Laws of Black Hole Mechanics from Holst Action
aa r X i v : . [ g r- q c ] F e b Laws of Black Hole Mechanics from Holst Action
Ayan Chatterjee ∗ and Amit Ghosh † Theory Division, Saha Institute of Nuclear Physics, Kolkata 700064, India
The formulation of Weak Isolated Horizons (WIH) based on the Isolated Horizon formulation ofblack hole horizons is reconsidered. The first part of the paper deals with the derivation of lawsof mechanics of a WIH. While the zeroth law follows from the WIH boundary conditions, first lawdepends on the action chosen. We construct the covariant phase space for a spacetime having an WIHas inner boundary for the Holst action. This requires the introduction of new potential functions sothat the symplectic structure is foliation independent. We show that a precise cancellation amongvarious terms leads to the usual first law for WIH. Subsequently, we show from the same covariantphase space that for spherical horizons, the topological theory on the inner boundary is a U (1)Chern-Simons theory. PACS numbers: 04070B, 0420
I. INTRODUCTION
In general relativity black holes are exact solutions of Einstein’s equations and are the simplest macroscopic objects of Nature [1]. Therefore, it is only natural that such objects are perfect laboratories for the search of the quantumtheory of general relativity (much like the way solitons are ideal laboratories of any non-perturbative quantum theoryof non-abelian gauge fields interacting with scalar fields). In the last century one of the major development in thestudy of black hole physics came through the realization that these solutions are analogous to some macrostates inthermal equilibrium. Dynamical processes involving only black holes obey laws that are qualitatively similar to thefour laws of thermodynamics - the zeroth, first, second and the third laws. The proofs of these laws, called the lawsof black hole dynamics, were first given by [2], where black hole spacetimes were supposed to contain some eventhorizons. However, it was soon realized that the notion of event horizons is too impractical. The global features ofsuch horizons (for example one needs the entire asymptotic future null infinity to know whether an event horizon ispresent) percolate everywhere in the derivation of the said four laws of black hole dynamics [2], making the laws tooabstract and unsuitable for use in many practical situations. notion of event horizons (see [3] for details). Killinghorizons, which were introduced as a practical alternative to the event horizons, are closer to reality. First of all,Killing horizons are almost local, requiring the Killing vector only on and in the vicinity of the black hole horizon. Thelaws of black hole mechanics were also proved in this setting [4, 5], although it had difficulties in handling the extremalblack holes at the same footing of the non-extremal ones. One difficulty being that the proofs for the laws of Killinghorizon dynamics crucially depends on the existence of a bifurcating two-sphere at the horizon, which are absent forthe extremal black holes. The precise dependence on the bifurcation two-sphere goes as follows: For non-extremalhorizons the zeroth law states that the the surface gravity is non-zero and constant on the Killing horizon. Such alaw holds if and only if the horizon is extendible to a bifurcating horizon. Furthermore, the Noether charge coming asa surface integral over a cross-section of the horizon, which is defined to be the entropy /2 π in this case, is expressedin terms of all dynamical fields, their derivatives and also on the Killing field χ a and its derivatives. However, theexplicit dependence on χ a can be eliminated provided one uses the cross-section to be the bifurcation two-sphere. Onecan also eliminate the second and the higher derivatives of χ a by using Killing vector identities, leaving the entropyas a function of χ a and ∇ a χ b . The contribution of the term linear in χ a vanishes at the bifurcation point since theKilling vector χ a vanishes at that point. Finally, at the bifurcation surface one has ∇ a χ b = ǫ ab , where ǫ ab denotes thebinormal. Thus, at the bifurcation point all explicit reference to the Killing field can be eliminated from the Noethercharge Q . Thus Q evaluated at the bifurcation two-sphere defines the entropy only as a function of local geometricquantities such as the metric, the matter fields and their derivatives [4]. Using the extension of the horizon one canthen evaluate the entropy as an integral over an arbitrary section of the Killing horizon rather than on the bifurcationtwo-sphere alone, provided the surface gravity is constant and nonvanishing somewhere on the Killing horizon.Isolated horizon [3, 6–8] is another local and practical alternative to a black hole horizon whose descriptions requirethe existence of some marginally trapped surfaces at the horizon. Since there is no explicit reference to any Killing ∗ Electronic address: [email protected] † Electronic address: [email protected] vector, isolated horizons claim to do a better job than the Killing ones (see [8–10] for a detailed comparison and therecent surveys). In fact, a large number of black hole horizons are isolated horizons but not Killing horizons. Thereason for this enhancement in the space of solutions is that the boundary conditions defining an isolated horizonare weaker compared to the ones defining a Killing horizon. For example, as a consequence of the isolated horizonboundary conditions it is seen that such horizons admit a Killing vector field only on the horizon, whereas a Killinghorizon requires such a Killing vector field in some neighborhood of the horizon. The isolated horizon formalismwas initially formulated in terms of canonical variables leading to the canonical phase space [3, 6]. This formulationalso used to show that the effective field theory on the isolated horizon is a Chern-Simons theory [11]. The detailedcalculation, extensions and other consequences are discussed in [13] This also led to the calculation of the entropydone first in [12]. The covariant formulation was discussed in [7]. The various extensions of the isolated horizons andits ramifications were discussed in a series of papers [14].One important advantage of the isolated horizons over the Killing ones is that it can deal with the extremal solutionsat the same footing of non-extremal ones [15]. This requires the standard isolated horizon boundary conditions tobe weakened enough so as to contain the extremal and non-extremal horizons as part of the same phase space. Thestandard formulation of isolated horizons takes a rigid class of null normals (which are Killing vectors only on thehorizon) where the null vector fields associated with a null surface are allowed to rescale only by a positive constant.While this is definitely a possibility, the isolated horizons may actually admit a much larger class of null vectors. Thenew formulation, called weak isolated horizons (WIH), proposed in [15], relaxes this rigidness and allows rescalingby a class of functions. This new rescaling opens up the possibility of extending the space of solution of an isolatedhorizon. The standard formulation of isolated horizons place extremal and non-extremal solutions in two distinctphase spaces, much like the way they have been treated in the Killing horizon formulations; whereas in WIH, onegets a single unified space of solutions that contain both types of solutions. This implies that the laws of mechanicsof WIHs, with these improved set of boundary conditions, encompass both extremal and non-extremal solutions atone go.In this paper, we shall derive the laws of mechanics of an WIH from Holst’s action [16] from a totally covariantframework. In the framework of Loop Quantum Gravity (LQG) this action is a natural starting point than the Palatiniaction. From this action we construct the phase space for a solution having a WIH as an inner boundary (this meansthat the spacetime admits an inner boundary which, in the present case, satisfies the WIH boundary conditions).The symplectic structure for this phase space is obtained, from which we prove the first law of black hole mechanics.From this covariant symplectic structure one then finds that the effective theory at the spherically symmetric WIH isprecisely a U (1) Chern-Simons theory. The Chern- Simons one- form gauge field is such that it does not depend onthe extremal or non- extremal nature of the horizon. This shows that the effective theory for the spherical symmetrichorizons, extremal or non- extremal, is a U (1) Chern-Simons theory. Correspondingly, the entropy of these horizonswill again be proportional to area of the horizon. That the effective topological theory on the horizon is a U (1)Chern-Simons theory was also shown in [12, 13]. However in the present calculation, we carefully derive the lawsof black hole mechanics from a completely covariant formulation taking into account the weakest possible boundaryconditions for a black hole horizon and then reinforce the claims that the surface symplectic structure of the WIH isthat of a Chern- Simons theory.The plan of the paper is as follows. We first recall the boundary conditions of a Weak Isolated Horizon (WIH).Then, we derive some key consequences of these boundary conditions. For example, the boundary conditions result inthe zeroth law of black hole mechanics provided we restrict the equivalence class of null normals on the WIH. Theseboundary conditions are equally applicable to extremal as well as non-extremal WIH. Then, using the Holst actionwe show that the principle of least action is well defined in presence of some appropriate boundary terms. In thenext step we construct the symplectic structure in the space of solutions in which each solution contains a WIH as itsinner boundary. We then derive the 1st law of black hole mechanics using this symplectic structure. We also derivethe Chern-Simons symplectic structure on the horizon from this formulation. II. WEAK ISOLATED HORIZONS
We now give an introduction to the idea of weak isolated horizons [15]. Let us consider M to be a four-manifoldequipped with a metric g ab of signature ( − , + , + , +). Our notations and conventions closely follow that of [7, 15]. ∆is a null hypersurface in M of which ℓ a is a future directed null normal. However, if ℓ a is a null normal, so is ξℓ a ,where ξ is any arbitrary positive function on ∆. Thus, ∆ naturally admits an equivalence class of null normals [ ξℓ a ].We denote by q ab , g ab ←− the degenerate intrinsic metric on ∆ induced by g ab (indices that are not explicitly intrinsicon ∆ will be pulled back and , means that the equality holds only on ∆). The tensor q ab will be an inverse of q ab ifit satisfies q ab q ac q bd , q cd . The expansion θ ( ℓ ) of the null normal ℓ a is then defined by θ ( ℓ ) = q ab ∇ a ℓ b , where ∇ a isthe covariant derivative compatible with g ab .The null surface ∆ introduced above is an arbitrary null surface equipped with an equivalence class of null normals[ ξℓ a ]. The conditions on ∆ are too general to make it resemble a black hole horizon. To enrich ∆ with useful andinteresting information, we need to impose some restrictions on this surface. The idea is that we endow a minimal setof conditions on the null hypersurface ∆ so that it behaves as a black hole horizon. As we shall see, the zeroth lawand the first law of black hole mechanics will naturally follow from these conditions. These definitions will be localand only provides a construction of black hole horizon and do not define a black hole spacetime which is a globalobject. However, if there is a global solution, like the Schwarzschild solution, then these conditions will be satisfied. A. The First Set of Boundary Conditions
