First law of black hole mechanics with fermions
P. B. Aneesh, Sumanta Chakraborty, Sk Jahanur Hoque, Amitabh Virmani
aa r X i v : . [ h e p - t h ] A p r First law of black hole mechanics with fermions
P. B. Aneesh ∗ , Sumanta Chakraborty † , ,Sk Jahanur Hoque ‡ , and Amitabh Virmani § Chennai Mathematical Institute,H1 SIPCOT IT Park, Kelambakkam, Tamil Nadu 603103, India School of Mathematical and Computational SciencesIndian Association for Cultivation of Science, Kolkata 70032, India School of Physical SciencesIndian Association for Cultivation of Science, Kolkata 70032, India Institute of Theoretical Physics, Faculty of Mathematics and Physics, Charles University,V Holeˇsoviˇck´ach 2, 180 00 Prague 8, Czech Republic
Abstract
In the last few years, there has been significant interest in understanding the stationarycomparison version of the first law of black hole mechanics in the vielbein formulation of grav-ity. Several authors have pointed out that to discuss the first law in the vielbein formulationone must extend the Iyer-Wald Noether charge formalism appropriately. Jacobson and Mohd[arXiv:1507.01054] and Prabhu [arXiv:1511.00388] formulated such a generalisation for sym-metry under combined spacetime diffeomorphisms and local Lorentz transformations. In thispaper, we apply and appropriately adapt their formalism to four-dimensional gravity coupledto a Majorana field and to a Rarita-Schwinger field. We explore the first law of black holemechanics and the construction of the Lorentz-diffeomorphism Noether charges in the presenceof fermionic fields, relevant for simple supergravity. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] ontents Iyer and Wald [1] gave a derivation of the stationary comparison version of the first law of blackhole mechanics for arbitrary perturbations around a stationary axisymmetric black hole in anydiffeomorphism covariant theory of gravity. The black hole entropy was identified with the integralover the bifurcation surface of the diffeomorphism Noether charge for the horizon generating Killingfield. The results of Iyer and Wald have found numerous applications over the years [2, 3, 4].However, as emphasised in detail in references [5, 6], there are situations of physical interestwhere Iyer-Wald analysis cannot be applied directly. An assumption that goes into their analysisis that matter fields, if present, are smooth tensor fields on the spacetime. Often in gauge theories,one cannot always make a gauge choice such that the gauge fields are smooth tensor fields. A similarsituation arises for gravity theories written in vielbein formulation [5], where the spin-connectionmight not be smooth in some chosen gauge (i.e., for a choice of vielbeins e aµ ). Since the coupling offermions to gravity is typically through the vielbeins, Iyer-Wald analysis cannot be directly appliedto gravity coupled to fermions. In the last few years, there has been interest in understanding thestationary comparison version of the first law of black hole mechanics in the vielbein formulationand renewed interest in the theory of Noether charges in the vielbein formulation of gravity, seee.g., [7, 8, 9, 10, 11].In order to discuss the first law in the vielbein formulation, one must extend the Iyer-Wald’sNoether charge formalism appropriately. Prabhu [6] formulated a generalisation for symmetryunder combined diffeomorphisms and internal gauge transformations in terms of fields living on aprincipal bundle over spacetime. Using this formalism he presented a derivation of the first lawof black hole mechanics in the vielbein formulation of gravity, for gravity coupled to Yang-Mills2eld and for gravity coupled to a Dirac fermion. Interestingly, he found that the contribution dueto the Dirac field to the Lorentz-diffeomorphism Noether charge vanishes on-shell for the horizongenerating Killing field at the bifurcation surface. Thus, we are led to the question: does thecontribution to the Lorentz-diffeomorphism Noether charge vanish generically for fermions at thebifurcation surface?The situation should be compared to bosonic fields. For a minimally coupled scalar too thecontribution to the diffeomorphism Noether charge due to the scalar vanishes. Though, this isnot the case for a vector field (see e.g., [12]). Thus, it is interesting to explore the case of theRarita-Schwinger field, which besides being fermionic also carries a spacetime index. In this paper,we apply and appropriately adapt the formalism of [5, 6] to four-dimensional gravity coupled to aMajorana field and to a Rarita-Schwinger field. We explore the first law of black hole mechanicsand the construction of Lorentz-diffeomorphism Noether charges. With the Rarita-Schwinger fieldone can in principle write down a few different diffeomorphism covariant Lagrangians. Perhapsthe most natural set-up to consider is the case of N = 1 , D = 4 supergravity, often called simplesupergravity.At this stage, it is important to note that local supersymmetry is not an internal gauge sym-metry; it is a spacetime symmetry. Prabhu’s formalism [6], although quite general, is restricted tothe cases of internal gauge symmetry. Thus, local supersymmetry cannot be properly taken intoaccount within the framework of references [5, 6]. For this reason, we ‘switch off’ supersymmetryin our study of the Lorentz-diffeomorphism Noether charges. Perhaps an independent formulationof the first law can be achieved by considering supergravity as a geometric theory on superspace,where the fermionic gauge symmetry is properly taken into account. We leave this for future work.The rest of the paper is organised as follows. In section 2 we summarise the Lorentz diffeomor-phism