Black hole entropy from the -invariant formulation of Type I isolated horizons
Abstract
A detailed analysis of the spherically symmetric isolated horizon system is performed in terms of the connection formulation of general relativity. The system is shown to admit a manifestly invariant formulation where the (effective) horizon degrees of freedom are described by an Chern-Simons theory. This leads to a more transparent description of the quantum theory in the context of loop quantum gravity and modifications of the form of the horizon entropy.
pacs:
04.60.-m, 04.60.Pp, 04.20.Fy, 11.15.YcI Introduction
Black holes are intriguing solutions of classical general relativity describing important aspects of the physics of gravitational collapse. Their existence in our nearby universe is by now supported by a great amount of observational evidence observ . When isolated, these systems are remarkably simple for late and distant observers: once the initial very dynamical phase of collapse is passed the system is expected to settle down to a stationary situation completely described (as implied by the famous results by Carter, Israel, and Hawking wald ) by the three extensive parameters (mass , angular momentum , electric charge ) of the Kerr-Newman family kerrnew .
However, the great simplicity of the final stage of an isolated gravitational collapse for late and distant observers is in sharp contrast with the very dynamical nature of the physics seen by in-falling observers which depends on all the details of the collapsing matter. Moreover, this dynamics cannot be consistently described for late times (as measured by the infalling observers) using general relativity due to the unavoidable development, within the classical framework, of unphysical pathologies of the gravitational field. Concretely, the celebrated singularity theorems of Hawking and Penrose hawking imply the breakdown of predictability of general relativity in the black hole interior. Dimensional arguments imply that quantum effects cannot be neglected near the classical singularities. Understanding of physics in this extreme regime requires a quantum theory of gravity. Black holes (BH) provide, in this precise sense, the most tantalizing theoretical evidence for the need of a more fundamental (quantum) description of the gravitational field.
Extra motivation for the quantum description of gravitational collapse comes from the physics of black holes available to observers outside the horizon. As for the interior physics, the main piece of evidence comes from the classical theory itself which implies an (at first only) apparent relationship between the properties of idealized black hole systems and those of thermodynamical systems. On the one hand, black hole horizons satisfy the very general Hawking area theorem (the so-called second law) stating that the black hole horizon area can only increase, namely
| (1) |
On the other hand, the uniqueness of the Kerr-Newman family, as the final (stationary) stage of the gravitational collapse of an isolated gravitational system, can be used to prove the first and zeroth laws: under external perturbation the initially stationary state of a black hole can change but the final stationary state will be described by another Kerr-Newman solution whose parameters readjust according to the first law
| (2) |
where is the surface gravity, is the electrostatic potential at the horizon, and the angular velocity of the horizon. There is also the zeroth law stating the uniformity of the surface gravity on the event horizon of stationary black holes, and finally the third law precluding the possibility of reaching an extremal black hole (for which ) by means of any physical process333The third law can only be motivated by a series of examples. Extra motivations comes from the validity of the cosmic censorship conjecture.. The validity of these classical laws motivated Bekenstein to put forward the idea that black holes may behave as thermodynamical systems with an entropy and a temperature where is a dimensionless constant and the dimensionality of the quantities involved require the introduction of leading in turn to the appearance of the Planck length , even though in his first paper beke Bekenstein states “that one should not regard as the temperature of the black hole; such identification can lead to all sorts of paradoxes, and is thus not useful”. The key point is that the need of required by the dimensional analysis involved in the argument called for the investigation of black hole systems from a quantum perspective. In fact, not long after, the semiclassical calculations of Hawking Hawking:1974sw —that studied particle creation in a quantum test field (representing quantum matter and quantum gravitational perturbations) on the space-time background of the gravitational collapse of an isolated system described for late times by a stationary black hole—showed that once black holes have settled to their stationary (classically) final states, they continue to radiate as perfect black bodies at temperature . Thus, on the one hand, this confirmed that black holes are indeed thermal objects that radiate at a the given temperature and whose entropy is given by , while, on the other hand, this raised a wide range of new questions whose proper answer requires a quantum treatment of the gravitational degrees of freedom.
Among the simplest questions is the issue of the statistical origin of black hole entropy. In other words, what is the nature of the the large amount of micro-states responsible for black hole entropy. This simple question cannot be addressed using semiclassical arguments of the kind leading to Hawking radiation and requires a more fundamental description. In this way, the computation of black hole entropy from basic principles became an important test for any candidate quantum theory of gravity. In string theory it has been computed using dualities and no-normalization theorems valid for extremal black holes string . There are also calculations based on the effective description of near horizon quantum degrees of freedom in terms of effective -dimensional conformal theories carlip . In loop quantum gravity the first computations (valid for physical black holes) were based on general considerations and the fact that the area spectrum in the theory is discrete bhe0 . The calculation was later refined by quantizing a sector of the phase space of general relativity containing a horizon in ‘equilibrium’ with the external matter and gravitational degrees of freedom bhe1 . In all cases agreement with the Bekenstein-Hawking formula is obtained with logarithmic corrections in .
In this work we concentrate and further develop the theory of isolated horizons in the context of loop quantum gravity. Recently, we have proposed a new computation of BH entropy in loop quantum gravity (LQG) that avoids the internal gauge-fixing used in prior works nous and makes the underlying structure more transparent. We show, in particular, that the degrees of freedom of Type I isolated horizons can be encoded (along the lines of the standard treatment) in an boundary connection. The results of this work clarify the relationship between the theory of isolated horizons and SU(2) Chern-Simons theory first explored in kiril-lee , and makes the relationship with the usual treatment of degrees of freedom in loop quantum gravity clear-cut. In the present work, we provide a full detailed derivation of the result of our recent work and discuss several important issues that were only briefly mentioned then.
An important point should be emphasized concerning the logarithmic corrections mentioned above. The logarithmic corrections to the Bekenstein-Hawking area formula for black hole entropy in the loop quantum gravity literature were thought to be of the (universal) form amit . In majundar Kaul and Majumdar pointed out that, due to the necessary gauge symmetry of the isolated horizon system, the counting should be modified leading to corrections of the form . This suggestion is particularly interesting because it would eliminate the apparent tension with other approaches to entropy calculation. In particular their result is in complete agreement with the seemingly very general treatment (which includes the string theory calculations) proposed by Carlip carlip-log . Our analysis confirms Kaul and Majumdar’s proposal and eliminates in this way the apparent discrepancy between different approaches.
The article is organized as follows. In the following section we review the formal definition of isolated horizons. In Section III we state the main equations implied by the isolated horizon boundary conditions for fields at a spherically symmetric isolated horizon. In Section IV we prove a series of propositions that imply the main classical part of our results: we derive the form of the conserved presymplectic structure of spherically symmetric isolated horizons, and we show that degrees of freedom at the horizon are described by an Chern-Simons presymplectic structure. In Section VI we briefly review the derivation of the zeroth and first law of isolated horizons. In Section V we study the gauge symmetries of the Type I isolated Horizon and explicitly compute the constraint algebra. In Section VII we review the quantization of the spherically symmetric isolated horizon phase space and present the basic formulas necessary for the counting of states that leads to the entropy. We close with a discussion of our results in Section VIII. The appendix contains an analysis of Type I isolated horizons from a concrete (and intuitive) perspective that makes use of the properties of stationary spherically symmetric black holes in general relativity.
