Barbero-Immirzi parameter, manifold invariants and Euclidean path integrals
aa r X i v : . [ g r- q c ] A p r Barbero-Immirzi parameter, manifold invariants andEuclidean path integrals
Tom ´ a ˇ s Liko ∗ Department of Mathematical and Statistical SciencesUniversity of AlbertaEdmonton, AB T6G 2G1, Canada
August 9, 2018
Abstract
The Barbero-Immirzi parameter γ appears in the real connection formu-lation of gravity in terms of the Ashtekar variables, and gives rise to a one-parameter quantization ambiguity in Loop Quantum Gravity. In this paper weinvestigate the conditions under which γ will have physical effects in EuclideanQuantum Gravity. This is done by constructing a well-defined Euclidean pathintegral for the Holst action with non-zero cosmological constant on a manifoldwith boundary. We find that two general conditions must be satisfied by thespacetime manifold in order for the Holst action and its surface integral tobe non-zero: (i) the metric has to be non-diagonalizable; (ii) the Pontryaginnumber of the manifold has to be non-zero. The latter is a strong topologicalcondition, and rules out many of the known solutions to the Einstein field equa-tions. This result leads us to evaluate the on-shell first-order Holst action andcorresponding Euclidean partition function on the Taub-NUT-ADS solution.We find that γ shows up as a finite rotation of the on-shell partition functionwhich corresponds to shifts in the energy and entropy of the NUT charge. Inan appendix we also evaluate the Holst action on the Taub-NUT and Taub-boltsolutions in flat spacetime and find that in that case as well γ shows up in theenergy and entropy of the NUT and bolt charges. We also present an exam-ple whereby the Euler characteristic of the manifold has a non-trivial effect onblack-hole mergers. PACS : 04.20.Cv;04.20.Fy;04.70.Dy
Since its discovery within the context of canonical gravity [1], the Barbero-Immirziparameter, denoted γ , has remained an elusive one-parameter quantization ambi- ∗ Electronic mail: [email protected] γ has been done by matching the Bekenstein-Hawking entropy of blackholes with spacetime topology R × C , with C a compact two-manifold, to thecorresponding isolated horizon quantum geometry [7–10]. It turns out that γ hasthe same value regardless of the topology of the black hole.The Einstein field equations admit more solutions in four dimensions than justthe black holes whose event horizons have topology R × C ; in particular, the NUT-charged spacetimes [11, 12] have topologies that cannot be foliated by a time func-tion and contain Misner strings with a non-zero entropy [13]. A comparison ofthe semi-classical and quantum geometry descriptions of NUT-charged spacetimeswould be of interest in order to provide an independent determination of γ that canbe compared to the previous results for black holes [7–10].Motivated by this interest, we want to know what effects, if any, γ may haveon the Euclidean path integral. In this paper we study the Euclidean Holst action[14] with non-zero cosmological constant, and derive the semi-classical energy andentropy of Taub-NUT-ADS spacetime in the presence of γ . This can only be donein first-order formalism. That said, we need to make sure that the first-order Holst-ADS action satisfies two important conditions in order that we may be able toevaluate the on-shell Euclidean Holst-ADS action: (I) The action has to have awell-posed action principle on a manifold with boundary ; and (II) the action has tobe finite .Condition I leads us to consider the Dirichlet boundary value problem for a generic first-order action that includes curvature and torsion as functionals of theconnection and coframe, and we present a general prescription for determining thecorresponding surface terms for which the first variation