Near Horizon Extremal Geometry Perturbations: Dynamical Field Perturbations vs. Parametric Variations
aa r X i v : . [ h e p - t h ] J u l IPM/P-2014/028July 31, 2018
Near Horizon Extremal Geometry Perturbations:
Dynamical Field Perturbations vs. Parametric Variations
K. Hajian † , ∗ , A. Seraj † , M. M. Sheikh-Jabbari † † School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O.Box19395-5531, Tehran, Iran ∗ Department of Physics, Sharif University of Technology,P. O. Box 11155-9161, Tehran, Iran
Abstract
In [1] we formulated and derived the three universal laws governing Near HorizonExtremal Geometries (NHEG). In this work we focus on the Entropy Perturbation Law(EPL) which, similarly to the first law of black hole thermodynamics, relates perturba-tions of the charges labeling perturbations around a given NHEG to the correspondingentropy perturbation. We show that field perturbations governed by the linearizedequations of motion and symmetry conditions which we carefully specify, satisfy theEPL. We also show that these perturbations are limited to those coming from differ-ence of two NHEG solutions (i.e. variations on the NHEG solution parameter space).Our analysis and discussions shed light on the “no-dynamics” statements of [2, 3]. [email protected] ali [email protected] [email protected] ontents δ Φ . . . . . . . . . . . . . . . . . . . . . 73.2 Further comments on entropy perturbation law for δ Φ . . . . . . . . . . . . . 11
It is well known that black holes obey laws of thermodynamics. A symmetry based covariantapproach to derivation of the laws of black holes mechanics was introduced in [4, 5]. In thisapproach, entropy and other extensive thermodynamic parameters of a black hole are shownto be the Noether-Wald conserved charges associated with the symmetries of the black holesolution. Specifically, entropy is the conserved charge corresponding to the generator ofblack hole horizon, which is a Killing vector constructed from the Killing symmetries of thegeometry and becomes null at the horizon. This relation between the symmetries, leads1o a relation between perturbations in the conserved charges, the “first law of black holethermodynamics”.The Noether conserved-charge based approach has two remarkable features: 1) It gives auniversal proof of the first law of black holes in any generally covariant theory of gravity inany dimension. 2) It provides a different interpretation and meaning to the first law of blackholes than was initially proposed in [6], where the perturbations/variations appearing in thefirst law are viewed as perturbations/variations in the parameters space of family of black holesolutions. In the Noether-Wald approach [5], however, the charge variations in the first laware attributed to generic perturbations (probes) on a given black hole background. Derivationin [4, 5] asserts that the perturbations, probe fields, which satisfy linearized equations ofmotion on the background black hole geometry with appropriate boundary conditions, arein thermal equilibrium with the thermal bath of background black hole geometry, whichspecifies the non-extensive quantities (like temperature and chemical potentials) appearingin the first law. In particular, one can associate entropy to these probes (as well as to theblack hole background [4]).A crucial assumption in the Wald’s approach to the first law is that it can only be appliedto geometries with a Killing horizon, this assumption is generically fulfilled by stationaryblack holes. Moreover, it requires the Killing horizon to be a bifurcate horizon, i.e. the blackhole should necessarily have a non-vanishing temperature. The existence of bifurcate horizonis required, as entropy (and its perturbations) are defined as integrals over the codimensiontwo bifurcation surface and the corresponding Killing vector is normalized by surface gravityof the black hole. The question which then arises naturally is the existence of a relationbetween conserved charges and their perturbations/variations, of extremal black holes whichhave zero surface gravity (Hawking temperature) and no bifurcation horizon.Even if one assumes that the general form of first law is valid for extremal black holes(e.g. using the physical expectation that the first law should be continuous in its parametersand in particular temperature), the first law at zero temperature reduces to a manifestationof extremality (BPS) relation and does not determine the entropy perturbations in terms ofother charge perturbations, simply because perturbation of the entropy is not present due tothe vanishing of temperature. Through a careful analysis of vanishing temperature limit ofthe first law together with generic properties of near extremal black holes the “entropy per-turbation law” for extremal black holes was obtained which relates variations/perturbationsof the entropy for extremal black holes to perturbations/variations of its other charges [7].A more concrete derivation of the Entropy Perturbation Law (EPL) for extremal blackholes was presented in [1], carrying out steps similar to Wald’s derivation [5] for the NearHorizon Extremal Geometries (NHEG). In [1], we focused on the NHEG as family of solu-tions to gravity theories (independently of extremal black holes) and showed that despite theabsence of Killing or event bifurcate horizon, one can still define an entropy as a conservedNoether-Wald charge of this space through integration of the appropriate entropy density2wo-form over a codimension two surface which can be unambiguously defined using theSL(2 , R ) isometry of the NHEG background. Using this approach we derived the “entropylaw” which is a universal relation between the entropy and other conserved Noether-Waldcharges associated with the NHEG. The entropy law is specific to NHEG and has no coun-terpart in the usual black hole mechanics.As mentioned above, in [1] we also derived an entropy perturbation law for NHEG. Whilein the derivation of entropy law we could completely rely on the SL(2 , R ) invariance of theNHEG background for defining the two-form conserved charge densities and the integrationsurface, the perturbations which satisfy EPL are not generically SL(2 , R ) invariant andthis may introduce a dependence on the integration surface for the charge perturbationsappearing in the EPL. In this paper we revisit the derivation of EPL, paying special attentionto this feature and show that one can conveniently derive the EPL which is independent ofthe surface of integration defining the charges, if we restrict the field perturbations to respecta part of SL(2 , R ) invariance of the background. We will argue that this restriction is verywell justified when we consider the extremal black hole leading to the NHEG in question inits near horizon limit.We then study which field perturbations satisfy the conditions required in the deriva-tion of EPL (these conditions are linearized field equations and invariance under the twodimensional subgroup of SL(2 , R )). Adding appropriate/necessary “boundary conditions”to these two conditions we show that these perturbations are uniquely determined by theircharges and can only be the perturbations which relate to nearby NHEG solutions in theparameters space of NHEG solutions. Our analysis here provides a new viewpoint on, aswell as an extension of, the results of [2, 3] where a “no dynamics” theorem in near horizonextremal Kerr (NHEK) geometry was presented. Our uniqueness theorem opens a new wayof studying boundary gravitons and the possibility of identification of microstates giving riseto extremal black hole or the corresponding NHEG entropy.Organization of this paper is as follows. In section 2, we will give a brief review ofNHEG geometry and its universal laws. In section 3, we summarize the conditions definingNHEG dynamical field perturbations and conditions for the entropy perturbation law beindependent of surface of integration over which the charge perturbations are defined. Insection 4, we show that field perturbations which correspond to the difference of two NHEGsolutions satisfy the conditions defining dynamical field perturbations discussed in section3, and that these perturbations satisfy the EPL. In section 5, we present the NHEG pertur-bations uniqueness theorem: The only field perturbations which satisfy the three conditionsdefining dynamical field perturbations outlined in section 3, are those discussed in section 4which correspond to the variations in the family of NHEG solutions. In the last section wesummarize our results and make concluding remarks. In three appendices we have gatheredsome more technical details of the computations.3 Review of NHEG’s and three laws of NHEG mechanics
Near Horizon Extremal Geometries (NHEG) are a generic family of solutions to (Einstein-Maxwell-Dilaton, EMD for short) gravity theory. As their name suggests, they have been firstobtained and studied in connection with extremal black holes and their near horizon limit[8, 9, 10, 11, 12, 13]. Given the metric of a stationary extremal black hole in d dimensions,with n axisymmetric coordinates and N − n U(1) gauge fields (producing U(1) N symmetry)and arbitrary numbers of dilaton fields, one can apply the near horizon limit which is aspecific coordinate transformation associated with near horizon expansion, accompanied byan appropriate scaling and limit, to obtain the NHEG.One can present NHEG metric by coordinates in which the SL(2 , R ) × U(1) n symmetry ismanifest: ds = Γ " − r dt + dr r + d − n − X α,β =1 Θ αβ dθ α dθ β + n X i,j =1 γ ij ( dϕ i + k i rdt )( dϕ j + k j rdt ) , (2.1)and a set of gauge fields A ( p ) A ( p ) = n X i =1 f ( p ) i ( dϕ i + k i rdt ) + e p rdt , (2.2)and dilatons: φ I = φ I ( θ α ) , (2.3)where i, j = 1 , · · · , n ( n ≤ d − p = n + 1 , · · · , N , and I counts arbitrary number ofdilatons. Γ , Θ αβ , γ ij , f ( p ) i are functions of the polar coordinates θ α whose explicit form canbe fixed using equations of motion. The constant t, r surfaces in the metric (2.1), which arespanned by θ α , ϕ i are chosen to be smooth and compact (finite volume) d − g µν , gauge fields A ( p ) and dilatons φ I will be collectively denoted by Φ.NHEG’s have some generic features [1, 13]: • They are solutions to the equations of motion of the same theory as the original ex-tremal black holes were and hence establish a new independent family of solutions.Unlike the original extremal black hole, the NHEG is not asymptotic to a maximallysymmetric geometry and also has not an event horizon. • NHEG’s have time-like Killing vector field, and may hence be regarded as a stationarygeometry. This time-like Killing vector field, however, is not generically or necessarilyglobally defined [14]. • They have an AdS factor and accordingly an SL(2 , R ) symmetry.4 It inherits the U(1) N symmetry from the extremal black hole; the NHEG solution hasthen SL(2 , R ) × U(1) N symmetry. • In the above coordinates, Killing vectors generating the SL(2 , R ) × U(1) n symmetry are: ξ = ∂ t ,ξ = t∂ t − r∂ r , (2.4) ξ = 12 ( t + 1 r ) ∂ t − tr∂ r − n X i =1 k i r ∂ ϕ i ,m i = ∂ ϕ i , (2.5)with the commutation relations:[ ξ , ξ ] = ξ , [ ξ , ξ ] = ξ , [ ξ , ξ ] = ξ , (2.6)[ ξ a , m i ] = 0 , a ∈ { , , } and , i ∈ { , . . . , n } . (2.7) • For n = d − uniquely determined by N conserved charges associated with U(1) N symmetry. That is, n angular momenta J i , i ∈ { , , . . . , n } and N − n electriccharges q p , p ∈ { n + 1 , . . . , N } . There could also be N − n magnetic charges, whichare generically topological, and not Noether charges and hence do not directly appearin our analysis and their presence will not change our results. • In NHEG’s, there are two independent vector fields which are null on the whole geom-etry. In Poincar´e coordinates (2.1), they are ℓ µ = ( 1 r , , , − k i r ) ,n µ = r