We shall now introduce the set of boundary conditions to be imposed on the null surface ∆ so that effectively thesurface behaves as a black hole horizon. The boundary conditions that are proposed here are the least number ofconditions that are necessary for a generic black hole horizon. Since the null surface has an equivalence class of nullnormals [ ξℓ a ] as its generators, it is natural to impose the boundary conditions on all of these null normals i.e. theboundary conditions has to hold for the entire equivalence class [ ξℓ a ].The null surface ∆ generated by the equivalence class [ ξℓ a ] will be called a non-expanding horizon (NEH) in ( M , g ab )if the following conditions are satisfied [15]:1. ∆ is topologically S × R .2. The expansion θ ( ξℓ ) , ξℓ a in the equivalence class.3. The equations of motion and energy conditions hold on the surface ∆ and the vector field − T ab ξℓ b is futuredirected and causal.There are some important points to note in the boundary conditions presented here. Firstly, all boundary conditionsare intrinsic to ∆. This implies that to describe NEH, one needs no reference to the spacetime in the exterior. Also,the definition doe not involve Killing vectors although, as we shall see, the boundary conditions imply the presence ofa Killing vector on the null surface ∆. Of these boundary conditions, the first one is just a topological restriction onthe horizon and has no reference to the equivalence class of null normals. The second boundary condition is the mostimportant of the conditions. The expansion freeness is a special requirement for any isolated black hole horizon i.e itholds only for those null surfaces which are black holes. Any null surface will not satisfy this condition. For example,the Minkowski light cone does not satisfy the expansion free condition. This is because the Minkowski light cone isnot a black hole horizon although it appears so for the Rindlar observers. One might be tempted to infer that thesecond condition implies infinite number of boundary conditions to be imposed on each of the infinite number of nullnormals in the equivalence class [ ξℓ a ]. This however is not true. In fact, it is enough that the expansion correspondingto any one null normal is zero. The quantities involved in the boundary conditions are such that once these conditionsare satisfied by one null vector ℓ a , then these are also obeyed by every null vector in the class [ ξℓ a ]. This is true alsofor the expansion-free condition, since θ ( ξℓ ) , ξθ ( ℓ ) . So, for all the conditions defining a NEH, it is sufficient thatthese are satisfied by only one normal vector field in the class [ ξℓ a ] The third boundary condition ensures that theequation of motion of all fields are satisfied on ∆. This condition only allows those fields which satisfy the dominantenergy condition. This requirement also holds true for all the null normals in the equivalence class [ ξℓ a ] if it holdstrue for one. B. Consequences of the Boundary Conditions
The above boundary conditions have important consequences for the kinematical structure of the horizon. Firstof all, note that one can have important simplifications for the null surface ∆ which are quite independent of theboundary conditions. We note them below. Since any ℓ a in [ ξℓ a ] is normal to ∆, these are twist free. Next, becausethe surface is null, the normal vector is also the tangent vector. The tangent vector fields are tangent to the generatorsof the surface. It can be shown that these generators are geodesics or in other words, each ξℓ a in [ ξℓ a ] is geodetic, i.e. ξℓ a ∇ a ( ξℓ b ) , κ ( ξℓ ) ξℓ b (2.1)where κ ( ξℓ ) is the acceleration of ξℓ a . It can be easily deduced from (2.1) that the acceleration varies in the equivalenceclass κ ( ξℓ ) = ξκ ( ℓ ) + £ ℓ ξ (2.2)We shall always work with the null tetrad basis ( ℓ, n, m, ¯ m ) such that 1 = − n · ℓ = m · ¯ m and all other scalar productsvanish. This is specially suited for the problem since one of the null normals ℓ a matches with one of the vectors inthe equivalence class [ ξℓ a ]. The spacetime metric is then given by g ab = − ℓ ( a n b ) + 2 m ( a ¯ m b ) .Given the simplification that any null normal ξℓ a is twist-free and the second boundary condition that any nullnormal ξℓ a is expansion- free, the Raychaudhuri equation becomes0 , £ ξℓ θ ( ξℓ ) , −| σ ( ξℓ ) | − ξ R ab ℓ a ℓ b (2.3)where, σ ( ξℓ ) = m a m b ∇ a ( ξℓ b ) is the shear of ξℓ a . By using the energy conditions and Einstein equations, it can beshown that both terms on the right hand side of (2.3) vanish independently on ∆. Therefore every null normal ξℓ a in the equivalence class is also shear-free. Again, note that if any one of the null normals ℓ a is shear free, all thenull normals in the equivalence class [ ξℓ a ] are shear free too. In short, every null normal in the equivalence class[ ξℓ a ] is twist- free, shear- free and expansion- free. These conditions imply that there exists an one-form ω ( ℓ ) a on ∆,depending on ℓ a such that ∇ a ←− ℓ b , : ω ( ℓ ) a ℓ b (2.4)The one-form defined in (2.4) plays an important role in the whole analysis. It is also clear that since the one- form ω ( ℓ ) a depends on the null normals, it varies in the class [ ξℓ a ]. The variation is ω ( ξℓ ) = ω ( ℓ ) + d ln ξ (2.5)where d is the exterior derivative in ∆. Since the pull-back of the one-form ξℓ a is zero on ∆, it follows that every ξℓ a in the class is a Killing vector on ∆, namely £ ( ξℓ ) q ab ,
0. A straightforward calculation (using results of appendix(IX A)) shows that the curvature of ω ( ξℓ ) dω ( ξℓ ) , ) ǫ , (2.6)where ImΨ = C abcd ℓ a m b ¯ m c n d is a complex scalar, C abcd is the Weyl-tensor and ǫ = im ∧ ¯ m is the area two- form onthe cross- sections of ∆. Again from equation (2.5), since the one form ω ( ξℓ ) varies like a U (1) gauge field, the equation(2.6) will hold true for all ω ( ξℓ ) corresponding to the vectors in the equivalence class. The Killing equations implythat the area two- form ǫ of the cross-section is preserved under Lie-flow of every ξℓ a in the class, £ ( ξℓ ) 2 ǫ ab , C. Weak Isolated Horizon and the Zeroth Law
Let us recall that from the point of view of boundary conditions, all the horizons generated by the null normals inthe equivalence class [ ξℓ a ] are equivalent. In other words, the boundary conditions are oblivious to the equivalenceclass and cannot prefer one horizon generated by say ℓ a , over another generated by ξℓ a , both null vectors being in theequivalence class. Again, recall from (2.2) that the accelerations κ ( ξℓ ) of the null normals ξℓ a vary in the equivalenceclass through κ ( ξℓ ) = ξκ ( ℓ ) + £ ℓ ξ . The point of view here is that the accelerations κ ( ξℓ ) just provides a nomenclaturewhich can be used as tags to the various surfaces generated by the corresponding null normals in the equivalence class[ ξℓ a ]. In short, all the null surfaces labelled by say κ ( ξ ℓ ) , κ ( ξ ℓ ) , κ ( ξ ℓ ) etc are on the equivalent footing from thepoint of view of boundary conditions. This exemplifies the claim that the WIH boundary conditions puts the non-extremal and extremal black hole horizons in the same footing.An noted several times, the acceleration κ ( ξℓ ) varies over the class [ ξℓ ]: κ ( ξℓ ) = ξκ ( ℓ ) + £ ℓ ξ and is not a constantin general. In order to obtain the zeroth law, which requires the acceleration for each normal vector in the class to bea constant, we need to restrict the NEHs further. Let us call the restricted horizon the weak isolated horizon (WIH),which is a NEH equipped with a class [ ξℓ a ] such that £ ( ξℓ ) ω ( ξℓ ) , . (2.7)Let us make a few comments here: First, as in Killing horizons, we will interpret the acceleration κ ( ξℓ ) as the surfacegravity of ξℓ a . However, since a global Killing field is absent, the value of the surface gravity cannot be uniquelydetermined. In isolated horizon formulation it is natural to keep this freedom. In fact, as we shall see below, thisfreedom will enable the extremal horizon with surface gravity κ = 0 and non- extremal horizons κ = 0 to be on thesame footing.Second, the boundary condition (2.7), unlike the previous ones, is not a single condition. Namely, if it is obeyedby one normal vector ℓ a ( £ ℓ ω ( ℓ ) , ξℓ a in the class willobey it. This pathology is eliminated by restricting the choice of ξ s ξ = c e − vκ ( ℓ ) + κ ( ξℓ ) /κ ( ℓ ) , (2.8)where, c is a nonzero function satisfying £ ℓ c = 0 and v is the affine parameter such that £ ℓ v = 1. For the rest ofthe paper, we choose c , constant. Having restricted ourselves to this specific class it is now easy to verify that (2.7)becomes just one condition: £ ( ξℓ ) ω ( ξℓ ) , d ( ξℓ · ω ( ξℓ ) ) , dκ ( ξℓ ) for every ξ belonging to the restricted class. From nowon the class of normal vectors will always follow this restriction. This class admits a ξ = c e − vκ ( ℓ ) , for which κ ( ξℓ ) , κ ( ℓ ) of ℓ a is nonzero. For obvious reasons such an isolated horizon, characterized by a normalvector of vanishing surface gravity, will be called an extremal horizon. Thus, our class of normal vectors contains bothextremal and non-extremal horizons, as opposed to the constant class of normal vectors [ cℓ a ]. In other words, theWIH boundary conditions cannot differentiate between the surfaces generated by null normals in the equivalence class[ ξℓ a ] with ξ = c e − vκ ( ℓ ) + κ ( ξℓ ) /κ ( ℓ ) and hence the extremal and non- extremal horizons become part of the WIH.As already noted, WIH boundary condition (2.7) is equivalent to the zeroth law : dκ ( ξℓ ) ,
0. Therefore, the surfacegravity corresponding to each ξℓ a in [ ξℓ a ] is constant on ∆, provided ξ s belong to the restricted class. III. THE HOLST ACTION
The Holst action ([16]) is a modification of the Palatini action where a term is added which has a property thatit does not contribute to the equation of motion. Let us recall that the Palatini action is constructed out of thebasic fields, the tetrads e Ia and a SO (3 ,
1) Lie algebra valued connection one form A IJ (see [17] and [18] for details).On shell, the connection A IJ equals the spin connection. The Legendre transformation of Palatini Lagrangian tothe Hamiltonian formulation introduces second class constraints. Solution of these constraints needs some gaugefixing which essentially reduces the theory to that of the standard metric variable theory and one looses the essentialadvantages of the connection formulation. One can however go to the self dual complex connections where the theoryis much easier but this also creates problems for the quantum theory. Indeed, the quantum theory based on LQGneeds background independent analysis on the connection space. However, such a theory is still to be constructedand in the meantime, the quantization programme is successful on the phase space of the real variables. This phasespace is the Barbero- Immirzi phase space constructed out of the original Palatini phase space by a one parametercanonical transformation labelled by the Barbero- Immirzi parameter γ . The Holst action is precisely the actionwhose Legendre transformation gives the Barbero- Immirzi phase space.Let us first begin with the Palatini action. We consider the spacetime M which is bounded by the the Cauchysurfaces M ± and intersecting at i . For now, we shall only deal with a spacetime without any inner boundary ∆. Theaction is given by: i M _ M + S + MS ∆ ∆ S_ FIG. 1: M ± are two partial Cauchy surfaces enclosing a region of space-time and intersecting ∆ in the 2-spheres S ± respectivelyand extend to spatial infinity i o . Another Cauchy slice M is drawn which intersects ∆ in S ∆ S P = 132 πG Z M ǫ IJKL e I ∧ e J ∧ F KL = 116 πG Z M Σ IJ ∧ F IJ (3.1)where, Σ IJ = ǫ IJKL e K ∧ e L is a two- form defined for future convenience. The variation of the integral with respectto A IJ will lead to : δS P = 116 πG Z M d (Σ IJ ∧ δA IJ ) − πG Z M D Σ IJ ∧ δA IJ (3.2)where we have used the fact that: D Σ IJ := d Σ IJ + A I K ∧ Σ KJ + A J K ∧ Σ IK (3.3)Thus, the equation of motion obtained from variation of the connection is : D Σ IJ = 0 (3.4)It is then easy to check from (3.4) that the connection A IJ is the spin connection. Then, the variation of the action(3.1) with respect to the tetrads e aI gives the Einstein equation. In the present case, observe that the boundary termgoes to zero at the outer boundary where the spacetime is flat and the variational principle is well defined. In caseswhere there is an inner boundary, we need to take care that the boundary term again vanishes.Now, let us consider the Holst action [16]: S H = S P − πGγ Z M e I ∧ e J ∧ F IJ (3.5)where, γ is a real constant called the Barbero -Immirzi parameter. Note that written in this form, the parametercannot be equal to zero. For γ = − i (+ i ), we get the self dual (anti self-dual) action. Then, the action S H canbe written in terms of ± F IJ ( A IJ ) = F IJ ( ± A IJ ). In these cases, it can be argued that the connection is actuallya self-dual (anti-self dual) part of the spin connection. For generic but other values of γ , it is also true that theconnection A IJ is actually a spin connection, completely determined by the tetrad. The variation of the action withrespect to the tetrad again reproduces the Einstein equation, the γ - dependent term, on shell, being zero by thealgebraic Bianchi identity.The extra γ - dependent term that appears in the action is important for quantum mechanical reasons. It is in somesense similar to the theta ( θ ) term in QCD. It is well known that in Yang- Mills theory, θ − term introduces inequivalent θ − sectors for the corresponding quantum theory. In the similar way, the extra term here introduces the inequivalent γ − sectors for quantum general relativity [19]. However, while the θ ( F Y M ∗ F Y M ) is a purely topological term , beingequal to a total divergence of a 3 − form, the γ − dependent term here vanishes because of the first Bianchi identity.One can show that the phase space corresponding to these two theories are equal. To see this one can construct thesymplectic structure for the Holst action and argue that if the phase space vectors satisfy the linearized version of thespin connection equation, the γ - dependent term disappears (see appendix IX B for the details). In other words, theHolst modification of the Palatini action implies a canonical transformation on the phase space. A. The Action and the Variational Principle
In this subsection, we will use the Holst action for a spacetime with inner boundary. In the present case, theinner boundary is a null surface which satisfies the WIH boundary conditions. To be more precise, the spacetimeunder consideration is a region bounded by the Cauchy surfaces M and M extending to spatial infinity and the nullsurface ∆ (see fig. 1). The variation of the fields will be between all those configurations which satisfy the boundaryconditions at infinity and at ∆. In particular, we consider those variations of ( e I a , A IJ ) which satisfy the standardfall off conditions at infinity and on ∆, satisfy the following conditions:1. each spacetime admits a null normal belonging to the equivalence class [ ξℓ a ].2. each pair (∆ , [ ξℓ a ]) is a WIH.We can always introduce a fixed set of internal null vectors ( ℓ I , n I , m I , ¯ m I ) on ∆ such that ∂ a ( ℓ I , n I , m I , ¯ m I ) = 0(this partially fixes the Lorentz frame). Given these internal null vectors and the tetrad e I a , we can construct thenull vectors ( ℓ a , n a , m a , ¯ m a ) through ℓ a = e I a ℓ I . In some String theories, like the E ⊗ E heterotic string theory, the θ − term arises from the Yang- Mills Chern-Simons term whichmakes the theory gauge anomaly free. In that case, derivative of θ is a field dual to the Kalb- Ramond three form. We consider thecase where θ in a constant just like in ordinary QCD. Tetards and Connection on ∆ To proceed further, we need the the expressions for the tetrad and the connection in terms of the null vectors. Theexpansion of tetrads in terms of the null vectors can be easily calculated from the expression of tetrads relating thespacetime metric g ab and internal flat metric η IJ . The expression is given by: e I a ←− , − n a ℓ I + m a ¯ m I + ¯ m a m I (3.6)The above equation (3.6) can be used to get an expression for the product of tetrads: e I a ←− ∧ e J b ←− , − n a ∧ m b ℓ [ I ¯ m J ] − n a ∧ ¯ m b ℓ [ I m J ] + 2 m a ∧ ¯ m b ¯ m [ I m J ] (3.7)= − n a ∧ m b ℓ [ I ¯ m J ] − n a ∧ ¯ m b ℓ [ I m J ] + 2 i m [ I ¯ m J ] 2 ǫ ab (3.8)Using this expression for the tetrad products (3.7), and the expansion for the internal epsilon tensor ǫ IJKL =4! ℓ [ I n J m K ¯ m L ] , we get Σ ab ←− IJ , ℓ [ I n J ] 2 ǫ ab + 2 n a ∧ ( im b ℓ [ I ¯ m J ] − i ¯ m b ℓ [ I m J ] ) (3.9)We are now in a position to calculate the connection A IJ . We will be using the Newman- Penrose formalism. Thedetails of the Newman- Penrose coefficients are given in the appendix IX A. Using those expansions, one gets thefollowing expression for the covariant derivatives of the null normals pulled back and restricted to ∆. ∇ a ←− ℓ b , ω ( ℓ ) a ℓ b (3.10) ∇ a ←− n b , − ω ( ℓ ) a n b + ¯ U ( l,m ) a m b + U ( l,m ) a ¯ m b (3.11) ∇ a ←− m b , U ( l,m ) a ℓ b + V ( m ) a m b (3.12) ∇ a ←− ¯ m b , ¯ U ( l,m ) a ℓ b − V ( m ) a ¯ m b (3.13)where, the superscripts for each of the one forms keep track of their dependencies on the rescaling of the correspondingnull normals. The expressions of the one forms ω ( ℓ ) , U ( l,m ) , ¯ U ( l,m ) and V ( m ) can be written in terms of the null normalsand are as follows: ω ( ℓ ) a , − ( ǫ + ¯ ǫ ) n a + ( ¯ α + β ) ¯ m a + (cid:0) α + ¯ β (cid:1) m a U ( l,m ) a , − ¯ πn a + ¯ µm a + ¯ λ ¯ m a V ( m ) a , − ( ǫ − ¯ ǫ ) n a + ( β − ¯ α ) ¯ m a + (cid:0) α − ¯ β (cid:1) m a (3.14)The part of the connection V ( m ) is purely imaginary. Let us at this stage point out the result of the rescaling of thenull normal ℓ a on the various quantities of interest. Firstly, for ℓ a −→ ξℓ a , we have: ω ( ℓ ) a → ω ( ξℓ ) a = ω ( ℓ ) a + ∇ a lnξ (3.15)Since the normalization of ℓ a and n a are connected, we must have n a −→ n a ξ when ℓ a −→ ξℓ a . Then the effect ofthe rescaling can be seen to be: ∇ a ←− (cid:18) n b ξ (cid:19) , − ω ( ξℓ ) a (cid:18) n b ξ (cid:19) + ¯ U ( ξℓ,m ) a m b + U ( ξℓ,m ) a ¯ m b (3.16)Thus, under this transformation, we have that the one- form ω ( ξℓ ) transforms in the usual way: ω ( ℓ ) → ω ( ξℓ ) a = ω ( ℓ ) a + ∇ a lnξ and the other one forms U ( l,m ) and ¯ U ( l,m ) transform as¯ U ( ℓ,m ) a −→ ¯ U ( ξℓ,m ) a = ¯ U ( ℓ,m ) a ξ (3.17) U ( ℓ,m ) a −→ U ( ξℓ,m ) a = U ( ℓ,m ) a ξ This rescaling is in the sector of ℓ, n . There can be another set of rescaling quite independent of the rescaling of ℓ, n . This concerns the transformation in the other set of null vectors m, ¯ m of the null tetrad. This transformationfunction will also be independent of the function ξ . Now, for m → f m and for ¯ m → ¯ mf , where f is any function on∆, we have the following transformations: ∇ a ←− (cid:0) f m b (cid:1) , U ( ℓ,fm ) a ℓ b + V ( fm ) a (cid:0) f m b (cid:1) (3.18) ∇ a ←− (cid:18) ¯ mf (cid:19) , ¯ U ( ℓ,fm ) a ℓ b − V ( fm ) a (cid:18) ¯ m b f (cid:19) (3.19)The transformation rules are as follows for the one forms U ( ℓ,m ) a , ¯ U ( ℓ,m ) a and V ( m ) a are as follows:¯ U ( ℓ,m ) a −→ ¯ U ( ℓ,fm ) a = ¯ U ( ℓ,m ) a f (3.20) U ( ℓ,m ) a −→ U ( ℓ,fm ) a = f U a V ( m ) a −→ V ( fm ) a = V ( m ) a + ∇ a lnf The part of the connection ω ( ℓ ) and V ( m ) transform as U (1) field whereas the the other parts of connections onlyrescale. We have constrained only one part of the connection while defining the Weak Isolated Horizon in the sensethat only the one form ω is constrained. The other part of the connection is left as it is. If we want to constrain moreof the parts of the connection then we get the definition of the Isolated horizon.We can use these information to find the connection. To do this, we first note that the internal null vectors arefixed such that ∂ a ( ℓ I , n I , m I , ¯ m I ) = 0. Then, we get ∇ a ←− ℓ I , A a ←− I J ℓ J (3.21)We can choose a tetrad e Ia which maps the vector ℓ I to ℓ a . This tetrad is annihilated by the covariant derivative, ∇ a e Ib = 0. Then the equation (3.21) gives: A a ←− I J ℓ J , ω ( ℓ ) a ℓ I . Written is a more compact form, this reduces to: A a ←− IJ , − ω ( ℓ ) a ℓ [ I n J ] + Q IJ (3.22)where, the one form Q IJ is such that Q IJ ℓ J , n I , m I and ¯ m I . For the null vector n I , that gives us: A a ←− IJ , − ω ( ℓ ) a ℓ [ I n J ] − U ( ℓ,m ) a m [ I ℓ J ] − U ( l,m ) a ¯ m [ I l J ] + R IJ (3.23)where, the one form R IJ is such that R IJ n J ,
0. For the null vector m I , we get: A a ←− IJ , − U ( ℓ,m ) a ¯ m [ I l J ] + 2 V ( m ) a m [ I ¯ m J ] + S IJ (3.24)where, the one form S IJ is such that S IJ m J ,
0. A similar construction for the null vector ¯ m I implies A a ←− IJ , − U ( ℓ,m ) a ¯ m [ I l J ] + 2 V ( m ) a m [ I ¯ m J ] + ¯ S IJ (3.25)such that ¯ S IJ ¯ m J , A a ←− IJ . These gives the connection to be: A IJ = − ω ( ℓ ) ℓ [ I n J ] + 2 U ( l,m ) ℓ [ I ¯ m J ] + 2 ¯ U ( l,m ) ℓ [ I m J ] + 2 V ( m ) m [ I ¯ m J ] (3.26)We define the following connection for ease of computation A ( H ) IJ := 12 (cid:16) A IJ − γ ǫ IJ KL A KL (cid:17) (3.27) This choice of the connection A ( H ) IJ with a factor in front is made to make our results conform to the standard results [12]. However,this factor can be arbitrarily chosen and the quantum implementation of the boundary conditions including state counting goes throughunchanged entirely. This leads to the following form of the connection: A ( H ) aIJ , ℓ [ I n J ] (cid:16) − ω ( ℓ ) a + iγV ( m ) a (cid:17) + m [ I ¯ m J ] (cid:16) V ( m ) a − iγω ( ℓ ) a (cid:17) + ℓ [ I ¯ m J ] (cid:16) U ( l,m ) a + iγU ( l,m ) a (cid:17) + ℓ [ I m J ] (cid:16) ¯ U ( l,m ) a − iγ ¯ U ( l,m ) a (cid:17) (3.28) Variation of the Action
The next step is to check the variational principle. The lagrangian that we are interested in is of the form: − πGγ L = γ Σ IJ ∧ F IJ − e I ∧ e J ∧ F IJ − γ d (Σ IJ ∧ A IJ ) + d ( e I ∧ e J ∧ A IJ ) . (3.29)where we have added the two boundary term just for convenience. These terms will not contribute to the equation ofmotion. The variation of the action on-shell will give two terms on the boundary ∆. They are: δS ( e, A ) = − πGγ Z ∆ ( iV ( m ) + γ ω ( ℓ ) ) ∧ δ ǫ (3.30)We can argue that the term is zero and hence the action principle is well defined. The argument goes as follows. Firstof all, the field configurations over which the variations are taken are such that they satisfy the standard boundaryconditions at infinity and the WIH boundary conditions at ∆. The weak isolation condition implies that £ ℓ ω ( ℓ ) , V ( m ) . However interestingly, dω ( ℓ ) and dV ( m ) are proportional to ǫ and henceinner product with ℓ a of these quantities are zero. This implies that for variations among field configurations withnull normals in the equivalence class, we have £ ξℓ ω ( ℓ ) , d ( ξκ ( ℓ ) ) and £ ξℓ V ( m ) , d ( ξ ( ǫ − ¯ ǫ )). This implies that on theapplication of £ ξℓ , the integral goes to the initial and the final cross section of ∆. However, the variation of the fieldsfor example δ ǫ is zero at the initial and final hypersurface by the standard rules of variational principle. Thus theintegral is lie dragged by any null normal in the equivalence class. In other words, the integral in zero at the initialand the final hypersurface and is lie dragged on ∆ . Thus, the entire integral is zero and the action principle is welldefined. B. The Symplectic Structure