of the action vanishesidentically. We then present, in an example, a derivation of the surface termsthat are necessary for a generalized Hilbert-Palatini action in four dimensions tobe functionally differentiable; the form of this functional is motivated by requiringconsistency with the Hamiltonian theory coupled to fermions when γ = i . A specialcase of this action is the Holst action with non-zero cosmological constant – the mainfocus of study in this paper. Some general properties of the Holst surface term arediscussed in Section 3. Condition II is also addressed in Section 3. The ADS andHolst-ADS actions are finite without the need of adding infinite counter-terms tothe boundary integral, as is normally done in the infinite subtraction method [15–17, 19–21].Once the two conditions are shown to be satisfied by the Holst-ADS action, weare able to evaluate the Euclidean Holst action on solutions to the field equations,and obtain the corresponding on-shell partition functions. We find that generically γ shows up in the Euclidean path integral as a finite rotation of the on-shell partitionfunction, and this rotation corresponds to shifts in the energy and entropy of the2pacetime. This property of the Holst-ADS action is a result of a strong topologicalcondition that we find: the Holst term and its surface term are non-zero if thePontryagin number of the corresonding spacetime manifold is non-zero. We confirmthese results by explicitly evaluating the Holst-ADS action and partition functionon the Taub-NUT-ADS solution; γ shows up as a finite shift in the energy andentropy of the NUT charge.We also include two appendices. In Appendix 1 we evaluate the Euclidean Holstaction on Taub-NUT spacetimes with zero cosmological constant. We show that γ shows up as a shift in the energy and entropy of the NUT and bolt charges, just asfor the Taub-NUT-ADS solution. In Appendix 2 we present an example which showshow a topological invariant of the manifold can have non-trivial physical effects ingravity. In particular, it is shown that the presence of the Euler characteristic ofthe manifold can lead to violations of the second law during a black-hole mergingprocess. In the first-order formulation of general relativity (see e.g. [4]), the configurationspace C is the pair { e, A } , consisting of the coframe e and a connection A valuedin SO ( D ) or SO ( D − ,
1) depending on the signature of spacetime. The coframedetermines the spacetime metric g ab = η IJ e Ia ⊗ e Jb and spacetime volume form ǫ = e ∧ · · · e D − , where ǫ I ··· I D is the totally antisymmetric Levi-Civita tensor.In this paper, spacetime indices a, b, . . . ∈ { , . . . , D − } are raised and loweredusing the metric g ab and internal indices I, J, . . . ∈ { , . . . , D − } are raised andlowered using the flat metric η IJ . The connection determines the curvature two-form Ω IJ = dA IJ + A IK ∧ A KJ = R IJKL e K ∧ e L , with R IJKL the Riemann tensor.The Lagrangian density is denoted L ; this is a functional , by which we mean amap from the space of functions ( e, A ) to R . The functional derivative of L withrespect to a function ϕ = ϕ ( x a ) is denoted Υ ϕ ≡ δ L /δϕ . The internal Hodge dualis denoted by ⋆ . In this paper, we write differential forms without indices. Let us begin with the following.
Proposition 1.
Let
Ω = dA + A ∧ A and T = de + A ∧ e be (resp.) the curvatureand torsion of a D -dimensional manifold M with boundary ∂ M , with A the con-nection and e the coframe. Let L [ e, A, Ω , T ] be the Lagrangian density, a D -form inspacetime. The first-order action for an arbitrary diffeomorphism invariant theoryof pure gravity on the configuration space C = { e, A } , I = 116 π Z M L [ e, A, Ω , T ] − ( − D π I ∂ M Υ Ω ∧ A + Υ T ∧ e , (1)3 s functionally differentiable. Proof.
Take the variation of L : δ L = Υ e ∧ δe + Υ A ∧ δA + Υ Ω ∧ δ Ω + Υ T ∧ δT = Υ e ∧ δe + Υ A ∧ δA + Υ Ω ∧ ( dδA + 2 A ∧ δA ) + Υ T ∧ ( dδe − e ∧ δA + A ∧ δe )= ( − D d ( Υ Ω ∧ δA ) + ( − D d ( Υ T ∧ δe ) + (cid:2) Υ e + Υ T ∧ A − ( − D d Υ T (cid:3) ∧ δe + (cid:2) Υ A + 2 Υ Ω ∧ A − ( − D d Υ Ω − Υ T ∧ e (cid:3) ∧ δA . The action will be functionally differentiable if the total derivative is cancelled, andthe equations of motion Υ e + Υ T ∧ A − ( − D d Υ T = 0 (2) Υ T ∧ e + ( − D d Υ Ω − Υ Ω ∧ A − Υ A = 0 (3)are satisfied. From the Fundamental Theorem of Exterior Calculus , the boundaryterm I ∂ M = ( − D π I ∂ M Υ Ω ∧ A + Υ T ∧ e (4)follows. This completes the proof. (cid:4) From the identity d = 0, terms involving higher derivatives in the configurationvariables e and A do not show up in the action (1). The action (1) is thereforegenerically first-order in derivatives of e and A . If we take the Wilsonian point of view on effective field theories, then we need toadd all possible D -forms to the action for gravity that are diffeomorphism invari-ant. In four dimensions, this implies that in addition to the Hilbert-Palatini andcosmological terms, we should include in the action the Holst term θ ∫ e ∧ e ∧ Ω, aswell as the characteristic classes: the Euler class θ ∫ ⋆ Ω ∧ Ω, the Pontryagin class θ ∫ Ω ∧ Ω, and the Nieh-Yan class θ ∫ T ∧ T − e ∧ e ∧ Ω. See e.g. [22, 23]. Here, θ = 1 /γ with γ a non-zero real constant – the so-called Barbero-Immirzi parameter[1]. It was pointed out in [24, 25], however, that in the case when fermions arecoupled to gravity, the Holst term needs to be replaced with the Nieh-Yan term.This extension of the Holst term is necessary in the presence of non-zero torsionso that the corresponding Hamiltonian with arbitrary γ reduces to the Ashtekar-Romano-Tate Hamiltonian [26] when γ is set equal to the imaginary unit. Thereforewe consider here the Nieh-Yan term in place of the Holst term in our action principle(with θ = 1 /γ ). When torsion is zero the Nieh-Yan term reduces to the Holst term.This makes the topological origin of the Holst term in the first-order action manifest!4rom Proposition 1, then, we have the following: Corollary 2.
The generalized Hilbert-Palatini action for general relativity withcosmological constant
Λ = 3 ε/ℓ ∈ R ( ε ∈ {− , } ), Barbero-Immirzi parameter γ ∈ R \ and θ , θ ∈ R on a four-dimensional manifold ( M , e, A ) with boundary ∂ M , I = 116 π Z M ⋆ ( e ∧ e ) ∧ Ω − ǫ + 1 γ ( T ∧ T − e ∧ e ∧ Ω) + θ ⋆ Ω ∧ Ω + θ Ω ∧ Ω − π I ∂ M ⋆ ( e ∧ e ) ∧ A − γ e ∧ de + 2 θ ⋆ Ω ∧ A + θ Ω ∧ A , (5) is functionally differentiable.
Proof.
For the Nieh-Yan-Holst term, we find that Υ T = T and Υ Ω = − e ∧ e so thatthe surface term has density T ∧ e − e ∧ e ∧ A = de ∧ e + A ∧ e ∧ e − A ∧ e ∧ e = de ∧ e .For the Euler term we find that Υ Ω = 2 ⋆ Ω so that the surface term has density2 ⋆ Ω ∧ A . For the Pontryagin term we find that Υ Ω = Ω so that the surface termhas density Ω ∧ A . The surface term in the action (5) follows. (cid:4) The action (6), with θ = θ = θ = θ = 0, was previously studied withinthe context of black-hole mechanics. In these works, the spacetimes under con-sideration include four-dimensional Einstein-Maxwell theory with zero cosmologicalconstant [27, 28], non-trivial matter couplings [29–31], higher-dimensional flat andADS spacetimes [32, 33], Gauss-Bonnet gravity [34] and supergravity with p formmatter couplings [35–37].In this formalism, the topology of the boundary in the action principle is takento be ∂ M ∼ = M ∪ M ∪ ∆ ∪ I , with ∆ a ( D − . . . +). M and M are partial Cauchy surfaces that extend from ∆ to the boundary I atinfinity; M and M intersect ∆ and I in ( D − C D − . Adding the surfaceterm H I ⋆ ( e ∧ e ) ∧ A to the action at I