2Γ ( 1 r , − , , − k i r ) , (2.8)the normalization is chosen such that ℓ · n = −
1. Note that ℓ · ∇ ℓ µ = 0 , n · ∇ n µ = − r Γ n µ . (2.9)This shows that ℓ, n are the generators of two null geodesic congruences ( ℓ , unlike n ,is affinely parameterized). Therefore, the near horizon geometry is a Petrov type Dspacetime. Moreover, these are null geodesics with vanishing expansion, rotation andshear, and hence NHEG is a Kundt spacetime [15]. • ℓ, n vector fields are normal to the vectors ∂ θ α and ∂ ϕ i . Therefore, the binormal toconstant t, r surfaces H is ǫ µν = ℓ [ µ n ν ] . (2.10)The normalization ℓ · n = − ǫ µν ǫ µν = − , R ) × U(1) N symmetry. The motivation for this proposal is twofold: 1) It is widely believed that mi-crostates of an extremal black hole reside somewhere near its horizon, so studying NHEGmight open a new insight to the unresolved problem of microstates of black holes. 2) Itextends the thermodynamic behavior observed in black hole solutions of gravity theories toanother family of solutions which do not have event horizons.Although NHEG’s do not have any bifurcate Killing horizon or event horizon, constant t and r surfaces, defined at arbitrary t = t H and r = r H in (2.1), provide an infinite set of d − , R ) invariant [1]. The NHEG’shave a Killing vector ζ H ζ H = n aH ξ a − k i m i , (2.11)where n aH is the unit vector of the SL(2 , R ), n = − t r − r , n = tr , n = − r , (2.12)computed at t = t H , r = r H . One can readily check that ζ H vanishes at H . The NHEGentropy S can hence be defined as the conserved charge associated with ζ H , as is done inWald’s formulation for black holes [4].The three laws of NHEG mechanics (paralleling those of black hole mechanics [6]) are [1]1. Zeroth law:
The coefficients k i and e p are constant, i.e. independent of the coordi-nates θ α . Entropy law:
For any given NHEG there is always the following relation: S π = k i J i + e p q p − I H √− g L (2.13)in which L is the Lagrangian density of the theory and H H √− g L is calculated on an H surface defined at arbitrary r H and t H .3. Entropy perturbation law:
For the perturbations (probes) around a given NHEG(satisfying some “appropriate conditions”) we have: δS π = k i δJ i + e p δq p . (2.14)The main goal of the next section is introducing and justifying the “appropriate conditions”for the perturbations of dynamical fields around NHEG leading to the EPL. These conditionswill be used to specify these perturbations. The n a and n aH should not be confused with the null vector n µ defined in (2.8). We note that NHEG solution is not necessarily completely or uniquely specified in terms of the k i and e p . We will discuss this point further in section 4.3. NHEG dynamical field perturbations
In this section we will study perturbations over NHEG geometries and derive a relationbetween the charges associated to these perturbations. If we denote the background fieldconfiguration of an NHEG solution by Φ , we consider field perturbations around this back-ground δ Φ. This section provides a more precise and detailed definition of dynamical fieldperturbations δ Φ and derivation of the entropy perturbation law given in [1].
Definition 3.1.
Dynamical field perturbations δ Φ are defined with the following properties.That is, δ Φ (I) satisfy linearized field equations,(II) are stationary and symmetric under scaling, i.e L ξ a δ Φ = 0 , a = 1 , ,(III) and asymptotically respect the isometries of the background. Explicitly lim r →∞ L ξ a δ Φ = 0 , a = 1 , , and lim r →∞ L m i δ Φ = 0 , i = 1 , · · · , n . Proposition 3.1.
The charge perturbations corresponding to any field perturbations satis-fying conditions (I) and (II) satisfy the EPL relation: δS π = k i δJ i + e p δq p . (3.1)Proof of the above proposition will be given in section 3.2. However, before giving theproof, we discuss physical meaning and justification of the conditions enumerated above. δ ΦThe fact that field perturbations δ Φ should satisfy linearized equations of motion is needed forthe (on-shell) conservation of the corresponding Noether-Wald charge densities [1, 5]. Below,we will discuss requirement of symmetry of perturbations under transformations generatedby ξ , ξ , i.e. L ξ δ Φ = L ξ δ Φ = 0 and the asymptotic symmetry of the perturbations. ξ , ξ invariance of perturbations ξ is the generator of translations along the time direction of NHEG geometry t and ξ isthe generator of scaling t → t/k , r → kr , (3.2)in the NHEG metric (2.1). Moreover, recalling their Lie-bracket [ ξ , ξ ] = ξ , they forma maximal subgroup of the SL(2 , R ) isometry group. Below, we provide two argumentsfor requiring invariance of perturbations δ Φ under this subgroup. One is based on the7ear horizon limit procedure which relates the NHEG perturbations to perturbations of theassociated extremal black hole. The other one follows from the physical requirement thatthe EPL and all charge perturbations should be independent of the choice of the surface H ,and that any given point on the AdS part of the NHEG metric (2.1) can be mapped to apoint with given t = t H , r = r H by diffeomorphisms generated by ξ , ξ . Argument 1:
Perturbations of an extremal black hole which survive the near horizon limitand are well-behaved under the limit, give rise to perturbations on NHEG which are invariantunder ξ and ξ diffeomorphisms. To see the above consider an extremal black hole with the following metric ds = − ˜ f dτ + ˜ g ρρ dρ + ˜ g αβ dθ α dθ β + ˜ g ij ( dψ i − ω i dτ )( dψ j − ω j dτ ) . (3.3)It is well known that this geometry has a well defined near horizon limit, defined throughthe coordinate transformations (e.g. see [1] for more on the conventions and notations) ρ = r e (1 + λr ) , τ = αr e tλ , ϕ i = ψ i − Ω i τ , λ → , (3.4)where r e is the horizon radius, Ω i = ω i ( r = r e ), and α is an irrelevant constant which wecan ignore in the computations. Also we set r e = 1.Next, we perturb the extremal black hole geometry ¯ g µν by a metric perturbation ˜ h µν , thatis the metric for perturbed geometry is g µν = ¯ g µν + ˜ h µν . We are searching for perturbationswhich have a well defined near horizon limit. That is, we are looking for ˜ h µν with finite˜ h µν dx µ dx ν in the near horizon limit. For the ease of notation let us focus on the 4d case:˜ h µν dx µ dx ν = ˜ h ττ dτ + 2 dτ (˜ h τθ dθ + ˜ h τψ dψ + ˜ h τρ dρ )+ ˜ h ρρ dρ + 2 dρ (˜ h ρθ dθ + ˜ h ρψ dψ )+ ˜ h θθ dθ + 2˜ h θψ dθdψ + ˜ h ψψ dψ . (3.5)Using dψ = dϕ + Ω dτ and collecting powers of dτ = dtλ and dρ = λdr yields˜ h µν dx µ dx ν = dt λ (cid:16) ˜ h ττ + 2Ω˜ h τψ + Ω ˜ h ψψ (cid:17) + 2 dtλ (cid:16) λdr (˜ h τρ + Ω˜ h ρψ ) + dθ (˜ h τθ + Ω˜ h ψθ ) + dϕ (˜ h τψ + Ω˜ h ψψ ) (cid:17) + λ dr ˜ h ρρ + 2 λ dr (cid:16) ˜ h ρθ dθ + ˜ h ρψ dϕ (cid:17) + (cid:16) ˜ h θθ dθ + 2˜ h θψ dθdϕ + ˜ h ψψ dϕ (cid:17) . Therefore perturbation induced on the NHEG (which we denote by h µν ) is h tt = ˜ h ττ + Ω˜ h τψ + Ω ˜ h ψψ λ , h rr = λ ˜ h ρρ h tr = ˜ h τρ + Ω˜ h ρψ , h tθ = ˜ h τθ + Ω˜ h θψ λ , h tφ = ˜ h τψ + Ω˜ h ψψ λ (3.6) h θθ = ˜ h θθ , h ϕϕ = ˜ h ψψ , h rθ = λ ˜ h ρθ , h rϕ = λ ˜ h ρψ , h θϕ = ˜ h θψ . ν : ˜ h µν ∼ f ( θ ) e − i ( ντ − mψ ) ( ρ − r h ) x = f ( θ ) e i ( ν − Ω mλ ) t e imϕ ( λr ) x . (3.7)It is argued in [7, 16] that ν − Ω m ∼ λ and the λ dependence comes from the radialdependence of the modes. Therefore, we see that L ξ h µν ∼ λ → h µν in the λ → r dependence of the perturbations as: h µν = r r r /r /r /r , (3.8)in the ( t, r, θ, ϕ ) basis. Note that higher orders of r lead to terms with positive powers of λ in h µν so that they disappear in the λ → r lead to divergencein h µν which is excluded. Therefore, (3.8) gives the exact r -dependence of components (andnot just a leading large r behavior). One may readily check that this r -dependence is exactlydictated by the condition L ξ h µν = 0 (see also [9, 13]). Similar argument may be repeatedfor the gauge and dilaton fields with a similar conclusion. To summarize, ν − Ω m ∼ λ leads to L ξ h µν ∼ λ →
0, i.e. to time-independenceof NHEG perturbations h µν ; and the r -dependence of NHEG perturbations is fixed by the L ξ h µν = 0. Argument 2:
As discussed, there is an arbitrariness in the choice of the point t H , r H defining the surface H . It was shown in [1] that the entropy of NHEG, S , and its othercharges and hence the entropy law, are independent of the choice of H . It is hence expectedthe value of charge perturbations , too, to be independent of H . As we will show below,the necessary and sufficient condition for this requirement is L ξ δ Φ = L ξ δ Φ = 0 . We start our argument by recalling that [1] S π = − I H ǫ H E µναβ ǫ µν ǫ αβ , (3.9)in which E µναβ = δ L δR µναβ (3.10) See, however, [17]. It is instructive to note the similarity and the differences between (3.8) and the Kerr/CFT boundaryconditions [16].
