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Quasilocal rotating conformal Killing horizons

The formulation of quasi-local conformal Killling horizons(CKH) is extended to include rotation. This necessitates that the horizon be foliated by 2-spheres which may be distorted. Matter degrees of freedom which fall through the horizon is taken to be a real scalar field. We show that these rotating CKHs also admit a first law in differential form.


I. INTRODUCTION
Black holes in general relativity behave like thermal objects. This analogy is based on number of facts. It is known that in general relativity, the surface gravity κ H of a stationary black hole must be a constant over the event horizon [1]. Moreover, the first law of black hole mechanics, which refers to stationary space-times admitting an event horizon and small perturbations about them, states that the differences in mass M , area A and angular momentum J to two nearby stationary black hole solutions are related through δM = κ H δA/8π + Ω H δJ. Additionally, according to the second law, area of black holes can never decrease in a classical process [2]. Hawking's proof that due to quantum processes, black holes radiate to infinity, particles of all species at temperature κ H /2π, implies that the laws of black hole mechanics are indeed the laws of thermodynamics [3][4][5].
These derivations, of laws of black hole mechanics, require that the spacetime admits an event horizon and hence the spacetime is assumed to be stationary. In that case, the event horizon of a stationary black hole is a Killing horizon. However, not all Killing horizons require that the entire spacetime be stationary. Indeed, one may have Killing horizons which has a timelike Killing vector field in the neighbourhood of the horizon only. Since Killing horizons give a local description of black hole horizons, one may enquire if the laws of black hole mechanics hold good for Killing horizons too. If it is, then the laws of black hole mechanics are specific to inner black hole boundaries. Remarkably, the laws of black hole mechanics hold good for bifurcate Killing horizons. The framework of Killing Horizon is also useful to study and unravel the origin of entropy and black hole thermodynamics [6][7][8][9][10][11][12][13]. Killing horizons are not the only local description of black hole boundaries, one can construct more. The notion of trapping horizons is one such description [14,15]. However, as it turned out, the formalism of isolated horizons and dynamical horizons [16][17][18][19][20][21][22][23][24][25][26][27][28] are better suited to address the questions regarding classical and quantum mechanics of black holes. Not only are the laws of black hole mechanics can be proved here, but the entropy of black holes can also be determined by counting the black hole microstates residing on the horizon only [29][30][31][32][33][34][35][36][37].
Another class of horizons that are of interest are conformal Killing horizons (CKH) [38][39][40][41][42]. These horizons capture the essence of dynamical situations. CKHs are null hypersurfaces whose null geodesics are orbits of a conformal Killing field. More precisely, if ξ a is a vector field satisfying £ ξ g ab = 2f g ab , and is null, it generates a CKH for the metric g ab . Since ξ a generates a null surface, and generates geodesics, one can define an acceleration given by the relation ξ b ∇ b ξ a = κ ξ ξ a . Then, it arises that the quantity (κ ξ − 2f ) which is a combination of the acceleration of the conformal Killing vector and the conformal factor, is Lie dragged along the horizon. Moreover, if the stress-energy tensor satisfies the strong energy condition, then this quantity is a constant on the horizon. It can therefore be interpreted as a temperature. Thus, a form of zeroth law holds for these horizons. One may enquire if a quasi-local formulation of conformal Killing horizons may be developed. This extension would be similar to the generalisation of Killing horizons to isolated horizons. A conformal Killing horizon is defined retroactively and one needs to know the full space-time history and a globally defined conformal Killing vector. In contrast, a quasi-local conformal Killing horizon only requires the existence of a null hypersurface generating vector field. 1 Indeed, it has been shown that it is possible to broaden the boundary conditions to construct a quasi-local conformal Killing horizon and that these horizons have a zeroth law and a first law [43]. The basic idea goes as follows: Consider a spacetime M having a null boundary ∆ with non-zero expansion (θ = −2ρ = 0) but the null generators of ∆ are assumed to be shear-free. These conditions guarantee that the null generators l a are conformal Killing vectors on ∆. Clearly, these null surfaces are not expansion free, they may be growing. Indeed, £ l 2 ǫ = θ 2 ǫ and hence, they are good candidates for growing horizons. It was further assumed in [43] that the horizons expand due to the reason that matter fields fall through these horizons. For definiteness, this matter field was taken to be a massless scalar field satisfying the condition This assumption is motivated by the fact that l a is a conformal Killing vector on ∆. The first law for quasi-local CKHs was shown to get the form dU = T dS along with a flux term arising due to matter fields falling through the horizon. Thus, it seems that in a particular class of dynamical situation, one may obtain a form of first law.
