ µ , Λ α , λ } constitute aset of reducibility parameters, they all transform covariantly with respect todiffeomorphisms but under the gauge transformations : A α (cid:55)→ A α + ∂ α ψ , (79) B αβ (cid:55)→ B αβ + ∂ [ α Ψ β ] − k g ∂ [ α ψA β ] . (80)26ccording to the rules : ξ µ (cid:55)→ ξ µ , (81) λ (cid:55)→ λ − ξ α ∂ α ψ + λ c , (82)Λ α (cid:55)→ Λ α + k g (cid:0) ψ ∂ α λ − ( ξ β ∂ β ψ ) ∂ α ψ (cid:1) − ξ β ∂ β Ψ α + Λ cα , (83)where λ c is an arbitrary constant (a closed zero-form) and Λ cα the com-ponents of an arbitrary closed one-form. Thus, without loss of general-ity, from now we may assume that the functions b φ u (0 , ( u, φ ) in eq. (25), b τ θ (0 , ( τ, φ ) in eq. (71g), λ ( u, φ ) in eq.(51) and λ ( τ, θ ) in eqs (71e, 71f) tobe zero, respectively.The black string configuration Eqs (5a–5d) admits such parameters { ζ } := { ξ µ , Λ α , λ } constituted by the Killing vector fields ∂ τ and ∂ θ of the metric,ensuring that δg µν = 0 and leaving the dilaton invariant δ Φ = 0, and gaugeparameters Λ α (given by eq. (32) and λ (constant) such that δA α = 0 and δB α β = 0 respectively (see eqs (3a–3d)). They lead to elementary chargesthat all appear to be integrable.Its Killing horizons, located at r = r + and r = r − [see eq. (6)], are generatedby the vector fields : γ ± = ∂ τ − ω Lr ± ∂ θ . (84)This structure allows us to obtain a first law relating variation of energy,angular momentum and charges. It requires that the gauge parameter arereducibility parameters, i.e. when, evaluated on the black string back-ground, ξ µ are the components of the Killing vector fields, λ is constantand Λ = dF ( τ, ρ, θ ) + Λ θ dθ . Associated to them are conserved charges: en-ergy, angular momentum and abelian (29) and Kalb-Ramond (34) chargesrespectively. A standard calculation, based on the surface charge densityprovided in Appendix (C) leads to : δ E = 18 π G κ ± δA ± − ω Lr ± δJ + Φ Max δ Q Max + Φ KR δ Q KR . (85)Here E = ( r + + r − ) / (2 G ) is the energy, κ ± = (cid:113) − / γ µ ± ; ν γ ν ± ; µ = 2 ( r ± − r ∓ ) /r ± is the surface gravity and A ± = 2 π r ± the length of the outer horizon( r = r + ) or the inner horizon ( r = r − ). Let us emphasize the contributionof the topological term introduced in the angular component of the abelianpotential (71f) : J := − L G (cid:16) ωQ + 2 c θ (cid:112) k g (cid:115) r + r − L − ω Q − k g c θ ω (cid:17) . (86)27he charges are given by eqs (29, 34) : Q Max = 18 G (cid:16) (cid:112) k g (cid:115) r + r − L − ω Q − k g c θ ω (cid:17) , Q KR = − ω G L . (87)The Nœther-Wald surface charge expression (153) provides those of the as-sociated potentials. For the abelian fields, it takes its usual expression :Φ ± Max = γ µ ± A µ (cid:12)(cid:12)(cid:12) r ± = L r ± (cid:16) (cid:112) k g (cid:115) r + r − L − ω Q − c θ ω (cid:17) . (88)However, due to a more complicated gauge transformation rule of the B field(3c), the corresponding potential linked to the Kalb-Ramond field reads :Φ ± KR = γ µ ± (cid:0) B µ θ − k g A µ A θ ) (cid:12)(cid:12)(cid:12) r ± = − L r ± (cid:16) ωQ +2 c θ (cid:112) k g (cid:115) r + r − L − ω Q − k g c θ ω (cid:17) . (89)Remarkably all these relations remain valid in the framework of thesolution described by eqs (71a–71h) if we lightly restrict the phase space byfixing c θ , i.e. restricting the solutions of the linearised equations by imposing δc θ = 0 . (90)Indeed while the vector fields ∂ τ and ∂ θ are no longer the generators ofisometries preserving the matter fields (even asymptotically), they neverthe-less constitute symplectic symmetries [17, 18] : the surface charge densityconstructed with them and the fields eqs (71a–71h) is conserved.We also may look for asymptotic symmetries. We restrict ourself to thesearch of such symmetries leading to asymptotic conserved charges. Suchasymptotic charges that may be linked to the field configurations describedin the previous section rest on the existence of