9s a tensor built from the background fields, ǫ µν denotes components of the SL(2 , R ) -invarianttwo-form Γ dt ∧ dr , and ǫ H is the d − H , ǫ H = Vol(H) ǫ α ,...,α d − ( d x α ∧ · · · ∧ d x α d − ) , (3.11)where ǫ α ,...,α d − is the Levi-Civita symbol defined on surface H .Consider the entropy perturbation associated with dynamical field perturbations δ Φaround the NHEG background denoted by field configuration Φ : δS π (cid:12)(cid:12)(cid:12)(cid:12) H = − I H δ ( ǫ H E µναβ ǫ µν ǫ αβ ) δ Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ δ Φ . (3.12)Next, recall that any two arbitrary H surfaces (defined at different values of t H , r H ) arerelated by a diffeomorphism generated by ξ , ξ . H -independence of δS then means thatthe integrand should be invariant under such diffeomorphisms. That is, L ξ a δ ( ǫ H E µναβ ǫ µν ǫ αβ ) δ Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ δ Φ ! = δ ( ǫ H E µναβ ǫ µν ǫ αβ ) δ Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ L ξ a ( δ Φ) = 0 , a = 1 , , (3.13)where in the second equality we used the fact that background fields Φ are SL(2 , R ) invariant.The above clearly states that L ξ ( δ Φ) = L ξ ( δ Φ) = 0.The above reasoning can be readily used for any generic conserved charge of NHEG.Explicitly, consider Q (cid:12)(cid:12) H = H H ǫ H Q, then δ Q H caused by the dynamical field perturbationsaround the NHEG background Φ , will be H -independent only if the integrand ǫ H Q isinvariant under ξ , ξ diffeomorphisms, L ξ a ( ǫ H Q) = δ ( ǫ H Q) δ Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ L ξ a ( δ Φ) = 0 ⇐⇒ L ξ a ( δ Φ) = 0 , a = 1 , . (3.14) After discussing physical meaning of condition (II), we now discuss and justify condition(III) which plays the role of boundary conditions for perturbations. To this end, we firstnote that in order to find solutions to the e.o.m of a given theory, some boundary conditionsare usually needed. Boundary conditions can usually be expressed in terms of asymp-totic isometries/symmetries. For instance, one can replace the asymptotic flatness in 4dby requesting asymptotic Poincar´e symmetry. Expressing boundary conditions in terms ofthe symmetries/isometries has the advantage that they could be presented in a covariant,coordinate independent manner. It may happen that the symmetry requirements we impose on a solution are so restrictive that theyuniquely specify the solution, without the need for a separate boundary conditions. An example of suchcases is the special class of NHEG for which we have uniqueness theorems [13]. In these cases the solutionis uniquely determined by requesting SL(2 , R ) × U(1) N symmetry and smoothness of H surface.
10n the same spirit, to completely specify solutions to the linearized equations of motion(l.e.o.m) we need to impose boundary conditions on field perturbations. The most naturalchoice for this boundary conditions is to require the perturbations to respect the symmetriesof the NHEG background. This is basically what we have required in (III). As another argument for the boundary conditions for perturbations (III), we recall discus-sions of [2, 3], where it is shown that asymptotic SL(2 , R ) × U(1) N invariance is the linearized-stability conditions for linearized perturbations δ Φ. We will discuss further this requirementin the end of next subsection 3.2. δ ΦThe proof of entropy perturbation law under conditions spelled out in proposition 3.1 wasgiven in [1]. In Appendix A we have reviewed the arguments of [1]. As reviewed in theappendix, direct Noether-Wald analysis leads to δS π (cid:12)(cid:12)(cid:12)(cid:12) H = k i δJ i + e p δq p (cid:12)(cid:12)(cid:12)(cid:12) H + n aH δ E a , (3.15)where we have explicitly put the subscripts H for charges defined as integrals over surface H at r = r H and t = t H and the two charges J i and E a are defined as integrals over thespace-like surface at r = ∞ ( cf. appendix A). Here we discuss further implications of theconditions (II) and (III) and show how condition (II) can remove the apparent H -dependencein (3.15), and more importantly condition (III) yields δ E a = 0.In section 3.1.1 we showed that δS π is independent of surface H and hence we may dropsubscript H on δS term. As for the angular momentum perturbation δJ i , we recall itsdefinition (A.16), δJ i ≡ − I ∞ δ Q m i . Since pullback of m i · Θ vanishes over any constant t, r surface on NHEG, one can show that δJ i (cid:12)(cid:12) ∞ has the same value once the integral at r = ∞ is replaced by any arbitrary r = r H surface. δq p is also independent of surface H . To see this, let us recall definition of theelectric charge, q p = − I H ǫ H ǫ µν ∂ L ∂F ( p ) µν . (3.16)Due to the argument above (3.14) we deduce δq p is independent of surface H .So we can rewrite (3.15) as: δS π − k i δJ i − e p δq p = n aH δ E a . (3.17) Another covariant boundary condition, besides (III), is the “subleading fall-off boundary condition”(used for instance in the work of Brown-Henneaux [18]). For the NHEG one can show that it leads to trivialset of perturbations [14]. t H , r H , the RHS should also be r H and t H independent.Noting that there is no r H dependence in the δ E a (because it is calculated at infinity) andrecalling (2.12), we learn that different powers of r H should vanish separately. That is, δ E = 0 , t H δ E − δ E = 0 . (3.18)Upon the above conditions, the proof of EPL is complete. In other words, for EPL to holdwe need to require (3.18) and δ E and δ E need not vanish independently.We now show that δ E , δ E vanish separately if we consider condition (III), the asymptoticSL(2 , R ) × U(1) N invariance. To this end, we recall the fact that E a are defined as integralsat infinity, explicitly t H δ E − δ E = I ∞ ǫ H ( t H δE − δE ) = I ∞ ǫ H L ξ ( t H δE − δE ) = 0 , (3.19)where δE , δE are scalars composed of (Φ , δ Φ , ξ ) and (Φ , δ Φ , ξ ) respectively and arebilinear in ξ a and δ Φ. In the second equality above we have used i) (asymptotic) U (1) n symmetry of Φ and δ Φ, which implies δE , δE are independent of coordinates ϕ i ; ii) the explicit form of ξ and that it is independent of θ α and does not have any componentin direction of ∂ θ α ; iii) and L ξ ( t H δE − δE ) = ξ µ ∂ µ ( t H δE − δE ). This latter, uponexpansion in powers of r implies that L ξ does not change θ α dependence of the integrand.Recalling the SL(2 , R ) algebra, L ξ δE = 0 and L ξ δE = − δE asymptotically. Therefore,we learn that I ∞ ǫ H L ξ ( t H δE − δE ) = − I ∞ ǫ H t H δE = − t H δ E = 0 . (3.20)Since t H is an arbitrary number, we learn that δ E = 0 and hence, δ E a = 0 , ∀ a. (3.21)We have then shown how all the three conditions (I), (II) and (III) are essential for vanishingof δ E a = 0, while arriving at the EPL (3.1), where each and every term in the EPL is H -independent, does not require (III).Before closing this section we also comment that, as is known from the canonical formu-lation of general relativity, δ E a are generators of asymptotic gauge transformation x → x + ξ a through the Poisson bracket, under the assumption of integrability, conservation and finite-ness of charges [19]. If one assumes that a symplectic current exists such that these assump-tions are satisfied, then: [ δ E a , Φ] = L ξ a δ Φ (cid:12)(cid:12) r →∞ . (3.22)Therefore the condition δ E a = 0 is equivalent to the statement that ξ a , a = 1 , , asymptotic symmetries of dynamical field perturbations L ξ a δ Φ (cid:12)(cid:12) r →∞ = 0 , a = 1 , , . (3.23)12 NHEG parametric perturbations
In this section we consider a specific set of perturbations around a given NHEG which areproduced through moving in the parameter space of NHEG solutions. These perturbationswill hence be called parametric perturbations . An NHEG is specified by a set of conservedcharges, angular momenta J i and electric charges q p . One may hence denote an NHEGsolution by fields Φ { J i ,q p } ( x ). A parametric perturbation, denoted by ˆ δ Φ, is defined asˆ δ Φ ≡ ∂ Φ { J i ,q p } ∂J i δJ i + ∂ Φ { J i ,q p } ∂q p δq p . (4.1)We start our analysis of parametric perturbations ˆ δ Φ by showing that they indeed fulfillthe three conditions stated in the definition 3.1. • Linearized equations of motion. ˆ δ Φ is the difference between two adjacent solutionsof field equations, and the conserved charges J i and q p do not appear in the equations ofmotion. Therefore, one can readily deduce that ˆ δ Φ solves the linearized field equations. • ξ , ξ invariance. The Killing vectors ξ , ξ , and also m i , do not involve any parametersof the NHEG solution (like k i and e p ). Therefore, if ˆ δ Φ = Φ ′ − Φ , L ξ ˆ δ Φ = L ξ Φ ′ − L ξ Φ = 0 , ξ = { ξ , ξ , m i } . (4.2)So, parametric perturbations ˆ δ Φ are not only ξ , ξ invariant, but also m i invariant.We also note that parametric perturbations preserve the null vectors fields ℓ, n (2.8),i.e. ˆ δ ( ℓ ) = ˆ δ ( n ) = 0. Among other things, this also implies that parametric pertur-bations, too, preserve constant t, r surfaces H . • Asymptotic SL (2 , R ) × U (1) N invariance. The Killing vector ξ (2.4) involves k i andhence ˆ δ Φ are not in general invariant under SL(2 , R ) × U(1) N symmetry of the NHEG.Nonetheless, the k i dependence of ξ is such that ˆ δ Φ are asymptotically ξ invariant.To see this explicitly, let us denote the corresponding Killing vectors of two NHEGsolutions Φ , Φ ′ by ξ and ξ ′ . Therefore, L ξ Φ = L ξ ′ Φ ′ = 0 = ⇒ L ξ ˆ δ Φ = −L ˆ δξ Φ , (4.3)where ˆ δξ = − ˆ δk i r m i , and ξ is not the symmetry of ˆ δ Φ. However, since ˆ δξ = − ˆ δk i r m i ,one can see that L ˆ δξ Φ ∼ O (1 /r n ), n ≥
1, i.e. ξ is an asymptotic symmetry of ˆ δ Φ. Note that, as we will discuss in section 4.3, specification in terms of charges is more precise than the onein terms of “conjugate parameters” k i and e p . Note also that NHEG uniqueness theorems has been sofarproven for a subset of all NHEG’s [13] and there might be NHEG’s which are not uniquely specified by theirconserved charges.