Noether charge formalism. For simple supergravity, the Rarita-Schwinger field is a Majoranavector-spinor. The Majorana condition brings in new elements in the computation. It is instruc-tive to study the simpler spin-1/2 Majorana field first, before diving into the case of the simplesupergravity. Therefore, in section 3 we study the Lorentz-diffeomorphism Noether charge for thesimpler case of spin-1/2 Majorana field. This section connects to the case of the Dirac field studiedin reference [6]. It also serves as a warm-up for simple supergravity considered in section 4. Thekey result of section 4 is an expression for the contribution to the Lorentz-diffeomorphism Noethercharge due to the Rarita-Schwinger field. Using this Noether charge we formulate a stationarycomparison version of the first law in section 5. We close with a brief discussion in section 6.3 A summary of the Lorentz-Diffeomorphism Noether charge for-malism
For theories with internal gauge symmetries there is no natural action of spacetime diffeomorphismson dynamical fields. We only have a notion of spacetime diffeomorphisms upto gauge transforma-tions. Without such a separation between spacetime diffeomorphisms and gauge transformations,the diffeomorphism
Noether charge is not an adequate concept to work with. This issue has beendiscussed in the context of the first law of black hole mechanics over the years [13, 14, 12, 15, 16]and, from our point of view, is satisfactorily addressed in [5, 6]. A different perspective on theseissues is presented in [10].Prabhu [6] formulates a given gravity theory in terms of fields living on a principal bundle overspacetime. Then he considers the full group of transformations, diffeomorphisms together withgauge transformations, viewed as automorphisms of the bundle. For a general such automorphismhe defines a notion of Noether charge and uses it to obtain a first law of black hole mechanics as avariational identity. A key idea in this construction is the fact that the variation of the fields undera general automorphism X is simply given by the standard Lie derivative with respect to vectorfield X on the bundle .Jacobson and Mohd in reference [5] take a more pragmatic approach. They propose a gener-alisation of the Iyer-Wald diffeomorphism Noether charge for a given spacetime diffeomorphism towhat they call Lorentz-Diffeomorphism
Noether charge. The two approaches have some similaritiesand some differences. In general there is no unique way to associate an automorphism of the bundleto a given spacetime diffeomorphism. However, the requirement that the vielbeins (co-frames) bepreserved by the corresponding Killing vector of a given spacetime metric, uniquely determinesthe infinitesimal bundle automorphism. A proof of this statement with references to the originalliterature can be found in [5, 6]. This uplift defines a Lorentz-Lie derivative. We use the notation K following [5] where the Lorentz-Lie derivative was defined for arbitrary spacetime vector fields bythe same formula as for the Killing vector fields. We call it the Kosmann derivative. On co-frames,for arbitrary ξ , K ξ e aµ = L ξ e aµ + (cid:16) E ν [ a L ξ e b ] ν (cid:17) e bµ , (2.1)where L ξ is the standard Lie derivative computed by ignoring the internal indices, and E µa arethe inverse vielbeins. Since in the later parts of the paper we suppress spacetime indices andwork in the form notation, a separate notation is required for the vielbeins and inverse vielbeins.The vielbeins and inverse vielbeins satisfy the usual properties, E a · e b := E µa e bµ = δ ba , and underarbitrary variation δE a · e b = − E a · δe b . 4n order to generalise the concept of Kosmann derivative to an arbitrary tensor-spinor objectcarrying both spacetime and/or Lorentz/spinor indices, we first rewrite Eq. (2.1) as: K ξ e aµ = L ξ e aµ + 12 (cid:0) ξ · ˚ ω cd + E ρc E σd ∂ [ ρ ξ σ ] (cid:1) (cid:16) η a [ c δ d ] b (cid:17) e bµ , (2.2)= L ξ e aµ + 12 (cid:16) ξ · ˚ ω cd + ˚ D [ c ξ d ] (cid:17) (cid:16) η a [ c δ d ] b (cid:17) e bµ , (2.3)where ˚ ω cd is the torsionless part of the spin-connection and ˚ D µ is the torsionless Lorentz covari-ant derivative. This rewriting makes the vector representation of the Lorentz group in equationEq. (2.1) manifest, and allows us to propose a Kosmann derivative for an arbitrary tensor-spinorobject [17]. For an arbitrary spacetime vector field ξ , we define, K ξ T µ ··· µ m ν ··· ν n ≡ L ξ T µ ··· µ m ν ··· ν n + 12 (cid:16) ξ · ˚ ω ab + ˚ D [ a ξ b ] (cid:17) Γ r ( M ab ) T µ ··· µ m ν ··· ν n , (2.4)where the Lorentz/spinor indices are suppressed and the Lie derivative is the usual Lie deriva-tive that only sees the spacetime indices, and where Γ r ( M ab ) are the representation matrices forthe Lorentz generators M ab in the representation r of the Lorentz tensor-spinor T . For vectorrepresentation, Γ vec ( M ab ) cd = 2 η c [ a δ b ] d , (2.5)and for the four-dimensional spinor representation,Γ spinor ( M ab ) = 12 γ ab = 14 (cid:16) γ a γ b − γ b γ a (cid:17) . (2.6)For spinors, gamma matrices, Majorana condition, Majorana flip conventions we exclusively followreference [18].It then follows that for spinors, the Kosmann derivative takes the form, K ξ Ψ = ξ µ ˚ D µ Ψ + 14 ∂ [ µ ξ ν ] γ µν Ψ , (2.7)where ˚ D µ is the torsionless spinor covariant derivative,˚ D µ Ψ( x ) = (cid:18) ∂ µ + 14˚ ω µab γ ab (cid:19) Ψ( x ) , (2.8)and where ˚ ω µab and the Christoffel symbol ˚Γ σµν are related by the vielbein postulate, ∂ µ e aν + ˚ ω µab e bν − ˚Γ σµν e aσ = 0 . (2.9)Along identical lines, it also follows that the Kosmann derivative for the Rarita-Schwinger field,5hich is a vector-spinor field, takes the form, K ξ ψ µ = L ξ ψ µ + 14 (cid:16) ξ · ˚ ω ab + ˚ D [ a ξ b ] (cid:17) γ ab ψ µ = ξ α ∂ α ψ µ + ψ α ∂ µ ξ α + 14 (cid:16) ξ · ˚ ω ab + ˚ D [ a ξ b ] (cid:17) γ ab ψ µ = ξ α (cid:18) ∂ α ψ µ − ∂ µ ψ α + 14˚ ω abα γ ab ψ µ − ω abµ γ ab ψ α (cid:19) + ∂ µ ( ξ α ψ α )+ 14˚ ω abµ γ ab ( ξ α ψ α ) + 14 ˚ D [ a ξ b ] γ ab ψ µ = ξ α (cid:16) ˚ D α ψ µ − ˚ D µ ψ α (cid:17) + ˚ D µ ( ξ α ψ α ) + 14 ˚ D [ a ξ b ] γ ab ψ µ , (2.10)where in the third line we have added and subtracted appropriate terms to bring it in the finalform. Later we will convert this final form in the index-free form notation.Jacobson and Mohd [5] proposed to use the Kosmann derivative to define a Lorentz-diffeomorphismassociated to ξ . Although, the association of the (projection of the) bundle automorphism K ξ tospacetime diffeomorphism L ξ is different from the spirit of [6], when restricted to Killing vectorsthe conclusions from the two formalisms are identical. In particular, one gets the same first law forstationary spacetimes using either of the two prescriptions. We make use of this fact and borrowconvenient notation from both these references to discuss the first law with the Majorana andRarita-Schwinger fields.A comment about notation is in order: Prabhu [6] uses underline to distinguish between fieldson the spacetime (base space) from those on the principle bundle. We do not use that notation. Forus all quantities are on the spacetime. The main technical point we use from [5, 6] is the modificationof the Lie derivative to the Kosmann derivative when evaluating the Noether currents.Let L denote the Lagrangian d -form. Spacetime differential forms are denoted with boldfaceletters and center dot is used to denote the interior derivative: for a p -form A , ξ · A = i ξ A . Weassume that the Lagrangian is diffeomorphism covariant and is a Lorentz scalar, and its variationsare the same when we vary the fields with the Kosmann derivative or with the Lie derivative, i.e., K ξ L = L ξ L = d ( ξ · L ) . (2.11)Let us denote the dynamical field collectively as ϕ α (and also simply as ϕ ), where α denotes anyinternal indices that the field may carry. The variation of the Lagrangian d -form induced by a fieldvariation δ ϕ α can be written as, δ L = E α ( ϕ ) ∧ δ ϕ α + d θ ( ϕ, δϕ ) . (2.12)The quantity E α ( ϕ ) defines the equations of motion, E α ( ϕ ) = 0, and the ( d −
1) form θ is constructedout of the dynamical fields ϕ and their first variations δϕ . Now let us consider the variation of the6agrangian induced by an arbitrary vector field ξ , with field variations, δ ξ ϕ α = K ξ ϕ α . (2.13)To such a variation we can associate an ( d − J ξ = θ ( ϕ, K ξ ϕ ) − ξ · L . (2.14)For all vector fields ξ , the Noether current is closed on-shell. This implies [19, 5, 6] that on-shell J ξ is an exact form J ξ = dQ ξ . In this work, we are concerned with the construction of theLorentz-diffeomorphism Noether charge Q ξ for the theories of interest.The Lagrangians considered in this paper are of the form, L = L grav + L matter , (2.15)where L grav depends on the vielbein one-forms (also called frame-field one forms or co-frames) e a = e aµ dx µ and the spin-connection one-forms ω ab = ω abµ dx µ . We exclusively work in the firstorder formalism, i.e., we treat e a and ω ab as independent fields. We write the equations of motionterms in the variation with respect to the co-frames e a and the spin-connection ω ab as,( δ e L grav ) eom = E a ∧ δ e a , ( δ e L matter ) eom = − T a ∧ δ e a , E a − T a = 0 , (2.16)( δ ω L grav ) eom = E ab ∧ δ ω ab , ( δ ω L matter ) eom = − S ab ∧ δ ω ab , E ab − S ab = 0 . (2.17)The above equations define the symbols E a , T a , E ab and S ab .For the first order formulation of gravity with a cosmological constant we use the followingLagrangian, L grav = ( R − πG ǫ , (2.18)often called the Einstein-Hilbert-Palatini Lagrangian. This theory was considered in detail inreferences [5, 6]. For this Lagrangian we have a symplectic potential boundary term, which takesthe form, θ = 132 πG ε abcd e a ∧ e b ∧ δ ω cd . (2.19)Now, in order to compute the Lorentz-diffeomorphism Noether current Eq. (2.14) we need theKosmann derivative of the spin-connection. This can be obtained in a number of ways. We follow Sometimes the d − θ ( ϕ, δϕ ) is called the presymplectic potential. Using this we construct the presymplectic current ω ( ϕ, δ ϕ, δ ϕ ) = δ θ ( ϕ, δ ϕ ) − δ θ ( ϕ, δ ϕ ). Integrating ω on a Cauchy surface gives a presymplectic 2-formon the configuration space F . The presymplectic 2-form is not a true symplectic 2-form as it is not non-degenerate.It is possible to construct a phase space with a non-degenerate symplectic form [20]. However, for our purposes thepresymplectic forms will be sufficient. Thus, to simplify terminology we use the names symplectic potential for θ andsymplectic current for ω . δ ω ab transformscovariantly under local Lorentz transformations. The first Cartan structure equation ( d e a + ω ab ∧ e b = T a , where T a is the torsion 2-form) allows us to write δ ω ab in terms of δ e a and δ T a . Since T a is a proper Lorentz-tensor, its Kosmann derivative is uniquely defined by the above considerations,cf. Eq. (2.4). Therefore, taking δ ω ab = K ξ ω ab , K ξ ω ab can be related to K ξ e a and K ξ T a . Acalculation gives, K ξ ω ab = L ξ ω ab − D (cid:16) e µ [ a L ξ e b ] µ (cid:17) , (2.20)where D is the standard local Lorentz covariant derivatives. When restricted to the torsionlesscase, this equation is same as the one given in [5].The Lorentz-diffeomorphism Noether current for the gravity sector Lagrangian Eq. (2.18) takesthe form J ξ = 132 πG ε abcd e a ∧ e b ∧ (cid:16) K ξ ω cd (cid:17) − ( ξ · L grav ) (2.21)= dQ ξ + E a ( ξ · e a ) + E ab (cid:26) E a · E b · [ d ξ − T c ( E c · ξ )] − ( E c · ξ ) (cid:0) E [ a · E c · T b ] (cid:1)(cid:27) , where E a = − πG ε abcd e b ∧ R cd + Λ8 πG ( ⋆ e a ) , (2.22) E ab = 116 πG ε abcd e c ∧ T d , (2.23)and the Noether charge Q ξ is Q ξ = 132 πG ε abcd e a ∧ e b (cid:26) E c · E d · [ d ξ − T e ( E e · ξ )] − ( E e · ξ ) (cid:0) E [ c · E e · T d ] (cid:1)(cid:27) . (2.24)In equation Eq. (2.22), ⋆ denotes the four-dimensional Hodge dual. Expression Eq. (2.24) can bewritten more conveniently using the contorsion tensor (see below) and can be compared with (theprojection of) equation (5.8) of reference [6]. In the torsionless case, the Noether charge Q ξ takesthe familiar form, Q ξ = − πG ( ⋆d ξ ) . (2.25)To get the first law for a stationary, axisymmetric, black hole solution we use the horizon generatingKilling field k in place of ξ . Since k vanishes at the bifurcation surface, the integration of the Noethercharge Eq. (2.24) over the bifurcation 2-sphere B gives Z B Q k = κ B π Area( B ) , (2.26)where κ B is surface gravity of the black hole horizon.8 Lorentz-Diffeomorphism Noether charge for Majorana field
In this section we study the Lorentz-diffeomorphism Noether charge for the spin-1/2 Majoranafield. The aim of this section is to first compute the symplectic potential ( d − θ and thenthe corresponding Lorentz-diffeomorphism Noether current J ξ .The Lagrangian four-form for a massive Majorana field takes the form, L matter = ǫ (cid:18) −
12 ¯Ψ /D Ψ + 12 m ¯ΨΨ (cid:19) , (3.1)where ¯Ψ is the Majorana conjugate and the derivative /D is defined below. For the benefit of thereader we review the definition of Dirac, Majorana, and charge conjugate following [18]. Let C bethe charge conjugation matrix and let Γ ( r ) be the rank r product of antisymmetrised γ matricesΓ ( r ) : γ µ ...µ r = γ [ µ . . . γ µ r ] . (3.2)The conventions reference [18] follows for four-dimensions (and Lorentzian signature) are( C Γ ( r ) ) T = − t r C Γ ( r ) , (3.3)with t = +1 , t = − , t = − , t = +1 , and t r +4 = t r . The Majorana conjugate is defined as,¯Ψ = Ψ T C. (3.4)The Majorana field by definition satisfies the Majorana condition, Ψ C = Ψ, where Ψ C is the chargeconjugate defined via Ψ ∗ = it Cγ Ψ C . These definitions imply that the Dirac conjugate ¯Ψ = i Ψ † γ is same as the Majorana conjugate for a Majorana spinor. Due to this equivalence, ¯Ψ in Eq. (3.1)can be taken to be the Dirac conjugate for the Majorana spinor Ψ. For explicit computations it iseasier to work with the Dirac conjugate.The covariant derivative /D used in the above Lagrangian is defined as /D Ψ = γ µ D µ Ψ = γ a E µa D µ Ψ = γ a ( E a · D Ψ) = γ µ (cid:18) ∂ µ + 14 ω abµ γ ab (cid:19) Ψ , (3.5)where the latin indices are Lorentz frame indices and the greek indices are spacetime indices. It isconvenient to define the derivative operator acting on ¯Ψ as well via /D Ψ = − ¯Ψ ←− /D = − ¯Ψ ←− D µ γ µ = − (cid:18) ∂ µ ¯Ψ −
14 ¯Ψ ω abµ γ ab (cid:19) γ µ . (3.6)Unlike the case of the Dirac field, where the reality condition on the Lagrangian requires us towork with the symmetrised derivative ( ¯Ψ /D Ψ + /D ΨΨ), Lagrangian Eq. (3.1) is automatically real9ue to the Majorana condition. To see this we note that,( ¯Ψ /D Ψ) † = ( /D Ψ) † ( i Ψ † γ ) † = i (cid:18) ∂ µ Ψ † + 14 ω µab Ψ † γ γ ab γ (cid:19) ( γ γ µ γ )( γ Ψ)= − (cid:18) ∂ µ ¯Ψ − ω µab ¯Ψ γ ab (cid:19) γ µ Ψ= ¯Ψ γ µ ∂ µ Ψ + 14 ω µab ¯Ψ γ µ γ ab Ψ= ¯Ψ /D Ψ , (3.7)where we have repeatedly used the relation ¯Ψ γ µ Ψ = 0 for a Majorana spinor. Note that thisproperty is not available for the Dirac field.The Lagrangian presented in Eq. (3.1) depends on three fields: the vielbein e a , the spin-connection ω ab and the Majorana field Ψ. The variation of the Lagrangian with respect to thevielbein takes the form, δ e L = ( δ ǫ ) (cid:18) −
12 ¯Ψ /D Ψ + 12 m ¯ΨΨ (cid:19) − ǫ δ ( ¯Ψ /D Ψ)= ( δ ǫ ) (cid:18) −
12 ¯Ψ /D Ψ + 12 m ¯ΨΨ (cid:19) − ǫ ¯Ψ γ a ( δE a · D Ψ)= ( δ ǫ ) (cid:18) −
12 ¯Ψ /D Ψ + 12 m ¯ΨΨ (cid:19) + 12 ǫ ¯Ψ γ a ( E b · D Ψ)( E a · δ e b )= − ( ⋆ e a ) (cid:18) −
12 ¯Ψ /D Ψ + 12 m ¯ΨΨ (cid:19) ∧ δ e a −
12 ( E b · ǫ ) ¯Ψ γ b ( E a · D Ψ) ∧ δ e a = − ( ⋆ e a ) (cid:18) −
12 ¯Ψ /D Ψ + 12 m ¯ΨΨ (cid:19) ∧ δ e a − ⋆ e b ¯Ψ γ b ( E a · D Ψ) ∧ δ e a := − T a ∧ δ e a . (3.8)The above manipulations have been performed as follows. In going from the second to the thirdstep we have related δE a to δ e b . In going from the third to the fourth step we have used the result,( δ ǫ ) = − ( ⋆ e a ) ∧ δ e a , where ⋆ denotes the Hodge star, and also the identity, V · ( A ∧ B ) = ( V · A ) ∧ B + ( − p A ∧ ( V · B ) , (3.9)where, V is a vector field, A is a p -form and B is q -form. In the present context we have usedidentity Eq. (3.9) with A = ǫ , B = δ e a and V = E a , such that, A ∧ B = 0. In going from thefourth to the fifth step we have used ( E b · ǫ ) = ⋆ e b . Finally, the last line of Eq. (3.8) defines theenergy-momentum three-form T a , T a = ( ⋆ e a ) (cid:18) −
12 ¯Ψ /D Ψ + 12 m ¯ΨΨ (cid:19) + 12 ( ⋆ e b ) ¯Ψ γ b E a · D Ψ . (3.10)10imilarly, the variation with respect to the spin connection gives, δ ω L = − ǫ ¯Ψ γ a E a · (cid:18) δ ω cd γ cd (cid:19) Ψ= − ǫ ¯Ψ γ a γ cd Ψ( E a · δ ω cd ) = 18 ( E a · ǫ ) ¯Ψ γ a γ cd Ψ ∧ δ ω cd := − S cd ∧ δ ω cd , (3.11)where we have again used identity Eq. (3.9) with A = ǫ , B = δ ω cd , and V = E a such that A ∧ B = 0. The last line of the above equation defines the spin-current three-form S cd , S cd = −
18 ( E a · ǫ ) ¯Ψ γ a γ cd Ψ . (3.12)Finally, we consider the variation of the Lagrangian with respect to Ψ. Unlike the case of Diracfield, where variations of Ψ and ¯Ψ are treated as independent, here the field variations must alsosatisfy the Majorana condition so that δ Ψ is related to δ ¯Ψ. In practice this condition is implementedthrough the Majorana flip relations. Let λ and λ be two arbitrary Majorana spinors (possiblywith other spacetime indices), then, ¯ λ Γ ( r ) λ = t r ¯ λ Γ ( r ) λ , (3.13)which for our applications read, δ ¯ΨΨ = ¯Ψ δ Ψ , δ ¯Ψ γ a Ψ = − ¯Ψ γ a δ Ψ , (3.14) δ ¯Ψ γ cd Ψ = − ¯Ψ γ cd δ Ψ , δ ¯Ψ γ abc Ψ = ¯Ψ γ abc δ Ψ , (3.15) δ ¯Ψ γ a ∂ µ Ψ = − ∂ µ ¯Ψ γ a δ Ψ , δ ¯Ψ γ a γ cd Ψ = ¯Ψ γ cd γ a δ Ψ . (3.16)These relations imply, δ ¯Ψ /D Ψ = δ ¯Ψ γ µ (cid:18) ∂ µ + 14 ω abµ γ ab (cid:19) Ψ= − ∂ µ ¯Ψ γ µ δ Ψ + 14 ¯Ψ ω µab γ ab γ µ δ Ψ= − ¯Ψ ←− D · E a γ a δ Ψ = − ¯Ψ ←− /D δ Ψ . (3.17)Using these relations, the variation of the Lagrangian with respect to the Majorana field yields, δ Ψ L = 12 ǫ (cid:0) − δ ¯Ψ /D Ψ − ¯Ψ /Dδ Ψ + m ¯Ψ δ Ψ + mδ ¯ΨΨ (cid:1) = 12 ǫ ¯Ψ ←− /D δ Ψ + 12 ( E a · ǫ ) ¯Ψ γ a ∧ D δ Ψ + ǫ ( m ¯Ψ δ Ψ)= ǫ (cid:18) ¯Ψ( ←− /D + m ) − T bab ( ¯Ψ γ a ) (cid:19) δ Ψ − d (cid:20)
12 ( E a · ǫ ) ¯Ψ γ a δ Ψ (cid:21) , (3.18)where T abc are the frame components of the torsion T abc = E b · E a · T c . In going from the first tothe second step we have used once again identity Eq. (3.9) with A = ǫ , B = D δ Ψ, and V = E a A ∧ B = 0. In going from the second to the third line we have used integration by partsand manipulations similar to the ones performed in [6] for the Dirac field.The contribution to the symplectic potential θ from the Majorana field can be read off fromEq. (3.18), θ (Ψ , δ Ψ) = −
12 ( E a · ǫ ) ¯Ψ γ a δ Ψ . (3.19)Given the symplectic potential, the Lorentz-diffeomorphism Noether current can be obtained in astraight forward manner. It takes the form, J ξ := θ (Ψ , K ξ Ψ) − ξ · L = −
12 ( E a · ǫ ) ¯Ψ γ a ( K ξ Ψ) − ( ξ · ǫ ) (cid:18) −
12 ¯Ψ /D Ψ + 12 m ¯ΨΨ (cid:19) . (3.20)From Eq. (2.7) we have, K ξ Ψ = ξ · d Ψ + 14 ξ · ˚ ω ab γ ab Ψ + 18 ( E b · E a · d ξ ) γ ab Ψ , (3.21)Inserting this expression in Eq. (3.20) and using the identity, V · A = ( V · e a )( E a · A ) , (3.22)we get J ξ = − T a ( ξ · e a ) − S ab (cid:18) E a · E b · ( d ξ − T c ( E c · ξ )) − ( E c · ξ )( E [ a · E c · T b ] (cid:19) , (3.23)where T a and S ab are defined in Eq. (3.10) and Eq. (3.12) respectively.Now, the full Lorentz-diffeomorphism Noether current of the gravity plus the matter takes theform, J ξ = J grav ξ + J matter ξ (3.24)= dQ ξ + ( E a − T a ) ( ξ · e a )+ (cid:16) E ab − S ab (cid:17) (cid:18) E a · E b · ( d ξ − T c ( E c · ξ )) − ( E c · ξ )( E [ a · E c · T b ] (cid:19) . (3.25)Therefore, on-shell we have J ξ = dQ ξ where the Noether charge Q ξ is Q ξ = 132 πG ε abcd e a ∧ e b (cid:18) E c · E d · ( d ξ − T e ( E e · ξ )) − ( E e · ξ )( E [ c · E e · T d ] (cid:19) , (3.26)and the torsion 2-form is fixed by the equations of motion E ab − S ab = 0 . At the bifurcation surface,for the horizon generating Killing field ξ µ = k µ = ( ∂ t ) µ + Ω( ∂ φ ) µ , ξ µ vanishes. As a result, theNoether charge Q ξ simplifies to Q k (cid:12)(cid:12)(cid:12)(cid:12) B = 132 πG ε abcd e a ∧ e b (cid:18) E c · E d · d k (cid:19) = − πG ⋆ d k . (3.27)12hus, we see that the Majorana fermion does not contribute to the full Noether charge for thehorizon generating Killing field at the bifurcation surface.Note that, through the change in torsion due to the Majorana field, in general both the Noethercurrent and the Noether charge are different from pure general relativity. However, when theNoether charge is evaluated for the horizon generating Killing field at the bifurcation surface allthose terms do not contribute. In this section we study the Lorentz-diffeomorphism Noether charge for the spin-3/2 Rarita-Schwinger field in four spacetime dimensions. The most natural set-up to consider is the caseof N = 1 , D = 4 supergravity, often called simple supergravity. With very little effort, this discus-sion can be generalised to N = 1 , D = 4 AdS supergravity. This is the set-up we work with. TheLagrangian 4-form for simple AdS supergravity is, L sugra = 14 κ ε abcd R ab ( ω ) ∧ e c ∧ e d + i κ ¯ ψ ∧ γ ∗ γ ∧ ˆ Dψ − κ Λ ǫ , (4.1)where the Rarita-Schwinger field ψ is a Majorana spinor valued one-form, γ is a matrix-valuedone-form γ = γ a e a , (4.2) γ ∗ = iγ γ γ γ , and the derivative operator on the Rarita-Schwinger field is defined as, ˆ Dψ = Dψ − L γ ∧ ψ = (cid:18) ∂ µ ψ ν + 14 ω cdµ γ cd ψ ν − L γ µ ψ ν (cid:19) d x µ ∧ d x ν . (4.3)Furthermore, κ = 8 πG and the cosmological constant Λ is related to the AdS radius L throughΛ = − L . Lagrangian Eq. (4.1) is presented in a slightly different form compared to standardreferences such as [18]. Since we use the form notation, the above structure of the Lagrangian iscomputationally simpler to work with.The gravity sector of the Lagrangian presented in Eq. (4.1) is the same as in the previoussections. Therefore, it is enough to consider the matter part of the Lagrangian Eq. (4.1), L = i κ ¯ ψ ∧ γ ∗ γ ∧ ˆ Dψ . (4.4)The variation with respect to e a yields the energy-momentum 3-form T a , and the variationwith respect to ω ab yields the spin-current 3-form S ab ,( δ e L ) eom = − T a ∧ δ e a , ( δ ω L ) eom = − S ab ∧ δ ω ab . (4.5)13t turns out that the variations δ e L and δ ω L do not give total derivative terms. The variation withrespect to ψ yields the equations of motion for the Rarita-Schwinger field and a total derivativeterm from where we read off the contribution to the symplectic potential 3-form θ .We have, δ L = + i κ (cid:0) δ ¯ ψ (cid:1) ∧ γ ∗ γ ∧ ˆ Dψ + i κ ¯ ψ ∧ γ ∗ γ a ˆ Dψ ∧ δ e a + i κ ¯ ψ ∧ γ ∗ γ ∧ ˆ D δ ψ + i κ ¯ ψ ∧ γ ∗ γ ∧ (cid:18) δ ω cd γ cd (cid:19) ∧ ψ + i κ ¯ ψ ∧ γ ∗ γ ∧ (cid:18) L γ a (cid:19) ψ ∧ δ e a . (4.6)The variation of ¯ ψ is related to the variation of ψ through the Majorana flip relations. The followingmore general version of the Majorana flip relation is very useful in actual computations:¯ λ Γ ( r ) Γ ( r ) λ = t r (cid:0) Γ ( r ) λ (cid:1) Γ ( r ) λ = t t r t r ¯ λ Γ ( r ) Γ ( r ) λ . (4.7)Using γ ∗ = iγ γ γ γ together with Eq. (3.13) and Eq. (4.7), we have for the first term in thevariation presented in Eq. (4.6), (cid:0) δ ¯ ψ (cid:1) ∧ γ ∗ γ a e a ∧ ˆ Dψ = (cid:16) δ ¯ ψ µ γ ∗ γ a ˆ D ρ ψ σ e aν (cid:17) d x µ ∧ d x ν ∧ d x ρ ∧ d x σ = (cid:16) ( ˆ D ρ ψ σ ) γ ∗ γ a δ ¯ ψ µ e aν (cid:17) d x µ ∧ d x ν ∧ d x ρ ∧ d x σ = (cid:26)(cid:18) ∂ ρ ¯ ψ σ − ω ρcd ¯ ψ σ γ cd + 12 L ¯ ψ σ γ ρ (cid:19) γ ∗ γ ν δψ µ (cid:27) d x σ ∧ d x ρ ∧ d x ν ∧ d x µ = − (cid:18) d ¯ ψ + 14 ¯ ψ ∧ ω cd γ cd − L ¯ ψ ∧ γ (cid:19) ∧ γ ∗ γ ∧ δ ψ = − ¯ ψ ←− ˆ D ∧ γ ∗ γ a e a ∧ δ ψ , (4.8)where the operation ←− ˆ D is defined as,¯ ψ ←− ˆ D ≡ d ¯ ψ + 14 ¯ ψ ∧ ω cd γ cd − L ¯ ψ ∧ γ = (cid:18) ∂ µ ¯ ψ ν − ω µcd ¯ ψ ν γ cd + 12 L ¯ ψ ν γ µ (cid:19) d x µ ∧ d x ν = ¯ ψ ν ←− ˆ D µ d x µ ∧ d x ν . (4.9)14dditionally, to simplify the second term in the variation Eq. (4.6), we note that¯ ψ ∧ γ ∗ γ ∧ ˆ D δ ψ = ¯ ψ ∧ γ ∗ γ ∧ d δ ψ + 14 ¯ ψ ∧ γ ∗ γ ∧ ω cd γ cd ∧ δ ψ − L ¯ ψ ∧ γ ∗ γ ∧ γ ∧ δ ψ = d (cid:0) ¯ ψ ∧ γ ∗ γ ∧ δ ψ (cid:1) − d (cid:0) ¯ ψ ∧ γ ∗ γ (cid:1) ∧ δ ψ −
14 ¯ ψ ∧ ω cd ∧ γ ∗ γ γ cd ∧ δ ψ + 12 L ¯ ψ ∧ γ ∧ γ ∗ γ ∧ δ ψ = d (cid:0) ¯ ψ ∧ γ ∗ γ ∧ δ ψ (cid:1) + ¯ ψ ∧ γ ∗ γ a de a ∧ δ ψ − (cid:18) d ¯ ψ − L ¯ ψ ∧ γ (cid:19) ∧ γ ∗ γ ∧ δ ψ −
14 ¯ ψ ∧ ω cd ∧ γ ∗ γ a γ cd e a ∧ δ ψ = d (cid:0) ¯ ψ ∧ γ ∗ γ ∧ δ ψ (cid:1) + ¯ ψ ∧ γ ∗ γ a (cid:16) de a + ω ab ∧ e b (cid:17) ∧ δ ψ − (cid:18) d ¯ ψ + 14 ¯ ψ ∧ ω cd γ cd − L ¯ ψ ∧ γ (cid:19) ∧ γ ∗ γ ∧ δψ = d (cid:0) ¯ ψ ∧ γ ∗ γ ∧ δ ψ (cid:1) + ¯ ψ ∧ γ ∗ γ a T a ∧ δ ψ − ¯ ψ ←− ˆ D ∧ γ ∗ γ ∧ δ ψ , (4.10)where we have separated out the total derivative term.Using both Eq. (4.8) and Eq. (4.10), the variation of the Rarita-Schwinger Lagrangian simplifiesto, δ L = dθ − T a ∧ δ e a − S cd ∧ δ ω cd − E ψ ∧ δ ψ , (4.11)with θ = i κ ¯ ψ ∧ γ ∗ γ ∧ δ ψ , (4.12) T a = − i κ (cid:20) ¯ ψ ∧ γ ∗ γ a Dψ − L ¯ ψ ∧ γ ∗ γ ab e b ∧ ψ (cid:21) , (4.13) S cd = i κ ¯ ψ ∧ γ ∗ γ γ cd ∧ ψ , (4.14) E ψ = iκ (cid:20) ¯ ψ ←− ˆ D ∧ γ ∗ γ −
12 ¯ ψ ∧ γ ∗ γ a T a (cid:21) . (4.15)Having obtained an expression for θ , we can now obtain the Lorentz-diffeomorphism Noethercurrent via our general formula, J ξ = θ ( ψ , K ξ ψ ) − ξ · L . (4.16)To compute this we need the Kosmann derivative K ξ ψ of the Rarita-Schwinger field. An expressionfor K ξ ψ was given in Eq. (2.10). Expression Eq. (2.10) can be written more concisely in the formnotation as follows, K ξ ψ = ξ · ˚ Dψ + ˚ D ( ξ · ψ ) + 14 (cid:18) − E a · E b · dξ (cid:19) γ ab ψ , (4.17)or equivalently, K ξ ψ = ξ · (cid:18) Dψ − K ab γ ab ∧ ψ (cid:19) + D ( ξ · ψ ) − K ab γ ab ( ξ · ψ ) + 14 (cid:18) − E a · E b · dξ (cid:19) γ ab ψ = ξ · ( Dψ ) + D ( ξ · ψ ) + 14 (cid:18) − E a · E b · dξ − ξ · K ab (cid:19) γ ab ψ , (4.18)15here we have replaced ˚ D with D . In the new form the spinor covariant derivative includes torsion.The contorsion tensor K ab is defined via ω ab = ˚ ω ab + K ab , and can be expressed in terms of thetorsion tensor as K αµν = −
12 ( T αµν − T µνα + T ναµ ) . (4.19)The contorsion tensor is antisymmetric in the last two indices, while torsion tensor is antisymmetricin the first two indices. We can write contorsion tensor term ξ · K ab in terms of the torsion twoform. We have, ξ · K ab = ξ α K αµν E µa E νb = − ξ α ( T αµν − T µνα + T ναµ ) E µa E νb = 12 ξ α T µνα E µa E νb − ξ α ( T αµν − T ανµ ) E µa E νb = −
12 ( ξ · e c ) [ E a · ( E b · T c )] − E [ a · (cid:0) ξ · T b ] (cid:1) (4.20)where T a = (1 / T µνα E αa d x µ ∧ d x ν . Thus, the Kosmann derivative of the spinor one-form can bewritten as: K ξ ψ = ξ · ( Dψ ) + D ( ξ · ψ ) + 14 (cid:20) − E a · ( E b · dξ ) + 12 ( ξ · e c ) [ E a · ( E b · T c )] + E [ a · (cid:0) ξ · T b ] (cid:1)(cid:21) γ ab ψ . (4.21)It is slightly easier to work with expression Eq. (4.18), as it is less cumbersome. However, tocompare with some expressions in [6] it is better to use the form Eq. (4.21). We continue to useexpression Eq. (4.18) and use Eq. (4.20) to convert contorsion tensor into the torsion tensor whenneeded.Now, from equations Eq. (4.12) and Eq. (4.15), it follows that θ ( ψ , K ξ ψ ) = i κ ¯ ψ ∧ γ ∗ γ ∧ ξ · Dψ + i κ ¯ ψ ∧ γ ∗ γ ∧ D ( ξ · ψ ) −S cd (cid:18) E c · E d · dξ + ξ · K cd (cid:19) . (4.22)16he ξ · L term in J ξ = θ ( ψ , K ξ ψ ) − ξ · L is more tedious. We find, ξ · L = i κ ( ξ · ¯ ψ ) γ ∗ γ ∧ ˆ Dψ − i κ ¯ ψ ∧ γ ∗ γ a ˆ Dψ ( ξ · e a )+ i κ ¯ ψ ∧ γ ∗ γ ∧ ξ · Dψ − i κ (cid:18) L (cid:19) ¯ ψ ∧ γ ∗ γ γ a ∧ ψ ( ξ · e a )+ i κ (cid:18) L (cid:19) ¯ ψ ∧ γ ∗ γ ∧ γ ( ξ · ψ )= i κ ( ξ · ¯ ψ ) γ ∗ γ ∧ ˆ Dψ + i κ ¯ ψ ∧ γ ∗ γ ∧ ξ · Dψ (cid:18) − i κ ξ · e a (cid:19) (cid:20) ¯ ψ ∧ γ ∗ γ a Dψ − L ¯ ψ ∧ γ ∗ ( γ a γ b − γ b γ a ) e b ∧ ψ (cid:21) + i κ (cid:18) L (cid:19) ¯ ψ ∧ γ ∗ γ ∧ γ ( ξ · ψ )= i κ ( ξ · ¯ ψ ) γ ∗ γ ∧ ˆ Dψ + i κ ¯ ψ ∧ γ ∗ γ ∧ ξ · Dψ + ( ξ · e a ) T a + i κ (cid:18) L (cid:19) ¯ ψ ∧ γ ∗ γ ∧ γ ( ξ · ψ ) . (4.23)The above manipulations are as follows. In the first step we have expanded ξ · ˆ Dψ in ξ · Dψ andthe remaining terms. In the second step we have regrouped various terms to factor out ( ξ · e a ). Inthe third step we have regrouped terms such that the coefficient of ( ξ · e a ) is the energy-momentumthree-form T a introduced in Eq. (4.13).Combining Eq. (4.22) and Eq. (4.23), the Lorentz-diffeomorphism Noether current for theRarita-Schwinger part of the Lagrangian can be expressed as, J RS ξ = θ ( ψ , K ξ ψ ) − ξ · L = i κ ¯ ψ ∧ γ ∗ γ ∧ ˆ D ( ξ · ψ ) − S cd (cid:18) E c · E d · dξ + ξ · K cd (cid:19) − T