asymptotic reducibility pa-rameters i.e. gauge transformations that preserve the asymptotic structureof the field configurations. In our case the asymptotic field behaviour is de-fined by eqs (26, 27) and the Bondi gauge conditions eqs (16). Such gaugetransformations are determined as follows (see ref. [10], appendix A.1).Imposing on the metric the Bondi gauge condition, we have to maintain thefollowing three conditions : g r r = 0 , g r φ = 0 , g φ φ = r , (91)28.e. in terms of the infinitesimal diffeomorphisms (3a) to impose that theirgenerators ξ α satisfy the three conditions L ξ g r r = 0 which implies ∂ r ξ u = 0i.e. ξ u = X ( u, φ ) , (92) L ξ g r φ = 0 which implies ∂ r ξ φ − L e β r ∂ φ X ( u, φ ) = 0i.e. ξ φ = Y ( u, φ ) + L ∂ φ X ( u, φ ) (cid:90) e β r dr (93)= ∂ φ X ( u, φ ) e β (0 , ln( r/L ) + O (1) , (94) L ξ g u φ = 0 which implies ξ r = − r (cid:0) U ∂ φ ξ u + ∂ φ ξ φ ) . (95)Moreover, imposing that g u r remains at order r : L ξ g u r = O ( r ) , (96)leads to an equation dictated by the term of order O (ln( r/L )) in eq. 94: ∂ φ ξ u ∂ φ e β (0 , ( u, φ ) = 2 ∂ φ (cid:0) ∂ φ ξ u e β (0 , ( u, φ ) (cid:1) (97)whose solution is ∂ φ ξ u = ∆( u ) e − β (0 , ( u, φ ) . (98)But, for ξ u to be periodic we have to require that ∆( u ) = 0. As a conse-quence we obtain : ξ u = X ( u ) , ξ r = − r ∂ φ Y ( u, φ ) , ξ φ = Y ( u, φ ) . (99)The remaining assumption on the asymptotic behaviour of the metric com-ponents : g u u = O ( r ) and g u φ = O ( r ) and the dilaton field does not implyany further condition. It is worthwhile to notice that in particular we obtain: L ξ Φ = O (1) (100)in accordance with the ansatz eq. (18).The Bondi gauge conditions eqs (16) on the abelian and the Kalb-Ramond fields are preserved by the infinitesimal diffeomorphisms definedby the vector field eq. (99). The residual gauge transformations Eqs(3c–3b)that preserve conditions (16) are : λ = (cid:96) ( u, φ ) , Λ = dL ( u, r, φ ) + P φ ( u, φ ) dφ . (101)29his expression of the one-form Λ as written is not canonical. It can as wellbe written in terms of three component functions :Λ = K u ( r, u, φ ) du + K r ( r, u, φ ) dr + K φ ( r, u, φ ) dφ (102)linked by the two integrability conditions : ∂ r K u ( r, u, φ ) − ∂ u K r ( r, u, φ ) = 0 , and ∂ r K φ ( r, u, φ ) − ∂ φ K r ( r, u, φ ) = 0(103)The asymptotic symmetry algebra is thus expressed in terms of triples s = ( ξ, (cid:96), Λ) of vectors ξ of the form eq. (99), function λ of two variablesand one-form obeying the conditions eq. (103).The algebra they satisfy is given, in terms of usual Lie derivatives L by:[( ξ , (cid:96) , Λ ) , ( ξ , (cid:96) , Λ )] = ( ξ, (cid:96), Λ) with (104) ξ = L ξ ξ − L ξ ξ , (cid:96) = L ξ (cid:96) − L ξ (cid:96) , Λ = L ξ Λ − L ξ Λ + k g (cid:16) (cid:96) d(cid:96) − (cid:96) d(cid:96) (cid:17) . (105)In particular the asymptotic Killing vector fields realise the semidirect prod-uct of two Wit algebras.But this is not the end of the computation of asymptotic reducibility pa-rameters. To simplify the analysis let us fix the background by imposing theextra conditions eqs (50). To obtain conserved asymptotic charges, remem-bering that we are considering polyhomogeneous asymptotic expansions, wealso have to require that : ∂ α (cid:0) √− g k α τ (cid:1) = O (cid:16) ln[ ρ/L ] n τ ρ (cid:17) , (106a) ∂ α (cid:0) √− g k α ρ (cid:1) = O (cid:16) ln[ ρ/L ] n ρ ρ (cid:17) , (106b) ∂ α (cid:0) √− g k α θ (cid:1) = O (cid:16) ln[ ρ/L ] n θ ρ (cid:17) , (106c)where n τ , n ρ and n θ are all three non-negative integers. These conditionsinsure both that the elementary charges defined by the integral (164) areindependent of the shapes of the loops on which the various integrals are Non conserved asymptotic charges and their algebra are considered in refs [10, 19, 4];see also appendix (C). ξ τ = X h , ξ θ = Y j , λ = (cid:96) , Λ = Λ θ dθ (107)constitute asymptotic reducibility parameters. Indeed we obtain that forthe