13o complete the above argument we need to discuss the cases involving gauge fieldsseparately. For the gauge fields L ξ A ( p ) is not zero, it is a pure gauge transformation: L ξ A ( p ) = d (cid:18) e p r (cid:19) , L ξ ′ A ′ ( p ) = d (cid:18) e ′ p r (cid:19) , L ξ ˆ δA ( p ) = −L δξ A ( p ) − ˆ δe p r dr. Hence, gauge fields also exhibit asymptotic ξ , and hence SL(2 , R ) × U(1) N invariance. So far we have introduced two classes of field perturbations, “dynamical field perturba-tions” and “parametric field perturbations”. While dynamical field perturbations act onlyon dynamical fields (governed by field equations), parametric perturbations act both on dy-namical and nondynamical parameters of an NHEG solution. For example, dynamical fieldperturbations do not affect the Killing vectors of the background NHEG. Despite the factthat parametric perturbations fulfill the three conditions of definition 3.1, our derivationand proof of EPL, reviewed in appendix A and discussed in section 3, does not immediatelyextend over the parametric perturbations. This is due to the fact that in the derivationof EPL, we have assumed the perturbations do not affect the Killing vectors associated tothe background geometry. This was explicitly used in the derivation of EPL for dynamicalfield perturbations, cf. (A.2). We should hence revisit derivation of the EPL for parametricperturbations. This is the task of this subsection.Consider a dynamical perturbation δ Φ and a parametric perturbation ˆ δ Φ with the samedynamical content, i.e. ˆ δ Φ = δ Φ. As noted above, charge perturbations corresponding tothese perturbations can in principle be different. However, we will show below that thisis not the case. To investigate this, we note that parametric perturbation of the chargeassociated to a Killing ξ can be expressed asˆ δ Q ξ = I Q ξ ′ (Φ ′ ) − I Q ξ (Φ ) = δ Q ξ + Q ˆ δξ , (4.4)where δ Q ξ ≡ I Q ξ (Φ ′ ) − I Q ξ (Φ ) (4.5)is the charge perturbation associated with “dynamical field perturbations” used in section3, and in its definition, unlike ˆ δ Q ξ , we do not vary the Killing vector. Since ˆ δm i = 0, (4.4)implies that ˆ δJ i = δJ i . (4.6)Recalling the definition of electric charges q p , and that it does not involve non-dynamicalfields (such as a Killing vector) we readily haveˆ δq p = δq p . (4.7)14ext, we consider parametric variations of the entropy ˆ δS , which using (4.4) can be writtenas ˆ δS = δS + I H Q ˆ δζ H , (4.8)where ˆ δζ H = − ˆ δk i ( r H r − m i . (4.9)According to Wald’s decomposition theorem [5], one can write the Noether charge corre-sponding to any diffeomorphism ζ in the form Q ζ = I d Σ µν Q µνζ (4.10)where Q µνζ = W µνα ζ α − E µναβ ∇ α ζ β + Y µν + ( dZ ) µν . (4.11)In this equation, the last two terms are ambiguities in the definition of charge which arelinear in the generator ζ and, E µναβ is defined in (3.10). For the diffeomorphism ˆ δζ H , notingthe fact that ˆ δζ H (cid:12)(cid:12)(cid:12) H = 0, we have S ˆ δζ H = − I H d Σ µν E µναβ ∇ α (ˆ δζ H ) β = I H (cid:16) X αβ ∇ α (ˆ δζ H ) β (cid:17) ǫ H , (4.12) ǫ H is the d − H , and have defined, X αβ = − ǫ µν E µναβ , (4.13)which is an antisymmetric rank two tensor defined on the background fields, and has sym-metries of the background. It can be easily checked that any such tensor has the followingform X µν = F tr ( θ ) 0 0 − F tr ( θ ) 0 0 rF rϕ i ( θ )0 0 0 F θ α ϕ i ( θ )0 − rF rϕ i ( θ ) − F θ α ϕ i ( θ ) 0 (4.14)with arbitrary functions F which only depend on θ α and have the condition that X rϕ i = − k i rX rt . (4.15)15n the other hand, it can be checked that on the surface H we have H αβ ≡ ∇ [ α (ˆ δζ H ) β ] = H tr = Γ γ ij k i ˆ δk j H rϕ i = − Γ γ ij ˆ δk j r (4.16)and zero otherwise, with the property H rt = r X i k i H rϕ i . (4.17)Using (4.17) and (4.15) , we have S ˆ δζ H = I H (cid:16) X rt H rt + X rϕ i H rϕ i (cid:17) ǫ H = 0 , (4.18)and therefore (4.8) yields ˆ δS = δS .Finally, let us consider ˆ δ E a :ˆ δ E a = δ E a + I ∞ ( Q ρ a − ρ a · Θ ) (4.19)where ρ a ≡ ˆ δξ a . ρ = ρ = 0 , ρ = − ˆ δk i r m i . (4.20)It is clear that ρ a → r → ∞ and a similar argument like above implies that at r → ∞ I ∞ ( Q ρ a − ρ a · Θ ) = 0 , (4.21)so ˆ δ E a = δ E a . In brief, we have shown thatˆ δJ i = δJ i , ˆ δq p = δq p , ˆ δS = δS , ˆ δ E a = δ E a = 0 , (4.22)and consequently, ˆ δS π = k i ˆ δJ i + e p ˆ δq p . (4.23)That is, EPL also holds for parametric perturbations. One of the universal laws of NHEG’s is the entropy law , which relates entropy to other chargesof NHEG. Considering the background NHEG and its adjacent NHEG (call it NHEG ′ ), used16o define the parametric perturbation ˆ δ , each of these geometries has its own constraint fortheir parameters, imposed by the entropy law: S = k i J i + e p q p − I √− g L , (4.24) S ′ = k ′ i J ′ i + e ′ p q ′ p − I p − g ′ L ′ . (4.25)Subtracting the above leads toˆ δS = k i ˆ δJ i + e p ˆ δq p + ( J i ˆ δk i + q p ˆ δe p − ˆ δ I √− g L ) . (4.26)Using (4.23), J i ˆ δk i + q p ˆ δe p = ˆ δ I √− g L . (4.27)This relation is a consistency relation for the NHEG perturbations. One can indeed showthat (4.27), once viewed asˆ δ H √− g L ˆ δk i = J i , ˆ δ H √− g L ˆ δe p = q p , (4.28)is basically (a part of) the equations of motion for the NHEG background ansatz (2.1), asis also pointed out in Sen’s entropy function formalism [8]. As discussed in the opening of this section parametric perturbations, except the ξ invariance,keep the rest of SL(2 , R ) × U(1) N symmetry of the NHEG background. Here, we investigatethe question whether there are a subset of parametric perturbations (which will be denotedby ˆˆ δ ) preserving the full SL(2 , R ) × U(1) N symmetry. To answer this question we start notingthat ˆ δξ = − ˆ δk i r m i ≡ ρ , L ξ ˆ δ Φ = −L ρ Φ . (4.29)ˆˆ δ perturbations are hence those generated by ρ ’s such that L ρ Φ = 0. In particular, L ρ g µν = 0 = ⇒ γ ij ˆˆ δk j = 0 , or ˆˆ δk i = 0 ∀ i. (4.30)In the last relation we have used smoothness of metric and H surface and that γ ij is anon-degenerate matrix.The question is then whether ˆˆ δ family of perturbations are non-empty. To answer thisquestion, let us first consider NHEG solutions to pure gravity theory. Recalling the basicproperty of vacuum Einstein equations one can show that17 i = k i ( J j ) is a homogeneous function of order zero. This implies that k i ( J j ) = k i (cid:0) (1 + λ ) J j (cid:1) , and thereforeˆˆ δJ i = λJ i (4.31)is a direction which leaves k i invariant. The above dovetails with the fact that if metric g µν is an NHEG solution to d dimensional pure Einstein gravity with angular momenta J i andentropy S , κ g µν is a different NHEG solution with angular momenta κ d − J i and entropy κ d − S , but with the same set of k i . Note that NHEG are not asymptotically flat or (anti)-deSitter. The discussion above also implies that there are n − k i ’s for an NHEGwith n independent angular momenta.A similar argument can also be made for the NHEG solutions to d dimensional Einstein-Maxwell-Dilaton theory, where the equations of motion are invariant under g µν → κ g µν accompanied by A µ → κA µ . Upon this scalings an NHEG solution with parameters k i , e p goes to another NHEG with parameters k i , κe p (that is, ˆˆ δk i = 0 , ˆˆ δe p = 0), while the chargestransform as J i → κ d − J i , S → κ d − S, q p → κ d − q p . This section contains our main result which is stated in the following proposition:
Proposition 5.1.