II Definition of isolated horizons
The standard definition of a BH as a spacetime region from which no information can reach idealized observers at (future null) infinity is a global definition. This notion of BH requires a complete knowledge of a spacetime geometry and is therefore not suitable for describing local physics. The physically relevant definition used, for instance, when one claims there is a black hole in the center of the galaxy, must be local. One such local definition was introduced in ack ; better ; ih_prl with the name of isolated horizons (IH). Here we present this definition according to ih_prl ; afk ; abl2002 ; abl2001 . This discussion will also serve to fix our notation. In the definition of an isolated horizon below, we allow general matter, subject only to conditions that we explicitly state.
Definition: The internal boundary of a history will be called an isolated horizon provided the following conditions hold:
-
i)
Manifold conditions: is topologically , foliated by a (preferred) family of 2-spheres and equipped with an equivalence class of transversal future pointing vector fields whose flow preserves the foliation, where is equivalent to if for some positive real number .
-
ii)
Dynamical conditions: All field equations hold at .
-
iii)
Matter conditions: On the stress-energy tensor of matter is such that is causal and future directed.
-
iv)
Conditions on the metric determined by , and on its levi-Civita derivative operator : (iv.a) The expansion of within is zero. This, together with the energy condition (iii) and the Raychaudhuri equation at , ensures that is additionally shear-free. This in turn implies that the Levi-Civita derivative operator naturally determines a derivative operator intrinsic to via , tangent to . We then impose (iv.b) .
-
v)
Restriction to ‘good cuts.’ One can show furthermore that for some intrinsic to . A 2-sphere cross-section of is called a ‘good cut’ if the pull-back of to is divergence free with respect to the pull-back of to . As shown in abl2002 , every horizon satisfying (i)-(iv) above possesses at least one foliation into ‘good cuts’; this foliation is furthermore generically unique. We require that the fixed foliation coincide with a foliation into ‘good cuts.’
Let us discuss the physical meaning of these conditions. The first two conditions are rather weak. The third condition is satisfied by all matter fields normally used in general relativity. The fifth condition is a partial gauge fixing of diffeomorphisms in the ‘time’ direction. The main condition is therefore the fourth condition. (iv.a) requires that be expansion-free. This is equivalent to asking that the area 2-form of the 2-sphere cross-sections of be constant along generators . This combined with the matter condition (iii) and the Raychaudhuri equation implies that in fact the entire pull back of the metric to the horizon is Lie dragged by . Condition (iv.b) further stipulates that the derivative operator be Lie dragged by . This implies, among other things, an analogue of the zeroeth law of black hole mechanics: conditions (i) and (iii) imply that is geodesic — . The proportionality constant is called the surface gravity, and condition (iv.b) ensures that it is constant on the horizon for any given . Furthermore, if we had not fixed , but only required that an exist such that the isolated horizon boundary conditions hold, then condition (iv.b) would ensure that this is generically unique abl2002 . From the above discussion, one sees that the geometrical structures on that are time-independent are precisely the pull-back of the metric to , and the derivative operator . In fact, the main conditions (iv.a) and (iv.b) are equivalent to requiring and . For this reason it is natural to define as the horizon geometry.
Let us summarize. Isolated horizons are null surfaces, foliated by a family of marginally trapped 2-spheres such that certain geometric structures intrinsic to are time independent. The presence of trapped surfaces motivates the term ‘horizon’ while the fact that they are marginally trapped — i.e., that the expansion of vanishes — accounts for the adjective ‘isolated’. The definition extracts from the definition of Killing horizon just that ‘minimum’ of conditions necessary for analogues of the laws of black hole mechanics to hold. Boundary conditions refer only to behavior of fields at and the general spirit is very similar to the way one formulates boundary conditions at null infinity.
Remarks:
-
1.
All the boundary conditions are satisfied by stationary black holes in the Einstein-Maxwell-dilaton theory possibly with cosmological constant. Note however that, in the non-stationary context, there still exist physically interesting black holes satisfying our conditions: one can solve for all our conditions and show that the resulting 4-metric need not be stationary on lew2000 .
-
2.
In the choice of boundary conditions, we have tried to strike the usual balance: On the one hand the conditions are strong enough to enable one to prove interesting results (e.g., a well-defined action principle, a Hamiltonian framework, and a realization of black hole mechanics) and, on the other hand, they are weak enough to allow a large class of examples. As we already remarked, the standard black holes in the Einstein-Maxwell-dilatonic systems satisfy these conditions. More importantly, starting with the standard stationary black holes, and using known existence theorems one can specify procedures to construct new solutions to field equations which admit isolated horizons as well as radiation at null infinity lew2000 . These examples, already show that, while the standard stationary solutions have only a finite parameter freedom, the space of solutions admitting isolated horizons is infinite dimensional. Thus, in the Hamiltonian picture, even the reduced phase-space is infinite dimensional; the conditions thus admit a very large class of examples.
-
3.
Nevertheless, space-times admitting isolated horizon are very special among generic members of the full phase space of general relativity. The reason is apparent in the context of the characteristic formulation of general relativity where initial data are given on a set (pairs) of null surfaces with non trivial domain of dependence. Let us take an isolated horizon as one of the surfaces together with a transversal null surface according to the diagram shown in Figure 1. Even when the data on the isolated horizon may be infinite dimensional (for Type II and II isolated horizons, see below), in all cases no transversing radiation data is allowed by the IH boundary condition. Roughly speaking the isolated horizon boundary condition reduces to one half the number of local degrees of freedom.
-
4.
Notice that the above definition is completely geometrical and does not make reference to the tetrad formulation. There is no reference to any internal gauge symmetry. In what follows we will deal with general relativity in the first order formulation which will introduce, by the choice of variables, an internal gauge group corresponding to local transformations (in the case of Ashtekar variables) or transformations (in the case of real Ashtekar-Barbero variables). It should be clear from the purely geometric nature of the above definition that the IH boundary condition cannot break by any means these internal symmetries.
Isolated horizon classification according to their symmetry groups
Next, let us examine symmetry groups of isolated horizons. A symmetry of
is a diffeomorphism on which preserves the horizon geometry
and at most rescales elements of by a positive constant. It is clear that
diffeomorphisms generated by are symmetries. So, the symmetry group is at
least 1-dimensional. In fact, there are only three possibilities for :
-
(a)
Type I: the isolated horizon geometry is spherical; in this case, is four dimensional ( rotations plus rescaling-translations444In a coordinate system where the rescaling-translation corresponds to the affine map with constants. along );
-
(b)
Type II: the isolated horizon geometry is axi-symmetric; in this case, is two dimensional (rotations round symmetry axis plus rescaling-translations along );
-
(c)
Type III: the diffeomorphisms generated by are the only symmetries; is one dimensional.