and fixing the geometry of ∆ are sufficientconditions for the action to be functionally differentiable and for the zeroth and firstlaws of black-hole mechanics to be satisfied. In particular, all the conserved chargesare defined locally at the horizon ∆; these include the non-monopolar (dipole) chargeof the five-dimensional black ring solution [37, 38].Subsequently it was shown that the action is finite on asymptotically flat space-times [39, 40], and that a partition function can be given for Euclidean metrics [41]without having to add infinite counter-terms to the boundary [41]. It was thenshown that the the Holst term and its surface term together are finite [42]. Inthis paper we further explore properties of the Holst action and corresponding Eu-clidean path integral on a manifold with boundary, in the presence of a non-zerocosmological constant. 5 Holst action with cosmological constant
In this paper we will focus particular attention to ADS spacetimes with Λ <
0, butthe mathematical results obtained also apply to de Sitter spacetimes with Λ > non-zero cosmologi-cal constant Λ = 3 εℓ − ∈ R \
0, with ε ∈ {− , } and ℓ is the de Sitter radius.Specifically, we consider the action (5) with T = 0 and with θ = θ = 0.The first-order Holst action with cosmological constant Λ = 3 εℓ − ∈ R \ ε ∈ {− , } ) and Barbero-Immirzi parameter γ ∈ R \ M , e, A ) with boundary ∂ M is: I = 116 π Z M (cid:20)(cid:18) ⋆ − γ (cid:19) ( e ∧ e ) (cid:21) ∧ Ω + 6 εℓ ǫ − π I ∂ M ⋆ ( e ∧ e ) ∧ A − γ e ∧ de . (6)From Corollary 2, this action is functionally differentiable. The surface term here isthe same as the one that was previously found for the Holst action in flat spacetime[42]. This is because when T = de + A ∧ e = 0 we have e ∧ de = e ∧ e ∧ A .An important property of the Holst surface term e ∧ de is that it is identicallyzero for spacetimes with metrics that can be put into diagonal form. To see this,consider the tetrad in components: e I = e Ia dx a . Differentiating the tetrad gives de I = ( ∂e Ia /∂x b ) dx b ∧ dx a . By direct substitution we get e ∧ de = (cid:20) e (cid:18) ∂e a ∂x b (cid:19) + e (cid:18) ∂e a ∂x b (cid:19) + e (cid:18) ∂e a ∂x b (cid:19) + e (cid:18) ∂e a ∂x b (cid:19)(cid:21) ∧ dx b ∧ dx a (7)If the metric is diagonal, then e ∧ de = e (cid:18) ∂e ∂x b (cid:19) ∧ dx b ∧ dx + e (cid:18) ∂e ∂x b (cid:19) ∧ dx b ∧ dx + e (cid:18) ∂e ∂x b (cid:19) ∧ dx b ∧ dx + e (cid:18) ∂e ∂x b (cid:19) ∧ dx b ∧ dx = 0 . (8)Of particular interest in general relativity are spacetimes with a ‘ t - φ ’ component intheir corresponding metrics. For a general metric with x - x cross term, e ∧ de canbe written in component form such that e ∧ de = (cid:26) e (cid:18) ∂e ∂x b (cid:19) − e (cid:18) ∂e ∂x b (cid:19)(cid:27) dx ∧ dx b ∧ dx . (9)In spherical coordinates x a ∈ { τ, r, θ, φ } , on a constant- r hypersurface, the boundaryterm is: e ∧ de = (cid:26) e (cid:18) ∂e ∂θ (cid:19) − e (cid:18) ∂e ∂θ (cid:19)(cid:27) dτ ∧ dθ ∧ dφ . (10)This expression can be used to evaluate the Holst surface term on solutions with a t - φ component in the metric.For spacetimes with non-zero cosmological constant, another condition can befound on the Holst surface term. Let us first substitute the equation of motion6 e + A ∧ e = 0 into the action (6) to eliminate the tetrad derivative in the surfaceterm: I = 116 π Z M (cid:20)(cid:18) ⋆ − γ (cid:19) ( e ∧ e ) (cid:21) ∧ Ω + 6 εℓ ǫ − π I ∂ M ⋆ ( e ∧ e ) ∧ A − γ e ∧ e ∧ A . (11)Then with the equation of motion e ∧ e = ( εℓ / ! = denoting equality on-shell ) − π I ∂ M ⋆ ( e ∧ e ) ∧ A − γ e ∧ e ∧ A ! = − εℓ · π I ∂ M ⋆ Ω ∧ A − γ Ω ∧ A . (12)From the Fundamental Theorem of Exterior Calculus, this boundary integral canbe written as a bulk integral: − εℓ · π I ∂ M ⋆ Ω ∧ A − γ Ω ∧ A ! = − εℓ · π Z M ⋆ Ω ∧ Ω − γ Ω ∧ Ω . (13)Putting this in (11), we see that the Holst action can be written as a bulk integral: I = 116 π Z M (cid:20)(cid:18) ⋆ − γ (cid:19) ( e ∧ e ) (cid:21) ∧ Ω + 6 εℓ ǫ − εℓ ⋆ Ω ∧ Ω + εℓ γ Ω ∧ Ω . (14)Written this way, the surface terms appearing in the action (6) are invariants of themanifold M . It follows from this form of the action that the Holst surface termis identically zero for manifolds that have zero Pontryagin number . In addition,the Holst term itself can be written purely in terms of the connection by using theequation of motion e ∧ e = ( εℓ / γ Z M e ∧ e ∧ Ω ! = εℓ γ Z M Ω ∧ Ω . (15)Therefore we see that on-shell the Holst term will be identically zero if the Pontrya-gin number of the manifold is zero.The first-order Holst action with negative cosmological constant is finite. Tosee this, consider first the Einstein-Hilbert and cosmological terms only. Then,action (14) with ε = − together are finite. Therefore, the action (14) is finite.Let us briefly summarize this section. The Holst-ADS action is functionallydifferentiable and finite. In order for the Holst term and its surface term, andtherefore γ to be present in the on-shell action, the corresponding solution has tohave a non-diagonalizable metric and a non-zero Pontryagin number. These are veryrestrictive conditions and rule out many of the known spacetimes that are criticalpoints of the action. The Taub-NUT spacetime is known to have Pontryagin number2 [44]. Therefore, in Section 4 and in Appendix 1, we evaluate the Euclidean on-shellHolst-ADS and Holst actions and partition functions (resp.) on the Taub-NUT-ADSspacetime with Λ < Euclidean path integrals and Taub-NUT-ADS space-time
Let us consider the formal path integral Z = Z D [Ψ]exp {− I [Ψ] } , (16)with I [Ψ] the Euclidean action and Ψ a generic field variable. Here, the measure D [Ψ] includes all fields and not just the classical fields e Ψ that satisfy the equationsof motion δI [ e Ψ] = 0. However, if the dominant contributions to the partitionfunction come from fields that are close to the classical fields, then the action canbe expanded in a Taylor series: I [ e Ψ + δ Ψ] = I [ e Ψ] + δI [ e Ψ , δ Ψ] + δ I [ e Ψ , δ Ψ] + . . . . (17)In order for the path integral Z to make sense, at least to second order in the Taylorseries, we require that the first term I [ e Ψ] be finite and that the linear term δI vanishidentically. If these conditions are satisfied, then the on-shell partition function isapproximated by e Z = exp n − I [ e Ψ] o ; (18)the average energy h E i and entropy S are then given by h E i = − ∂ ln e Z ∂β and S = β h E i + ln e Z . (19)The physical meaning of the energy may differ based on the boundary conditionsthat are used, i.e. holding the pressure or volume constant.From Section 3, we know that the Holst-ADS action (6) is functionally differen-tiable and finite. Therefore we may consider the Holst-ADS partition function e Z = exp (cid:26) − π Z M (cid:20)(cid:18) ⋆ − γ (cid:19) ( e ∧ e ) (cid:21) ∧ Ω + 6 εℓ ǫ − εℓ ⋆ Ω ∧ Ω + εℓ γ Ω ∧ Ω (cid:27) (20)for spacetimes with negative cosmological constant and non-zero Pontryagin num-ber. Let us therefore proceed by evaluating the partition function (20) on theTaub-NUT-ADS spacetime. Here we consider the Taub-NUT-ADS spacetime. The metric for four-dimensional
Euclidean
Taub-NUT spacetime, with negative cosmological constant Λ = − ℓ − ,has line element ds = V ( r ) [ dτ + 2 N cos θdφ ] + dr V ( r ) + ( r − N )( dθ + sin θdφ ) ,V ( r ) = r − M r + N + ( r − N r − N ) ℓ − r − N , (21)8ith N the NUT parameter. Regularity of the metric requires that the Euclideantime τ have a period β = 8 πN .A suitable tetrad of coframes for this spacetime is given by e = √ V dτ + 2 √ V N cos θdφ , e = 1 √ V dr , e = p r − N dθ ,e = p r − N sin θdφ . (22)The Euclidean action for the Taub-NUT-ADS spacetime is then given by I = I + 64 π N γ (cid:18) − N ℓ (cid:19) , (23)with I the on-shell action of the Taub-NUT-ADS solution without the Holst term(i.e. contributions from the Einstein-Hilbert and cosmological terms only) [20, 45].Substituting this in (16) then gives the on-shell partition function e Z = exp (cid:26) − I − π N γ (cid:18) − N ℓ (cid:19)(cid:27) . (24)Whence the thermodynamic quantities are given by h E i = h E i + N ( ℓ − N ) γℓ and S = S + 4 πN γ (cid:18) − N ℓ (cid:19) , (25)with h E i and S denoting (resp.) the average energy and entropy of the Taub-NUT-ADS solution without the Holst term [20, 45].We conclude that γ appears in the thermodynamics of the Taub-NUT-ADSspacetime as a shift in the energy and entropy of the NUT charge (provided that γ is finite and real). The quantities derived here are in agreement with previousresults found by Mann [20] and Chamblin et al [45] but with a finite shift in theenergy and entropy of the Taub-NUT-ADS spacetime; these shifts vanish in thelimit when γ is taken to infinity. Let us briefly summarize the main results that are presented in this paper. Westudied the properties of the first-order Holst action with non-zero cosmologicalconstant. In particular, we showed that the spacetimes for which γ will be presentin the on-shell action (14), and hence in the partition function (20), are the onesthat have non-diagonalizable metrics and non-zero Pontryagin number. This led usto evaluate (20) on the Taub-NUT-ADS spacetime. It was found that γ shifts theenergy and entropy of the NUT charge. The analogous results for the case whereΛ = 0 are presented in Appendix 1. Some results regarding the Euler characteristicand black-hole mechanics are presented in Appendix 2.The results found in this paper agree with recent results found by Durka andKowalski-Glikman [46, 47]. Durka and Kowalski-Glikman derived the Noether-Wald9harges for solutions of a constrained BF theory with SO (3 ,
2) symmetry, firstintroduced by Freidel and Starodubtsev [48]. They found that γ shows up in theNoether-Wald charges of solutions that have ∂ θ g tφ = 0. For the Taub-NUT-ADSsolution, the energy and entropy are shifted by the same factor as found here, butwith 1 /γ → γ . This is because, in the action that they studied, the Holst andNieh-Yan terms appear seperately with different weight factors: the Holst term hascoefficient 1 /γ while the Nieh-Yan term has coefficient ( γ + 1) /γ , so their actionis fundamentally different. In particular, the limit γ → ∞ cannot be taken in thisaction to recover the Einstein-Hilbert action. Apart from this minor difference, thetwo approaches are equivalent: γ contributes to the on-shell partition function andto the Noether-Wald charges through the Pontryagin number of the spacetime.In this paper we considered initially the Holst action with a generic cosmologicalconstant, and in particular focused on the case when Λ <
0. However, in light ofcosmological data, it would be of interest to also study in detail the Holst actionwith Λ >
0. The expression (20) is mathematically valid for any sign ε = sign(Λ)of the cosmological constant. An important step toward determining how γ affectse.g. the NUT-charged DS spacetimes [49, 50], is to first prove the finiteness of theHolst-DS action with ε = 1. Then (20) will be a well-defined partition functionfor DS spacetimes with non-vanishing Pontryagin number. This approach may alsoreveal new insights regarding the Kodama wavefunction with arbitrary real valuesof γ [51, 52].In this paper we looked at the torsion-free case. This field will be non-zero inthe presence of fermion