Perturbations around any given NHEG solution to d dimensional EMDtheory with SL (2 , R ) × U (1) d − isometry, subject to the conditions of definition 3.1 and withgiven charge perturbations δJ i , δq p , are restricted to the NHEG parametric perturbations ˆ δ Φ .In other words, the only solution to the EPL subject to the three conditions of definition 3.1are parametric perturbations ˆ δ Φ . Note that the SL(2 , R ) × U(1) d − isometry condition has been imposed, because these arethe only NHEG backgrounds for which we have uniqueness theorems [13], and of coursethe above mentioned “NHEG perturbation uniqueness theorem” holds only when we have asimilar uniqueness at the background level. Idea of the proof.
In the previous section, we explicitly showed that parametric pertur-bations ˆ δ Φ satisfy the conditions in the definition 3.1, and therefore { ˆ δ Φ } ⊂ { δ Φ } . So, ourproof will be complete if we show that the converse is also true, i.e. { δ Φ } ⊂ { ˆ δ Φ } . Tothis end we first parameterize field perturbations and simplify them using the symmetryconditions we have assumed and then impose linearized equations of motion.We have given an alternative argument in Appendix C using the gauge invariant analysisof perturbations proposed first by Teukolsky [20].18 .1 Parameterizing field perturbations In the EMD theory we are interested in, there are metric, Maxwell gauge fields and dilatons.Here we discuss their perturbations separately. δg µν Requiring δg µν to have ξ and ξ symmetries fixes δg µν to the form (see analysis of appendixB) δg µν = r h tt h tr rh tθ α rh tϕ i h rr r h rθα r h rϕi r h θ α θ β h θ α ϕ i h ϕ i ϕ j (5.1)in which h µν = h µν ( θ α , ϕ i ). Discussions in the appendix B imply that requesting asymptoticSL(2 , R ) × U(1) n symmetry makes h tr = h rθ α = h rϕ i = 0 and, that the asymptotic U (1) n isometry is extended to the whole bulk, removing the ϕ i dependence of the remaining h ’s,except for h rr .Hereafter, we restrict to cases with U(1) d − isometry, i.e. to the cases where there isonly one θ -type coordinate. In these cases h θϕ i can be removed by the diffeomorphism ϕ i → ϕ i + f i ( θ ) and h θθ may be removed by the remaining diffeomorphism θ → θ + g ( θ ),and therefore, δg µν = r h tt rh tθ rh tϕ i h rr r h ϕ i ϕ j , (5.2)where h rr = h rr ( θ, ϕ i ) and h = h ( θ ) for all the other components. Therefore, imposing ξ , ξ and asymptotic SL(2 , R ) × U(1) N invariance, we remain with ( d − d − / h tθ , h tϕ i using θ, ϕ i diffeo-morphisms and remain with metric perturbations block diagonal in t, r and θ, ϕ i parts alongcodimention two surface H .) We also note that the parametric NHEG metric perturbationsˆ δh µν can be brought to the form (5.2) with h tθ = 0. δA ( p ) µ Let us denote the N − ( d −
3) gauge fields in Einstein-Maxwell theory by A ( p ) . Symmetryconditions of definition 3.1 for perturbations δA ( p ) then imply that ( cf. appendix B) δA ( p ) = ( rh ( p ) t , , h ( p ) θ , h ( p ) ϕ i ) (5.3)19n which h ( p ) ’s are only functions of θ . h ( p ) θ are simply removed by gauge transformations δA ( p ) → δA ( p ) + dΛ ( p ) ( θ ), so δA ( p ) can be chosen to be: δA ( p ) = ( rh ( p ) t , , , h ( p ) ϕ i ) , (5.4)which parameterize ( d − N − ( d − d − H . Moreover,(5.4) implies that r∂ θ δF tr = δF tθ , where F µν is the gauge field strength. This latter iscompatible with the parametric field strength perturbation which satisfy r∂ θ ˆ δF tr = ˆ δF tθ . δφ I Finally let us consider the dilaton field perturbations δφ I . Requesting (II) and (III) forvariations of these fields δφ I also fixes them via lemma in appendix B to be δφ I = δφ I ( θ ). In the cases with SL(2 , R ) × U(1) d − symmetry constant t, r H -surfaces are d − θ coordinate we find a d − H surface is ds H = Γ( θ ) (cid:2) dθ + γ ij ( θ ) dϕ i dϕ j (cid:3) , ϕ i ∈ [0 , π ] . (5.5)The smoothness condition then implies that Γ cannot have any zeros. If we denote the eigen-value of the matrix γ ij by γ i ( θ ), they should be such that (1) when one of these eigenvaluesvanish, the others remain finite, e.g. if γ i ( θ = 0) = 0, then γ j ( θ = 0) = 0 , j = i ; (2) Firstderivative of γ i should also vanish at θ = 0 but its second derivative should remain finite, ex-plicitly, around roots of γ i (assuming its located at θ = 0), γ i = θ + O ( θ ) , γ j = γ j (0) , j = i .Considering the whole geometry, the smoothness conditions in the basis where γ ij is diagonal,and around the root of ii component of metric (at θ = 0) take the form: g ϕ i ϕ i g θθ ∼ g ϕ i t g θθ ∼ θ , g ϕ j ϕ j g θθ ∼ g ϕ j t g θθ = finite j = i . (5.6)It may of course happen that γ i has more than one roots. Then, the above conditions shouldhold for all roots.The above smoothness condition was for the metric itself. It is then readily seen smooth-ness conditions (5.6) should also be extended to the metric perturbations allowed by oursymmetry requirements given in (5.2). To see this point it is enough to recall that metricperturbations should be relating two metrics g µν and g µν + δg µν , while these two metrics are20oth smooth. In particular, since we have adopted a gauge in which h θθ = h θϕ i = 0 then,smoothness implies h ϕ i ϕ j (cid:12)(cid:12) θ =0 ∼ θ , h tϕ i ∼ θ , ∂ θ h tϕ j (cid:12)(cid:12) θ =0 = 0 , j = i , (5.7)where θ = 0 is the locus h ϕ i ϕ i vanishes. Note that for deriving the behavior of h tϕ k ’s wehave used the fact that a constant piece in these h ’s can be absorbed into a shift in ϕ k , atconstant r = r H on a given H surface. Gauge field perturbations:
To analyze implications of smoothness on the gauge fieldwe consider its field strength δF ( p ) = h ( p ) t dr ∧ dt + r∂ θ h ( p ) t dθ ∧ dt + ∂ θ h ( p ) ϕ i dθ ∧ dϕ i . (5.8)Requiring absence of forces perpendicular to any one of axis of rotations for a chargedparticle, leads to ∂ θ h ( p ) t ∼ ∂ θ h ( p ) ϕ i ∼ h ( p ) ϕ i does not change the field strength, i.e. there aregauge freedoms for adding constants to h ( p ) ϕ i functions. Dilaton perturbations:
We note that δφ I should be smooth and single-valued over the d − H , this explicitly means that that regularity at “the pole” θ = 0 fixes ∂ θ δφ I = 0. Having imposed conditions (II) and (III) on perturbations, we are now ready to imposecondition (I), the linearized equations motion. We need to consider equations of motion formetric, gauge fields and dilaton perturbations.