Note that these symmetries refer only to the horizon geometry. The full space-time metric need not admit any isometries even in a neighborhood of the horizon. In this paper, as in the classic works bhe1 ; ack , we restrict ourselves to the Type I case. Although a revision would be necessary in light of the results of our present work, the quantization and entropy calculation in the context of Type II and Type III isolated horizons has been considered in jon .
III Some extra details for Type I isolated horizons
In this section we first list the main equations satisfied by fields at an isolated horizon of Type I. The equations presented here can be directly derived from the IH boundary conditions implied by the definition of Type I isolated horizons given above. Most of the equations presented here can be found in ack . For completeness we prove these equations at the end of this section. As we shall see in Subsection III.2, some of the coefficients entering the form of these equations depend on the representative chosen among the equivalence class of null generators . Throughout this paper we shall fix an null generator by the requirement that the surface gravity matches the one corresponding to the stationary black hole with the same macroscopic parameters as the Type I isolated horizon under consideration. This choice makes the first law of IH take the form of the usual first law of stationary black holes (see Section VI).
III.1 The main equations
When written in connection variables, the isolated horizon boundary condition implies the following relationship between the curvature of the Ashtekar connection at the horizon and the -form (in the time gauge)
| (3) |
where is the area of the IH, the double arrows denote the pull-back to with a Cauchy surface with normal at , and null and normalized according to . Notice that the imaginary part of the previous equation implies that
| (4) |
Another important equation is
| (5) |
The previous equations follow from equations (3.12) and (B.7) of reference ack . Nevertheless, they also follow from the abstract definition given in the introduction. From the previous equations, only equation (5) is not explicitly proven from the definition of IH in the literature. Therefore, we give here an explicit prove at the end of this section. For concreteness, as we think it is helpful for some readers to have a concrete less abstract treatment, another derivation using directly the Schwarzschild geometry is given in Appendix A. The previous equations imply in turn that
| (6) |
where is the Ashtekar-Barbero connection 555In our convention the isomorphism is defined by which implies that and . .
III.2 Proof of the main equations
In this subsection we use the definition of isolated horizons provided in the previous section to prove some of the equations stated above. We will often work in a special gauge where the tetrad is such that is normal to and and are tangent to . This choice is only made for convenience, as the equations presented in the previous section are all gauge covariant, their validity in one frame implies their validity in all frames.
Lemma 1: In the gauge where the tetrad is chosen so that (which can be completed to a null tetrad , and ), the shear-free and vanishing expansion (condition () in the definition of IH) imply
| (7) |
Proof: The expansion and shear of the null congruence of generators of the horizon is given by
| (8) |
This implies
| (9) | |||||
| (10) |
where we have used the definition of the spin connection . Similarly we have
| (11) | |||||
| (12) |
As and form a non degenerate frame for , and from the definition of pull-back, the previous two equations imply the statement of our lemma.
The previous lemma has an immediate consequence on the form of equation (4) for the component in the frame of the previous lemma. More precisely it says that . The good-cut condition () in the definition implies then that
| (13) |
Another important consequence of the previous lemma is equation (3), also derived in ack . We give here for completeness and self consistency a sketch of its derivation. This equation follows from identity
| (14) |
where is the Riemann tensor and , which can be derived using Cartan’s structure equations. A simple algebraic calculation using the null tetrad formalism (see for instance chandra page 43) with the null tetrad of Lemma 1, and the definitions and , where is the Ricci tensor and the Weyl tensor, yields
| (15) |
where . An important point here is that the previous expression is valid for any two sphere embedded in spacetime in an adapted null tetrad where and are normal to . However, in the special case where (where with a Type I isolated horizon) it follows from spherical symmetry that with a constant on the horizon . Moreover, in the gauge defined in the statement of Lemma 1, the only non vanishing component of the previous equation is the component for which (using Lemma 1) we get
| (16) |
with the area element of . Integrating the previous equation on one can completely determine the constant , namely
| (17) |
from where equation (3) immediately follows.
Lemma 2: For Type I isolated horizons
| (18) |
for some constant . One can choose a representative from the equivalence class of null normals to the isolated horizon in order to fix by making use of the translation symmetry of IH along . By studying the stationary spherically symmetric back hole solutions one can show that this corresponds to the choice where the surface gravity of the IH matches the stationary surface gravity (see Appendix A).
Proof: In order to simplify the notation all free indices associated to forms that appear in this proof are pulled back to (this allows us to drop the double arrows from equations). In the frame of lemma 1, where is normal to , the only non trivial component of the equation we want to prove is the component, namely:
| (19) |
where and . Now, in that gauge, we have that for some matrix of coefficients . Notice that the left hand side of the previous equation equals . We first prove that is time independent, i.e. that . We need to use the isolated horizon boundary condition
| (20) |
where is the derivative operator determined on the horizon by the Levi-Civita derivative operator . One important property of the commutator of two derivative operators is that it also satisfy the Leibnitz rule (it is itself a new derivative operator). Therefore, using the fact that the null vector is normalized so that we get
| (21) |
where we have also used that . Evaluating the equation on the right hand side explicitly, and using the fact that 666 Here we used that which comes from the restriction to ‘good cuts’ in definition of Section II. More precisely, if one introduces a coordinate on such that and on some leaf of the foliation, then it follows—from the fact that is a symmetry of the horizon geometry , and the fact that the horizon geometry uniquely determines the foliation into ‘good cuts’—that will be constant on all the leaves of the foliation. As must be normal to the leaves one has , whence . we get
where in the second line we have used the fact that plus the fact that as the Lie derivative for some (moreover, one can even fix if one wanted to by means of internal gauge transformations). Then it follows that
| (22) |
a condition that is also valid for the so called weakly isolated horizons better . A similar argument as the one given in equation (21)—but now replacing by for —leads to
where, in addition to previously used identities, we have made use of lemma 1, eq. (7). The previous equations imply that the left hand side of equation (19) is Lie dragged along the vector field , and since is also Lie dragged (in this gauge), all this implies that
| (23) |
Now we must use the rest of the symmetry group of Type I isolated horizons. If we denote by () the three Killing vectors corresponding to the symmetry group of Type I isolated horizons. Spherical symmetry of the horizon geometry implies
| (24) |
which,through similar manipulations as the one used above, lead to
| (25) |
which completes the prove that is constant on . We can now introduce the dimensionless constant . Finally one can fix by choosing the appropriate null generator from the equivalence class .
IV The conserved presymplectic structure
In this section we show in detail how the IH boundary condition implies the appearance of an Chern-Simons boundary term in the symplectic structure describing the dynamics of Type I isolated horizons. This result is key for the quantization of the system described in Section VII.