stationary black string configuration eqs (7a, 7b) the relevant parts ofthe components of the divergence of the charge density are given by : ∂ α (cid:0) √− g k α τ (cid:1) =0 , (108a) ∂ α (cid:0) √− g k α ρ (cid:1) = 116 π G (cid:18) ρ (cid:16)(cid:0) Q Y (cid:48)(cid:48)(cid:48) ( τ, θ ) − ˙ Y (cid:48) ( τ, θ ) (cid:1) δQQ + k g Q (cid:0) ˙ (cid:96) (cid:48) ( τ, θ ) + c θ ˙ Y (cid:48) ( τ, θ ) (cid:1) δc θ (cid:17) + 8 G (cid:16) ˙ X ( τ ) δ E − ˙ Y ( τ, θ ) δJ + L ˙ (cid:96) ( τ, θ ) δ Q Max − L ˙ K θ ( τ, θ ) δ Q KR (cid:17) + (cid:112) k g L Q Y (cid:48)(cid:48) ( τ, θ ) (cid:32) (cid:115) r + r − L − ω Q δc θ − c θ δ (cid:32)(cid:115) r + r − L − ω Q (cid:33)(cid:33) − (cid:112) k g L Q (cid:96) (cid:48)(cid:48) ( τ, θ ) δ (cid:32)(cid:115) r + r − L − ω Q (cid:33)(cid:33) + O [1 /r ] , (108b) ∂ α (cid:0) √− g k α θ (cid:1) = 116 π G (cid:18)(cid:0) ˙ Y ( τ, θ ) − Q Y (cid:48)(cid:48) ( τ, θ ) (cid:1) δQQ − k g Q (cid:0) ˙ (cid:96) ( τ, θ ) + c θ ˙ Y ( τ, θ ) (cid:1) δc θ − Lρ ˙ X ( τ ) δω (cid:19) . (108c)However, if we restrict the phase space by fixing Q and c θ we obtain a largerset of reducibility parameters : ξ τ = X h , ξ θ = Y [ τ, θ ] ( arbitrary ) , λ = − c θ Y [ τ, θ ] + (cid:88) n (cid:96) ( n ) e i n θ − Q n τ , Λ = (cid:18) − k g c θ Q Q Y [ τ, θ ] + Λ θ (cid:19) dθ . (109)But if we only impose c θ constant on the all phase space, the reducibilityparameter set becomes more restricted. Instead to have ξ θ arbitrary it mustbe taken constant : ξ θ = Y j constant to define conserved asymptotic charges.Let us now consider the conditions eqs (106a–106c) applied on the moregeneral asymptotic solutions obtained in section (3) we are led to restrict31he covariant phase space as follows. To satisfy the first of the conditionseq. (106a) we have to require : f (0 , ( τ, θ ) = f (0)(0 , , a θ (0 , ( τ, θ ) = a (0) θ (0 , + 4 τ a (0) θ (1 , , a (0) θ (1 , δc θ = 0 . (110)For instance, if the conditions (110) are not satisfyed, we obtain that, atfixed value of τ = τ (cid:63) , the integral of the surface charge density, on a circleof large radius ρ = r (cid:63) , behaves as : (cid:90) π k τ,ρ { X h ,Y j ,(cid:96), Λ θ } [ τ (cid:63) , r (cid:63) , θ ] dθ =4 π X h L Q k g δc θ a θ (1 , (cid:16) ln( r (cid:63) /L ) + 4 τ (cid:63) (cid:17) + τ (cid:63) and r (cid:63) independent terms . (111)The three conditions eq. (110), combined with eqs (63, 65, 67) implythat : δ (cid:16) Q (cid:0) ˙ V (0 , ( τ, θ ) + Q V (cid:48)(cid:48) (0 , ( τ, θ ) (cid:1)(cid:17) = 0 , (112)and similar relations for m (0 , ( τ, θ ) and n (cid:48) (0 , ( τ, θ ). Using them, we seethat to verify condition (106b), we also have to impose that : n (cid:48) (0 , ( τ, θ ) δQ = 0 , ˙ m (0 , ( τ, θ ) δc θ = 0 . (113)Then the third condition (106c) implies no more restriction.Thus in order to obtain conserved, well defined elementary charges, wehave, together with the first two conditions (110), four options to definecovariant phase spaces: to assume c θ fixed or a (0) θ (1 , = 0 and m (0 , ( τ, θ ) = m (0)(0 , and Q fixed or n (0 , ( τ, θ ) = n (0)(0 , . According to the choice, we obtainas elementary charges : δ E = 18 G (cid:16) δV (0)(0 , − f (0)(0 , Q δQ + 2 k g Q a (0) θ (0 , δc θ (cid:1) , (114) δJ = 18 G δ (cid:16) n (0)(0 , − k g c θ m (0)(0 , − ω c θ (cid:17) , (115) δ Q Max = 116
G k g δ (cid:16) m (0)(0 , − ω c θ (cid:17) , (116) δ Q KR = − G L δω . (117)In the above expressions we have to set δc θ = 0 if we choose the phase spacesector to have a (0)(1 , (cid:54) = 0. All these elementary charge remain integrable,32xcepted for the energy (the one linked to the asymptotically reducibilityparameter X h , corresponding to τ translation), unless we assume Q and c θ fixed, or more generally appropriate special Q and c θ dependences for the f (0)(0 , and a (0) θ (0 , integration constants. We have obtained a (formal) generalisation of our previous results of blackstring configurations, established