Black Hole Entropy from complex Ashtekar variables
aa r X i v : . [ g r- q c ] D ec Black Hole Entropy from complex Ashtekar variables
Ernesto Frodden, Marc Geiller ∗ , Karim Noui,
3, 2 and Alejandro Perez Centre de Physique Th´eorique † , Campus de Luminy, 13288 Marseille, France. Laboratoire APC – Astroparticule et Cosmologie,Universit´e Paris Diderot Paris 7, 75013 Paris, France. Laboratoire de Math´ematique et Physique Th´eorique ‡ , 37200 Tours, France. In loop quantum gravity, the number N Γ ( A, γ ) of microstates of a black hole for a given discretegeometry Γ depends on the so-called Barbero-Immirzi parameter γ . Using a suitable analytic con-tinuation of γ to complex values, we show that the number N Γ ( A, ± i ) of microstates behaves asexp( A/ (4 ℓ )) for large area A in the large spin semiclassical limit. Such a correspondence withthe semiclassical Bekenstein-Hawking entropy law points towards an unanticipated and remarkablefeature of the original complex Ashtekar variables for quantum gravity. PACS numbers: 04.70.Dy, 04.60.-m
The Barbero-Immirzi parameter γ was originally intro-duced [1] as a way to circumvent the problem of imposingthe reality constraints in the complex (self-dual) Ashtekarformulation of gravity [2]. Historically, γ appeared asa parameter labeling a family of canonical transforma-tions turning the ADM phase space into the so-calledAshtekar-Barbero phase space, parametrized by a real su (2) connection and its conjugate momentum. Lateron, Holst [3] realized that this Hamiltonian formulationof gravity could be obtained by adding to the standardHilbert-Palatini Lagrangian a topological term with γ as a coupling constant. This term vanishes due to theBianchi identities when one resolves the spin connectionin terms of the tetrad, and for this reason γ is not relevantat the classical level.In the quantum theory however, the Barbero-Immirziparameter plays a crucial role since the spectrum of thegeometric operators is discrete in units of the loop quan-tum gravity (LQG) scale ℓ LQG = √ γG ~ = √ γℓ Pl , where ℓ Pl is the Planck length [4]. Moreover, this γ -dependencyof the fundamental physical cut-off is inherited by thevalue of the black hole entropy in the LQG calcula-tion. Compatibility with the expected semiclassical value S = A/ (4 ℓ ) (where A is the area of the horizon) requiresthat γ be fixed to a particular real value. In fact, a lotof different techniques have been developed in order toobtain the value of γ [5, 6] (see also [7]).In LQG, the horizon of a black hole has the topology ofa 2-sphere, with colored punctures coming from the spinnetwork links that cross the horizon. Each puncture car-ries a quantum of area, and the sum of these microscopicareas gives the macroscopic area A of the horizon. In the ∗ New address: Institute for Gravitation and the Cosmos & PhysicsDepartment, Penn State, University Park, PA 16802, U.S.A. † Unit´e Mixte de Recherche (UMR 6207) du CNRS et des Uni-versit´es Aix-Marseille I, Aix-Marseille II, et du Sud Toulon-Var;laboratoire afili´e `a la FRUMAM (FR 2291). ‡ F´ed´eration Denis Poisson Orl´eans-Tours, CNRS/UMR 6083. microcanonical ensemble, the entropy of the black hole isgiven by the logarithm of the number N ( A, γ ) of micro-scopic states compatible with the macroscopic area A ofthe horizon. As suggested by the notation, the quantity N ( A, γ ) depends on A but also on γ , since admissible mi-crostates are sets Γ = { j , · · · , j p } of punctures labelledby spins, where p is the number of punctures, satisfyingthe quantum area constraint A − ǫ < πγℓ p X ℓ =1 p C ( j ℓ ) < A + ǫ, (1)for some small coarse graining ǫ >
0, and C ( j ) = j ( j + 1).It has been shown recently [6] that there is a close rela-tionship between black holes in LQG and SU(2) Chern-Simons theory. In this framework, the number N ( A, γ )of microstates can be expressed as N ( A, γ ) = X Γ w Γ N Γ ( A, γ ) , (2)where N Γ ( A, γ ) is the dimension of the Hilbert spaceof SU(2) Chern-Simons theory on a punctured 2-sphere,and w Γ is a weight assigned to each Γ.However, the present state of development of LQG isnot conclusive about the precise form of the weights w Γ ,and one must admit that this remains to a large extendan open issue. It seems clear to us that these weightsshould be fixed by dynamical considerations, in