IV.1 The action principle
The conserved pre-symplectic structure in terms of Ashtekar variables can be easily obtained in the covariant phase space formalism. The action principle of general relativity in self dual variables containing an inner boundary satisfying the IH boundary condition (for asymptotically flat spacetimes) takes the form
| (26) |
where and is the self-dual connection, and a boundary contribution at a suitable time cylinder at infinity is required for the differentiability of the action. No boundary term is necessary if one allows variations that fix an isolated horizon geometry up to diffeomorphisms and Lorentz transformations. This is a very general property and we shall prove it in the next section as we need a little bit of notation that is introduced there.
First variation of the action yields
| (27) |
from which the self dual version of Einstein’s equations follow
| (28) |
as the boundary terms in the variation of the action cancel.
IV.2 The classical results in a nutshell
In the following subsections a series of technical results are explicitly proven. Here we give an account of these results. The reader who is not interested in the explicit proofs can jump directly to Section V after reading the present subsection. In this work we study general relativity on a spacetime manifold with an internal boundary satisfying the isolated boundary condition corresponding to Type I isolated horizons, and asymptotic flatness at infinity. The phase space of such system is denoted and is defined by an infinite dimensional manifold where points are given by solutions to Einstein’s equations satisfying the Type I IH boundary condition. Explicitly a point can be parametrized by a pair satisfying the field equations (28) and the requirements of Definition II. In particular fields at the boundary satisfy Einstein’s equations and the constraints given in Section III. Let denote the space of variations at (in symbols ). A very important point is that the IH boundary conditions severely restrict the form of field variations at the horizon. Thus we have that variations are such that for the pull back of fields on the horizon they correspond to linear combinations of internal gauge transformations and diffeomorphisms preserving the preferred foliation of . In equations, for and we have that
| (29) |
where the arrows denote pull-back to , and the infinitesimal transformations are explicitly given by
| (30) |
while the diffeomorphisms tangent to H take the following form
| (31) |
where for any -form , and the first term in the expresion of the Lie derivative of can be dropped due to the Gauss constraint .
So far we have defined the covariant phase space as an infinite dimensional manifold. For it to become a phase space it is necessary to provide it with a presymplectic structure. As the field equations, the presymplectic structure can be obtained from the first variation of the action (27). In particular a symplectic potential density for gravity can be directly read off from the total differential term in (27) cov . The symplectic potential density is therefore
| (32) |
and the symplectic current takes the form
| (33) |
Einstein’s equations imply . Therefore, applying Stokes theorem to the four dimensional (shaded) region in Fig. 1 bounded by in the past, in the future, a timelike cylinder at spacial infinity on the right, and the isolated horizon on the left we obtain
| (34) |
Now it turns out that the horizon integral in this expression is a pure boundary contribution: the symplectic flux across the horizon can be expressed as a sum of two terms corresponding to the two-spheres and . Explicitly (see Proposition 1 proven below), the symplectic flux across the horizon factorizes into two contributions on given by Chern-Simons presymplectic terms according to
| (35) |
Thus
| (36) |
which implies that the following presymplectic structure is conserved
| (37) |
or in other words is independent of . The presence of the boundary term in the presymplectic structure might seem at first sight peculiar; however, we will prove in the following section that the previous symplectic structure can be written as
| (38) |
where we are using the fact that, in the time gauge where is normal to the space slicing, when pulled back on . The previous equation is nothing else but the familiar presymplectic structure of general relativity in terms of the Palatini variables. In essence the boundary term arises when connection variables are used in the parametrization of the gravitational degrees of freedom.
Finally, as shown in Section IV.4, the key result for the quantization of Type I IH phase space: the presymplectic structure in Ashtekar Barbero variables takes the form
| (39) |
The above equation is the main result of the classical analysis of this paper. It shows that the conserved presymplectic structure of Type I isolated horizons aquires a boundary term given by an Chern-Simons presymplectic structure when the unconstrained phase space is parametrized in terms of Ashtekar-Barbero variables. In the following subsection we prove this equation.
IV.2.1 On the absence of boundary term on the internal boundary
Before getting involved with the construction of the conserved presymplectic structure let us come back to the issue of the differentiability of the action principle. In the isolated horizon literature it is argued that the IH boundary condition guaranties the differentiability of the action principle without the need of the addition of any boundary term (see afk ). As we show here, this property is satisfied by more general kind of boundary conditions. As mentioned above, the allowed variations are such that the IH geometry is fixed up to diffeomorphisms of and gauge transformations. This enough for the boundary term arising in the first variation of the action (26) to vanish. The boundary term arising on upon first variation of the action is
| (40) |
First let us show that for as given in (30). We get
| (41) |
where we integrated by parts in the first identity, the first term in the second identity vanishes due to Eisntein’s equations while the second term vanishes due to the fact that fields are held fixed at the initial and final surfaces and and so when evaluated at . Similarly we can prove that for as given in (31) with (this is the only difference) . We get
| (42) |
where in the last line the first and second terms vanish due to Einstein’s equations, and the last term vanishes because variations are such that the vector field vanishes at . Notice that we have not made use of the IH boundary condition.
IV.3 The presymplectic structure in self-dual variables
In this section we prove a series of propositions implying that the presymplectic structure of Type I isolated horizons is given by equation (37). In addition, we will prove that the symplectic structure is real and takes the simple form (38) in terms of Palatini variables. Proposition 1: The symplectic flux across a Type I isolated horizon factorizes into boundary contributions at and according to
| (43) |
Proof: On all variations are linear combinations of transformations and tangent diffeos as stated in (29), (30), and (31).
for and . Lets start with transformations. Using (30) we get
| (44) | |||||
where in the first line we used the equations of motion and in the second line we used the IH boundary condition (3). We have therefore shown that
Similarly, for diffeomorphisms we first notice that (31) implies that
where and the explicit form of is defined as
We have that
| (45) | |||||
where in the third line we used the vector constraint , while in last line we have used the equations of motion and equation (3). Notice that the calculation leading to equation (44) is also valid for a field dependent such as . This plus the linearity of the presymplectic structure lead to
| (46) |
and concludes the proof of our proposition .
The previous proposition implies that the presymplectic structure (37) is indeed conserved by evolution in . Now we are ready to state the next important proposition.
Proposition 2: The presymplectic form given by
is independent of and real. Moreover, the symplectic structure can be described entirely in terms of variables and taking the familliar form
| (47) |
which is manifestly real and has no boundary contribution.
Proof: The independence of the sysmplectic structure on follows directly from Proposition 2 and the argument presented at the end of the previous section. Now let us analyze the reality of the presymplectic structure. The symplectic potential for written in terms of self dual variables is
| (48) |
Using we get
| (49) |
Using a well known property of the spin connection thiemann , and denoting by the term in parenthesis in the previous equation, we have
The proposition follows from that fact that vanishes as proven in the following Lemma .
Lemma 3: The phase space one-form defined by
| (50) |
is closed.
Proof: From the definition of the phase space given in Section IV.2, in particular from Eqs. (29) we know that
| (51) |
Let us denote by
the exterior derivative of . For infinitesimal transformations we have
| (52) |
from which it follows
| (53) | |||||
where in the first line we have integrated by parts, and in the second line we used the IH boundary condition. The action of diffeomorphisms tangent to H on the connection and triad take the following form
| (54) |
Now we have
| (55) | |||||
where in addition to integrating by parts and using that , we have used the identity for a -form and a -form in a two dimensional manifold, and Cartan’s structure equation . In the last line we used eq. (3), and eq. (5)—which implies that .