in ref. [3], that describes a general so-lution of the Einstein-dilaton-abelian-Kalb-Ramond field equations. Thisdynamical system depends on two degrees of freedom. The solution we ob-tain depends on an infinite set of arbitrary functions of τ and θ that wemay reinterpret as Cauchy data expressed as series involving power of ρ andlog[ ρ/L ]. Surprisingly we have obtained a closed form solution of the prob-lem without isometry–gauge invariance : a solution describing a dynamicalblack string configuration. Nevertheless this solution exhibits a hidden sym-plectic symmetry that allows us to define for it charges which are similarto that of the stationary black string configuration. We also have shown,that restricting some of the arbitrary functions occurring in the expansionof the general asymptotic solution to constants, we may still define fourconserved elementary charges. Three of these charges are integrable. Onlythe energy requires some extra conditions to be defined. Actually, that theabelian and Kalb-Ramond charges always exist results from eqs (29, 34).Less expected, on the contrary to what is necessary for the asymptotic en-ergy, the asymptotic angular momentum didn’t require extra conditions tobe defined, .A few other points deserve attention.Firstly : the algebra of the asymptotic reducibility parameters may seemfrustrating: in general it coincides with the four-dimensional algebra ofisometry–gauge transformations preserving the stationary configuration.Secondly : the black string configuration is proved to be unstable, as conjec-tured by Horne and Horowitz on the basis of to the work of Chandrasekharand Hartle. Of course the link between these instabilities and their CFTinterpretation in terms of deformations by marginal operators of the under-lying sigma-model will require extra work.Thirdly : the subdominant terms occurring in the asymptotic expansionof the general solution have to be restricted in order to obtain conservedcharges or even only finite non conserved charges [see appendix (C) ]Fourthly : the asymptotic solutions are mainly determined by the two func-33ions eqs (35, 36). Their expansion in terms of inverse powers or r andpower of ln[ r/L ] involves arbitrary (complex) constants determining the co-efficients f ( q,p ) ( u, φ ) and a ( q,p ) ( u, φ ) and their time derivatives. Roughlyspeaking they corresponds to the initial data of these two degrees of freedomand thus, for each choice of them define inequivalent physical configurations. A Geometrical description of the asymptotic ge-ometry
On a four dimensional Minkowski space, with metric : dS = dU dV + dX + dY (118)we have to consider the half-cones with equation : U > , p U V − q ( X + Y ) = 0 , p + q = 14 p, q ∈ ]0 , [ , (119)parametrised by : U = q Q L ( ρ/L ) q − q e − q τ , V = q Q L ( ρ/L ) q +12 q e q τ , X = p Q ρ cos( θp Q ) , Y = p Q ρ sin( θp Q ) . (120)The p and q parameters have to satisfy the rationality condition ( T beingthe period of the cyclic coordinate θ , choosen to be 2 π in the main text): T π p Q ∈ Q , (121)in order for the cones given by eq. (119) to constitute finite coverings of themanifold we are looking for. The induced metric on this surface is given byeq. (8a). B Special solution
We start from the special field configuration ansatz :Φ = −
12 ln (cid:18)
Q ρL (cid:19) (122) A µ = { A τ ( τ, ρ, θ ) , , L c θ ) } (123)34ut the general metric components (in Bondi gauge) : g µν = g τ τ ( τ, ρ, θ ) g τ ρ ( τ, ρ, θ ) g τ θ ( τ, ρ, θ ) g τ ρ ( τ, ρ, θ ) 0 0 g τ θ ( τ, ρ, θ ) 0 ρ . (124)The H µνρ field components are given by eq. (15) : H τ ρ θ = L ω | g τ ρ ( τ, ρ, θ ) | r . (125)The Einstein equation E ρ ρ = 0 [see eqs 9] implies that : g τ ρ ( τ, ρ, θ ) = rL γ τ ρ ( τ, θ ) , (126)while E ρ θ = 0 provides : g τ θ ( τ, ρ, θ ) = L ρ (cid:16) γ (1) τ θ ( τ, θ ) + ( r/L ) (cid:104) γ (2) τ θ ( τ, θ ) + γ (cid:48) τ ρ ( τ, θ ) (cid:16) − ln ( r/L ) (cid:17)(cid:105)(cid:17) . (127)The abelian field equation J τ = 0 [see eqs (12)] leads to the expected radialdependence of the potential : A τ ( τ, ρ, θ ) = L ρ m ( τ, θ ) + L a τ (0 , ( τ θ ) . (128)The equation J θ = 0 involves three terms, that are respectively proportionalto ρ − , ρ − and ρ − . The first two imply that : a τ (0 , ( τ θ ) = λ ( τ ) , (129) γ τ ρ = − b ( τ ) , (130)where λ ( τ ) and b ( τ ) are arbitrary functions. Thanks to an appropriatechoice of gauge we may assume that λ ( τ ) = 0 and by redefining the τ variable that b ( τ ) = Q . With this choice of the τ variable the term of order ρ − in the equation J θ = 0 fixes γ (1) τ θ : γ (1) τ θ = ω . (131)Let us now turn to the dilaton-field equation (11). It implies that : g τ τ = L ρ γ τ τ ( τ, θ ) − k g L Q m ( τ, θ ) + ρ (cid:16) γ (2) τ θ ( τ, θ ) + γ (2) (cid:48) τ θ ( τ, θ ) − (cid:17) , (132)35n expression that the gravitational equation Eqg v r = 0 simplify as it re-quires that : γ (2) τ θ ( τ, θ ) = a ( τ ) =: ˙ A ( τ ) . (133)Here we have introduce a primitive A ( τ ) of the arbitrary function a ( v ).The last abelian field equation J ρ = 0 leads us to the evolution equation ofthe m ( τ, θ ) function :˙ m ( τ, θ ) = − Q m (cid:48)(cid:48) ( τ, θ ) + a ( τ ) m (cid:48) ( τ, θ ) (134)whose general solution reads m ( τ, θ ) = (cid:88) n m ( n )(0 , e i n ( A ( τ )+ θ )+ n Q τ with m ( − n )(0 , = ( m ( n )(0 , ) (cid:63) . (135)Similarly the last Einstein equation E τ τ = 0 leads to an equation for thefunction γ τ τ ( τ, θ ) :˙ γ τ τ ( τ, θ ) = − Q γ (cid:48)(cid:48) τ τ ( τ, θ ) + a ( τ ) γ (cid:48) τ τ ( τ, θ ) + 2 L ω ˙ a ( τ ) (136)whose solution reads : γ τ τ ( τ, θ ) =2 ω a ( τ ) + Q V (0 , ( τ, θ ) (137)with : V (0 , ( τ, θ ) = (cid:88) n V ( n )(0 , e i n ( A ( τ )+ θ )+ n Q τ and V ( − n )(0 , = ( V ( n )(0 , ) (cid:63) . (138)Then the remaining two gravitational equations are identically satisfied.To make an end let us notice that by redefining the angular variable as : θ + A ( τ ) (cid:55)→ θ (139)the metric simplifies into : g ( B ) µν = − Q ρ + L Q ρ V (0 , ( τ, θ ) − k g L Q m , ( τ, θ ) − Q ρ L ω ρ − Q ρ L ω ρ ρ (140)and, thanks to a residual gauge transformation, the abelian potential maybe written as : A µ = (cid:26) L ρ m (0 , ( τ, θ ) , , L c θ (cid:27) . (141)36 Surface charges
A covariantisation lemma
In this appendix we want to show that for second order diffeomorphisminvariant theories, the surface charge density build by ”cohomological meth-ods” (see for instance [14, 16]) even if not explicitly covariant actually iscovariant.Let us denote tensorial background fields as φ A and their variations as δφ A .The on-shell vanishing Noether current density can be expanded as : · s µ = (cid:88) n =0 · s µ, ( ν ) n ,B ∇ ( ν ) n δφ B = (cid:88) n =0 ˜ · s µ, ( ν ) n ,B ∂ ( ν ) n δφ B , (142)where all the coefficients · s µ, ( ν ) n ,B are tensor densities, depending on thebackground fields and the reducibility parameters. The indices in the mul-tisum are totally symmetrized. On-shell conservation ∂ µ · s µ = 0 implies : · s ( µ,ν ν ) ,B = 0 . (143)The two expansion coefficients occurring in eq. 142 are related by :˜ · s µ,ν ν ,B = · s µ,ν ν ,B , ˜ · s µ,ν,B = · s µ,ν,B − · s µ,ν ν ,C (Γ νν ν δ BC + 2Γ BC ( ν δ νν ) ) . (144)The not manifestly covariant expression for the surface charge density isgiven by : · k µν = ˜ · s [ µ,ν ] ,B δφ B − δφ B ∂ ρ ˜ · s [ µ,ν ] ρ,B + 43 ∂ ρ δφ B ˜ · s [ µ,ν ] ρ,B , (145)while the manifestly covariant one reads : · κ µν = · s [ µ,ν ] ,B δφ B − δφ B ∇ ρ · s [ µ,ν ] ρ,B + 43 ∇ ρ δφ B · s [ µ,ν ] ρ,B . (146)In order to check that on-shell · k µν = · κ µν , on one hand we expand · k µν using(144), and on the other hand we expand the covariant derivatives in · κ µν andcompare the two expressions.The terms proportional to Γ BCα are givenin · k µν by : − · s [ µ,ν ] ν ,C Γ BCν δφ B , (147a)in · κ µν by : (cid:18) − − (cid:19) δφ B ˜ · s [ µ,ν ] ρ,C Γ BCρ , (147b)37nd therefore are identical.The terms proportional to Γ ραβ are givenin · k µν by : − · s [ µ, | ν ν | ,B Γ ν ] ν ν δφ B , (148a)in · κ µν by : − δφ B Γ [ µρν · s | ν | ,ν ] ρ,B − δφ B Γ [ νρν · s µ ] ,ν ρ,B (148b)= + 23 δφ B Γ [ νρν · s | ν | ,µ ] ρ,B − δφ B Γ [ νρν · s µ ] ,ν ρ,B . (148c)Therefore, their difference is given by : δφ B Γ [ νν ν (cid:18) − · s µ ] ,ν ν ,B − · s | ν | ,µ ] ν ,B (cid:19) (149)= − δφ B Γ νν ν · s ( µ,ν ν ) ,B − ( µ ↔ ν ) (150)= 0 (151)in virtue of the conservation equation (143).Let us mention that this elementary proof does not straightforwardly gen-eralises to higher order theories. Explicit expressions
In this appendix we write an explicit covariant expression of the surfacecharge density used in section (4), based on the previous lemma. The dif-feomorphism and gauge invariant action given by eq. (1) defines the presym-plectic potential : Θ µ = ∇ α δg µ α − ∇ µ δg − ∇ µ Φ δ Φ − k g e − F µ α δA α − e − H µ α β (cid:0) δB αβ + k g A α δA β ) , (152)where δg µν , δA α and δB αβ are arbitrary variation of the various fields( δg αβ := g α µ g β,ν δg µν , δg =: g µν δg µν ). Following standard techniques (i.e.using the contracting homotopy operator; see for instance refs [14, 16]), weobtain the Nœther-Wald surface charge : Q µν = 116 π G (cid:16) ∇ µ ξ ν + ∇ ν ξ µ + k g e − F µν ( ξ α A α + λ )+ e − H µ ν α (cid:0) ξ β B β α − k g ξ β A β + 2 λ ) A α + Λ α (cid:1)(cid:17) , (153)38here { ζ } := { ξ µ , Λ α , λ } are gauge parameters. The surface charge density may be written as the sum of four contributions : · k µν { ζ } = √− g π G (cid:16) k µνE + k µνT + k µνH + k µνF (cid:17) . (154)A suffisant condition to be conserved ( ∂ µ · k µν { ζ } = 0) is that the backgroundconfiguration is left invariant with respect to the gauge transformation gen-erated by ξ µ , Λ α and λ (in which case they constitute a set of reducibilitygauge parameters) and the field variations δg µν , δA α and δB αβ are solutionsto the field equations linearised around this background.Defining D σµων := g σω g µν − g σν g µω , the general covariant expression of thesurface charge is given by the sum of : • Einstein tensor contribution k µνE = 2 (cid:16) ξ [ ν ∇ µ ] δg − ∇ ρ δg ρ [ µ ξ ν ] + ξ ρ ∇ [ ν δg µ ] ρ (cid:17) − (cid:16) ∇ [ µ ξ ν ] δg − ∇ ρ ξ [ ν δg µ ] ρ + δg [ µρ ∇ ν ] ξ ρ (cid:17) (155) • Energy-momentum tensor contribution k µνT = 8 ( ξ ν ∇ µ Φ − ξ µ ∇ ν Φ) δf − e − ξ [ µ η ν ] ρσ (cid:63) H (cid:18) δB ρσ + k g A ρ δA σ (cid:19) + k g e − (cid:16) ξ σ F σ [ µ g ν ] ρ − ξ [ ρ F µν ] (cid:17) δA ρ (156)Here (cid:63)H := η αβγ H αβγ and the Schouten identity is used to simplifythe second term. • Scalar field contribution
As there is no derivative of the gauge parameter in case of scalar fields,these fields only contribute via the other field equations We adopt here a commonly used terminology. Actually, what we are considering hereis an ”elementary surface charge density”, an object that when integrated will provide,in principle, a one-form on the phase space that still must be exact in order to allow todefine true charges by an integration along a path, starting from a configuration wherethe values of the charges are prescribed by convention. ( λ := ξ ρ A ρ + λ ) k µνF = k g (cid:26) λ e − D ρσνµ ∇ ρ δA σ − D ρσνµ ∇ ρ ( λ e − ) δA σ + λ e − (cid:20)(cid:18) g ρ [ ν F µ ] σ − F µν g ρσ (cid:19) δg ρσ + 2 D µνσρ ( ∇ σ Φ) δA ρ + 4 F µν δf (cid:21) + λ e − (cid:20) g ρ [ ν F µ ] σ δB ρσ + H µνρ δA ρ + k g g ρ [ µ F ν ] σ δA [ ρ A σ ] (cid:21)(cid:27) (157) • (Λ α := ξ ρ B ρα − k g ( ξ ρ A ρ + 2 λ ) A α + Λ α ) k µνH = e − (cid:26) −
13 Λ β (cid:16) η µβ ( ν η ρ ) σω − η νβ ( µ η ρ ) σω (cid:17) (cid:18) ∇ ρ δB σω − k g A ω ∇ ρ δA σ (cid:19) + 4 ∇ ω Φ Λ β η βω [ µ η ν ] ρσ (cid:18) δB ρσ − k g A σ δA ρ (cid:19) + k g β ∇ σ A ω (cid:16) η µβ [ ν η σ ] ωρ − η νβ [ µ η σ ] ωρ (cid:17) δA ρ + 12 Λ ω H µνω hδg + 8 Λ β H µνβ δf (cid:27) + 16 ∇ ρ (cid:104) e − Λ β (cid:16) η µβ ( ν η ρ ) σω − η νβ ( µ η ρ ) σω (cid:17)(cid:105) δB σω − k g ∇ ρ (cid:104) e − Λ β A ω (cid:16) η µβ ( ν η ρ ) σω − η νβ ( µ η ρ ) σω (cid:17)(cid:105) δA σ (158) Non conserved charges