However, since the most useful application of these geometrical structures are in the dynamical evolution of black holes, one must address some further issues left out in [43]. Suppose that the horizon ∆ is rotating with some angular momentum J. What are the boundary conditions which will ensure a zeroth law? Are these boundary conditions enough to construct the space of solutions of general relativity? How would one define an angular momentum? By how much do the first law change in the presence of rotation ? Can the first law be written in a differential form ? In this paper, we answer these questions.
The plan of the paper is as follows. We start by developing the geometry of a rotating quasi-local conformal Killing horizon. To account for rotation, we assume that there is a spacelike conformal Killing vector φ a on ∆ such that it commutes with l a . We show that the zeroth law is valid. Using the first order action, we construct the space of solutions of Einstein's theory and show that a well-defined hypersurface independent symplectic structure exits. In the next section, we construct angular momentum as a Hamiltonian corresponding to the axial conformal Killing vector field. We derive the first law for rotating horizons and show that it may be written in a differential form. We will closely follow the formalism already laid down in [43]. However, there are some crucial changes since we allow for rotating cross-sections.

II. BOUNDARY CONDITIONS FOR ROTATING QUASILOCAL CKH
Let M be a 4-manifold equipped with a metric g ab of signature (−, +, +, +). We assume that all fields on M are smooth. Let ∆ be a null hypersurface of M. On this hypersurface, we construct the Newman-Penrose basis (l, n mm), where l a is the future directed null normal and n a the transverse and future directed null vector field to ∆. The set of complex null vector field (m,m) are taken to be tangential to ∆. This null tetrad (l, n mm) satisfy the condition l.n = −1 = −m.m, while all other scalar products vanish. The degenerate metric on this hypersurface ∆ is denoted by q ab . The expansion θ l of the null normal is defined by q ab ∇ a l b . In terms of the Newman-Penrose formalism, θ l = −2ρ (see appendix A of [43] or [44] for details). The accelaration of l a can be obtained from the expression l a ∇ a l b = (ǫ +ǭ)l b and is given by κ l := (ǫ +ǭ). 2 One may further define an equivalance class of null normals [l a ] such that l and l ′ belong to the same equivalance class if l ′ = cl where c is a constant on ∆.
Definition: A null hypersurface ∆ of M will be called quasi-local conformal horizon if the following conditions hold.: 1. ∆ is topologically S 2 × R and null.
2. The shear σ of l a vanishes on ∆ for any null normal l a .
3. All equations of motion hold at ∆ and the stress-energy tensor T ab on ∆ is such that −T a b l b is future directed and causal.
4. If ϕ is a matter field then it must satisfy £ l ϕ = −2ρ ϕ on ∆ for all null normals l a .
6. There is a spacelike axial conformal Killing vector φ a on S ∆ such that £ φ 2 ǫ = −2g 2 ǫ and it commutes with the l a viz. [φ, l] a = 0 7. φ a has closed circular orbits of length 2π and vanishes on exactly two generators of ∆.
8. For the matter field ϕ, £ φ ϕ = −2gϕ for some smooth function g on ∆ The conditions 1 − 5 are already described in [43] where it has been established that they describe a quasi-local conformal horizon. To include rotations, we must take into account a description of the vector fields that generate the 2-spheres of the horizons and incorporate the boundary conditions on these fields. Thus, for a rotating CKH, apart from the above conditions (1 − 5), the conditions (6 − 8) are also assumed to be true. The first condition imposes restrictions on the topology of the hypersurface. The cross-sections of such quasi-local horizons may admit other topologies but we do not include such generalities here.