IV.4 Presymplectic structure in Ashtekar-Barbero variables
In the previous section (Proposition 2) we have shown how the presymplectic structure
| (56) |
is indeed preserved in the presence of an IH. More precisely in the shaded space-time region in Fig. 1 one has
| (57) |
That is, the symplectic flux across the isolated horizon vanishes due to the isolated horizon boundary condition ack . We will show now, how the very same presymplectic structure takes the form (39) when written in terms of the Ashtekar-Barbero connection variables. For this we need to prove the following lemma:
Lemma 4: The phase space one-form defined by
| (58) |
is closed.
Proof: from the definition of the phase space (Section IV.2) we have
| (59) |
Let us first study where the infinitesimal transformation are explicitly given by
| (60) |
We have
| (61) | |||||
where in the first line we have integrated by parts, and in the second line we used the IH boundary condition. The proof that the presymplectic potential vanishes for mimics exactly the corresponding part of the proof of Lemma 3 .
The next step is to write the symplectic structure in terms of Ashtekar-Barbero connection variables necessary for the quantization program of LQG. When there is no boundary the connection
| (62) |
is canonically conjugate to where is the so-called Immirzi parameter. In the presence of a boundary the situation is more subtle: the symplectic structure acquires a boundary term.
Proposition 3: In terms of Ashtekar-Barbero variables the presymplectic structure of the spherically symmetric isolated horizon takes the form
| (63) |
where .
Proof: The result follows from the variation of the presymplectic potential
which is simply the presymplectic potential leading to the conserved presymplectic structure in variables (in equation (47)) to which we have added a term proportional to ; a closed term which does not affect the presymplectic structure according to Lemma 4 . Remark: Notice that one could have introduced a new connection with a new parameter independent of the Immirzi parameter. The statement of the previous lemma would have remained true if on the right hand side of equation (58) one would have replaced by and by . Consequently, the presymplectic structure can also be parametrized in terms of the analog of equation (63) with a boundary term where is replaced by and on the prefactor of the boundary term is replaced by . This implies that the description of the boundary term in terms of Chern-Simons theory allow for the introduction of a new independent parameter which has the effect of modifying the Chern-Simons level. This ambiguity in the description of the boundary degrees of freedom has however no effect in the value of the entropy.
IV.5 A side remark on the triad as the boundary degrees of freedom
Here we show that one can write the presymplectic structure
| (64) |
in a way such that a surface term depending only on the pull back of the triad field while the bulk term coincides with the one obtained in the previous section in terms of real connection variables. In order to do this we rewrite the symplectic potential as follows:
| (65) | |||||
As a result the symplectic structure becomes noni (and independently wi )
| (66) |
The previous equation shows that the boundary degrees of freedom could be described in terms of the pull back of the triad on the horizon. One could try to quantize the IH system in this formulation in order to address the question of black hole entropy calculation. Such project would be certainly interesting. However, the treatment is clearly not immediate as it would require the background independent quantization of the triad field on the boundary for which the usual available techniques do not seem to naturally apply. Nevertheless, the previous equations provides an interesting insight already at the classical level, as the boundary symplectic structure, written in this way, has a remarkable implication for geometric quantities of interest in the first order formulation. To see this let us take and so that we can introduce the fluxes according to
| (67) |
where . Now (66) implies the Poisson bracket from which the following remarkable equation follows:
| (68) |
The Poisson brackets among surface fluxes is non vanishing and reproduces the Lie algebra! This is an interesting property that we find entirely in terms of classical considerations using smooth field configurations. However, compatibility with the bulk fields seems to single out the treatments of kinematical degrees of freedom in terms of the so called holonomy-flux algebra of classical observables for which flux variables satisfy the exact analog of (68) as described in zapata . This fact strengthens even further the relevance of the uniqueness theorems lost , as they assume the use of the holonomy-flux algebra as the starting point for quantization.
V Gauge symmetries
In this section we rederive the form of the presymplectic symplectic structure written in Ashtekar-Barbero variables by means of gauge symmetry argument. The idea is to first study the gauge symmetries of the presymplectic structure when written in Palatini variables, as in Equation (38). We will show that, due to the nature of variations at the horizon, the boundary term in Equation (39) is completely fixed by the requirement that the gauge symmetry content is unchanged when the presymplectic structure is parametrized by Ashtekar-Barbero variables. This argument is completely equivalent to the content of the previous section and was used in nous as a shortcut construction of the presymplectic structure for Type I isolated horizons in terms of real connection variables. Another important result of this section is the computation of the classical constraint algebra in Subsection V.1 which are essential for clarifying the quantization strategy implemented in Section VII
The gauge symmetry content of the phase space is implied by the following proposition.
Proposition 4: Phase space tangent vectors of the form
| (69) |
for and tangent to the horizon, are degenerate directions of .
Proof: The proof follows from manipulations very similar in spirit to the ones used for proving the previous propositions. We start with the transformations , and we get
| (70) |
where we used the Gauss constraint . In order to treat the case of the infinitesimal diffeomorphims tangent to the horizon it will be convenient to first write the form of the vector constraint in terms of variables tate . We have
| (71) |
variations of the previous equation yields
| (72) | |||||
where in the second line we have put all the ’s to the right, and modified the second and third terms using the identities that is valid for any two -forms and on a -manifold, and for a -form and a -form on a -manifold respectively. We are now ready to show that is a null direction of . Explicitly:
| (73) |
where in the last line we have used the identity valid for an arbitrary -form and arbitrary -form on a -manifold, the fact that is tangent to , and the IH boundary condition Eq. (5) implying when pulled back on .
The previous proposition shows that the IH boundary condition breaks neither the symmetry under diffeomorphisms preserving nor the internal gauge symmetry introduced by the use of triad variables.
The gauge invariances of the IH system have been made explicit in the parametrization of the presymplectic structure. However, due to the results of Propositions 2 and 3, these can also be made explicit in the parametrization of the presymplectic structure using either self-dual connection variables or real Ashtekar-Barbero variables. It is in fact possible to uniquely determine the horizon contributions to the presymplectic structure in connection variables entirely in terms of the requirement the transformations (69) be gauge invariances of the standard bulk presymplectic contribution plus a suitable boundary term. More precisely, the requirement of local invariance becomes
| (74) |
for an (in principle) unknown horizon contribution to the presymplectic structure . This gives some information about the nature of the boundary term, namely
where we used the Gauss law , condition (6), and that boundary terms at infinity vanish. A similar calculation for diffeomorphisms tangent to the horizon gives an equivalent result. This together with the nature of variations at the horizon (see Eqs. (29)) provides an independent way of establishing the results of Proposition 3. This alternative derivation of the conserved presymplectic structure was used in nous .