Linearized Einstein equations takes the form G ( lin ) µν = T ( lin ) µν , (5.9)where the LHS in the Einstein-Hilbert theory is G ( lin ) µν = ∇ α ∇ ν ( δg ) αµ + ∇ α ∇ µ ( δg ) αν − ✷ ( δg ) µν −∇ ν ∇ µ ( δg ) − g µν [ ∇ α ∇ β ( δg ) αβ − ✷ ( δg )] , (5.10)and the RHS in a general EMD theory is T ( lin ) µν = δT µν δA ( p ) α (cid:12)(cid:12)(cid:12)(cid:12) bg. δA ( p ) α + δT µν δg αβ (cid:12)(cid:12)(cid:12)(cid:12) bg. δg αβ + δT µν δ Φ I (cid:12)(cid:12)(cid:12)(cid:12) bg. δ Φ I , (5.11)21here variations are computed on the NHEG background.We may now plug the field perturbations discussed in previous subsection into (5.9). Asexpected ( cf . discussions of appendix B) the linearized Einstein equation takes the form r E tt E tr rE tθ rE tϕ i E rr r E rθ r E rϕi r E θθ E θϕ i E ϕ i ϕ j = 0 . (5.12)The main feature of these equations is that because both background and field perturbationshave { ξ , ξ } symmetry, there is not any ( t, r ) dependence in the coefficients in E µν above,nor are any derivatives w.r.t these coordinates. It can be checked (using the generic shapeof background fields and their perturbations discussed in previous sections) that E tr = E tθ = E rϕ i = E θϕ i = 0 leads to h tθ = 0 and ∂ ϕ i h rr = 0, removing the only ϕ i dependence inequations. Therefore, the above simply means E µν = 0 are second order ordinary differentialequations in θ . Moreover, these equations are homogeneous linear differential equations forremaining h µν and h ( p ) µ ’s and δφ I , which are of course only functions of θ . Note that theabove are showing only a part of l.e.o.m associated with Einstein equations, and there areother equations for gauge field and dilaton perturbations which will come next.Linearized gauge field equations in an EMD theory take the form ∇ µ δF µ ( p ) ν + δ ( ∇ µ F µ ( p ) ν ) δg αβ δg αβ − α I F µ ( p ) ν (cid:12)(cid:12)(cid:12)(cid:12) bg. ∇ µ δφ I − α I ∇ µ φ I (cid:12)(cid:12)(cid:12)(cid:12) bg. δF µ ( p ) ν = 0 , (5.13)where δF = d δA is field strength of the gauge field perturbations, and α I are constantsassociated with dilaton-gauge field coupling, through terms like e − α I φ I F for each of thegauge fields in the action. Since background and field perturbations have { ξ , ξ }× U(1) d − isometry, (5.13) is structurally of the form( rE ( p ) t , E ( p ) r r , E ( p ) θ , E ( p ) ϕ i ) = 0 , (5.14)where there is no ( t, r, ϕ i ) dependence in coefficients of operators in E ( p ) = 0’s above. Re-moving redundant r ’s in (5.14) we remain with the following system of equations,( E ( p ) t , E ( p ) r , E ( p ) θ , E ( p ) ϕ i ) = 0 , (5.15)where E ( p ) µ = 0’s are ordinary linear second order homogeneous differential equations with θ -dependent coefficients. To arrive at this conclusion we have crucially used the form of gauge and dilaton field perturbations andtheir contribution to the perturbed energy momentum tensor (5.11). δφ I : ✷ δφ I + δ ( ✷ ) φ I δg µν δg µν + α J δφ J ✷ φ I − α I e − α J φ J δ ( F µν ) = 0 . (5.16)These unknowns and equations are added to the system of differential equations in (5.9) and(5.13). As discussed above, linearized equations for all perturbations reduce to some (at most)second order ordinary differential equations with respect to coordinate θ . Moreover, theseequations are linear in the perturbation fields. In addition, there are smoothness conditionswhich these solutions should also satisfy. Dealing with a set of ordinary linear differentialequations, if the equations are all consistent with each other (note that number of equationsin a crude counting is more than unknown functions), then the solutions are unique up toinitial conditions.On the other hand, as we discussed, these equations do have a set of smooth and regularsolutions, the parametric perturbations ˆ δ Φ. In other words, as we already pointed out, allparametric perturbations ˆ δ Φ are of the form of (5.2) and (5.4) and satisfy the correspondingl.e.o.m. So it just remains to show that for any chosen initial conditions for a member of theset { δ Φ } , there is a member of { ˆ δ Φ } which matches that initial conditions. Then uniquenessof the solutions finishes the proof of { δ Φ } ⊂ { ˆ δ Φ } .To this end, we need to investigate the linearized equations more closely. Below, webring the analysis in sentences and words. These sentences are of course based on explicitcomputations and cross-checks for four and five dimensional cases. We have not added theequations to avoid cumbersome, not so illuminating differential equations.Let us start with linearized Einstein equations (5.9) or (5.12) focusing on E tr = E tθ = E rϕ i = E θϕ i = 0 components of equations. As mentioned in the previous subsection, theseequations lead to h tθ = 0 and also removes the ϕ i dependence of h rr . We next note that E rθ and E θθ components of equations only involve first order differential equations in θ ; theyare “constraint equations” among the initial conditions. So from the Einstein equations, weremain with d ( d − / d ( d − / ,plus the unknowns of gauge fields and dilatons h ( p ) µ and δφ I .Similarly, one may consider the gauge field equations (5.13). Noting the allowed form ofgauge field perturbations (5.4), one can readily see that r, θ components of linearized equation(5.13) is satisfied leading to no extra constraints. Therefore, the number of unknown gauge These equations are E tt = E rr = E tϕ i = E ϕ i ϕ j = 0 and the unknowns are similar components of h µν ’s. to d − δφ I , which are again subject tosecond order ordinary differential equations (5.16), one equation per each δφ I .As discussed above, number of dynamical equations and unknowns match. Therefore,a member of { δ Φ } is uniquely determined if the initial conditions (which are twice thenumber of the unknowns, as we are dealing with linear second order ordinary differentialequations) are completely specified too. Some of the initial conditions are pre-determinedby smoothness conditions, therefore it remains to show that the remaining initial conditionsare either constrained to other ones or can be reproduced by labels δJ i and δq p : • For the metric perturbations, two of the initial conditions (which can be chosen to be ∂ θ h tt and ∂ θ h rr ), are constrained to other ones by E θθ = E rθ = 0. Also, we note thatone can still use gauge freedom (diffeomorphisms) generated by ξ = t∂ t to subtract offa constant piece from h tt . The d ( d −
3) + 1 initial conditions for the other componentsof metric perturbations are completely fixed by the ( d − smoothness conditions (5.7)and importantly by the values of d − δJ i . • For the gauge fields perturbations, initial condition for h ( p ) t (cid:12)(cid:12) θ =0 is fixed by the charges δq p and other initial conditions are fixed using discussion in subsection 5.1.4. • Dilaton fields in the EMD theory has a shift symmetry φ I → φ I + a I for any constant a I .This removes half of the required initial conditions. Recalling our earlier discussions,the regularity and smoothness provides the other half of initial conditions and hencethe solutions for dilaton perturbations are also uniquely specified.To conclude this section, perturbations of an NHEG with SL(2 , R ) × U(1) d − isometry andrequirements (I), (II) and (III), with a given set of charge perturbations δJ i , δq p are uniquelyspecified by the smoothness conditions. On the other hand, we already know one suchsolution, the parametric perturbations ˆ δ Φ. Therefore, we have proved the proposition statedin the beginning of this section. In the Appendix C we have given an alternative argumentfor our uniqueness theorem.
In this work we continued the analysis of our earlier paper [1] where we had formulated lawsof NHEG mechanics. We focused on the entropy perturbation law and tackled the ques- They are equations E ( p ) t = E ( p ) ϕ i = 0 and unknowns h ( p ) t , h ( p ) ϕ i . Note that these gauge transformations do not change the ( t, r, ϕ i ) structure of ˆ δ (or δ ) which has beencrucially used in our arguments. They are ∂ θ h ( p ) t , h ( p ) ϕ i and ∂ θ h ( p ) ϕ i around the pole θ = 0. ∂ t and t∂ t − r∂ r Killing vector fields of the background as well as asymptotically keeping theSL(2 , R ) × U(1) N symmetry of the NHEG background. In section 3 we gave various justify-ing arguments for these symmetry assumptions on the perturbations. As discussed in 3.1,these symmetry assumption are required if we want to relate NHEG perturbations to theperturbations of an extremal black hole, yielding to the NHEG in consideration in the nearhorizon limit. Therefore, our analysis uncovers a class of perturbations of an extremal blackhole which satisfy first law of black hole thermodynamics. Of course, we can only specifythe near horizon behavior of these perturbations from our analysis. It would be interestingto study how these perturbations can be extended to the whole bulk of the extremal blackhole.Our main result in this work is the NHEG perturbation uniqueness theorem. We showedby explicit computations that the NHEG perturbations subject to the three conditions dis-cussed above ( cf . definition 3.1) is limited to the NHEG parametric perturbations denotedby ˆ δ Φ discussed in section 4. ˆ δ Φ corresponds to the difference of two NHEG solutions whichhave slightly different conserved charges than J i , q p of the background. We proved our NHEGperturbation uniqueness theorem for a class of NHEG solutions with SL(2 , R ) × U(1) d − sym-metry. These are the NHEG solutions for which we have background NHEG solutionsuniqueness theorems (see [13] for a review on NHEG uniqueness theorems). The fact thatour proof covers all NHEG’s for which the background is unique within the given set ofcharges, is quite natural. Based on the arguments we gave in our proof, we expect that ourNHEG perturbation uniqueness theorem can be extended to possible future extensions tothe background NHEG uniqueness analysis. Moreover, in our proof we replaced the U (1)symmetry requirements of NHEG background uniqueness theorems [13], with “asymptotic U (1)” symmetries. This may also show a way to extend such theorems for other NHEG withpossibly less symmetries. Our uniqueness theorem also dovetails with, and in a sense extends, completes and gen-eralizes the “no dynamics” statements of the NHEK background [2, 3]. We have proved thatNHEG perturbations are only limited to those which change an NHEG to another NHEG(near-by in the parameter space). In other words, NHEG cannot be dynamically excitedwith perturbations which remain normalizable and asymptotically small compared to thebackground NHEG. In light of the above discussion and our uniqueness results, one maythen revisit the statement of Kerr/CFT correspondence [16, 14] and explore what is thekinematical and dynamical content of the chiral 2d CFT proposed to be dual to the NHEG.This is what we will discuss in our upcoming paper and here we just discuss our perspective We would like to thank Harvey Reall for a comment on this point.
25n the issue [22]: We have shown that any field perturbation subject to the two conditions(among three) of definition 3.1 is necessarily an NHEG parametric perturbation which satis-fies the EPL and is definitely not among the states identified in Kerr/CFT. The Kerr/CFTperturbations should hence be solutions subject to other conditions (than these three). Inparticular, one can show that Kerr/CFT perturbations are solution to conditions (I) and(II), but not (III), so we need to replace the asymptotic symmetry requirement with some-thing more appropriate. Moreover, the Kerr/CFT perturbations should all have vanishingentropy and charge perturbations, and hence satisfy EPL in a trivial way. This latter isexpected, because Kerr/CFT perturbations should parameterize “microstates” accountingfor the entropy of a given NHEG.
Acknowledgements
We would like to thank Geoffrey Comp`ere, Mahdi Godazgar, Harvey Reall and HosseinYavartanoo for the comments. We would like to thank the workshop “Recent Developmentsin Supergravity Theories”, June 2014, Istanbul and the ICTP Network Scheme Net-68 forproviding the stimulating discussion venue. M.M.Sh-J would like to thank Kyung-Hee Uni-versity, Seoul, Korea, under the international visiting scholar program.