The second boundary condition concerns shear which measures the amount of gravitational flux flowing across the surface. We assume that the gravity flux be vanishing. This boundary condition on the shear σ of null normal l a has several consequences. First, since l a is hypersurface orthogonal, the Frobenius theorem implies that ρ is real and κ = 0. Secondly, l a is twist-free and a geodetic vector field. The acceleration of l a is given through expression l a ∇ a l b = (ǫ +ǭ)l b where, κ l := (ǫ +ǭ). The acceleration varies in the equivalence class [cl a ] since in the absence of the knowledge of asymptotics, the acceleration cannot be fixed. Thirdly, a Ricci identity is given by where D = l a ∇ a , δ = m a ∇ a , Ψ 0 is one of the Weyl scalars and the other quantities are the Newman-Penrose scalars (see [44] for details). For σ ∆ = 0, it implies Ψ 0 ∆ = 0. Further, it can be seen that the null normal l a is such that which implies that l a is a conformal Killing vector on ∆. Moreover, the Raychaudhuri equation implies that R ab l a l b = 0 and hence −R a b l b can have components which are tangential as well as transverse to ∆. The third boundary condition only implies that the field equations of gravity be satisfied and that the matter fields be such that In the fourth boundary condition, we have kept open the possibility that matter fields may cross the horizon and the horizon may grow. The matter field is taken to be a massless scalar field ϕ which behaves in a certain way which mimics it's conformal nature. The fifth condition is motivated if the fact that (2ρ + ǫ +ǭ) remains invariant under conformal transformations [40,41]. This can also be shown as follows. A conformal transformation of the metric amounts to a conformal transformation of the two-metric on ∆. Under a conformal transformation g ab → Ω 2 g ab one needs a new covariant derivative operator which annihilates the conformally transformed metric. Under such a conformal transformation l a → l a , l a → Ω 2 l a , n a → Ω −2 n a , n a → n a , m a → Ω −1 m a , m a → Ωm a . The new derivative operator is such that it transforms as If one defines a one-form as ω a It follows that the Newman-Penrose scalars ρ = −m amb ∇ a l b and σ = −m amb ∇ a l b transform in such a way that (2ρ + ǫ +ǭ) remains invariant under a conformal transformation. Further, since the Weyl tensor is invariant under a conformal rescaling, it follows that Ψ 1 ∆ = 0 in this case. The sixth and the eighth conditions on the vector field φ a are motivated from the similar conditions on l a and that the vector field must preserve the geometric structures on ∆. The seventh condition is just the statement that the integral curves of φ a cover the sphere.

Gauge choices
Since the null tetrad is typically not a coordinate basis, it leads to non-trivial commutation relations [45]. The following two commutation relations are useful to choose a gauge so as to make the commputations simpler.
where D = l a ∇ a , ∆ = n a ∇ a , δ = m a ∇ a . Since l a is geodetic one can choose a coordinate v on ∆ such that £ l v = 1.
If we choose the function f = v,then from the above commutation relations it follows that µ =μ and π = α +β.

Calculation of dω
Since we deal with rotating horizons it is useful to obtain an expression for dω. This is because dω contains the imaginary part of the Weyl scalar Ψ 2 . The information of angular momentum is contained in this scalar. In the case of an isolated horizon dω IH = 2(ImΨ 2 ) 2 ǫ. However unlike an isolated horizon the induced connection (ω) on a CKH is not Lie-dragged along the horizon but is however related to ω IH through a conformal transformation as demonstrated in eq (4). So one needs to check if the induced connection on a CKH also contains the information of angular momentum. Further it would also be a check if the results of isolated horizon can be recovered if one goes from ω to ω IH via a conformal transformation (4). We start with the definition of Riemann tensor given as Consider the left hand side of the above equation. Using the expression for ∇ a l b given in [43] we get, Contracting the above by n c , pulling back the expression on to ∆ gives and using the gauge choices, gives, Now the Weyl tensor can be written in terms of the curvatures as follows: We also expand the Ricci tensor and the Weyl tesor in a Newman-Penrose basis and obtain the following results: Combining the above expressions one gets, Using the gauge choices and the Ricci identity Which immediately implies that, Now we can again rewrite this expression in terms of the conformally transformed connectionω with the choice £ l log Ω = ρ to get, which is exactly the espression obtained for an isolated horizon.

Symmetries of Quasi-local conformal horizons
In this section we discuss the infinitesimal symmetries of a quasi-local conformal horizon. In the absence of asymptotic infinity, symmetry in our case would mean preserving the relevant geometric structures of the horizon. It is at once clear that only vector fields tangent to ∆ boundary conditions preserving. Let us consider a general case first. Since ∆ is a hypersurface it follows that the vector fields tangent to ∆ form a closed lie algebra. Let ξ a be a vector field tangent to the horizon. Then ξ a will be said to be a symmetry generating vector field if the following conditions hold 1. It preserves the equivalence class of null normals i.e [ξ, l] a ∆ = cl a where c is a constant on ∆ 2. ξ a is a conformal Killing vector on ∆. If q ab is the degenerate metric on and h is smooth function on ∆ then, 3. It Lie drags the conformally transformed connectionω on ∆ with the conformal factor satisfying £ l log Ω ∆ = ρ.