V.1 On the first class nature of the IH constraints
The previous equation above can be written as
| (75) |
or equivalently
| (76) |
where
| (77) |
In the canonical framework Equation 76 implies that local transformations are Hamiltonian vector fields generated by the “Hamiltonian” . It follows immediately from the definition of Poisson brackets that the Poisson algebra of closes. More precisely, one has
| (78) |
from where we get
| (79) |
Not surprisingly we get the Lie algebra a local transformations. In the previous section we showed that these local transformations are indeed gauge transformations. This implies, in the canonical picture, that canonical variables must satisfy the constraints
| (80) |
Now let us look at diffeomorphisms. A calculation based on the analog of equation (74) for an infinitesimal diffeormorphism preserving yields
| (81) |
where
| (82) |
Finally, a simple calculation as the one leading to (79), leads to the following first-class constraint algebra
| (83) |
where we have ignored the Poisson brackets involving the scalar constraint777Recall that the smearing of the scalar constraints must vanish on and hence the full constraint algebra including the scalar constraint will remain first class.. Using and with support only on the horizon we can now conclude that the IH boundary condition is first class which justifies the Dirac implementation that will be carried our in the quantum theory.
VI The zeroth and first laws of BH mechanics for (spherical) isolated horizons
The definition given in Section II implies authomaticaly the zeroth law of BH mechanics as is constant on . In turn, the first law cannot be tested unless a definition of energy of the IH is given. Due to the fully dynamical nature of the gravitation field in the neighbourhood of the horizon this might seem problematic. Of course one can in addition postulate an energy formula for the IH in order to satisfy de facto the first law. Fortunately, there is a more elegant way. This consists in requiring the time evolution along vector fields which are time translations at infinity and proportional to the null generators at the horizon to correspond to a Hamiltonian time evolution afk . More precisely, denote by the phase space tangent vector field associated to an infinitesimal time evolution along the vector field (which we allow to depend on the phase space point). Then is Hamiltonian if there exists a functional such that
| (84) |
This requirement fixes a family of good energy formula and translates into the first law of isolated horizons
| (85) |
where we have put the explicit expression of the electromagnetic work term where is the electromagnetic potential (constant due to the IH boundary condition) and is the electric charge. The above equation implies that and to be functions of the IH area and charge alone. A unique energy formula is singled out if we require to coincide with the surface gravity of Type I stationary BHs, i.e., those in the Reissner-Nordstrom family:
| (86) |
Here we can explicitly prove the above statement in terms of our variables. We shall make here the simplifying assumption that there are no matter fields, i.e. , we work in the vacuum case. The explicit form of is given by
| (87) |
We can now explicitly write the main condition, namely
| (88) |
Where we have used the same kind of manipulations used in equation (73) paying special attention to the fact that the relevant vector field is (at the horizon) no longer tangent to the horizon cross-section, and the fact that the first term in the third line vanishes due to the IH boundary condition 888This follows from equations (5), (175), and (178) implying that Integrating the previous expression on the horizon gives zero..
The condition is solved by with
| (89) |
Demanding time evolution to be Hamiltonian singles out a notion of isolated horizon energy which automatically satisfies, by this requirement, the first law of black hole mechanics (now extended from the static or locally static context to the isolated horizon context). The general treatment and derivation of the first law can be found in afk ; abl2001 .
VII Quantization
The form of the symplectic structure motivates one to handle the quantization of the bulk and horizon degrees of freedom (d.o.f.) separately. We first discuss the bulk quantization. As in standard LQG [8] one first considers (bulk) Hilbert spaces defined on a graph and then takes the projective limit containing the Hilbert spaces for arbitrary graphs. Along these lines let us first consider for a fixed graph with end points on , denoted . The quantum operator associated with in (6) is
| (90) |
where at each . Consider a basis of of eigenstates of both as well as for all with eigenvalues and respectively. These states are spin network states, here denoted , where and are the spins and magnetic numbers labeling the edges puncturing the horizon at points (other labels are left implicit). They are also eigenstates of the horizon area operator
Now substituting the expression (90) into the quantum version of (6) we get
| (91) |
As we will show now, the previous equation tells us that the surface Hilbert space, that we are looking for is precisely the one corresponding to (the well studied) CS theory in the presence of particles. More precisely, consider the Chern-Simons action
to which we couple a collection of particles by adding the following source term:
where are the particle world-lines, coupling constants, and are group valued d.o.f. of the particles. The particle d.o.f. being added will turn out to correspond precisely to the d.o.f. associated to the bulk appearing in (90). The horizon and bulk will then be coupled by identifying these d.o.f. The gauge symmetries of the full action are
| (92) |
where , and
| (93) |
where .
In order to perform the canonical analysis we assume that (). Under the left action of the group we have
| (94) |
for a function whose explicit form will not play any role in what follows. The infinitesimal version of the previous action is
| (95) |
If we define the (spin) momentum as
| (96) |
where are the conjugate momenta of then it is easy to recover the following simple Poisson brackets
| (97) |
where the last equation follows from the Jacobi identity. Explicit computation shows that . Therefore; we have three primary constraints per particle
| (98) |
The primary Hamiltonian is simply given by
the requirement that the constraints be preserved by the time evolution reads
| (99) |
and the constraint algebra is
If we write , where is the component normal to while is the parallel one, equations (99) completely fix the Lagrange multipliers . This means that, per particle, two (out of three) constraints are second class. The fact that remains unfixed by the equations of motion implies the presence of first class contraints which are in fact given by
| (100) |
This constraint generates rotations which conserve the quantity . Now, the fact that there are secons class constraints implies that in order to quantize the theory one has to either work with Dirac brackets, solve the constraints classically before quantizing, or parametrize the phase space in terms of Dirac observables. In this case the third option turns out to be immediate. The reason is that the turn out to be Dirac observables of the particle system as far as the constraints is concerned, namely
| (101) |
This implies that the Poisson bracket relations (97) remain unchanged when one replaces the brackets by Dirac brackets . Due to this fact and for notational simplicity we shall keep using the standard Poisson bracket notation.
In summary, the phase space of each particle is , where the momenta conjugate to are given by satisfying the Poisson bracket relations
| (102) |
Explicit computation shows that are primary constraints (two of which are second class). In the Hamiltonian framework we use , and the symmetries (92) and (93) arise from (and imply) the following set of first class constraints on :
| (103) | |||||
| (104) |
The first constraint tells us that the level of the Chern-Simons theory is 999 If we use the connection introduced in the remark below (63) then the level takes the form .
| (105) |
and that the curvature of the Chern-Simons connection vanishes everywhere on except at the position of the defects where we find conical singularities of strength proportional to the defects’ momenta.
The theory is topological which means in our case that non trivial d.o.f. are only present at punctures. Note that due to (102) and (104) the are quantized according to where is a half integer labelling a unitary irreducible representation of .