The definition of physically relevant charges in the framework of gravityis a subtle question. Inserting the expressions of the asymptotic solutionsobtained in sect. (3) and asymptotic symmetries (99, 101) in the previousexpression of the surface charge density, will in general leads to ill defined(divergent) expressions. Nevertheless we may a priori restrict the phasespace by conditions insuring that the charges so obtained are finite, even ifthey remain in general dependent on the curves on which the charge densityis integrated and thus, in particular, time dependent. The relevance of suchtime dependent charge is particularly well illustrated by the Bondi massformula that describes the conversion of mass into gravitational radiation(seefor instance [20, 21, 22]).Expanding the charge density (165) in terms of the radial coordinate ρ we40btain : (cid:63)k = (cid:0) ρ k τ (0 , − ( τ, θ ) + ln [ ρ/L ] k τ (2 , ( τ, θ ) + ln[ ρ/L ] k τ (1 , ( τ, θ ) (cid:1) dτ + (cid:0) ln [ ρ/L ] k θ (2 , ( τ, θ ) + ln[ ρ/L ] k θ (1 , ( τ, θ ) (cid:1) dθ + 1 ρ (cid:0) ln[ ρ/L ] k ρ (1 , ( τ, θ ) + k ρ (0 , ( τ, θ ) (cid:1) dρ + O [1] . (159)This charge density is defined up to an exact one-form dF ( τ, ρ, θ ). Usingthis freedom we obtain the following conditions of finiteness. • To avoid linear divergences :16 π G k τ (0 , − ( τ, θ ) = (cid:16) ˙ Y ( τ, θ ) − Q Y (cid:48)(cid:48) ( τ, θ ) (cid:17) δQQ − k g Q (cid:16) ˙ (cid:96) ( τ, θ ) + c θ ˙ Y ( τ, θ ) (cid:17) δc θ =0 (160a)and some squared logarithmic ones :16 π G (cid:0) k θ (2 , ( τ, θ ) − ∂ θ k ρ (1 , ( τ, θ ) (cid:1) = − (cid:16) (cid:96) (cid:48) ( τ, θ ) + c θ Y (cid:48) ( τ, θ ) (cid:17) δ (cid:16) Q a (0) θ (1 , (cid:17) = 0 . (160b)From these two conditions we infer that : Y ( τ, θ ) = (cid:88) n Y ( n ) e i n θ − n Q τ , (cid:96) ( τ, θ ) = − c θ Y ( τ, θ ) + λ c . (161)Let us emphasise the asymptotic behaviour of Y ( τ, θ ) that goes toa constant when τ goes to infinity, on the contrary to the varioussolutions of the heat equation met in the main text. • There remains a squared logarithmic divergent term. Inserting in itsexpression the previous conditions, we obtain :16 π G (cid:0) k τ (2 , ( τ, θ ) − ∂ τ k ρ (1 , ( τ, θ ) (cid:1) = − L ˙ X ( τ ) δ (cid:16) Q f (cid:48) (0 , ( τ, θ ) (cid:17) − L X ( τ ) δ (cid:16) Q f (cid:48)(cid:48)(cid:48) (0 , ( τ, θ ) (cid:17) =0 (162)41hich imposes for being satisfied that we restrict f (0 , ( τ, θ ) to be aconstant f (0)(0 , and as a consequence the constant f (0)(1 , to be zero (seeeqs (52b, 52a). • Finally, to avoid logarithmic divergences two more conditions have tobe satisfied : π G (cid:16) k θ (1 , ( τ, θ ) − ∂ θ k ρ (0 , ( τ, θ ) (cid:17) = k g L Q (cid:16) X ( τ ) (cid:0) a (0)(1 , ( τ, θ ) − Q a (cid:48)(cid:48) (0 , ( τ, θ ) (cid:1) − (cid:0) Y (cid:48) ( τ, θ ) a (0)(1 , − Y ( τ, θ ) a (cid:48) (0 , ( τ, θ ) (cid:1)(cid:17) δc θ − LQ Y (cid:48) ( τ, θ ) δ (cid:16) Q f (0)(0 , (cid:17) =0 (163a)16 π G (cid:16) k τ (1 , ( τ, θ ) − ∂ τ k ρ (0 , ( τ, θ ) (cid:17) = L ˙ X ( τ ) δ (cid:16) Q n (0 , ( τ, θ ) (cid:17) − L Q Y (cid:48)(cid:48) ( τ, θ ) δ (cid:16) Q f (0)(0 , (cid:17) + 14 L (cid:16) X ( τ ) n (cid:48)(cid:48) (0 , ( τ, θ ) − k g a (cid:48)(cid:48) (0 , ( τ, θ ) λ c (cid:17) δ (cid:16) Q (cid:17) − k g L Q (cid:16)(cid:0) Y ( τ, θ ) a (cid:48) (0 , ( τ, θ ) (cid:1) (cid:48) + (cid:0) X ( τ ) a (cid:48) (0 , ( τ, θ ) (cid:1) ˙+ Y (cid:48)(cid:48) ( τ, θ ) a (0)(1 , (cid:17) δc θ = 0 (163b) We shall not pursue the discussion of the various restrictions that maybe imposed on the phase space to satisfy these conditions. The importantpoint to notice is that subdominant terms of the asymptotic expansionscontribute to the divergent part of the asymptotic charges.