Note that the third condition £ φω is analogous to the one for a weakly isolated horizon [? ] It is immediately clear that φ a satisfies the first two conditions. However, to qualify as a symmetry vector field, the third condition on φ a must also be met. It implies certain conditions on fields. First, we note the following: Together the above two equations can be rewritten together as, Since l a commutes with φ it follows that £ l g = £ φ ρ. The symmetry vector field puts the following restriction on the conformal factor Ω, given by £ l log Ω = ρ. It immediately follows that, Since log Ω is not completely determined by the condition £ l log Ω = ρ one still has the liberty of choosing the spatial dependendence of log Ω such that £ φ log Ω = g. Then condition (26) ensures that it holds everywhere on ∆. The above calculation shows that one can choose log Ω such that both l a and φ a are Killing vectors of the conformally transformed metric.

III. ACTION PRINCIPLE, PHASE SPACE AND THE FIRST LAW
We are interested in constructing the space of solutions of general relativity, and we use the first order formalism in terms of tetrads and connections to construct the covariant phase-space. For the first order theory, we take the fields on the manifold to be (e a I , A aI J , ϕ), where e a I is the co-tetrad, A aI J is the gravitational connection and ϕ is the scalar field. The Palatini action in first order gravity with a scalar field is given by: where Σ IJ = 1 2 ǫ IJ KL e K ∧ e L , A IJ is a Lorentz SO(3, 1) connection and F IJ is a curvature two-form corresponding to the connection given by F IJ = dA IJ + A IK ∧ A K J . The action might have to be supplemented with boundary terms to make the variation well defined. We now check that the variational principle is well-defined if the boundary conditions on the fields, as given in the previous section, hold.
The Lagrangian 4-form for the fields (e a I , A aI J , ϕ) is given in the following way.
The first variation of the action leads to equations of motion and boundary terms. The equations of motion consist of the following equations. First, variation of the action with respect to the connection implies that the curvature F IJ is related to the Riemann tensor R cd , through the relation F ab IJ = R ab cd e I c e J d . Second, variation with respect to the tetrads lead to the Einstein equations and third, the first variation of the matter field gives the equation of motion of the matter field (details can be found in the appendix of [43]). On-shell, the first variation is given by the following boundary terms The quantity Θ(δ) is called the symplectic potential and the boundary terms are to be evaluated on the initial and final spacelike boundaries M − , M + , asymptotic infinity and the internal boundary ∆. However, since fields are set fixed on the initial and the final hypersurfaces, they vanish. The boundary conditions at infinity are assumed to be appropriately chosen and they can be suitably taken care of. The only terms which are of relevance for this case are the terms on the internal boundary. On the conformal horizon ∆, the Σ IJ is given by [43] Σ and the connection is given by [43] A Consider the gravity terms first 3 By using the Ricci identities in terms of Newman-Penrose coeffecients, we find from equations (29), (30) and (31) that Note that this expression is analogous to the one in [43] but have few differences. Here, we have terms like 2 ǫ ∧ δ[(α +β)m + (ᾱ + β)m] were ignored in [43] owing to the fact that the horizon 2-surfaces were assumed to be round 2-spheres. Here, we have to take them into account and hence these terms have been retained throughout. The matter Lagrangian leads to the following variation: Adding everything up, one finds that a total derivative term survives and can be written as: Since Einstein's equations give R 11 = 8πG T 11 , R 13 = 8πG T 13 and R 14 = 8πG T 14 only − 2ρ n ∧ 2 ǫ survives. Thus, if one adds the term 16πG S ′ = ∆ 2ρ n ∧ 2 ǫ to the action, it is well defined for the set of boundary conditions on ∆. As we shall see below, since this is a boundary term, it does not contribute to the symplectic structure.