relationin particular to the requirement that a semiclassical ge-ometry plus field configuration (the suitable low energyvacuum state) be recovered near the horizon. In fourdimensions, some first steps in this direction have beenexplored in [8]. In fact, perhaps the most transparent ex-ample of the dynamical nature of the weights w Γ is thethree-dimensional pure gravity description of the BTZblack hole [10]. In four dimensions, we will show in thisletter that the key result concerning the analytic con-tinuation, together with the assumption that puncturesare indistinguishable [9], lead to a complete semiclassicalagreement with Bekenstein-Hawking thermodynamics.More precisely, we are going to propose an expressionfor N Γ ( A, ± i ), which is the number of states for a the-ory defined in terms of the complex self-dual Ashtekarconnection, i.e. for γ = ± i . Our proposal is based ona certain analytic continuation of the formula for the di-mension of the Hilbert space of SU(2) Chern-Simons the-ory from SU(2) representations to suitable SL(2 , C ) rep-resentations satisfying self-duality constraints. The strik-ing result is that after analytic continuation the asymp-totic behavior of the Chern-Simons Hilbert space is holo-graphic, i.e. satisfieslog (cid:0) N Γ ( A, ± i ) (cid:1) s.c. ∼ A ℓ , (3)where s.c. ∼ denotes that the result is valid in the large spin( semiclassical ) asymptotic regime. This result suggeststhat the complex formulation in terms of the Ashtekarvariables (which is to be put in parallel with the so-calledLorentz-covariant formulation [13]) could lead to a clear-cut derivation of the black hole entropy in the frameworkof quantum gravity. Analytic continuation
Black holes in LQG can be described in terms of anSU(2) Chern-Simons theory S CS [ A ] = k π Z ∆ tr (cid:18) A ∧ d A + 23 A ∧ A ∧ A (cid:19) , (4)where k ∝ A (the precise form of the level will not beimportant), and ∆ is the black hole horizon with spatialarea A . This area A is obtained as the sum of the fun-damental contributions carried by the p links ℓ crossingthe horizon, according to the formula A = 8 πℓ γ p X ℓ =1 p C ( j ℓ ) , (5)where p C ( j ℓ ) is the quantum of area associated to thepuncture ℓ ∈ J , p K , which is colored with an SU(2) rep-resentation of spin j ℓ and of finite dimension d ℓ = 2 j ℓ +1.Usually, C ( j ) = j ( j + 1) is the quadratic Casimir oper-ator evaluated in the representation j , but there existsalso a linear model with C ( j ) = j , which correspondsto a different regularization of the area operator. Thesetwo expressions obviously agree for large spins.The Hilbert space associated to a black hole whosehorizon is crossed by p colored links, is exactly theHilbert space of the SU(2) Chern-Simons theory (4) with p punctures colored with the representations of spins ( j , . . . , j p ). Its dimension is therefore given by the fi-nite sum [14, 15] D k ( j , . . . , j p )= 2 k + 2 k +1 X d =1 sin (cid:18) πdk + 2 (cid:19) p Y ℓ =1 sin (cid:18) πdd ℓ k + 2 (cid:19) sin (cid:18) πdk + 2 (cid:19) . (6)A configuration for a black hole of horizon area A isan assignment of p spins ( j , . . . , j p ) compatible with A in the sense that the relation (5) is satisfied. As a con-sequence of this restriction, the number of microstatesfor a given puncture configuration is a function of theBarbero-Immirzi parameter as well. To remind us of thisfact we introduce the notation N Γ ( A, γ ) = D k ( j , . . . , j p ) . (7)The total number of microstates is given by a sum of thetype (2) over the colored graphs Γ.The analytic continuation that we propose consists inreplacing the spins j by complex values according to therule j ℓ → is ℓ − , (8)where s ∈ R . As we are going to show below, this corre-sponds in a precise sense to choosing self-dual representa-tions, i.e., solutions of the self-duality constraints whichin addition satisfy suitable reality conditions . Using theanalyticity of D k ( j , . . . , j p ) in its arguments, the abovetransformation defines uniquely the number density N A Γ that we are looking for. Namely, D k ( j , . . . , j p ) → i p D k ( is − , . . . , is p −