The phase space for the system can be constructed. We recall that the variation of the Lagrangian produces thethree form Θ( δ ), such that δL =: d Θ( δ ). In the present case, we have:16 πGγ Θ( δ ) = γ δ Σ IJ ∧ A IJ − δ ( e I ∧ e J ) ∧ A IJ = − δ ( e I ∧ e J ) ∧ A ( H ) IJ (3.31)The construction of the symplectic current from here is standard. The current is J ( δ , δ ) := δ Θ( δ ) − δ Θ( δ ). Thecurrent is closed on- shell i.e. dJ = 0. The resulting Symplectic Current is : J ( δ , δ ) := 18 πGγ (cid:8) δ [1 ( e ∧ e ) (cid:9) ∧ n δ (cid:16) A IJ − γ ǫ IJ KL A KL (cid:17)o (3.32)Integrating the symplectic current over M , we get the contribution of the symplectic current from the boundaries ofthe spacetime region under consideration: Z M + ∪ M − ∪ ∆ ∪ i J ( δ , δ ) = 0 (3.33)The boundary conditions at infinity ensure that the integral of the symplectic current at i vanishes. However, toconstruct the symplectic structure, we must be careful that no data flows out of the phase space because of our choiceof foliation. To ensure this, we must check that the symplectic structure is independent of the choice of foliation. Forthat, we introduce potentials:1. £ ( ξℓ ) ψ ( ξℓ ) , ξℓ a ω ( ξℓ ) a , κ ( ξℓ ) £ ( ξℓ ) µ ( m ) , iξℓ a V ( m ) a , iξ ( ǫ − ¯ ǫ )0along with the boundary condition that they are zero at one of the cross- sections of ∆ so as to fix the additiveambiguities. We choose ψ ( ξℓ ) = 0 and µ ( m ) = 0 at S − .The basic idea now is to write the symplectic current J ( δ , δ ) on ∆ in terms of these potentials and see that J ( δ , δ ) , dj ( δ , δ ). With this simplification, the integrals of J ( δ , δ ) on ∆ will be taken to the boundaries S ± of∆. To see this, first note that the expression of symplectic current J ( δ , δ ) on ∆ is given by: J ( δ , δ ) | ∆ , − πGγ [ δ ǫ ∧ δ ( iV + γω ( ℓ ) ) − δ ǫ ∧ δ ( iV + γω ( ℓ ) )] (3.34)The potential for κ ( ℓ ) is ψ ( ℓ ) . It can be seen from the definition that ψ ( ℓ ) is a function of v only. The potential for i ( ǫ − ¯ ǫ ) is µ ( m ) and it can be seen again that the wedge product of the variation of ǫ and variation of dµ ( ℓ ) + £ ℓ µ ( m ) n vanish. These two results imply that J ( δ , δ )) | ∆ , d (cid:20) − πGγ (cid:0) δ ǫ δ ( µ ( m ) + γψ ( ℓ ) ) − (1 ↔ (cid:1)(cid:21) (3.35)We take a particular orientation of the spacetime foliation into account. That gives us the result that the symplecticcurrent is independent of the foliation.( Z M + − Z M − ) J ( δ , δ ) , πGγ ( Z S − − Z S + ) { δ ǫ δ ( µ ( m ) + γψ ( ℓ ) ) − (1 ↔ } (3.36)The construction of symplectic current is independent of our choice foliation and hence all the phase space informationcan be obtained from this symplectic current by staying on any arbitrary foliation. We choose a particular Cauchysurface M which intersects ∆ in the sphere S ∆ . The symplectic structure is then given by :Ω( δ , δ ) := 18 πGγ Z M h δ ( e I ∧ e J ) ∧ δ A ( H ) IJ − δ ( e I ∧ e J ) ∧ δ A ( H ) IJ i + 18 πGγ Z S ∆ (cid:2) δ ǫ δ ( µ ( m ) + γψ ( ℓ ) ) − δ ǫ δ ( µ ( m ) + γψ ( ℓ ) ) (cid:3) (3.37) IV. THE FIRST LAW
The first law requires defining an energy. Since the WIH is a local definition of a horizon, the first law should involveonly locally defined quantities. To be more precise, the first law is expected to relate variations of local quantitiesthat are defined only at the horizon without any reference to the rest of the spacetime. We already have the surfacegravity κ ( ξℓ ) defined only locally at the horizon and the other quantity that we require now is a locally defined energy(for horizons carrying other charges, such as angular momentum, electric potential etc., we must also provide localdefinitions for them). To proceed, it should be noted that in spacetime, energy is associated with a timelike Killingvector field. Given any vector field W in spacetime, it naturally induces a vector field δ W in the phase space. Thephase space vector field δ W is the generator of time translation in the phase space. If time translation is a canonicaltransformation in the phase space then δ W defines a Hamiltonian function H W for us.So to find out the Hamiltonian function associated with energy, we must look for phase space transformations thatkeep the symplectic structure invariant, in other words the canonical transformations. The vector fields tangent tothese canonical flows are the Hamiltonian vector fields. To check wheather a vector field δ t in the phase space isHamiltonian, one constructs a one-form X t where X t ( δ ) := Ω( δ, δ t ), where δ t is the lie flow £ t generated by thespacetime vector field t a when tensor fields are varied. The necessary and sufficient condition for the vector field δ t to be a globally Hamiltonian vector field is that the one-form X t is to be exact , X t = d H t where d is the exteriorderivative in phase space and H t is the corresponding Hamiltonian function. In other words, the vector field δ t isglobally Hamiltonian if and only if X t ( δ ) = δH t for any vector field δ in the phase space. Because of the presence ofthe boundary, the WIH, The vector fields t a are also restricted by the condition that it should be tangential on ∆.Now being a null surface, the WIH has only three tangential directions, one null and the two other spacelike. Theclosest analog of ‘time’ translation on WIH is therefore translation along the null direction. It is generated by thevector field [ ξℓ a ] (one can take any but fixed ξ belonging to the equivalence class). For global solutions this null normalvector field becomes timelike outside the horizon and is expected to match with the asymptotic time-translation forasymptotically flat spacetimes. So in this sense, the local energy on the horizon is the local snapshot of ADM orBondi energy defined at spacelike/null asymptotic infinity.The relevant question is: Is the flow generated by the phase space vector field δ ξℓ Hamiltonian? To find that, wecalculate the symplectic structure for any arbitrary live vector field δ ξℓ . It is useful to recall that the action of the phase1space vector field δ ξℓ on tensor fields is the lie flow £ ξℓ generated by the vector field ξℓ a . For the above symplecticstructure, X ( ξℓ ) ( δ ) gets contribution from both the bulk and the surface symplectic structure. The bulk term, thanksto the equation of motion satisfied by the fields and their variations, contributes only through the boundaries of theCauchy surface M , which are the 2 − spheres S ∆ and S ∞ respectively: X ξℓ ( δ ) | M = − πG ξκ ( ℓ ) δ A ∆ − i πGγ Z S ∆ ξ ( ǫ − ¯ ǫ ) δ ǫ + δE ( ξℓ ) (4.1)where, A ∆ = R S ∆ ǫ is the area of S ∆ and E ( ξℓ ) is the ADM energy arising out of the integral at S ∞ , assuming thatthe asymptotic time translation matches with the vector field ξℓ a at infinity.The X ( ξℓ ) ( δ ) also gets contribution from the surface symplectic structure. For t = ξℓ a , we must be careful withthe evaluations of the action of δ ( ξℓ ) on the potentials µ ( m ) and ψ ( ℓ ) . The action of δ ( ξℓ ) cannot be interpreted as £ ( ξℓ ) when acting on potentials. To determine the action, we proceed as follows. For the case of ψ ( ℓ ) , it is clearthat since variation of ψ ( ξℓ ) is completely determined by κ ( ξℓ ) , δ ( ξℓ ) ψ ( ξℓ ) = 0. However, ψ ( ξℓ ) = ψ ( ℓ ) + ln ξ impliesthat δ ( ξℓ ) ψ ( ℓ ) = − £ ℓ ξ . For the other potential, observe that δ ( ξℓ ) µ ( m ) − i ( ǫ − ¯ ǫ ) satisfies the differential equation £ ξℓ ( δ ( ξℓ ) µ ( m ) − i ( ǫ − ¯ ǫ )) = 0 with the boundary condition that µ ( m ) = 0 at the point v = 0. This implies that because( ǫ − ¯ ǫ ) = 0 at v = 0, the action is δ ( ξℓ ) µ ( m ) = i ( ǫ − ¯ ǫ ). The considerations above leads to: X ξℓ ( δ ) | S ∆ = 18 πGγ Z S ∆ [ δ ǫ δ ξℓ ( µ ( m ) + γψ ( ℓ ) ) − £ ξℓ ǫ δ ( µ ( m ) + γψ ( ℓ ) )] (4.2)= − πG £ ℓ ξ δ A ∆ + i πGγ Z S ∆ ξ ( ǫ − ¯ ǫ ) δ ǫ Combining the two equations (4.1) and (4.2), we get: X ξℓ ( δ ) , − πG κ ( ξℓ ) δ A ∆ + δE ( ξℓ ) (4.3)This is a fundamental result of the generalization to the most general class of null normals [ ξℓ a ]. In the constant classof null normals, there is no contribution from the surface symplectic structure. In the generalized class of null normals,the precise contribution from the bulk (4.1) and the boundary (4.2) leads to the physically meaningful variation.The condition that δ ξℓ is a Hamiltonian implies that the surface gravity κ ( ξℓ ) is a function of area A ∆ only. Tosee that note that to check that δ ξℓ is a Hamiltonian, we must check that d X ξℓ = 0. In other words, this implies d X ξℓ ( δ , δ ) = 0. A simple calculation gives: d X ξℓ ( δ , δ ) = δ ( κ ( ξℓ ) δ A ∆ ) − δ ( κ ( ξℓ ) δ A ∆ ) (4.4)= δ κ ( ξℓ ) δ A ∆ − δ κ ( ξℓ ) δ A ∆ (4.5)This can be written in a more suggestive form as d κ ( ξℓ ) ∧ d A ∆ ( δ , δ ) = 0. This implies that since δ and δ arearbitrary, the wedge product is zero by itself i.e. we get: d κ ( ξℓ ) ∧ d A ∆ = 0 (4.6)which implies that the surface gravity κ ( ξℓ ) is a function of area A ∆ only. The exact functional form however remainsundetermined. This also implies that there exists a locally defined function E ∆ . Defining the total Hamiltonian X ξℓ ( δ ) = δH ξℓ =: E ( ξℓ ) − E ∆ , we get the first law of weak isolated horizons: δE ∆ , πG κ ( ξℓ ) δ A ∆ (4.7)This is consistent since the previous condition imples the existence of a locally defined energy such that d κ ( ξℓ ) ∧ d A ∆ ( δ , δ ) = 0. With the present first law, we see that this relationship holds. V. CHERN-SIMONS THEORY FROM SYMPLECTIC STRUCTURE