A comment about asymptotic charges and diffeomorphisms
Practically, on 3-dimensional spaces as such considered in the main text, theelementary charges at infinity are given by integrals on closed loops locatedat infinity : ¯ δ Q = (cid:90) C (cid:63)k µ dx µ (164)of a one-form (cid:63)k = k µ dx µ whose differential goes to zero as well as itsintegral on two-dimensional surfaces (denoted S ) located near infinity. Hereby infinity, we refer to the region parametrised by a radial variable ( ρ or r )42hat grows without limit.More precisely, on the τ, ρ, θ patch the one form (cid:63)k reads (cid:63)k = k τ ( τ, ρ, θ ) dτ + k ρ ( τ, ρ, θ ) dρ + k θ ( τ, ρ, θ ) dθ . (165)Let us consider the loops given by the circles : C [ τ , ρ ] = { τ = τ , ρ = ρ , θ ∈ [0 , π ] } . (166)Elementary charges are given by the integrals :¯ δ Q = lim ρ →∞ (cid:90) C [ τ , ρ ] (cid:63)k µ dx µ = lim ρ →∞ (cid:90) π k θ ( τ , ρ , θ ) dθ . (167)Performing a diffeomorphism like those considered in the main text [eqs (46,48)] : τ = T ( u ) = (cid:90) u e b υ ) dυ , ρ = r/ ( Q ∂ φ F ( u, φ )) = r e − f (0 , ( u, φ ) /Q ,θ = Q F ( u, φ ) = Q (cid:16) (cid:90) φ e f (0 , ( u, ξ ) dξ + h ( u ) (cid:17) , (168)the one-form (cid:63)k transforms covariantly and reads : (cid:63)k = k u ( u, r, φ ) du + k r ( u, r, φ ) dr + k φ ( u, r, φ ) dφ . (169)On the new patch so defined, let us assume that elementary charges ¯ δ ˜ Q are obtained from integrations on the circles :˜ C [ u , r ] = { u = u , r = r , φ ∈ [0 , π ] } (170)in the limit r → ∞ :¯ δ ˜ Q = lim r →∞ (cid:90) π k φ ( u , r , φ ) dφ (171)= lim r →∞ (cid:90) π (˜ k θ e f (0 , ( u , φ ) − ˜ k ρ r e − f (0 , ( u , φ ) ∂ φ f (0 , ( u , , φ ) (cid:1) dφ (172)where ˜ k θ, ( resp. ρ ) ( u, r, φ ) = k θ, ( resp. ρ ) (cid:0) T ( u ) , rQ e − f (0 , ( u,φ ) , Q F ( u, φ ) (cid:1) . These circles does not coincide with the previous ones [eq.(166)] but nev-ertheless ¯ δ Q will be equal to ¯ δ ˜ Q if asymptotically d (cid:63) k and its integral43n asymptotic surfaces S such that their boundaries are ∂ S = ˜ C − C alsovanishes : lim r →∞ (cid:90) S d (cid:63) k = 0 implies that ¯ δ ˜ Q = ¯ δ Q . (173)Moreover in such case the elementary charges are conserved.However, we also have to take into account that the gauge parametersbeing covariantly transformed the two charges so obtained, while numericallyequal didn’t refer to the same physical quantities . Indeed let us supposethat we are considering, using coordinates { x α } := { τ, ρ, θ } , an elementarycharge with respect to the vector field : ξ α = { X ( τ ) , − ρ ∂ θ Y ( τ, θ ) , Y ( τ, θ ) } . (174)Once having performed the coordinate change defined by eqs (168), thatmaps { x α } on { y β } := { u, r, φ } , the charge given by eq. (171) refers toanother configuration, physically distinct, of the system (the previous onetransformed by a large diffeomorphism) but with respect the another vectorof components ζ β obtained by acting with the Jacobian ∂y β /∂x α on ξ α : ζ u = 1 /∂ u T ( u ) X ( T ( u )) , . . . ,ζ φ = ( e − f (0 , ( u, φ ) /Q ) ξ θ − ( e − f (0 , ( u, φ ) /∂ u T ( u )) ∂ φ F ( u, φ ) ξ τ . Thus the obtainment, for a black string configuration with an arbitraryasymptotic expression f ( u, φ ) of the dilaton field, of the charge associatedto an asymptotic Killing vector ξ such that ξ φ = e i n φ require the knowledgeof the inverse of the transformation eqs (168). A task that in general isdifficult to do analytically.But as in the context of the field configurations discussed in the main textthe charges provided by general ξ α (174) is not conserved, and becomesdependent (in order to be finite) on the choice of the asymptotic curves onwhich the density is integrated, we shall not pursue their discussion. Acknowledgments
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