12 )= 2 k + 2 k +1 X d =1 sin (cid:18) πdk + 2 (cid:19) p Y ℓ =1 sinh (cid:18) πds ℓ k + 2 (cid:19) sin (cid:18) πdk + 2 (cid:19) , (9)where the factor i p comes from the Jacobian d j / d s nec-essary when dealing with densities. When the spins s ℓ are large, this sum is dominated by the exponential withthe largest argument (obtained for d = k + 1), which isgiven by N Γ ( A, ± i ) = i p D k ( is − , . . . , is p −
12 ) ≈ k sin (cid:18) π ( k + 1) k + 2 (cid:19) − p p Y ℓ =1 sinh(2 πs ℓ ) . (10)This is the main equation of the paper . In a previous version of this work, a different version of the
It is important to point out that our procedure of ana-lytic continuation (9) would be a rigorous mathematicalfact if the starting point was a function of p real variables.In this case the dimension would extended uniquely to afunction of p complex variables. However, as the input isa function of p integer variables d ℓ , the analytic continua-tion is ambiguous. For instance, a different result wouldbe obtained by multiplying the input by any functionequal to one when the entries are integers. What makesour result unique is that only the expression (6) withcomplex variables as entries can be given a field theoreti-cal description in the context of the analytic continuationof Chern-Simons theory [12]. Self dual representations and reality conditions
Now, we clarify the meaning of the representations s ℓ appearing in formula (10). In the first place, they are self-dual or anti self-dual representations of SL(2 , C ) solutionsto the constraints ~L ± i ~K = 0 , (12)where ~L and ~K are the generators of rotations and boostin the internal SL(2 , C ) group. For simplicity, let us focusfrom now on on the self-dual representations, i.e., with aplus sign in the previous constraint. From the Lie algebra sl (2 , C ), one can immediately show that the three equa-tions in (12) are first class constraints. Thus, it makes analytic continuation has been proposed, where instead of thepresent procedure the replacement k → ik in performed (6). Inorder for this to make mathematical sense, one would need tojustify several issues. First, the upper bound in the sum over thedummy variable d has to remain real if the dummy variable isto remain real. For this one should argue that the upper boundis actually given by | k | + 1 instead of k + 1. Second, in ordernot to get an obviously complex result for the number of states,one would also need to replace the overall factor of 1 / ( k + 2) by1 / ( | k | + 2). With this, the result would still be complex becauseof the k +2 dependence of the sin functions in (6). Only the lead-ing order would be real and positive in the large | k | asymptoticexpansion. With all these ingredients, the result was D k ( j , . . . , j p ) → D ik ( j , . . . , j p )= 2 | k | + 2 | k | +1 X d =1 sin (cid:18) πdik + 2 (cid:19) p Y ℓ =1 sin (cid:18) πdd ℓ ik + 2 (cid:19) sin (cid:18) πdik + 2 (cid:19) ≈ | k | | k | X d =1 sinh (cid:18) πdk (cid:19) p Y ℓ =1 sinh (cid:18) πdd ℓ k (cid:19) sinh (cid:18) πdk (cid:19) . (11)Interestingly, this formula has appeared in the description of theentropy of a BTZ black hole from the canonical point of view [10]and from the point of view of spin foam models [11], where inorder to get a negative cosmological constant it is indeed the levelof Chern-Simons theory which has to be analytically-continued. sense to look for solutions of these constraints among rep-resentations of SL(2 , C ). If we label, as usual, by ( p, k )the irreducible representations of the Lorentz group, thenthe previous condition (12) is solved strongly in differentways.One has on the one hand solutions which are finitedimensional and simply correspond to the usual spinorrepresentations of the Lorentz group. These solutionsare found by expanding SL(2 , C ) representations in termsof unitary irreducible representations of the subgroupSU(2). They are the ones used in standard field theory todescribe matter fields. However, they must be discardedin the present gravitational context because they fail tosatisfy some necessary reality requirements imposed byAshtekar’s self-dual formulation. More precisely, a sim-ple analysis shows that the spectrum of the square of theflux operators is negative-definite (the area spectrum isimaginary). This is because [ Area | j i = b Σ ± · b Σ ± | j i = − (8 πℓ ) j ( j + 1) | j i , (13)where | j i denotes a flux excitation, and Σ ± is the (anti)self-dual component of the densitized triad field (Σ = e ∧ e ). The appearance of the uncomfortable minus signin (13) is a well known fact since the very early stages ofconstruction of the LQG formalism [18], and one of thereasons for using the real connection variables with a realBarbero-Immirzi parameter.However, there are other solutions of (12) which be-come apparent when expanding SL(2 , C ) representationsin terms of unitary irreducible representations of the sub-group SU(1 ,