a ( ξ · e a ) + i κ (cid:0) ξ · ψ (cid:1) γ ∗ γ ∧ ˆ Dψ . (4.24)The first term in the above equation can be rewritten as a sum of a total derivative and other terms(cf. Eq. (4.10)), i κ ¯ ψ ∧ γ ∗ γ ∧ ˆ D ( ξ · ψ ) = d (cid:18) i κ ¯ ψ ∧ γ ∗ γ ( ξ · ψ ) (cid:19) − i κ ¯ ψ ←− ˆ D ∧ γ ∗ γ ( ξ · ψ )+ i κ ¯ ψ ∧ γ ∗ γ a T a ( ξ · ψ ) , (4.25)while the last term in Eq. (4.24) can be Majorana flipped (cf. Eq. (4.7)) to get, i κ (cid:0) ξ · ψ (cid:1) γ ∗ γ ∧ ˆ Dψ = − i κ ¯ ψ ←− ˆ D ∧ γ ∗ γ ( ξ · ψ ) . (4.26)We finally have, J RS ξ = d (cid:20) i κ ¯ ψ ∧ γ ∗ γ ( ξ · ψ ) (cid:21) − T a ( ξ · e a ) − E ψ ( ξ · ψ ) − S cd (cid:18) E c · E d · dξ + ξ · K cd (cid:19) . (4.27)17he full Lorentz-diffeomorphism Noether current is therefore, J ξ = J grav ξ + J RS ξ = dQ grav ξ + d (cid:20) i κ ¯ ψ ∧ γ ∗ γ ( ξ · ψ ) (cid:21) + ( E a − T a )( ξ · e a ) + E ψ ( ξ · ψ )+ (cid:16) E cd − S cd (cid:17) (cid:18) E c · E d · dξ + ξ · K cd (cid:19) . (4.28)On-shell we have J ξ = dQ ξ where the total Noether charge Q ξ is Q ξ = Q grav ξ + Q RS ξ , (4.29)with Q RS ξ = i κ ¯ ψ ∧ γ ∗ γ a e a ( ξ · ψ ) , (4.30)and where we recall that, Q grav ξ = 132 π ε abcd e a ∧ e b (cid:18) E c · E d · dξ + ξ · K cd (cid:19) (4.31)= 132 π ε abcd e a ∧ e b (cid:18) E c · E d · ( d ξ − T e ( E e · ξ )) − ( E e · ξ )( E [ c · E e · T d ] (cid:19) . (4.32)Expressions Eq. (4.12), Eq. (4.13), and the expression for the Noether charge Eq. (4.30) are themain results of this section. We now have all the ingredients necessary to formulate a first law for simple AdS supergravity. Werecall that a first law is an identity relating the perturbed Hamiltonians for the horizon generatingKilling field evaluated at the bifurcation surface and at spatial infinity. The first variations of theHamiltonians are constructed out of the Noether charge 2-form Q ξ and symplectic potential 3-form θ as, δH ξ = Z Σ ω ( ϕ, δϕ, K ξ ϕ ) = Z ∂ Σ ( δ Q ξ − ξ · θ ) . (5.1)In this expression Σ is a Cauchy surface in the spacetime and ∂ Σ is its boundary, ω is the symplecticcurrent ω ( ϕ, δ ϕ, δ ϕ ) = δ θ ( ϕ, δ ϕ ) − δ θ ( ϕ, δ ϕ ) . (5.2)The Noether charge 2-form Q ξ and symplectic potential 3-form θ for both the gravitational La-grangian and the Rarita-Schwinger Lagrangian have been obtained in the previous sections.A first step to discuss the first law is to specify the boundary conditions for the gravitationaland Rarita-Schwinger fields. Asymptotically, we demand the metric to behave as global AdS, ds = − (cid:18) L r (cid:19) dt + (cid:18) L r (cid:19) − dr + r (cid:0) dθ + sin θdφ (cid:1) . (5.3)18 set of co-frames that capture the above metric is simply, e = (cid:18) L r (cid:19) / dt, e = (cid:18) L r (cid:19) − / dr, (5.4) e = rdθ, e = r sin θdφ. (5.5)The “Dirichlet” boundary conditions that define asymptotically AdS spacetimes in simple super-gravity were worked out in [21]. They are, e = e + O ( r − ) dr + O ( r − ) dt + O ( r − ) dθ + O ( r − ) dφ, (5.6) e = e + O ( r − ) dr + O ( r − ) dt + O ( r − ) dθ + O ( r − ) dφ, (5.7) e = e + O ( r − ) dr + O ( r − ) dt + O ( r − ) dθ + O ( r − ) dφ, (5.8) e = e + O ( r − ) dr + O ( r − ) dt + O ( r − ) dθ + O ( r − ) dφ. (5.9)and ψ t = (1 − γ ) O ( r − / ) , ψ r = (1 + γ ) O ( r − / ) , (5.10) ψ θ = (1 − γ ) O ( r − / ) , ψ φ = (1 − γ ) O ( r − / ) . (5.11)These boundary conditions ensure that the symplectic current is finite at the boundary and thatthe symplectic flux through the boundary vanishes [22]. The corresponding boundary conditionsin the asymptotically flat setting were first discussed in [23, 24].Let us assume that there is a stationary axisymmetric black hole solution to the field equations.Let the black hole horizon be a bifurcate Killing horizon generated by the Killing field k µ . Fromgeneral results on the bifurcate Killing horizon it follows that the Killing field is a linear combinationof time translation t µ = ( ∂/∂t ) µ and rotation φ µ = ( ∂/∂φ ) µ , k µ = t µ + Ω H φ µ , (5.12)where Ω H is a constant representing horizon angular velocity and that k µ vanishes at the bifurcation2-sphere.Let us also assume that the Rarita-Schwinger field is smooth on the relevant part of the space-time, i.e., in the neighbourhood of the future and past horizons, at the bifurcation 2-sphere, and inthe spacetime region outside the horizon all the way to infinity. Moreover, let us assume that theRarita-Schwinger field is stationary and axisymmetric, i.e., K t ψ = 0 and K φ ψ = 0 = ⇒ K k ψ = 0 . (5.13)These conditions ensure stationarity and axi-symmetry of the solution to the field equations. Sincethe vector k µ vanishes at the bifurcation 2-sphere, the contribution from the k · θ term in the19erturbed Hamiltonian δH k = Z B ( δ Q k − k · θ ) , (5.14)is zero. Our boundary conditions at infinity are such that there exists [22] a 2-form Θ , such that Z ∂ Σ ∞ k · θ = δ Z ∂ Σ ∞ k · Θ . (5.15)Hence, boundary Hamiltonians exist.For the gravity sector, at the bifurcation 2-sphere, the contribution to the perturbed Hamilto-nian δH k becomes T H δS . Here T H = ( κ B / π ) and S = (A / κ B is the surface gravity and A is the area of the bifurcation 2-sphere. This is because with k µ = 0 on the bifurcation surfaceand δk µ = 0 everywhere, one can argue that the variation of the temperature term is zero [1, 6].Since the Rarita-Schwinger field is assumed to be smooth on the horizon, the contribution ofthe Rarita-Schwinger field to the Noether charge Eq. (4.30) at the bifurcation 2-sphere is zero:since k µ = 0 at the bifurcation 2-sphere, k · ψ vanishes. Hence on the bifurcation surface, thecontribution from simple AdS supergravity is simply T H δS .At infinity, as is well known the contribution to the Hamiltonian from the gravitational