Once the four laws of black hole mechanics are established one wonders whether their resemblances with the fourlaws of thermodynamics are pure coincidences? In a brilliant paper, Bekenstein argued that this is not a mere2resemblance but black holes indeed may have entropy proportional to the classical area of the horizon. Shortly after,by analyzing quantum fields in a collapsing spacetime Hawking showed that a black hole has a temperature of (inunits of ~ = k B = 1) κ/ π , where κ is the surface gravity at the horizon. It readily implies that under semiclassicalapproximation the entropy of a black hole is equal to the one-quarter of its horizon area. This interpretation ofentropy as area obviously begs for its derivation from first principles, namely the Boltzmann definition of entropyarising out the microstates of the system. However, in the present case the system being a black hole spacetime, thequestion is what are its microstates? In other words, what are the microscopic constituents of a black hole spacetimeto which such microstates are to be assigned? This is indeed a deep question and its answer lies beyond the generaltheory of relativity. Since the laws of the microscopic world are quantum mechanical, one naturally asserts that suchmicroscopic constituents of spacetime obey the laws of quantum mechanics, rather than the classical laws of generalrelativity. This therefore, warrants for a quantum theory of spacetime or a quantum theory of gravity. Such a theory,to our complete satisfaction, is simply not available at present.One general approach towards a statistical interpretation of black hole entropy is the loop approach which flourishedunder the umbrella of loop quantum gravity. In this approach, the statistical analysis of the microscopic constituentsfor a generic black hole was first presented in a series of papers [11–13]. The general idea was to not to find out themicroscopic constituents of an entire black hole spacetime, but rather their imprints on the classical horizon of theblack hole, which when treated as a boundary of the spacetime outside of a black hole, gets excited by some effectivedegrees of freedom that arise due to a delicate but well-defined interaction between the boundary and the bulk ofthe spacetime. One assumes that these effective degrees of freedom capture the bare minimal features of a black holespacetime, thereby it is only natural that such effective states are localized only at the horizon rather than spreadout all over the spacetime. The isolated horizons become relevant in this context because such surfaces are tailoredto capture the essential features of a black hole spacetime. One then quantizes this effective theory induced at anisolated horizon and and count the appropriate quantum states. This turn out to be consistent with the semiclassicalestimates made by Bekenstein and Hawking. Furthermore, the effective theory on the horizon can only be a theory ofthe topological kind, namely it must be insensitive to the metric on the horizon. This is because the horizon is a nullsurface and therefore cannot support a physical particle. The above papers show, through a detailed canonical phasespace analysis, that the effective theory on the horizon is Chern-Simons type, more precisely a U (1) Chern-Simonstheory.The main objective of the section is to find out the effective field theory on a spherically symmetric WIH, startingfrom the Holst action, in a completely covariant framework. It will not only reinforce the claims made in the abovework but also at the same time will make the results independent of any slicing. A. Spherical Horizons
The plan is to use these considerations to find the toplogical theory on the inner boundary called WIH. To proceedfurther, we consider the following case: the fields on the boundary are such that the energy mommentum tensor is ofthe form − T ba ℓ a = eℓ b , where e is spherically symmetric. Then, using Einstein equation, we get:Φ + 18 R −
12 Λ , πGe (5.1)This condition implies that for spacetimes with cosmological constant zero, the term Φ + R is spherically symmetricas e is spherically symmetric. From the above energy conditions with the above form of − T ba ℓ a it also follows simplyfrom the Einstein equations that Φ , Φ , Φ ,
0. This implies that no flux of radiation falls through thehorizon. This is expected as the spherical symmetric energy condition is a special case of the more general casesconsidered so far.We need one further expression to proceed further. First, see that for spherical symmetric horizons π , λ , µ is a real and spherically symmetric function. In fact, in canonical formulation, this function measures theexpansion of the null vector n . In general black hole horizons, the function µ is positive but might not be sphericallysymmetric. Indeed, for distorted black hole horizons, it is true that µ is not spherically symmetric and π is not zero.The Riemann tensor is then calculated for the null vector n a by using (3.11) as follows: R abcd n d = 2 ∇ [ a ∇ b ] n c , ∇ [ a ω b ] n c + 2 ∂ [ a µ ¯ m b ] m c + 2 µ ∇ [ a ¯ m b ] m c + 2 µ ¯ m [ b ∇ a ] m c + 2 ∂ [ a µ m b ] ¯ m c + 2 µ ∇ [ a m b ] ¯ m c + 2 µ m [ b ∇ a ] ¯ m c + 2 µω [ a ¯ m b ] m c + 2 µω [ a m b ] ¯ m c (5.2)We can use the expansion of the Riemann tensor in terms of the Weyl tensor, the Ricci tensor and the Ricci scalar.3Transvecting with ℓ a m b ¯ m c , and using the expansion of the scalar Φ , we get that: Ψ + 112 R , £ ℓ µ + κ ( ℓ ) µ (5.3)This equation then implies that since the all the terms on the right hand side are real and R is real, the termIm Ψ = 0, i.e. , dω ( ℓ ) = 0. Also, the equation implies that the term ( Re Ψ + R ) is again spherically symmetricas µ is spherically symmetric.To proceed further, we want to restrict ourselves to the fixed area phase space. This needs an expression of the areatwo- form to be fixed on the phase space. We have already noticed that for spherical horizons, dω ( ℓ ) = 0. However,the curvature of the one form V ( m ) involves the area form. The details of the calculation is given in the appendix (seeappendix IX C). Using the equations (5.3) and (5.1), we see that the term F := ( Ψ ( H ) − Φ − R ) in the equation(9.27) is again spherically symmetric. This property of F remains to be spherically symmetric throughout ∆. To seethis, note that the Bianchi identity for V ( H ) in equation (9.27) implies: d F ∧ ǫ , d ǫ ,
0. Transvected by ℓ a , the above equation shows that the value of ( Ψ ( H ) − Φ − R )is lie dragged by ℓ a and hence, remains fixed over ∆. We want to find a value for F in (9.27). Then, we first evaluatethe curvature of the connection V ( H ) = − im [ I ¯ m J ] A ( H ) IJ by using the Gauss- Bonnet theorem. The connection iV ( m ) is precisely the connection on the sphere S and then its field strength will be the curvature. Integrating both sidesof (9.27), using Gauss- Bonnet theorem and remembering that F is independent of sphere coordinates, we get:( Ψ ( H ) − Φ − R
24 ) = − π A s (5.5)This equation (5.5) and the equation (9.27) together imply that the two- form ǫ can be written in terms of thecurvature of the U (1) field V ( H ) : ǫ = − A s ∆ π dV ( H ) s (5.6)The superscripts and subscripts s in A s ∆ and V ( H ) s denote that we are in spherically symmetric case. As we shallsee, this term will give the requisite level for the U (1) Chern- Simons theory. One can envisage another approach.Consider all spacetimes which provide different set of values for h Ψ ( H ) − Φ − R i . Now, we shall consider only theaverage values. Then integrating the Eq.. (9.27), we get an average value of h Ψ ( H ) − Φ − R i to be − π A s . Whilethis is a possibility, We shall use (5.6) in future. This is the most important expression in the derivation of the U (1)Chern -Simons theory. We shall use this expression for the surface contribution to symplectic current for the casewhen the the areas are fixed. This will imply that we will always remain in the fixed area phase space. The symplecticstructure will not be able to give any first law as the variation of the area is zero on this phase space. We shall seehowever that the surface contribution to the symplectic structure acquires the form of a U (1) Chern- Simons theory.To prove that claim, we go back to the expression for the symplectic current. We have already seen that thesymplectic current on the spacetime region bounded by the Cauchy surfaces M + , M − and ∆ is given by:( Z M + − Z M − ) J ( δ , δ ) = − Z ∆ J ( δ , δ ) (5.7)In the expression for the symplectic current on ∆, i.e. R ∆ J ( δ , δ ), the potentials ψ ( ℓ ) and µ ( m ) come into play(see(3.35)). The potential ψ ( ℓ ) is a function of v only while µ ( m ) is still a function of ( θ, φ ). Then on the fixed areaphase space, the contribution to the symplectic current comes only from the terms involving the potential µ ( m ) . Thecontribution to ∆ for the spherical horizons can be calculated for fixed area horizon. A simple calculation gives: Z ∆ J ( δ , δ ) = 18 πGγ ( Z S − − Z S + ) { δ µ ( m ) δ ǫ − (1 ↔ } (5.8)The one- forms m have a gauge freedom. This is given by m ( v, θ, φ ) = e − iµ ( m ) ( v,θ,φ ) m (0 , θ, φ ). From now on, weshall only indicate the v dependence of µ ( m ) . This gives that V ( m ) → V ( m ) g = V ( m ) − i dµ ( m ) (see (3.20)). Theconnection V ( H ) then transforms as V ( H ) g = V ( H ) + 12 dµ ( m ) ( v ) (5.9)4To proceed further, we use the expression (5.6) in the symplectic current and integrate by parts and again use (5.9).This gives the following expression for the current Z ∆ J ( δ , δ ) = 28 πGγ A s ∆ π ( Z S − − Z S + ) { δ V ( H ) g ∧ δ V ( H ) g − (1 ↔ } (5.10) − πGγ A s ∆ π ( Z S − − Z S + ) { δ V ( H ) g ∧ δ V ( H ) − (1 ↔ } The further evaluation of the symplectic current is based on the idea that the expression for the connection V ( H ) issuch that it cannot be subject to any variation as we move along the direction of v . This is because, this connectionhas V ( H ) has only the dependence on ( θ, φ ). However, the information of the v dependence is carried by the field µ ( m ) . This implies that V ( H ) g has all the information of the v dependence. In short, the v dependence of V ( H ) g hasbeen transferred to µ ( m ) leaving V ( H ) only with the angular dependence. Then we can reduce the expression for thesymplectic current: Z ∆ J ( δ , δ ) = 18 πGγ A s ∆ π ( Z S − − Z S + ) { δ V ( H ) g ∧ δ V ( H ) g } (5.11)In other words, we get , that the symplectic structure of the topological theory on WIH is precisely the symplecticstructure of Chern- Simons theory. We will refer the theory on ∆ to be a U (1) Chern- Simons theory. The fullsymplectic structure for the spherically symmetric phase space of black hole spacetimes is:Ω( δ , δ ) = 116 πGγ Z M (cid:2) δ ( e I ∧ e J ) ∧ δ A HIJ − δ ( e I ∧ e J ) ∧ δ A HIJ (cid:3) − πGγ A s ∆ π Z S { δ V ( H ) g ∧ δ V ( H ) g } (5.12)The level of the Chern- Simons theory is − πGγ A s ∆ π . It is known that the level of the U (1) Chern- Simons theoryis an integer. So, we shall take πGγ A s ∆ π to be a positive integer. The result is highly non trivial considered in thebackdrop of the WIH formlation. What we have shown is that for all spherical horizons, extremal or non- extremal,the topological theory is still the Chern- Simons theory. The Chern- Simons gauge field does not see the ξ scaling ofthe null normal ℓ a , which controls the value of surface gravity for the horizon. This simply implies that whatever bethe null normal or whatever be the value of the surface gravity, the effective symplectic structure on the horizon isstill the Chern- Simons theory. VI. MINIMALLY COUPLED MAXWELL FIELDS ON WIH
A horizon should also be able to hold matter fields on it. In this section, we will give a general treatment of matterfields on WIH and find out the constraints that might be placed on the matter fields. We will take the example ofelectromagnetic field to analyse the situation. There will be some degree of simplification for the use of Maxwell fieldsrather than any arbitrary matter field, but the treatment will be general in the sense that the main results will be samefor all other matter fields minimally coupled to gravity. The reason we are interested in the Einstein-Maxwell systemis because our main objective is analysing the nonextremal and extremal horizons through a unified formulation andthe Einstein-Maxwell system provides the finest set of examples of these types of horizons. It is thus imperative tocheck that the WIH boundary conditions have sufficient structure to enable the existence of electromagnetic zerothlaw and a first law.This subsections are arranged in the following way. In the first, we will recall the boundary conditions of WIHwhich will put restrictions on the matter fields (Maxwell field), study its consequences and introduce conserved chargesdefined on the cross-section of the WIH. We will in the process show the main result that electromagnetic field canflow only along the horizon and none can cross it. This is a general result and can be shown to be true for allmatter fields on ∆. In the next subsection, we will prove that zeroth law and the go on to prove the first law for theEinstein-Maxwell system in the third subsection.