A Review of proof of EPL
This appendix is a review of the discussions in [1] leading to the entropy perturbation law.Starting from the Noether current corresponding to the diffeomorphism generated by ζ H : J ζ H = Θ (Φ , L ζ H Φ) − ζ H · L , (A.1)we consider variations in (A.1) associated with Φ → Φ + δ Φ: δ J ζ H = δ [ Θ (Φ , L ζ H Φ)] − ζ H · δ L . (A.2)We assume that the variations do not alter the quantities attributed to the background. Inparticular, this means that δζ H , δξ a , δm i are all vanishing (as they do in the case of blackholes). In this sense these variations are considered as perturbations or probes over theNHEG. Let us start our analysis from the last term in (A.2): δ L = E i δ Φ i + d Θ (Φ , δ Φ) . (A.3) E i is the equation of motion for the field Φ i . The first term vanishes due to the on-shellcondition and the second term is simplified recalling the identity ξ · d Θ = L ξ Θ − d( ξ · Θ )which is valid for any diffeomorphism ξ , therefore, ζ H · δ L = L ζ H Θ (Φ , δ Φ) − d( ζ H · Θ (Φ , δ Φ)) . (A.4)26nserting the above into (A.2) we obtain δ J ζ H = ω (Φ , δ Φ , L ζ H Φ) + d( ζ H · Θ (Φ , δ Φ)) . (A.5)where ω (Φ , δ Φ , δ Φ) ≡ δ Θ (Φ , δ Φ) − δ Θ (Φ , δ Φ) (A.6)is the symplectic current [4, 5]. The current J ζ H is conserved on-shell , i.e d J ζ H = 0, so onecan associate a conserved charge d − Q ζ H , J ζ H = d Q ζ H , to the symmetry generatedby ζ H . Moreover, when the solution is deformed by a perturbation which is a solution to thelinearized equations of motion, one can take the variation of the relation J ζ H = d Q ζ H andarrive at δ J ζ H = δ d Q ζ H = d δ Q ζ H . (A.7)Using (A.7) in (A.5) yields ω (Φ , δ Φ , L ζ H Φ) = d (cid:16) δ Q ζ H − ζ H · Θ (Φ , δ Φ) (cid:17) . (A.8)We integrate the above “conservation equation” over a timelike hypersurface Σ boundedbetween two radii r = r H , r = ∞ . The hypersurface Σ can be simply chosen as a constanttime surface t = t H . Integrating (A.8) over Σ then yields:Ω(Φ , δ Φ , L ζ H Φ) = I ∂ Σ (cid:16) δ Q ζ H − ζ H · Θ (Φ , δ Φ) (cid:17) = I ∞ (cid:16) δ Q ζ H − ζ H · Θ (Φ , δ Φ) (cid:17) − I H δ Q ζ H , (A.9)in which we used the definition of symplectic form associated with Σ asΩ(Φ , δ Φ , δ Φ) ≡ Z Σ ω (Φ , δ Φ , δ Φ) , (A.10)and in the first line we have used the Stokes theorem to convert the integral over Σ to anintegral over its boundary ∂ Σ and in the second line, we used the fact that ζ H = n aH ξ a − k i m i (A.11)vanishes on H . Since the charge perturbation δ Q ζ H is linear in the vector ζ H , one can expandthe first term on RHS of (A.9)Ω(Φ , δ Φ , L ζ H Φ) = n aH I ∞ (cid:16) δ Q ξ a − ξ a · Θ (cid:17) − k i I ∞ (cid:16) δ Q m i − m i · Θ (cid:17) − I H δ Q ζ H . (A.12)27 i is tangent to the boundary surface and hence the pullback of m i · Θ over the surface r = ∞ vanishes. It was shown in [1] that Ω(Φ , δ Φ , L ζ H Φ) = − e p δq p , where q p is the electriccharge of the gauge field A ( p ) q p = − I H ǫ µν ∂ L ∂F ( p ) µν . (A.13)Therefore we arrive at − e p δq p = n aH δ E a − k i I ∞ δ Q m i − I H δ Q ζ H , (A.14)where δ E a is the canonical generator of the SL(2 , R ) symmetry x → x + ξ a δ E a ≡ I ∞ ( δ Q ξ a − ξ a · Θ ) , (A.15)As usual to Noether-Wald charges [4, 5], there are ambiguities with definition of charges.These ambiguities were dealt with in [1] where it was shown that δS π = I H δ Q ζ H , δJ i = − I ∞ δ Q m i , (A.16)where δS and δJ i respectively denote the entropy and angular momenta perturbations. Plug-ging these into (A.14) we obtain δS π = k i δJ i + e p δq p + n aH δ E a . (A.17)Note that S, q p (and their perturbations) are defined on the surface H . B Extension of axisymmetry to the bulk
Lemma:
Considering field perturbations δφ I , δA µ and δg µν in the definition 3.1, then U (1) n isometry of these perturbations is extended to all r and is not limited to asymptotic r → ∞ region.Proof. We will consider three different field perturbations separately:
Dilaton: L ξ δφ = ∂ t δφ = 0 which means δφ is independent of t . L ξ δφ = r∂ r δφ = 0which means δφ is independent of r . Therefore δφ = δφ ( θ α , ϕ i ). Requesting condition (III),i.e lim r →∞ L m i δφ (cid:12)(cid:12) ∞ = 0, leads to δφ = δφ ( θ α ) as desired. Vector:
For a covariant vector A µ , L ξ δA µ = ∂ t δA µ + δA ν ∂ µ ξ ν = ∂ t δA µ = 0, thereforeits components are independent of t . Also the symmetry ξ fixes the r dependence of thecomponents as: δA µ = ( rh t , h r r , h θ α , h ϕ i ) (B.1)28n which h ’s are some functions of ( θ α , ϕ i ). Now assuming the asymptotic U (1) n symmetryleads to ∀ i r →∞ L m i δA ν (cid:12)(cid:12) ∞ = m µi ∂ µ δA ν + δA µ ∂ ν m νi (cid:12)(cid:12)(cid:12)(cid:12) ∞ = m µi ∂ µ δA ν (cid:12)(cid:12)(cid:12)(cid:12) ∞ = ∂ ϕ i δA ν (cid:12)(cid:12)(cid:12)(cid:12) ∞ . (B.2)The above leads to ∂ ϕ i h t = ∂ ϕ i h θ = ∂ ϕ i h ϕ j = 0. Then, asymptotic ξ symmetry leads to h r = 0. This, together with the the general form of gauge field δA µ = ( rh t , , h θ α , h ϕ i ), leadsto the result that δA µ is axisymmetirc everywhere. Metric:
Considering metric perturbation δg µν as a symmetric second rank tensor, L ξ δg µν =0 leads to independence of all components from t . L ξ δg µν = 0 fixes the r dependence as: δg µν = r h tt h tr rh tθ α rh tϕ i h rr r h rθα r h rϕi r h θ α θ β h θ α ϕ i h ϕ i ϕ j (B.3)in which all of the h ’s are functions of ( θ α , ϕ i ). Now assuming the asymptotic axisymmetryleads to 0 = L m i δg µν (cid:12)(cid:12)(cid:12)(cid:12) ∞ = ( m αi ∂ α δg µν + δg µν ∂ ν m αi + δg µν ∂ µ m αi ) (cid:12)(cid:12)(cid:12)(cid:12) ∞ = ∂ ϕ i δg µν (cid:12)(cid:12)(cid:12)(cid:12) ∞ , (B.4)which shows that all component of δg µν are axisymmetric ( ϕ i independent), except for h rr , h rθ , h rϕ i components which are accompanied by powers of 1 /r . Assuming asymptotic ξ symmetry in (III), i.e lim r →∞ L ξ δg µν = 0 leads to h tr = h rθ α = h rϕ i = 0. In summary,all remaining components of h ’s are ϕ i independent, except h rr . However, in section 5.2.2we have discussed that this component is also axisymmetic as a result of linearized fieldequations. C An alternative argument for the uniqueness theorem
In this appendix we give an alternative argument for proving the NHEG perturbation unique-ness proposition. The main point in this approach is that perturbations of metric and gaugefields are gauge dependent quantities. So while one can solve the linearized field equationsin a fixed gauge (this is what we have done in section 5), a more systematic approach isto work with gauge invariant quantities which contain the information about field pertur-bations, similarly to what is usually done in cosmic perturbation theory, using the gaugeinvariant quantities (e.g. see [23]).In the context of Petrov type D spacetimes, Teukolsky [20] introduced a set of gaugeinvariant scalars built from Weyl tensor and used them to discuss perturbations of Kerrgeometry in a series of papers [24, 25]. It was shown that stability of black hole, interaction29ith gravitational/electromagnetic waves, and superradiance effects could be studied usingthese scalars. Teukolsky formulation is based on the Newman-Penrose tetrad [26], and thecorresponding directional derivative and spin coefficients, which are briefly explained below.The basic vectors of Newman-Penrose tetrad are the four null vectors ℓ, n, m, m ∗ withthe following properties ℓ = n = m = m ∗ = 0 ,ℓ · n = − , m.m ∗ = 1 . (C.1)In the NHEK geometry (2.1) in four dimensions, the ℓ, n, m vectors are explicitly: ℓ = 1 r ∂ t + ∂ r − kr ∂ ϕ , (C.2) n = r θ ) (cid:18) r ∂ t − ∂ r − kr ∂ ϕ (cid:19) , (C.3) m = 1 p θ ) (cid:18) ∂ θ + iγ ( θ ) ∂ ϕ (cid:19) . (C.4)Using these vectors we can define directional derivative operators D = ℓ µ ∇ µ , ∆ = n µ ∇ µ ,δ = m µ ∇ µ , ¯ δ = m ∗ µ ∇ µ (C.5)and construct the spin coefficients [3] κ = − ℓ a ; b m a ℓ b ν = n a ; b m ∗ a n b ρ = − ℓ a ; b m a m ∗ b µ = n a ; b m ∗ a m b σ = − ℓ a ; b m a m b λ = n a ; b m ∗ a m ∗ b τ = − ℓ a ; b m a n b π = n a ; b m ∗ a ℓ b (C.6) ǫ = −
12 ( ℓ a ; b n a ℓ b − m a ; b m ∗ a ℓ b ) γ = −
12 ( ℓ a ; b n a n b − m a ; b m ∗ a n b ) α = −
12 ( ℓ a ; b n a m ∗ b − m a ; b m ∗ a m ∗ b ) β = −
12 ( ℓ a ; b n a m b − m a ; b m ∗ a m b ) . Teukolsky method derives a master equation for the Weyl scalars constructed using the Weyltensor and the null vectors [2]. It was shown that these scalars contain useful informationabout the metric and electromagnetic perturbations.