1) (as in [19]). The group SU(1 ,
1) has twotypes of unitary representations: those that are discreteand labelled by j ∈ N /
2, and those in the so-called prin-cipal series and labelled by j = is − /
2. There aresolutions of the self-duality constraints (12) in one-to-one correspondence with both types of solutions. Theones associated with the discrete series fail (just like thefinite-dimensional ones) the reality condition (13). How-ever, those associated with the principal series lead to a real (yet continuous) area spectrum: [ Area | s i = 8 πℓ p s + 1 / | s i ≈ πℓ s | s i , (14)where in the last line we have written a large s approxi-mation of the eigenvalues.The members of the principal series are the solutionswe are looking for. From the point of view of therepresentations of SL(2 , C ), they correspond to labels χ = ( p, k ) such that p = − i ( j + 1) and k = ∓ ij ,or p = 2 ij and k = ± j + 1) [17], where j = is − / s ∈ R . All this justifies the analytic continuation ofthe previous section and provides a clear interpretationof equation (10). Assuming that k is a non dynamicalregulator, then in the large s ℓ limit we havelog (cid:0) N Γ ( A, ± i ) (cid:1) ≈ A ℓ , (15)where A in N Γ ( A, ± i ) is the area eigenvalue of the statewith p punctures. The special role of SU(1 ,
1) representa-tions in the principal series in three dimensions is clarified[23].The above formula exhibits a holographic propertyof the number of states N Γ ( A, ± i ), at least in thelarge s ℓ limit. Remarkably, this is enough to recoverthe Bekenstein-Hawking entropy formula when summingover states Γ, if one adds the additional (but very natu-ral) assumption that punctures are indistinguishable ex-citations of the gravitational field [9]. It follows fromsimple statistical mechanical considerations that suchsystem can only be macroscopic if its temperature is T = T Hw (cid:0) o ( ℓ Pl / √ A ) (cid:1) (with T Hw the Hawking temper-ature), and that the number of punctures is p ≈ √ A/ℓ Pl ,which in turn implies that the system is dominated bylarge quantum numbers s ℓ , and finally that the entropyis S BH ≡ log (cid:0) N Γ ( A, ± i ) (cid:1) = A ℓ (cid:0) o ( ℓ Pl / √ A ) (cid:1) . (16)Notice that the entropy is finite despite the fact that thelabels s ℓ are continuous variables to be integrated over. Discussion
Our result suggests a possible way towards the quanti-zation of Ashtekar gravity, that would mimic our proce-dure for the quantum black hole. Such a recipe could con-sist in first starting from the kinematical quantum theoryin terms of the real Ashtekar-Barbero connection (leadingto a well understood Hilbert space structure based on thenotion of spin network states and quantum geometry) inorder, in a second step, to produce physical quantities bymeans of an analytic continuation of suitable quantitiesalong the lines presented here. A particularly interestingquestion is that of the definition of spin foam transitionamplitudes by means of such a procedure [17].Our results open important and (probably quite dif-ficult) mathematical questions that require a system-atic analysis. In particular, the representations selectedby the self-duality condition (12) and the reality condi-tions are exotic representations from the point of viewof SL(2 , C ). The representations χ = ( − i ( j + 1) , ± ij )and χ = ( − ij, ± i ( j +1)), with j = is − / f ( z , z ) which are non-analytic (they have branch-cut singularities that allow k to be different from an in-teger). We believe that a deeper understanding of theserepresentations will clarify further the geometrical mean-ing of our analytic continuation. This is certainly not aneasy task and it represents a longer term objective ofour work. Another important question to be studied inthe future is the relationship of our results with those ofBianchi [22] and Pranzetti [24]. Acknowledgements
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