A. Constraint on fields from boundary conditions
Let us begin by recalling the boundary conditions of WIH which are of importance here. The only way that WIHboundary conditions can restrict matter is through conditions on the stress-energy tensor T ab . Thus, constraints on5matter fields will essentially come from the NEH boundary condition since further restriction of NEH to WIH onlyrestricts the class of functions that can multiply the null normal ℓ a , so that the zeroth law is obeyed, and hence cannotput further restrictions on matter. Now, the basic result that was obtained by the use of NEH boundary conditionsand Raychaudhuri equation is R ab ℓ a ℓ b , ξℓ a ]. We had already pointedout that this implies, R a ←− b ℓ b ,
0, Eq.. for any null normal in [ ξℓ a ]. The consequences of these areΦ = 12 R ab ℓ a ℓ b , = 12 R ab ℓ a m b , = 12 R ab ℓ a ¯ m b , ξℓ a ]. However,it is useful to study these on a case by case basis for each of the matter fields since, as will be shown below, the formof the Maxwell energy -momentum tensor introduces further simplifications. To check those, let us notice that theEinstein field equation and Eq.. (6.1) implies that: T ab ℓ a ℓ b , ℓ a in [ ξℓ a ]. An immediate consequence of this and Eq.. is that T ab ℓ b , − eℓ a , (6.4)for some non negative function e on ∆ and any ℓ a in [ ξℓ a ] . As we shall explicitly show and also is clear, this resultphysically implies that there is no flux of radiation crossing the horizon, implying isolation.Let us now concentrate on the case of electromagnetic field. We will denote the electromagnetic counterparts bybold letters. The main condition on the field is thus T ab ℓ a ℓ b , ℓ a in [ ξℓ a ]. The stress-energy tensor for electromagnetic fields is given in terms of the field strength F = d A as T ab = 14 π [ F ac F bc − g ab F cd F cd ] . (6.6)Let us now argue what to expect. Contracting Eq..(6.6) with ℓ a on both the free indices, Eq. (6.4) implies that thevector ℓ a F ac is null. Moreover, since F is antisymmetric, the vector is also normal to ℓ a . Thus, we can conclude that ℓ a F ac is proportional to ℓ a and hence ℓ y F ←− ,
0. This result will obviously be true for any ℓ a in [ ξℓ a ]. To check thisresult explicitly, we contract the expression of T ab with ℓ a ℓ b for a fixed ℓ a in [ ξℓ a ] and check consequences for F . WithEq.. (6.5) in mind, the first term in Eq..(6.6) can be written as F ac F db ℓ a ℓ b g cd = F ac F db ℓ a ℓ b (cid:0) m c ¯ m d − ¯ m c m d (cid:1) = 2 ( F ac ℓ a m c ) ( F ac ℓ a ¯ m c ) (6.7)We have used the fact that the metric at the horizon can be expressed in terms of a null-tetrad as g ab = − ℓ ( a n b ) +2 m ( a ¯ m b ) in the first step and the anti-symmetry of F in the second. Similarly, it can be checked that the second termin the Eq..(6.6) vanishes resulting in 0 , T ab ℓ a ℓ b , | ℓ a m b F ab | , (6.8)An immediate consequence of eqn. (6.8) is that F ab ℓ a = aℓ b + bn b , where a and b are some arbitrary functions.Contraction with ℓ b and use of antisymmetry property of F implies that b = 0 and hence, we get ℓ a F ab ←−−− , . (6.9)6for any ℓ a in [ ξℓ a ]. In order to obtain a similar expression for ∗ F recall that the stress energy tensor can be written as T ab = − π [ ∗ F ac ∗ F bc − g ab ∗ F cd ∗ F cd ] (6.10)Using arguments which led to eqn. (6.9), we obtain a similar restriction on ∗ F : ℓ a ∗ F ab ←−−−− , . (6.11)It is straightforward to show that eqn. (6.9) and eqn. (6.11) puts further constraints on the electromagnetic fieldtensor. To observe this, note that one can write F ab ℓ a , aℓ b , for any ℓ a in [ ξℓ a ] and a similar one for the dual ∗ F .Then using the expressions (6.6) and (6.10), the following conditions can be easily checked. T ab ℓ a m b , , T ab ℓ a ¯ m b T ab m a m b , , T ab ¯ m a ¯ m b . (6.12)It is interesting to observe that the first two set of conditions are none other than the ones we already had obtainedin eqn. (6.2) and thus are universal for any matter field on WIH. The second set of conditions however are specialfor the electromagnetic fields. In terms of Newman-Penrose components, these imply the following restrictions on theRicci tensor: Φ := 12 R ab m a m b , := 12 R ab ¯ m a ¯ m b , isolation of the WIH. Given a global timelike Killing vector field τ a ,the Poynting vector, describing the direction of energy flow, is defined as T ab τ b . On a WIH, the corresponding termcan be defined as T ab ℓ b which by the boundary conditions is future directed and causal (null). Using eqn. (6.5) andthe first set in eqn. (6.13), we can safely say that T ab ℓ b is proportional to ℓ a . In a local coordinate system adapted toWIH, ℓ a = (cid:0) ∂∂v (cid:1) a . Thus the direction of the energy flow is along the horizon and nothing can cross the WIH thoughthere is no restriction on the presence of radiation even arbitrary close to the horizon. For e.g., in a local advancedEddington -Finkelstein coordinates ( v, r, θ, φ ), it is easy to show that the components F rv , F rθ , F rφ may exist closeto the horizon and are unrestricted in the values but have no contribution when restricted and pulled back to thehorizon.Let us now define the electric and magnetic flux density two forms directed outwards. The electric flux two formis given by E ∆ , − ∗ F and the magnetic one by B ∆ , − F . The signatures have been taken such so as to take theorientation of S ∆ ( S ∆ is a cross-section of ∆, see fig. 1) into account which is defined with respect to normal pointinginto the horizon. Let us first evaluate £ ξℓ F ←− , ξℓ · d F ←− + d ( ξℓ · F ) ←−−−−− . (6.14)The first term on the right hand side vanishes due to Maxwell’s equations on ∆, while the second term is zero dueto the previous restriction on F , eqn. (6.9). Therefore we conclude that F ←− is Lie dragged by any ℓ a in [ ξℓ a ]. Anidentical argument for ∗ F leads to the analogous conclusion. Therefore we obtain £ ξℓ F ←− , £ ξℓ ∗ F ←− , . (6.15)These results imply that the 2 − forms E ∆ and B ∆ are “time-independent”. However, these do not restrict the formsof these fields otherwise.We can now define the electric charge of the horizon (we assume that the magnetic charges are zero, which ifpresent, can be analogously defined). Since the horizon is an inner boundary of spacetime, the normal to a 2-spherecross section of the horizon will naturally be inward pointing. Bearing this in mind, we define the electric charge ofthe horizon as Q ∆ : , − π I S ∆ ∗ F (6.16)For the definition to be meaningful,we should ensure that the values of Q ∆ should be independent of the cross sectionof the horizon S ∆ . This result can be anticipated since the NEH boundary conditions imply that a Killing vectorfield exists on ∆, one expects that the charge to be independent of cross-section. Since the 2 − forms E ∆ and B ∆ are“time-independent”, this guarantees that Q ∆ is independent of the choice of cross section S ∆ of the horizon. Notethat this result was obtained using only the boundary conditions; equations of motion in the bulk are not needed.7 B. Electromagnetic Zeroth Law
The zeroth law for electromagnetic field states that one can define a scalar potential on the horizon that is constantthroughout the horizon. Thus, to establish the zeroth law for the electromagnetic case, we need to define an electricpotential Φ at the horizon. For this, the electromagnetic potential A is gauge fixed on ∆ such that £ ( ξℓ ) A a ←− , ∇ a ←− £ ( ξℓ ) χ ( ξℓ ) . (6.17)where χ ( ξℓ ) is arbitrary non-zero but a fixed function of v alone. Following eqn. (6.17), given such an electromagneticpotential A we can now define the scalar potential Φ ( ξℓ ) at the horizon asΦ ( ξℓ ) , − ξℓ · A + £ ( ξℓ ) χ ( ξℓ ) (6.18)In a flat spacetime, the scalar potential is defined as the time component of the gauge potential one form A .However, this requires a gauge fixing since A takes values in the gauge equivalence class of addition of an exact form, A → A + dλ , λ being some arbitrary function. So the scalar function also suffers from a gauge ambiguity of addinga total time derivative.Recall that the Einstein-Maxwell case (static space-times) involves the electro-static potential Φ which one typicallysets Φ = − τ a A a where τ a is the static Killing field and the gauge is chosen such that the vector potential A tendsto zero at infinity and satisfies L τ A = 0 everywhere in space-time . Note that under the electromagnetic gaugetransformation δ λ A a = ∇ a λ , the eqn. (6.17) reduces to £ ( ξℓ ) ¯A a ←− , £ ( ξℓ ) ∇ a ←− (cid:2) χ ( ξℓ ) − λ (cid:3) (6.19)If we gauge fix χ ( ξℓ ) = λ , the definition used in static spacetime is obtained.Thus the definition we are proposing here is completely consistent and more general than is usually used. Theneqn. (6.18) is just the standard definition of scalar potential, ξℓ a playing the role of ”time” on ∆ and the additionalterm in eqn. (6.18) is just a total time derivative. It follows immediately that d Φ ( ξℓ ) , ( ξℓ ) is constant on the horizon which essentially is the electromagnetic zeroth law . In order that this istrue for the entire equivalence class [ ξℓ a ] requires the gauge fixing functions to vary in the class in a specific way, £ ( ξℓ ) χ ( ξℓ ) − Φ ( ξℓ ) = ξ [ £ ℓ χ ( ℓ ) − Φ ( ℓ ) ]. This restriction is to be viewed as follows: it is always possible to choose χ ( ℓ ) = 0 for one ℓ a such that Φ ( ℓ ) is a constant on ∆. Then for each null vector ξℓ a , the above restriction fixes thegauge in eqn. (6.18) such that Φ ( ξℓ ) remains a constant on ∆. For constant rescaling of ℓ a , it is consistent to choose χ ( ℓ ) = 0 for all ℓ a (like one does for the flat spacetime) but is not true for the generalized class [ ξℓ a ]. It is a nontrivialfact that even for the generalized class [ ξℓ a ], a constant potential such as in eqn. (6.18) exists making use of the gaugeambiguity (which always exists for scalar potentials) and the boundary conditions alone. C. Electromagnetic First Law
The electromagnetic part of the Lagrangian four-form is given by 8 πL = − F ∧ ∗ F . The variation of this Lagrangianis carried out over all A s that have the expected asymptotic fall-offs and are gauge fixed on ∆ as in (6.17). Thekey point to note is that although a surface term is needed in the gravitational part of the action, thanks to theelectromagnetic zeroth law, such a term is not needed for the electromagnetic part. Proceeding as before we find abulk and a surface symplectic structure. To extract the surface term we introduce a potential for Φ ( ξℓ ) (just likewhat we did for κ ( ξℓ ) ): £ ( ξℓ ) ϕ ( ξℓ ) , − Φ ( ξℓ ) . It also suffers from an additive ambiguity which is removed by choosing ϕ ( ξℓ ) | S − ,
0. Then the electromagnetic part of the symplectic structure becomesΩ em ( δ , δ ) = − π Z M (cid:2) δ ∗ F ∧ δ A − (1 ↔ (cid:3) +14 π I S ∆ (cid:2) δ ∗ F δ ( χ ( ξℓ ) + ϕ ( ξℓ ) ) − (1 ↔ (cid:3) . (6.21)Again, we wish to evaluate X ( ξℓ ) from the electromagnetic part of the symplectic structure. Making use of the fieldequations we find that the bulk symplectic structure gets contributions only through the boundaries, which equals8( £ ( ξℓ ) χ ( ξℓ ) − Φ ( ξℓ ) ) δQ ∆ . To evaluate the contribution from the surface symplectic structure care should be takennot to equate δ ( ξℓ ) with £ ( ξℓ ) for the potential ϕ . It turns out that δ ( ξℓ ) ϕ ( ξℓ ) , − δ ( ξℓ ) χ ( ξℓ ) . Combining contributions from the bulk and the surface, we find X em( ξℓ ) ( δ ) , − Φ ( ξℓ ) δQ ∆ . (6.22)Thus, the combined first law for the gravitational and electromagnetic fields agrees with the standard first law ofblack hole thermodynamics. D. E.M. Contribution to Surface Symplectic Structure
We have already obtained that F ab ℓ a ←−−− ,
0. This implies that: F ab , ¯ α [ b m a ] + α [ b ¯ m a ] + β [ b ℓ a ] (6.23)where, α is a complex one form and β is a real one form such that F ab is real. However, it is also imperative that oneneeds more conditions on the one forms α and β to match the degree of freedom of the F tensor.The conditions arealso obtained from the previous condition eqn (6.23). They are: α a ℓ a , , ¯ α a ℓ a β a ℓ a , α a = am a + b ¯ m a and β a = ¯ cm a + c ¯ m a . Then, we write the expansion of F in terms of the geometricforms ( ℓ, n, m, ¯ m ). Then, we obtain that: F ab ←−− , ¯ a ¯ m [ b m a ] + a m [ b ¯ m a ] (6.25)Now, note that in NP formalism, the six components of F are expressed in terms of the three complex scalars φ , φ and φ . It can be observed that φ ,
0. The other scalar is φ = F ab (cid:0) m a ¯ m b − ℓ a n b (cid:1) . We can express the aboveequation eqn.(6.25) in terms of the complex scalars. This gives ( a − ¯ a ) = ( φ − ¯ φ ). Then, the expansion is : F ab ←−− , − (2 Imφ ) ǫ (6.26)Similarly, it can be argued that the expansion of ∗ F is : ∗ F ab ←−− , (2 Reφ ) ǫ (6.27)We now argue that on the phase space of fixed parameters, the symplectic structure of Maxwell fields does notcontain any surface term. To prove this, we first see that the symplectic current is given by: J ( δ , δ ) = 14 π ( δ ∗ F ∧ δ A − δ ∗ F ∧ δ A ) (6.28)The next crucial step that one should take is to check wheather the integral of the symplectic current, integratedover ∆ (as in fig. 1) is zero on the space of solutions. In general, this is not true. However, when we consider thosehistories where the parameters are held fixed, the symplectic current goes to zero when integrated over ∆.This meansthat on the parameter-fixed phase space, the Maxwell theory does not contribute to the surface symplectic structure. VII. DISCUSSIONS
The Weak Isolated Horizon boundary conditions had been shown to be weak enough to include the extremal andnon-extremal horizons in the same phase space [15]. This extension opens up the possibility of an understanding ofentropy of extremal black holes in supergravity and string theory. Extremal black holes play a fundamental role insupergravity and the string theories (see [22] for discussions and other references). These solutions possess a highdegree of supersymmetry as isometries and due to some non-renormalization theorems one expects the counting ofdegeneracies of the associated quantum state to be protected over a range of string coupling constant which can varyfrom small to large values. Thus, the results obtained perturbatively, at small values of the parameter, continue to9hold for other large values. The popular choice for calculating the entropy of such black holes is the Killing horizon(KH) framework and the classical entropy formula suggested in [4]. Inspite of its wide use, it is to our opinion highlyunsatisfactory: Firstly, the derivation requires the existence of bifurcation two-spheres and it is not clear how togeneralise the framework to extremal black holes which admit no bifurcation two-sphere. In other words, the phase-space of non-extremal black holes does not include extremal solutions. Secondly, the extremal solutions arising insuch theories have curvature singularities at the horizon when the extremal limit is taken. This necessarily asks forhigher order stringy corrections to be taken into account [23]. To find the entropy of these extremal black holes instring theory, it is argued that entropy of these solutions are to be defined only through limits from their non-extremalcounterparts (see [24] for the arguments). However, since the phase-space of non-extremal Killing horizons do notcontain extremal horizons, it becomes ambiguous how such limits are to be taken. The original formulation of isolatedhorizons (IH) [7] bypass some of these difficulties but still it is not enough. The extremal and non-extremal black holescontinue to remain in different phase spaces so far as the validity of the first law is concerned and hence, extremallimit doesn’t make much sense. A new framework, called weak isolated horizons (WIH), presented in [15], removesthis difficulty allowing one to take limits in the same phase space. Since in WIH, the extremal horizons are in thesame class of the non-extremal ones, the entropy of an extremal black hole is automatically determined when onequantizes this WIH. Expectedly, the result is proportional to the area of the horizon. This is consistent with the claimmade in [24] that the entropy of an extremal black hole must be proportional to the area of the horizon, providedone uses a phase space that contains both extremal and non-extremal global solutions. A recent paper [28] revisitssome of these arguments. The basic idea behind this paper was to check wheather such expectations are borne out ina completely covariant manner.In this paper we extended the formulation of Weak Isolated Horizons to Holst’s action. The main reason for takingup this exercise is to make WIHs applicable in the framework of loop quantum gravity which makes essential use ofthe Holst action (See [27] for other applications of Holst’s action). Through our analysis we show that the essentialstructures of the phase space remain unaltered from the Palatini phase space once we rewrite everything in termsof the Holst connection one-form A ( H ) . Interestingly, there still exists a boundary symplectic structure, althoughnew potentials are needed here. Both the bulk and boundary symplectic structures conspire in such a way that thefirst law of WIH mechanics still holds. Although the results are very similar in spirit with the ones obtained fromthe Palatini action, the two cases differ substantially in details which have been elaborated in this paper. Anotherhallmark of our approach is that compared to the earlier results which used canonical phase space our approach iscompletely covariant. As a by-product we find that the effective theory at the horizon is a U (1) Chern-Simons theorywhich is obtained here from a completely covariant framework (this is to our knowledge has not been derived earlier).We also argued that the presence of electromagnetic fields on WIH does not affect the boundary symplectic structureand hence does not have any affect on the topological theory on the boundary. VIII. ACKNOWLEDGMENTS
We gratefully acknowledge discussions with Parthasarathi Majumdar and Parthasarathi Mitra in our group meet-ings. We especially thank the former for continuous encouragement and many stimulating remarks that helped toimprove our work.