couplings, and therefore should be included in the action.Torsion-squared Lagrangian densities in the first-order action have recently beenstudied in [53, 54]; these terms are all consistent with Corollary 2. Note that inthe presence of fermion couplings, γ will appear in the chiral anomaly [55]. Ofparticular interest would be to extend the formalism here to supergravity. To studythe effects of γ in supergravity, the supersymmetric Holst actions found by Kaul[56] have to be generalized to a manifold with boundary. In practice, however,finding supersymmetric boundary terms without imposing any boundary conditionson the fields is difficult. See [57–59] for details. Ideally we would like to have anaction principle for supergravity that is invariant under the off-shell supersymmetryalbegra; this suggests that we extend the supergravity action to a manifold withboundary in superspace; such an action without boundary has been recently foundby Gates Jr. et al [60] where it was found that γ shows up in superspace as thecomplex component of the gravitational constant.It would also be of interest to determine the effects of γ in quantum gravity bystudying more general Euclidean path integrals. Because any topology may occurin classical and quantum gravity, one may have in general a partition function thatsums over all possible inequivalent topologies [61–63]. In Section 3 we found thatin order for γ to be present in the partition function, the Pontryagin number of the10anifolds have to be non-zero, so only those manifolds will contribute to the formalsum. In the case of supergravity with fermion couplings, the existence of a spinorstructure on M requires that the second Stiefel-Whitney class of the manifold benon-zero [64]. This condition places further restrictions on the formal sum. Acknowledgements
The author wishes to thank Remigiusz Durka and Jerzy Kowalski-Glikman for veryuseful correspondence, and for commenting on a draft of the manuscript. Theauthor also thanks Abhay Ashtekar and David Sloan for discussions during theearly stages of this project. This work was supported by NSERC. While at PennState University, the author was also supported in part by NSF grant PHY0854743,The George A. and Margaret M. Downsbrough Endowment and the Eberly researchfunds of Penn State.
A Taub-NUT and Taub-bolt spacetimes
Here we consider the Taub-NUT spacetime. The metric for four-dimensional
Eu-clidean
Taub-NUT spacetime, with zero cosmological constant, has line element ds = V ( r ) [ dτ + 2 N cos θdφ ] + dr V ( r ) + ( r − N )( dθ + sin θdφ ) ,V ( r ) = r − M r + N r − N , (26)with N the NUT parameter. Regularity of the metric requires that τ have a period β = 8 πN .A suitable tetrad of co-frames for this spacetime is given by e = √ V dτ + 2 √ V N cos θdφ , e = 1 √ V dr , e = p r − N dθ ,e = p r − N sin θdφ . (27)Using (10), we find that the Euclidean action for the Taub-NUT spacetime is givenby I = 4 πM N − πN γ . (28)Substituting this in (18) then gives the on-shell partition function e Z = exp (cid:18) − πM N + 2 πN γ (cid:19) . (29)The thermodynamic quantities can now be calculated. In particular, M = N for theNUT charge and substituting this into (29) gives the average energy and entropy h E i = N (cid:18) − γ (cid:19) and S = 4 πN (cid:18) − γ (cid:19) , (30)11hile M = 5 N/ h E i = 5 N (cid:18) − γ (cid:19) and S = 5 πN (cid:18) − γ (cid:19) . (31)Therefore, γ appears in the thermodynamics of Taub-NUT and Taub-bolt solutionswith zero cosmological constant as shifts in the energies and entropies of the NUTand bolt charges, just as we found for the NUT charge in the Taub-NUT-ADSsolution. B Euler characteristic and the second law