fieldyields the ADM mass M ADM and angular momentum J ADM . Thus the variation of the gravita-tional Hamiltonian at infinity yields the variation of the ADM mass and angular momentum. Whiledepending on the nature of the solution, the Hamiltonian for the Rarita-Schwinger field may ormay not contribute at infinity. The boundary conditions we mentioned above are such that the supercharges are finite. With these boundary conditions contributions to the energy and angularmomentum from the Rarita-Schwinger field (which depends quadratically on the spinor field) van-ish. Combining these elements, the stationary comparison version of the first law for black holeswith bifurcate Killing horizons in simple AdS supergravity takes the form, T H δS = δ M ADM − Ω H J ADM . (5.16)In summary, we found that smooth, stationary, axisymmetric Rarita-Schwinger field does notexplicitly contribute to the black hole entropy. The extra term in the Noether charge vanishes atthe bifurcation surface. Near infinity, Rarita-Schwinger field falls-off sufficiently fast that it doesnot contribute to the integrals for the energy and angular momentum. Thus, the first law of blackhole mechanics in simple supergravity retains the same form as in pure general relativity. In this work we have applied and appropriately adapted the Lorentz-diffeomorphism Noether chargeformalism of references [5, 6] to four-dimensional gravity coupled to a Majorana field and to a Rarita-Schwinger field. In section 3 we studied the Lorentz-diffeomorphism Noether charge for a spin-1/220ajorana field. The Majorana condition brings in certain new elements in the computation. Itserved as a warm-up for the Rarita-Schwinger field in the context of simple supergravity consideredin section 4. As we saw in that section the Majorana nature of the Rarita-Schwinger field playedan important role in the computations. A key result of our work is expression Eq. (4.30) forthe contribution to the Lorentz-diffeomorphism Noether charge due to the Rarita-Schwinger field.Using this Noether charge we formulated a stationary comparison version of the first law in section5. In our analysis of the first law with the Rarita-Schwinger field we made two important as-sumptions: (i) The Rarita-Schwinger field is smooth everywhere in the region of interest, (ii) TheRarita-Schwinger field is annihilated by the Kosmann derivative with respect the horizon generat-ing Killing field. Using these assumptions, we concluded that the Rarita-Schwinger field does notcontribute to the first law at the bifurcation surface. Perhaps these assumptions are too restric-tive. This situation should be compared to the analysis of the Yang-Mills field by Sudarsky andWald [13, 14]. Under similar assumptions, namely (i) a smooth Yang-Mills field can be chosen onthe spacetime, and (ii) it is annihilated by the Lie derivative with respect the horizon generatingKilling field, they also concluded that the Yang-Mills field does not contribute to the first law at thebifurcation surface. Over the years, this conclusion has been refined. In 2003 Gao [12] argued thatthe Yang-Mills field does contribute to the first law at the horizon, but he was not able to write thecontribution as a potential times the perturbed charge without making additional assumptions. In2015 Prabhu [6] by formulating the problem in terms of the principal bundle gave a satisfactorydiscussion of the first law for gravity coupled to a Yang-Mills field. He showed that the Yang-Millsfield contributes to the first law both at the bifurcation surface and at infinity. The contributionsare of the form potential times the perturbed charge, and generically it is not possible to write thetwo terms as the ‘difference in the potential between infinity and the bifurcation surface’ times theperturbed charge.It is natural to speculate that similar refinements are to be found with the Rarita-Schwingerfield. A reason our analysis is ill-equipped to address this question is that we have ignoredthe fermionic gauge symmetry of the Rarita-Schwinger field. The fermionic gauge symmetry issupersymmetry—a spacetime symmetry. It changes the frame field as well. The principal bundleformalism of Prabhu [6], though quite general, is not equipped to handle supersymmetry. Perhapsa formulation of the first law is possible using the superspace formalism of supergravity. The su-perspace is discussed at length in the mathematical physics literature [25, 26, 27, 28]. In such aformulation, we expect that the above mentioned shortcomings can be addressed and that super-charges may feature in the first law. Such a discussion would shed further light on black holes in21upergravity. We leave this for future work.For a class of supersymmetric black holes it is known that smooth (at least at the futurehorizon) normalisable linearised fermionic hair modes exist [29, 30]. It will be very interesting tounderstand how these modes appear in the first law for the corresponding black holes.We hope to return to these questions in our future work.
Acknowledgements
We thank Alok Laddha, Bindusar Sahoo, and Simone Speziale for discussions. SC thanks AEIPotsdam and CMI Chennai for hospitality towards the initial and final stages of this project re-spectively. The work of PBA, SkJH, and AV is supported in part by the Max Planck Partnergroup“Quantum Black Holes” between CMI Chennai and AEI Potsdam and by a grant to CMI from theInfosys Foundation. The research of SkJH is also supported in part by the Czech Science Foun-dation Grant 19-01850S. The work of SC is supported in part by the INSPIRE Faculty fellowship(Reg. No. DST/INSPIRE/04/2018/000893).
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