Covariant phase-space and the symplectic structure The symplectic potential Θ(δ) in eqn. (29) is a 3-form in space-time and a 0-form in phase space. Given the symplectic potential, one can construct the symplectic current J(δ 1 , δ 2 ) = δ 1 Θ(δ 2 ) − δ 2 Θ(δ 1 ), where S B is the 2-surface at the intersection of any hypersurface M with the boundary B. The quantity j(δ 1 , δ 2 ) is called the boundary symplectic current. In this case, the boundaries are the null boundary ∆ and a boundary at infinity. We shall assume that the fall-off on the fields are such that the integrals over the surface at infinity vanishe. Indeed, if the asymptotic is flat the fall-off on the fields can be set such that the contribution from the cylinder at infinity is zero. Thus, the contributions to the symplectic structure would come from the spheres on the inner boundary ∆. Our strategy shall be to construct the symplectic structure for the action given in eqn. (27). Let us first look at the Lagrangian for gravity. The symplectic potential in this case is given by, 16πGΘ(δ) = −Σ IJ ∧ δA IJ . The symplectic current is therefore given by, The above expression eqn. (37), when pulled back and rescticted to the surface ∆ gives where we have used eqns. (29), (30) and (31) in the first line and eqns. (32) in the second line. The first term in the above expression is exact but not others. However, we show that the contribution of the scalar field is such that the symplectic current is exact and one may obtain a boundary symplectic current. The symplectic current for the real scalar field is given by, J M (δ 1 , δ 2 ) = 2 δ [1 ϕ δ 2] ⋆ dϕ. The symplectic current on the hypersurface ∆ can be obtained as where D = l a ∇ a . The boundary condition on the scalar field implies Dϕ = −2ρ ϕ and hence, we get that The combined expression obtained by using the Einstein field equations, is then given by: It follows that the hypersurface independent symplectic structure is given by: In the next section, we shall use this expression to derive the first law of mechanics for the conformal Killing horizon.

Angular momentum as Hamiltonian
Angular momentum is usually defined as a conserved charge for an axial Killing vector. However, in dynamic situations charges may not be conserved. One requires a way to define angular momentum which would remain valid in the time-independent case too. Consider that angular momentum to be the Hamiltonian function corresponding to a space-like rotational vector-field. If this vector field is also Killing, the Hamiltonian function will match with the angular momentum obtained as a conserved charge of the Killing vector field. would match with the it would In our case it turns out that there is a choice of such a vector field, as discussed. We will therefore define the angular momentum to be the Hamiltonian corresponding to the spacelike conformal Killing vector φ a . We need to impose a few conditions on the fields to make a well defined Hamiltonian. These conditions are required since the action of δ φ on some phase-space fields in not like £ φ . First, we note the following equalities We assume that δ φ acts on (2ρ + ǫ +ǭ) and 2 ǫ ϕ like £ φ . Moreover, since δ φ £ l (2ρ + ǫ +ǭ) = 0 it immediately implies that £ l δ φ (2ρ + ǫ +ǭ) = 0. Hence, one may choose the variables in such a way that δ φ (2ρ + ǫ +ǭ) = 0 This implies that if we set δ φ 1 4πG log ρ − 1 8πG log ϕ − ϕ 2 =0 at the initial cross-section, it remains zero everywhere on ∆ and so, Another condition can be derived from the equation above The variations δ φ satisfy the following differential equations, which can be checked to be consistent with each other: Putting condition (46) in (48), we get where C(θ, φ), is a constant of integration. If we choose this constant C(θ, φ) = 0, it immediately implies that δ φ ϕ = 0 = δ φ 2 ǫ. With the choice of δ φ only the bulk symplectic structure survives and one getsIt therefore follows that: Note that the matter field part in the above expression will not contribute. We have also assumed that at infinity the contribution to the symplectic structure is a total variation δJ ∞ . It follows that Now we define J φ ∆ = H φ − J ∞ to be the horizon angular momentum. This is consistent with the fact that the angular momentum at ∆ is the difference of the contribution at infinity and the total angular momentum. Then . Note that while defining a temperature we used the conformally transformed connectionω (with the choice £ l log Ω = ρ). So ideally one should be defining an angular momentum to be 1 8πG S∆ (φ ·ω) 2 ǫ. In the next few lines we will show that these two definitions are equivalent. The transformation of the pull-back of the connection on to to ∆ has already been discussed. It follows that, It therefore follows that Since φ a is purely tangential to ∆, it follows that the last integration is zero. Morever since we are integrating over a compact surface, the first integral is also zero. It therefore follows that.
Hamiltonian evolution and the first law Given the symplectic structure, we can proceed to study the evolution of the system. We assume that there exists a vector which gives the time evolution on the spacetime. Given this vector field, one can define a corresponding vector field on the phase-space which can be interpreted as the infinitesimal generator of time evolution in the covariant phase-space. We assume there is a vector field t a in space-time which generates time evolution. It then follows that (t a + Ωφ a ) ǫ [l a ], where Ω is a constant on ∆ but may vary over the space of solutions. Therefore t a = l a − Ωφ a is a live vector field.