Hertz Potential.
In our problem we need to know the exact form of metric (and gaugefield) perturbations. It was shown in [27, 28, 29] (see [30] for a review) how to constructfield perturbations using a gauge invariant scalar, called the Hertz potential Ψ H which is asolution of Teukolsky master equation. Given the Hertz potential one can construct fieldperturbations in a specific gauge called ingoing radiation gauge (IRG). For this gauge, the30ertz potential for gravitational and Maxwell field perturbations is a solution of Teukolskymaster equation with spin s = − s = − h µν = (cid:16) ℓ ( µ m ν ) [( D + 3 ǫ + ¯ ǫ − ρ + ¯ ρ )( δ + 4 β + 3 τ ) + ( δ + 3 β − ¯ α − τ − ¯ π )( D + 4 ǫ + 3 ρ )] − ℓ µ ℓ ν ( δ + 3 β + ¯ α − ¯ τ )( δ + 4 β + 3 τ ) − m µ m ν ( D + 3 ǫ − ¯ ǫ − ρ )( D + 4 ǫ + 3 ρ ) (cid:17) Ψ g ( x ) + c.c,δA µ = (cid:16) ℓ µ ( δ + 2 β + τ ) + m µ ( D + 2 ǫ + ρ ) (cid:17) Ψ A ( x ) . (C.7) Indeed the Hertz map (C.7) is a map from the solutions of the Teukolsky equation, to thesolutions of the linearized field equations for metric (or gauge field) perturbations. Now thequestion is whether all solutions of the linearized field equations can be constructed usingthe Hertz map.For the case of Kerr black hole, Wald proved [31] that there are only specific type ofperturbations that lie out of this procedure. Assuming some regularity conditions, he showedthat they are restricted to perturbations to a nearby Kerr black hole with slightly differentparameters. In the terminology we used in section 5, the only solutions to the linearizedfield equations that cannot be reproduced with the Hertz map are parametric perturbations ˆ δ Φ. They are perturbations preserving the type D property of the geometry to first order[31] (see also section 4 of [2]). Noticing the argument given in [36], we assume that this isalso the case for NHEK geometry, i.e. the only solutions that cannot be constructed usingthe Hetrz map are parametric perturbations ˆ δ Φ.Therefore the outline of the proof is as follows: As we discussed the solutions to thelinearized field equations can be divided into two sets: (I) Those corresponding to a Hertzpotential, and (II) parametric perturbations. The next step is to show that the masterequation for the Hertz potential has no solution with the conditions given in the definition3.1, i.e. no member in set (I) has our desired conditions. On the other hand, since weshowed in the opening of section 4 that parametric perturbations satisfy the conditions ofdefinition 3.1, then we have shown that the only solution with these conditions are parametricperturbations.Note that the Hertz map formalism is generically developed in the case of vacuum back-ground, therefore we assume in this appendix that the background is a vacuum
NHEG. Thisassumption is also necessary in the Kaluza-Klein reduction used in solving the Teukolskyequation in the following. However we did not need this assumption in the arguments ofsection 5.In the following we only need to show that the master equation governing the Hertzpotential has no solution compatible with our conditions. It is shown [2] that the masterequation for the Hertz potential corresponding to metric perturbations is the Teukolsky31quation with spin s = −
2, and the master equation for the Hertz potential correspondingto gauge field perturbations is the Teukolsky equation with spin s = − AdS space. More precisely, the field equation forthe separable ansatz Ψ = R ( t, r ) Y ( θ, ϕ i ) reduces to the equation of a charged massive scalar R ( t, r ) over the AdS with a homogeneous electric field, and an eigenvalue equation for Y overthe compact surface H (covered by coordinates θ, ϕ i ). Noting isometries of the background,one can further expand the solution in eigenstates of ∂∂t , ∂∂ϕ i :Ψ = e i ( ωt − m i ϕ i ) R ω,λ,m ( t, r ) Y λ,m i ( θ ) , (C.8)and the corresponding field equations become (cid:0) D − µ (cid:1) R ( r ) = 0 , (C.9) O ( s ) Y λ,m i ( θ ) = λY λ,m i ( θ ) , (C.10)where D µ = ∇ µ − iqA µ , µ ∈ { t, r } µ = λ + q ,q = k i m i − is , (C.11)and s is the spin of perturbations ( − − hypergeometric functions [2, 19]. The value of λ is constrained by the equation on compact space H . The regularity of solutions at polesrestricts the eigenvalues λ to discrete values with a lower bound depending on the field spin.It is shown that the operator O ( s ) is self adjoint, therefore its eigenvalues are real. The mostgeneral solution is hence Ψ = X ω,λ,m i e i ( ωt − m i ϕ i ) R ω,λ,m ( t, r ) Y λ,m i ( θ ) . (C.12)Since we assume that perturbations are stationary and axisymmetric, solution reduces toΨ = X ,λ, R ,λ, ( r ) Y λ, ( θ ) . (C.13)According to (C.7), requiring h µν to be symmetric under ξ exactly fixes the r dependenceof Ψ to be Ψ H = C ( θ ) r . (C.14)However, such a radial behavior cannot be constructed using the hypergeometric functions.At large r , hypergeometric functions have the asymptotic behavior R λ ( r ) = X λ (cid:16) c + λ ( r √ λ + subleading) + c − λ ( r −√ λ + subleading) (cid:17) . (C.15)32quating R = r yields: c + λ = 0 , λ > c +0 = 1 . (C.16)The regularity of the eigenstates of (C.10), however, implies that λ > c +0 leads to a divergent behavior of the angular part. Therefore, a smoothperturbation with the radial behavior R = r cannot be constructed, i.e. no perturbationwith the specified symmetries can be constructed using the Hertz map. Hence, the onlyperturbation with our conditions are the perturbation missed by the Hertz map, i.e. theparametric perturbations ˆ δg µν . Gauge field perturbations.
The Hertz map for constructing gauge perturbation is δA µ = ( ℓ µ ( δ + 2 β + τ ) + m µ ( D + 2 ǫ + ρ )) Φ , (C.18)imposing the ξ symmetry yields Φ = rF ( θ ). Therefore according to (C.15) it requires c + λ = 0 , λ ≥ c − λ =0 = 1 , (C.19)which is again violating the regularity constraint λ > c − λ =0 = 0. Thereforeno perturbation with the specified symmetries can be constructed using the Hertz map andand the only perturbation with our conditions are the parametric perturbations ˆ δA µ with avariation in NHEG charge ˆ δq . Dilaton field.
Assuming the symmetry conditions (II), (III) implies that a scalar can onlydepend on the polar coordinate θ . Now for a vacuum background, one can directly solvethe linearized field equation, which results that the scalar should be constant allover thespacetime. Remembering the shift symmetry in dilaton field, it is clear that this solution isequivalent to the parametric perturbation of dilaton field. Generalizations.
For a generic NHEG, there are some steps to be passed in order to provethe uniqueness theorem. Some of these step are not present in the literature and filling thesegaps are not in the scope of this work. However, we give the outline and leave them asconjectures. For the NHEK geometry, the solutions to (C.10) are spin-weighted spheroidal harmonics analyzed in[20]. It turns out that the eigenvalues are λ = l ( l + 1) , l ≥ M ax {| s | , | m |} (C.17)therefore the condition λ ≥ λ min > λ also holdin NHEK-AdS geometry [32] and near horizon geometry of cohomogeneity-1 extremal Myers Perry blackholes [15]. One can argue that the positivity of λ is strongly related to the stability of NHEG geometry. Inother words, our NHEG perturbation uniqueness holds for stable NHEG geometries.
33. Teukolsky equations are written for d = 4. In higher dimensions, a generalizationof Teukolsky formalism is presented in [15]. It is shown that requiring the equationsto be decoupled, it is not sufficient that the space is Petrov type D (the concept ofPetrov classification is extended to d > Kundt space. Fortunately NHEG is anexample of Kundt spacetimes. In [35], the Hertz map for gravitational and gauge fieldperturbation of a higher dimensional Kundt background was given using the decoupledequation of Reall [36].2. In a more general NHEG, one should prove the positivity of O ( s ) . No general argumentstill exists, and this is shown to be valid for different examples individually. Since thepositivity of O ( s ) is related to the stability of the corresponding NHEG, therefore weexpect that this argument is valid only for a stable NHEG, which seems reasonable.3. It should be checked that the only regular perturbation which is missed in the Hetrzmap are parametric perturbations ˆ δ Φ. This is only proved for Kerr geometry [31].However an argument is given in [36] that this result may extend to NHEG geometries.This is also a gap in the literature.4. In this appendix, we assumed that the background NHEG is a vacuum solution. Weneeded this assumption in the construction of field perturbations using the Hertz po-tential, as well as in the Kaluza-Klein reduction of Teukolsky equation to
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