IX. APPENDICESA. The Newman-Penrose formalism
In this part of the appendix, we give a brief summary of the Newman- Penrose (NP) formalism ([25]). The detailsof the formalism, notation and other technicalities are given in ([1, 26]). Though we will follow [1] so far as thenotation is concerned, the signature ( − , + , + , +) , is different in our case (and is same as [7, 15]) and hence one needsto recalculate some of the results.The NP formalism relies on the fact that one is allowed to have a basis with set of 4 null vectors ( ℓ , n , m , ¯ m ).The pair ℓ and n are real while pair m and ¯ m are complex conjugates of each other. These satisfy the following orthonormality conditions: ℓ.n = − m. ¯ m = 1 , (9.1)the rest being equal to zero. The next step is to define the Newman-Penrose spin coefficients (also called the Riccirotation coefficients in the tetrad formalism). All the information that the connection provides is encoded in the 120independent complex scalars. These are designated by special symbols as is given below: κ := − m a ℓ b ∇ b ℓ a = − m a Dℓ a ρ := − m a ¯ m b ∇ b m a = − m a ¯ δm a σ := − m a m b ∇ b ℓ a = − m a δℓ a µ := − n a m b ∇ b ¯ m a = − n a δ ¯ m a λ := − n a ¯ m b ∇ b ¯ m a = − n a ¯ δ ¯ m a τ := − m a n b ∇ b ℓ a = − m a ∆ ℓ a ν := − n a n b ∇ b ¯ m a = − n a ∆ ¯ m a π := − n a ℓ b ∇ b ¯ m a = − n a D ¯ m a ǫ := −
12 ( n a ℓ b ∇ b ℓ a + m a ℓ b ∇ b ¯ m a ) = −
12 ( n a Dℓ a + m a D ¯ m a ) γ := −
12 ( n a n b ∇ b ℓ a + m a n b ∇ b ¯ m a ) = −
12 ( n a ∆ ℓ a + m a ∆ ¯ m a ) α := −
12 ( n a ¯ m b ∇ b ℓ a + m a ¯ m b ∇ b ¯ m a ) = −
12 ( n a ¯ δℓ a + m a ¯ δ ¯ m a ) β := −
12 ( n a m b ∇ b ℓ a + m a m b ∇ b ¯ m a ) = −
12 ( n a δℓ a + m a δ ¯ m a ) . where, the symbols D, ∆ , δ, ¯ δ are the directional derivatives along the basis vectors ℓ, n, m, ¯ m , i.e. D = ℓ a ∇ a ∆ = n a ∇ a δ = m a ∇ a ¯ δ = ¯ m a ∇ a . (9.2)In other words, the derivative operator written in terms of these symbols turn out to be: ∇ a = − n a D − ℓ a ∆ + ¯ m a δ + m a ¯ δ. The ten independent components of the Weyl tensor are expressed in terms of five complex scalars Ψ , Ψ , Ψ , Ψ and Ψ . The ten components of the Ricci tensor are defined in terms of four real and three complex scalars Φ , Φ ,Φ , Λ, Φ , Φ and Φ . These scalars are defined as follows:Ψ = C abcd ℓ a m b ℓ c m d Φ = 12 R ab ℓ a m b Φ = 12 R ab ℓ a ¯ m b Ψ = C abcd ℓ a m b ℓ c n d Φ = 12 R ab m a m b Φ = 12 R ab ¯ m a ¯ m b Ψ = C abcd ℓ a m b ¯ m c n d Φ = 12 R ab ¯ m a n b Φ = 12 R ab m a n b Ψ = C abcd ℓ a n b ¯ m c n d Φ = 12 R ab ℓ a ℓ b Φ = 14 R ab ( ℓ a n b + m a ¯ m b )Ψ = C abcd ¯ m a n b ¯ m c n d Φ = 12 R ab n a n b Λ = R {} . For example, { A a B b C c D d } := A a B b C c D d − B a A b C c D d − A a B b D c C d + B a A b D c C d + C a D b A c B d − D a C b A c B d − C a D b B c A d + D a C b B c A d (9.3)There is subtlety in this construction of {} . In case all the terms in the braces are different from each other, justas in the above example all the terms are distinct, the {} is of the form as given above. In case the term is like { A a B b A c B d } , having some indistinguishable terms, only the first set of four terms as given above will suffice and itwill satisfy the criteria of being antisymmetric in the first two indices and the last two indices and should remainunchanged under simultaneous interchange of the first two and the last two indices. It is then trivial to check thatthe components of the Weyl tensor C abcd can be expanded as C abcd = F { ℓ a n b ℓ c n d } + F { m a ¯ m b m c ¯ m d } + F { l a n b m c ¯ m d } + F { ℓ a m b ℓ c ¯ m d } + F { n a m b n c ¯ m d } + [ F { ℓ a m b ℓ c m d } + F { n a m b n c m d } + F { ℓ a n b ℓ c m d } + F { ℓ a n b n c m d } + F { ℓ a m b n c m d } + F { ℓ a m b n c ¯ m d } + F { ℓ a m b m c ¯ m d } + F { n a m b m c ¯ m d } + c.c. ] (9.4)1where, F , · · · are expansion coefficients to be determined. The c.c refers to the comples conjugates. Note that theterms outside the square braces are manifestly real whereas the terms inside are complex. The complex conjugatescan be written down by simultaneous interchange of m to ¯ m and index number 3 to 4.We now determine the terms in the expansion of C abcd i.e. F , · · · . It is easy to check that: F = Ψ , F = Ψ , F = − Ψ , F = − Ψ and F = Ψ . The determination of other terms in the expansion of C abcd requires some other relations. Firstly, cyclicity requires that: C + C + C = 0 (9.5)The notation used here is the following: we mark l = 1 , n = 2 , m = 3 , ¯ m = 4 so that whatever index is contractedgets the above assigned value, for e.g. , C = C abcd ℓ a n b m c ¯ m d . Now, since C abcd is trace free, it implies the followingresults. Firstly, C = C = C = C = 0 (9.6)and secondly, using eqn.(9.5) we get: C = C C = C C = C C = C C = C C = 12 ( C − C ) (9.7)Now, we will use these relations. First, note that F = − C = − C . Thus, − C = F = Ψ . Secondly,note that Ψ ∗ = C = − F . Using eqn. (9.5), C = − (Ψ − Ψ ∗ ). Thus, F = C = − (Ψ − Ψ ∗ ). Also,using the last two equation in eqn. (9.7), we get, F = C = (Ψ + Ψ ∗ ) and F = C == (Ψ + Ψ ∗ ). Therest, F = F = F = F = 0, follows from the eqn. (9.6).Putting all these values of F abcd in the eqn. (9.4), we get C abcd = (Ψ + Ψ ∗ ) [ { ℓ a n b ℓ c n d } + { m a ¯ m b m c ¯ m d } ] − (Ψ − Ψ ∗ ) { ℓ a n b m c ¯ m d } + [Ψ { ℓ a m b ℓ c m d } + Ψ { n a ¯ m b n c ¯ m d } − Ψ { ℓ a m b n c ¯ m d } + complex conjugates ]+ [Ψ ( { ℓ a n b n c ¯ m d } + { n a ¯ m b ¯ m c m d } ) + complex conjugates ]+ [Ψ ( { ℓ a m b m c ¯ m d } − { ℓ a n b ℓ c n d } ) + complex conjugates ] (9.8)These has been used used in finding the expression for dω ( ℓ ) and dV ( m ) . B. Equivalence of Palatini and Holst Symplectic Structure
Consider the case when the manifold has no boundary. Then, the symplectic structure is given by:Ω ( δ , δ ) := 116 πGγ Z M (cid:8) δ [1 ( e ∧ e ) (cid:9) ∧ n δ (cid:16) A IJ − γ ǫ IJ KL A KL (cid:17)o (9.9)Now, the crucial point to note is the following. When viewed from the perspective of phase-space, the Holst term isa canonical transformation on the phase space. We shall show that γ dependent term will vanish. even in presence ofinner boundaries. We then must show that: δ (cid:0) e I ∧ e J (cid:1) ∧ δ A IJ = δ (cid:0) e I ∧ e J (cid:1) ∧ δ A IJ ⇒ δ (cid:0) e I (cid:1) ∧ (cid:0) e J ∧ δ A IJ (cid:1) = δ (cid:0) e I (cid:1) ∧ (cid:0) e J ∧ δ A IJ (cid:1) (9.10)To prove the equality, we use the equation of motion. Firstly, from the equation of motion, we get that e J ∧ δ A IJ = dδ e I + A IJ δ e J (9.11)Putting this equation eqn. (9.11) in the L.H.S. of (9.10), we get, δ (cid:0) e I (cid:1) ∧ (cid:0) e J ∧ δ A IJ (cid:1) = δ (cid:0) e I (cid:1) ∧ d ( δ e I ) − A IJ ∧ e I ∧ δ e J . (9.12)Also, the first term in (9.12) can be further reduced as: δ e I ∧ dδ e I = − d (cid:0) δ e I ∧ δ e I (cid:1) + dδ e I ∧ δ e I (9.13)2Thus, using the equation eqn. (9.13), the equation (9.12) becomes: L.H.S. = − d (cid:0) δ e I ∧ δ e I (cid:1) + dδ e I ∧ δ e I − A IJ ∧ e I ∧ δ e J = − d (cid:0) δ e I ∧ δ e I (cid:1) + δ e I ∧ (cid:0) dδ e I + A IJ ∧ δ e J (cid:1) = − d (cid:0) δ e I ∧ δ e I (cid:1) + δ e I ∧ (cid:0) e J ∧ δ A IJ (cid:1) . (9.14)where in the third line in the above equation, we have used the equation of motion, i.e. , the δ version of eqn. (9.11).Thus, the term in the L.H.S. is equal to the term on the R.H. S. In other words, the Palatini and the Holst symplecticstructure are equivalent when there is no boundary in the spacetime.If the spacetime has an inner boundary which is the present case of interest, it is instructive to check wheather theequivalence still holds. To check that, we go back to the construction of the symplectic current J ( δ , δ ). The resulthas been calculated before and gives J ( δ , δ ) := 116 πGγ (cid:8) δ [1 ( e ∧ e ) (cid:9) ∧ n δ (cid:16) A IJ − γ ǫ IJ KL A KL (cid:17)o (9.15)Integrating the symplectic current over M , taking the orientation into account, we get:( Z M + − Z M − ) J ( δ , δ ) + Z ∆ J ( δ , δ ) = 0 (9.16)We have already seen that the γ - dependent term in the symplectic current gives a total derivative term. In thefollowing steps, we will only concern ourselves with γ - dependent term in the symplectic current since the other γ -independent term is the standard Palatini symplectic current.The first integration of the γ - dependent term over M + will go to the boundaries of M + and thence leave an integralover S + (and at infinity which goes to zero by asymptotic boundary conditions). Similarly, the integration of the γ -dependent term over M − will leave an integral over S − . The integral over ∆ will give two boundary integrals, one at S + and another at S − . Taking the orientations of the surfaces, we get that the γ - dependent symplectic current with J γ ( δ , δ ) = dα ( δ , δ ) gives:( Z S + − Z S − ) α ( δ , δ ) + ( Z S − − Z S + ) α ( δ , δ ) + Z M + ∪ M − ∪ ∆ J P ( δ , δ ) = 0 (9.17)where, J P ( δ , δ ) is the symplectic current for the Palatini action. The above equation shows that the contributionsfrom the γ - dependent terms cancel. This implies that even in presence of boundaries, the Holst symplectic currentis equivalent to that of Palatini. In other words, even in presence of boundaries, the canonical transformation holdsgood. C. Calculation of dV Let us consider the definition of the Riemann tensor:[ ∇ a ∇ b − ∇ b ∇ a ] X c = − R cabd X d (9.18)Now, consider the case when the vector X a = m a . Then, we have:[ ∇ a ∇ b − ∇ b ∇ a ] m c , ( ∇ a U b − ∇ b U a ) ℓ c + (cid:16) ∇ a V ( m ) b − ∇ b V ( m ) a (cid:17) m c + ( ω a U b − ω b U a ) ℓ c + (cid:16) U a V ( m ) b − U b V ( m ) a (cid:17) ℓ c , − R abdc m d (9.19)Multiply both sides by ¯ m c , we get: (cid:16) ∇ a V ( m ) b − ∇ b V ( m ) a (cid:17) , − R abdc m d ¯ m c , R abcd m d ¯ m c (9.20)Now, we use the expansion of Riemann tensor in terms of the Ricci, the Weyl tensor and the Ricci Scalar R abcd = C abcd + 2 R [ a [ c g b ] d ] − Rg [ ac g b ] d (9.21)3Then, using the expansion of the the Weyl tensor in terms of the Newman-Penrose scalars (see appendix (IX A)), weget the following result: R ab ←− cd m d ¯ m c , Ψ m [ a ¯ m b ] + 12 (cid:0) R bd ¯ m a m d − R bc ¯ m c m a − R da m d ¯ m b + R ca ¯ m c m b (cid:1) + 13 Rm [ a ¯ m b ] (9.22)To simplify the second term in the term in the above expression of eqn. (9.22) , we consider the following. Let, R b ←− d m d , Am b + B ¯ m b + Cn b . (9.23)where, A, B and C are to be determined. This implies that A = R bd ¯ m b m d , B = R bd m b m d and C = 0 as R ab ℓ a m b , R b ←− d m d , ( R pq ¯ m p m q ) m b + ( R pq m p m q ) ¯ m b (9.24)Putting these expressions in the eqn. (9.22), we get dV ( m ) , i (cid:20) Ψ − ( R pq m p ¯ m q ) + R (cid:21) ǫ (9.25)Using the expression Φ = R ab ( ℓ a n b + m a ¯ m b ), it is easy to check that Φ = R + R ab ℓ a n b . Then, defining thereal connection iV ( m ) =: ¯ V ( m ) = − im [ I ¯ m J ] A IJ , we get the following expression: d ¯ V ( m ) , Ψ − Φ − R
24 ) ǫ (9.26)Let us now look at the expression for the connection A HIJ . We shall define a new connection for this total connectionby projecting A HIJ as follows. Define the following connection V ( H ) , − im [ I ¯ m J ] A ( H ) IJ and the curvature of thisconnection becomes : dV ( H ) = (cid:20) Re Ψ + γ Im Ψ − Φ − R (cid:21) ǫ = (cid:20) Ψ ( H ) − Φ − R (cid:21) ǫ . (9.27)where, we have defined Ψ ( H ) = Re Ψ + γ Im Ψ . If we are in the vacuum (Φ ) = 0 and the cosmological constantis zero, then we have: dV ( H ) , Ψ ( H ) 2 ǫ (9.28) D. Going back to Canonical Phase Space
We now use the foregoing results of the covariant phase space to get the results of the canonical phase space tomake comparisons. To set the stage, we recapitulate the following conventions and results [18]:The indices i, j, · · · take values of the subspace orthogonal to the internal fixed vector τ I . If the projection vectoris denoted by q iI , then the induced internal metric on the subspace is given by: η ij = q Ii q Jj η IJ . (9.29)The internal 4 − dimensional antisymmetric tensor ǫ IJKL naturally induces the completely antisymmetric tensor onthe subspace and will be denoted by ǫ ijk such that: ǫ ijk = q Ii q Jj q Kk τ L ǫ LIJK . (9.30) See the appendix IX D for details and other comparisons. A ( H ) IJ , then we can define a connection one form Γ i := q iI ǫ IJ KL τ J A ( H ) KL and theextrinsic curvature K := q iI A ( H ) IJ n J . Let us now define the connection A ( H ) i := Γ i − γK i . It then follows that A ( H ) i = − ǫ iKL A ( H ) KL . (9.31)To see further, let us now consider the way to define the unit time- like normal τ I . The obvious way is to use thenull- normals ℓ a and n a . We define: τ I := ( ℓ I + n I ) √ r I := ( ℓ I − n I ) √ r i is such that it picks out the connection intrinsic to the two sphere S ∆ : A i r i = − √ q Ii ( ℓ I − n I ) ǫ iKL A ( H ) KL (9.33)= i ℓ K n L − ℓ L n K ) ǫ KLIJ A ( H ) IJ A i r i = − im [ I ¯ m J ] A ( H ) IJ =: ˜ V .
We can also define momentum P ai = πGγ e jb e kc ǫ ijk η abc . The 2 form dual to the momentum pulled back to S ∆ isgiven by Σ ←− iab = η abc P aj η ij . Then, we have:Σ ipq r i = 116 πGγ im [ j ¯ m k ] ǫ ijk r i ǫ pq (9.34)= 18 πGγ ǫ pq . This leads to the following form of the curvature on the sphere S ∆ dV ( H ) s = − π A s ∆ πGγ Σ i r i (9.35) E. Spherical Symmetry
We consider the spherical symmetric metric and prove the claims that λ = π , µ is spherically symmetric.Consider the following most general spherically symmetric metric: ds = − f ( r, t ) dt + g ( r, t ) dr + r d Ω (9.36)where, d Ω is the 2 − sphere metric. The null normals can be calculated to as follows: ℓ a , √ − f ∂∂t + 1 g ∂∂r ) n a , √ − f ∂∂t − g ∂∂r ) m a , r ∆ √ ∂∂θ − iSinθ ∂∂φ ) (9.37)where, the normalizations have been fixed so that ℓ.n = − m. ¯ m = 1. The covariant derivative of the null normalturns out to be: ∇ a n b = ∂ a f ( dt ) b − f Γ tab + ∂ a g ( dr ) b − g Γ rab (9.38)Then, it follows simply that π = 0 and λ = 0. For µ , observe that µ = ( f Γ tab + g Γ rab ) m a ¯ m b will only contribute for a, b = θ, φ . Then, µ = r g . Since g is only a function of ( t, r ), µ is spherically symmetric. [1] S. Chandrasekhar, The Mathematical Theory of Black Holes , Clarendon Press, Oxford, (1992). [2] J.W. Bardeen, B. Carter and S.W. Hawking, Commun. Math. Phys. , 161 (1973).[3] A. Ashtekar, C. Beetle and S. Fairhurst, Class. Quant. Grav. , 253 (2000).[4] R. M. Wald Phys. Rev.
D48 , 3427 (1993) and V. Iyer and R. M. Wald
Phys. Rev.
D50 , 846 (1994).[5] R. M. Wald and I. Racz
Class. Quant. Grav. , 539 (1996).[6] A. Ashtekar, C. Beetle and S. Fairhurst, Class. Quant. Grav. , L1 (1999).[7] A. Ashtekar, C. Beetle, O. Dreyer, S. Fairhurst, B. Krishnan, J. Lewandowski and Jacek Wisniewski Phys. Rev. Lett. ,3564 (2000); A. Ashtekar, B. Krishnan and S. Fairhurst, Phys. Rev.
D62 , 104025 (2000).[8] A. Ashtekar and B. Krishnan
Liv. Rev. Rel.
10 (2004).[9] R. M. Wald
Liv. Rev. Rel. , 6 (2001).[10] E. Gourgoulhon and J. L. Jaramillo Phys. Rept. , 159 (2006).[11] A. Ashtekar, A. Corichi and K. Krasnov,
Adv. Theo. Math. Phys. , 418 (1999).[12] A. Ashtekar, J. C. Baez, A. Corichi and K. Krasnov, Phys. Rev. Lett. , 904 (1998).[13] A. Ashtekar, J. C. Baez and K. Krasnov, Adv. Theo. Math. Phys. , 1 (2000); A. Ashtekar and A. Corichi, Class. Quant.Grav. , 4473 (2003); A. Ashtekar, J. Engle and C. van der Broeck, Class. Quant. Grav. , L27 (2004).[14] A. Ashtekar, C. Beetle and J. Lewandowski, Phys. Rev.
D64
Class. Quant. Grav. , 1195 (2002); A. Ashtekar, A. Corichi and D. Sudarsky, Class. Quant. Grav. , 3413 (2003); A.Ashtekar, J. Engle, T. Pawlowski and C. van der Broeck, Class. Quant. Grav. , 2549 (2004).[15] A. Chatterjee and A. Ghosh, Class. Quant. Grav. , 7521 (2006); A. Chatterjee AIP Conf. Proc. , 96 (2007).[16] S. Holst,
Phys. Rev.
D56 , (2002).[17] A. Ashtekar
Lectures on Non-perturbative Canonical Gravity (notes prepared in collaboration with R Tate), World Scien-tific, Singapore, (1991).[18] A. Ashtekar and J. Lewandowski,
Class. Quant. Grav. , , R53R152 (2004)[19] C. Rovelli, T. Thiemann, Phys.Rev.
D57
Phys. Rev. D7 , 2333 (1973).[21] S. W. Hawking, Commun. Math. Phys. , 199 (1975).[22] B. de Wit B Proc. Symp. in Honor of Julius Wess on the Occasion of his 70th Birthday, (Munich, Germany) Fortschr.Phys.
183 (2006)[23] A. Sen
J. High Energy Phys.
059 (2005);
J. High Energy Phys.
008 (2006).[24] A. Ghosh and P. Mitra
Phys. Rev. Lett. Fortschr. Phys.
49 (2000); K. Kiefer and J. Louko
Ann. Phys.
67, (1999).[25] R. Penrose and E. T. Newman
J. Math. Phys.
58 (1977).[26] J. Stewart
Advanced General Relativity
Cambridge Univ Press, Cambridge (1990).[27] L. Fatibene, M. Ferraris, M. Francaviglia, G. Pacchiella arXiv arXivarXiv