Where we have redefined our Hamiltonian H t =H t + S∆ ρ 2 ǫ. This redefination is possible since the definition of the Hamiltonian is ambiguous upto a total variation. Further, as expected Ω(δ t , δ t ) = 0. Next we define, E t ∆ = E ∞ − H t , as the horizon energy. It is clear from above that for ρ → 0 (i.e in the isolated horizon limit) it matches with the definition in [23,24] if asymptotics is flat and E ∞ = E ADM . It therefore follows that: To recover the the more familiar form of first law known for a dynamical situation, we assume there is a vector fieldδ on phase space which acts only on the fields on ∆ (and not in the bulk) such that it's action on the boundary variables is to evolve the boundary fields along the affine parameter v (it may be interpreted to be a time evolution, like £). Now demanding thatδ to be Hamiltonian would give an integrability condition which also ensures that δ t is Hamiltonian. So one can calculate Ω(δ, δ t ) :=δH t to get which can be written in the following form 4 where dots imply changes in the variables produced by the action ofδ. Note that ifδ = £ t , then,δϕDϕ gives the expression T ab t a l b . Equation (61) is the form of evolution for the conformal Killing horizons. The first term in the above expression is the usual T dS term while the second term is a flux term which takes into account the non-zero matter flux across ∆.

IV. DISCUSSIONS
In this paper, we have developed the formalism of rotating quasi-local conformal Killing horizon. We have constructed the phase-space, the symplectic structure on the space of solutions and showed that a form of first law arises which can be written in the differential form. More specifically, the first law can be written as dU = T dS + ΩdJ+ flux terms, where the flux arises due to the matter fields falling in through the horizon. The phase-space of the above construction consists of all those solutions which, on the horizon, satisfy the boundary conditions of a CKH as mentioned in the section II. These solutions, which constitute the phase-space, are in a conformal class. In other words, the space of solutions constitute points which are such that their geometrical quantities are covariant under a specific conformal transformation. For example, the connections induced on ∆, given by ω andω ( see eqn. (4)) and the surface gravity obtained from it are in the same equivalence class. As a result of this construction, the first law holds good for all solutions in phase-space which are related by a specific conformal transformation.
Some comments on the angular momentum are as follows. Generally, angular momentum is defined to be the conserved charge corresponding to an axial Killing vector. However, in absence of such a Killing vector, one may define angular momentum to be a Hamiltonian function corresponding to an angular vector field. This is precisely what we have done here. However, in doing so, one must keep in mind that a conformal Killing vector field of a generic sphere may also be considered as a conformal Killing vector of a round metric. The conformal transformations of a two-sphere is the Möbius group SL(2, C)/Z 2 . Also, since the homomorphism SL(2, C) to the restricted Lorentz group SO + (1, 3) is surjective with kernel 1 and − 1, it is possible to study the possible and relevant generators on the two-sphere from the generators of the restricted Lorentz group. There are three types of restricted Lorentz transformations: rotations in spacelike 2-plane, boosts in a timelike two plane and null rotations. Out of these, the only non-trivial vector field on the sphere is the rotation generator. We have shown that one can obtain a conformal rotation vector field whose Hamiltonian is the angular momentum. Further, in case ρ → 0, the expression for angular momentum exactly matches with those obtained for isolated horizons (which again is exactly same as the Komar integral).
Given a form of the first law, it may be compared to the first law of thermodynamics. However, since the horizon is growing, it describes a non-equilibrium situation. Still one gets a differential form of first law. This is due to the reason that in the case of conformal Killing horizons, the horizon identifies a 'time' and hence a meaningful notion of time translation arises. This leads to a definite identification of temperature and entropy. One may then enquire if that entropy can arise from some counting of microstates. In case of isolated horizons, the boundary symplectic structure has a natural interpretation of being the symplectic structure of a field theory residing on the boundary. A quantization of the boundary theory therefore provides a microscopic description of the entropy of the isolated horizon. The situation is similar here and hence it remains to see if such an interpretation may be given for conformal Killing horizons too. which implies DA m − A(ᾱ + β −π) + A(ρ + ǫ −ǭ) m + c.c = 0 (A2) To find relations for φ to be a conformal Killind vector on ∆. We consider the conformal Killing equation for φ.