From now on we denote the Hilbert space of the CS theory associated with a fixed choice of spins at each puncture . This will be a proper subspace of the ‘kinematical’ Hilbert space . In particular there is an important global constraint that follows from (103) and the fact that the holonomy around a contractible loop that goes around all particles is trivial. It implies
| (106) |
Moreover, the above containment becomes an equality in the limit witten , i.e. in the large BH limit. In that limit we see that the constraint (103) has the simple effect of projecting the particle kinematical states in into the singlet.
To make contact with the bulk theory, we first note that the bulk Hilbert space can be written
| (107) |
for certain spaces , and where, for each , the generators appearing in (90) act on the representation space . If we now identify with the ‘kinematical’ Chern-Simons Hilbert space , the operators in (91) are identified with the of (103). The constraints of the CS theory then restrict to yielding
| (108) |
as the full kinematical Hilbert space for fixed .
So far we have dealt with a fixed graph. The Hilbert space satisfying the quantum version of (6) is obtained as the projective limit of the spaces . The Gauss and diffeomorphism constraints are imposed in the same way as in ack ; bhe1 . The IH boundary condition implies that lapse must be zero at the horizon so that the Hamiltonian constraint is only imposed in the bulk.
VII.1 State counting
The entropy of the IH is computed by the formula where the density matrix is obtained by tracing over the bulk d.o.f., while restricting to horizon states that are compatible with the macroscopic area parameter . Assuming that there exist at least one solution of the bulk constraints for every state in the CS theory, the entropy is given by where is the number of horizon states compatible with the given macroscopic horizon area . After a moment of reflection one sees that
| (109) |
where the labels of the punctures are constrained by the condition
| (110) |
Similar formulae, with a different value, were first used in majundar .
Notice that due to (110) we can compute the entropy for (not necessarily infinite). The reason is that the representation theory of —describing for finite —implies
| (111) |
as long as all the as well as the interwining internal spins are less than . But for Immirzi parameter in the range this is precisely granted by (110) according to the Lemma below. All this simplifies the entropy formula considerably. The previous dimension corresponds to the number of independent states one has if one models the black hole by a single intertwiner!
Lemma 5: The Hilbert spaces of Chern-Simons theory with level selected by the restriction
| (112) |
are isomorphic to .
Proof: The Chern-Simons Hilbert space will be isomorphic to if for instance all element of a given basis (of intertwiners) of if (voir for instance babaez )
| (113) |
and
| (114) |
Equation (113) is immediately implied by (112) as the latter implies
| (115) |
The condition (114) requires a more precise analysis. Notice the fact that, being intertwining spins, the satisfy the following set of nested restrictions which imply the result:
where in each line we have used (115) .
Remark: An interesting point can be made here as a further developement of the remark below equation (63). Notice that if we had worked with the connection as our boundary field degree of freedom—corresponding to the choice in the notation of the remark below (63)—then the boundary Chern-Simons level would be (see Footnote 9). This implies that the condition (113) imposed on representations labelling the punctures would take the simple form
| (116) |
or equivalently
| (117) |
where is the maximum single-puncture eigenvalue allowed. Our effective treatment depends on a classical input: the macroscopic area. One would perhaps hope that this effective treatment would only allow for states where the microscopic area is close to , unfortunately such a strong requirement is not satisfied as the allowed eigenvalues can be very far away from . However, the effective theory at least forbids quantum states where individual area quanta are larger than . This is a nice interplay between the classical input and the associated effective quantum description. Of course this interplay is still qualitatively valid for the case in which one works with the Ashtekar-Barbero connection on the boundary (i.e., ).
VIII Conclusion
We have shown that the spherically symmetric isolated horizon (or Type I isoated horizon) can be described as a dynamical system by a pre-symplectic form that, when written in the (connection) variables suitable for quantization, acquires a horizon contribution corresponding to an SU(2) Chern-Simons theory. There are different ways to prove this important statement. In nous we first observed that gauge transformations and diffeomorphism preserving are not broken by the IH boundary condition. Moreover, infinitesimal diffeomorphisms tangent to and local transformations continue to be degenerate directions of on shell. This by itself is then sufficient for deriving the boundary term that arises when writing the symplectic structure in terms of Ashtekar-Barbero connection variables. Here we have reviewed this construction in Section V. A result that was not explicitly presented in nous is the precise form of the constraint algebra found in Subsection V.1. There we see in a precise way how the canonical gauge symmetry structure of our system is precisely that of an Chern-Simons theory: in particular, at the boundary, infinitesimal diffeomorphisms, preserving , form a subalgebra of gauge algebra, as in the topological theory.
A different, more direct approach is based on a subtle fact about the canonical transformation that takes us from the Palatini phase space parametrization to the Ashtekar-Barbero connection formulation, in the presence of an internal boundary. In the case of Type I isolated horizons, the term to be added to the symplectic potential producing the above transformation gives rise to a boundary contribution that eventually leads to a boundary Chern-Simons term in the presymplectic structure. This is the content of Section IV.4. The boundary Chern-Simons term appears due to the use of connection variables which in turn are the ones in terms of which the quantization program of loop quantum gravity is applicable.
Finally, at a fundamental level, what actually fixes the surface term in the symplectic structure is the requirement that it be conserved in time. The above mentioned proofs show that the various expressions for the symplectic structure using different variables are in fact one and the same symplectic structure. That this symplectic structure is preserved in time was proven in Section IV.
There is a certain freedom in the choice of boundary variables leading to different parametrizations of the boundary degrees of freedom. The most direct description would appear, at first sight, to be the one defined simply in terms of the triad field (pulled back on H) along the lines exhibited in Section IV.5. Such parametrization is however less preferable from the point of view of quantization as one is confronted to the background independent quantization of form fields for which the usual techniques are not directly applicable. In contrast, the parametrization of the boundary degrees of freedom in terms of a connection directly leads to a description in terms of Chern-Simons theory which, being a well studied topological field theory, drastically simplifies the problem of quantization. However, such description comes with the freedom of the introduction of an extra dimensionless parameter (as pointed out in the Remark below equation (63)). Such appearance of extra parameters is very much related to what happens in the general context of the canonical formulation of gravity in terms of connections (see Appendix in rezendeyo ). Therefore, this observation is by no means a new feature proper of IHs. The existence of this extra parameter has a direct influence on the value of the Chern-Simons level; however, the value of the entropy is independent of this extra parameter future2 .
Note that no d.o.f. is available at the horizon in the classical theory as the IH boundary condition completely fixes the geometry at (the IH condition allows a single (characteristic) initial data once is fixed (see fig. 1)). Nevertheless, non trivial d.o.f. arise as would be gauge d.o.f. upon quantization. These are described by SU(2) Chern-Simons theory coupled to (an arbitrary number of) defects through a dimensionless parameter proportional to , where is the basic quantum of area carried by the defect. These would be gauge excitations are entirely responsible for the entropy in this approach 101010More insight on the nature of these degrees of freedom could be gained by studying simpler models. In liu a theory with no local degrees of freedom has been introduced. The attractive feature of this model is that it admits an (unconstrained) phase parametrization in terms of the same field content as gravity. Moreover, one can argue that it contains the minimal structure to serve as a toy model to study some generic features of the Type I isolated horizon quantization..