In this Appendix, let us present an example that illustrates how a topological in-variant of the spacetime manifold can have non-trivial physical effects on black-holethermodynamics. In particular, we will derive an upper bound on θ that must besatisfied in order for the second law to hold when two black holes merge.The bound presented here is general and holds for solutions to the Gauss-Bonnetfield equations in arbitrary dimensions. We will consider four-dimensional asymp-totically flat black holes as a special case; in this case the black holes must satisfycertain theorems and these can be used to put a tight upper bound on θ for thearea theorem to hold when two Schwarzschild black holes merge [74].For black holes of Gauss-Bonnet gravity with generic cosmological constant Λ,the first law of black-hole mechanics holds with an entropy that is given by [34, 65,66] S = 14 π I C ˜ ǫ (1 + 2 θ R ) ; (32)here R is the Ricci scalar of the horizon cross section C and ˜ ǫ is the area ( D − C .Let us consider the merging of two black holes – one with mass m and the otherwith mass m . The entropies of these black holes are (resp.) S = 14 [ A + 2 θ X ( C )] and S = 14 [ A + 2 θ X ( C )] ; (33)here we have defined the surface area A and correction term X ( C ) via A = I C ˜ ǫ and X ( C ) = I C ˜ ǫ R . (34)Before the black holes merge, the total entropy is S = S + S = 14 [ A + A + 2 θ ( X ( C ) + X ( C ))] . (35)After the black holes merge, the total entropy of the resulting black hole is S ′ = 14 [ A ′ + 2 θ X ( C ′ )] . (36)12he area theorem will hold if and only if S ′ > S . Thus we have the followingbound: θ < − ( A + A − A ′ )2[ X ( C ) + X ( C ) − X ( C ′ )] . (37)This bound for θ is general, and holds for spacetimes in all dimensions with genericvalues of Λ.Without knowing more about the details of the black holes that are merging,nothing further can be said about the bound (37) because the topological struc-ture of black holes is much richer in D > ≥ D ≥ any productmanifold R × C D − , with C D − a space of positive Yamabe type . For example, infive dimensions the topology of the event horizon has to be (a connected sum of) athree-sphere C ∼ = S or three-ring C ∼ = S × S . A complete discussion of blackholes in higher dimensions is presented in [67]; topological properties are presentedin [68–73].For concreteness, then, let us consider the merging of two non-rotating blackholes in four dimensions, with Λ = 0. First, the Gauss-Bonnet theorem says that X ( C ) = I C ˜ ǫ R = 4 πχ ( C ) , (38)with χ ( C ) the Euler characteristic of C . Then the Hawking topology theorem saysthat the horizon cross sections can only be two-spheres so that C ∼ = S , and then χ ( S ) = 2. It follows that X ( C ) = X ( C ) = X ( C ′ ) = 8 π . (39)Finally, the Birkhoff theorem says that the only static asymptotically flat solutionto the field equations is the Schwarzschild solution. Since the surface area of aSchwarzschild black hole is related to its mass m via A = 16 πm , the surface areasof the initial and final black-hole states are A = 16 πm , A = 16 πm , and A ′ = 16 π ( m + m − α ) . (40)In the above definition for A ′ the parameter α ≥ θ : θ < m m − α [2( m + m ) − α ] . (41)Therefore, in four-dimensional asymptotically flat spacetimes, the second law willbe violated if θ is greater than twice the product of the masses of two Schwarzschildblack holes before merging minus a correction due to gravitational radiation. Thisis an important property because it shows that a non-zero Euler characteristic ofthe manifold with boundary can have physical effects in four dimensions.13 eferences [1] Barbero G J F 1995 Phys. Rev. D Class.Quant. Grav. L177[3] Rovelli C and Thiemann T 1997 Immirzi parameter in quantum general rela-tivity
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