We obtain a remarkably simple formula for the horizon entropy: the number of states of the horizon is simply given in terms of the (well studied) dimension of the Hilbert spaces of Chern-Simons theory with punctures labeled by spins. In the large limit the latter is simply equal to the dimension of the singlet component in the tensor product of the representations carried by punctures. In this limit the black hole density matrix is the identity on for admissible . Similar counting formulae have been proposed in the literature models ; majundar . Our derivation from first principles clarifies these previous proposals.
Remarkably, the counting of states necessary to compute the entropy of the above Type I isolated horizons can be exactly done barberos using the novel counting techniques introduced in barba . It turns out to be , where . However, the subleading corrections turn out to have the form (instead of the that follows the classic treatment bhe1 ; amit ) matching other approaches majundar . This is due to the full nature of the IH quantum constraints imposed here. We must mention that the proposal of Majumdar et al. majundar is most closely related to our result. Their intuition was particularly insightful as it yielded a universal form of logarithmic corrections in agreement with those found in different quantum gravity formulations carlip-log . Our work clarifies the relevance of their proposal.
We have concentrated in this work on Type I isolated horizons. The natural question that follows from this analysis is whether we can generalize the invariant treatment in order to include distortion. The classical formulation and quantization of Type II isolated horizons in the (gauge fixed) treatment has been studied in jon . Work in progress future1 shows that, in the invariant formulation, it is possible to include distortion in a simple way as long as the isolated horizon is non rotating (i.e., when ). The rotating case is more subtle but we believe that there are no insurmountable obstacles to its invariant treatment (this will be studied elsewhere).
IX Acknowledgements
We thank A. Ashtekar, C. Beetle, E. Bianchi, K. Krasnov, M. Montesinos, M. Reisenberger, and C. Rovelli for discussions. This work was supported in part by the Agence Nationale de la Recherche; grant ANR-06-BLAN-0050. J.E. was supported by NSF grant OISE 0601844 and the Alexander von Humboldt Foundation of Germany, and thanks Florida Atlantic University for hospitality during his visit there. A.P. was supported by l’Institut Universitaire de France. D.P. was supported by Marie Curie EU-NCG network.
Appendix A Type I Isolated Horizons: Horizon geometry from the Reissner-Nordstrom family
The spherically symmetric isolated horizons or Type I isolated horizons are easy to visualise in terms of the characteristic formulation of general relativity with initial data given on null surfaces lewa . This observation is very useful if one is looking for a concrete visualisation of the horizon geometry and properties of the matter fields at the horizon. In this appendix we chose to derive the main properties of Type I isolated horizons by studying their geometry in the context of Einstein-Maxwell theory (which is general enough for the most relevant applications of the formalism). An additional motivation for the explicit approach presented here is its complementarity with more abstract discussions available in the literature ack ; better ; ih_prl . In the context of Einstein-Maxwell theory, spacetimes with a Type I IH are solutions to Einstein-Maxwell equations where Reissner-Nordstrom horizon data are given on a null surface and suitable free radiation is given at the transversal null surface for both geometric as well as electromagnetic degrees of freedom. This allows to derive the main equations of IH directly from the Reissner-Nordstrom geometry as far as we are careful enough only to use the information that is intrinsic to the IH geometry.
A.1 The Reissner-Nordstrom solution in Kruskal-like coordinates
The Reissner-Nordstrom metric can be written in Kruskal-like coordinates chandra as
| (118) |
where
| (119) |
with , , and the function is determined by the following implicit equation:
| (120) |
The previous Kruskal-like coordinates are valid for the external region . The metric is smooth at the horizon which in the new coordinates corresponds to the null surface . An important identity is:
| (121) |
where denotes that the equality holds at the horizon for which . Here we are interested in the first order formalism. Thus we are interested in an associated tetrad with . It is immediate to verify that a possible such tetrad is given by
| (126) |
We now want to compute the components of the spin connection at the horizon. Therefore, we will use Cartan’s first structure equations at . The solution is (all details are given in Section A.3)
| (133) |
At this stage we consider a Lorentz transformation of the form
| (134) |
where and . It is immediate to see that under such transformation the connection above transforms to
| (141) |
where the arrows below the components denote the pull-back of the one forms to , and . We can obviously chose this Lorentz transformation in order for with an arbitrary constant. We have
| (142) |
This can be made compatible with the time gauge by changing the spacetime foliation just at the intersection with the horizon so that is the new normal111111 Recently, a similar analysis as the one presented here—and also in beigin —has been done maju . In that reference the authors derive a result which is compatible with the above equations in the singular vanishing extrinsic curvature slicing . Such (null) slicing is however inconsistent with the canonical formulation that is necessary for the LQG quantization of the bulk degrees of freedom. . Now we are ready to write the quantities we were looking for
| (149) |
where and . The self dual connection and the Ashtekar-Barbero connection become
| (156) |
The curvature of the self-dual and Ashtekar-Barbero connections is (when pulled back to the cross sections )
| (163) |
Using that we can write the previous equations as
| (164) |
and
| (165) |
In the following subsection we will show that defines the frame where the IH surface gravity matches the stationary black hole one. With this value of , the previous two equations and equation (149) imply eqs. (3), (5), and (6) respectively. For completeness we write the componets of
| (175) |
where on the right we have written the corresponding self-dual components.
A.2 Surface gravity and the value of
For stationary black holes, the surface gravity is defined by the equation
| (176) |
where is the Killing vector field tangent to the horizon. For isolated horizons there is no unique notion of . We shall define in terms of the tetrad in the usual way with 121212The future pointing null generators of the horizon are such that This implies that from wich we get and so that .. However, this definition still allows the freedom associated to the Lorentz transformations (134) which send . We can fix this freedom by demanding the surface gravity to match that of a Reissner-Nordstrom black hole with mass and charge for which
| (177) |
Indeed this choice is the one that makes the zero, and first law of IH look just as the corresponding laws of stationary black hole mechanics.
This choice is then physically motivated. In turn this will fix the value of in (165). If we define then we have that (176) implies
| (178) |
Notice that after the Lorentz transformation (134) we have
| (179) | |||||
Now we can fix so that takes the RN value. Recalling equation (142) and using the above equations, a simple calculation shows that this happens for
| (180) |
which implies the desired result
| (181) |
Notice that
| (182) |
and that (according to (141)) we also have
| (183) |
All this implies that as expected from (general proof in Lemma 2). In other words, the we have chosen by means of fixing the boost freedom is a member of the equivalence class in Definition II.
A.3 Solving Cartan’s equation
For this we first compute , namely:
| (184) |
Now we are ready to explicitly write Cartan’s first structure equations. They are
| (185) |
Let us now study the previous equation individually. The six components of the first, equation (A.3) become
| (186) |
The six components of the second, equation (LABEL:2), become
| (187) |
The six components of the third, equation (A.3), become
| (188) |
Finally, the six components of the fourth, equation (A.3), become
| (189) |
At this point we make the following ansatz . From which we get the solution (133).
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