Statistical and entanglement entropy for black holes in quantum geometry Alejandro Perez1 1 Aix Marseille Universit, CNRS, CPT, UMR 7332, 13288 Marseille, and Universit de Toulon, CNRS, CPT, UMR 7332, 83957 La Garde, France. We analyse the relationship between entanglement (or geometric) entropy with statistical me- chanical entropy of horizon degrees of freedom when described in the framework of isolated horizons in loop quantum gravity. We show that, once the relevant degrees of freedom are identified, the two notions coincide. The key ingredient linking the two notions is the structure of quantum geometry at Planck scale implied by loop quantum gravity, where correlations between the inside and outside of the black hole are mediated by eigenstates of the horizon area operator. PACS numbers: 04.70.Dy, 04.60.-m In the semiclassical regime where an approximate no- the renormalization group flow of gravity. However, due tion of black hole (BH) makes sense, horizon area is quan- to the completely combinatorial way in which γ0 arises tized in loop quantum gravity (LQG) [1–3]. As a conse- (which does not make reference to any dynamical notion) quence, the dimensionality of the surface Hilbert space, it is so far unclear how such scenario could be realized. compatible with a maximum macroscopic area, is finite There is however a recent new perspective that might and grows exponentially with the area in the large area be helpful in resolving this question. The idea is that vac- limit [4, 5]. In other words, if one treats the horizon itself uum fluctuation in the non geometric sector (including as a thermodynamical system in equilibrium—justified in matter degrees of freedom) should be appropriately taken the case of large semiclassical black holes—then the mi- into account in the computation of black hole entropy (as cro canonical entropy, in the ensemble of surface states, is we will argue in Section I B all degrees of freedom close proportional to the macroscopic black hole area. The rel- to the horizon must be accounted for). This is not so in evant ensemble is the one defined by the horizon surface the treatment leading to (4) as only pure-geometry area states. eigenvalues are counted while ignoring the degeneracy as- The standard loop quantum gravity counting yields sociated to other degrees of freedom (including matter). [6, 7] Indeed by taking these degrees of freedom suitably into account one obtains[11], for arbitrary values of γ, γ0 A Sstat = G (1) γ 4GN ~ GN A √A Sstat = + η (3) where G is the UV value of the gravitational constant at 4GN ~ √γG~ the fundamental scale, GN is the low energy Newton’s constant, γ is the Immirzi parameter, and γ0 is a numer- where η a dimensionless constant that depends on the ical constant that appears in the asymptotic expression punctures statistics. As expected the fundamental scale of the number of states. γG~ appears only in the quantum corrections to the The above result is compatible with Bekenstein- Bekenstein-Hawking entropy. Hawking’s entropy only if one fixes The fundamental surface degrees of freedom associated to matter fields are hard to describe in LQG. Thus vac- GN arXiv:1405.7287v2 [gr-qc] 4 Jun 2014 γ = γ0 . (2) uum fluctuations in the analysis leading to the previous G equation are accounted for by using (qualitative) infor- The necessity of fixing γ in order to achieve consistency mation provided by standard QFT on the black hole with the semiclassical regime is puzzling. The reason is background (in a spirit analogous to what is done in that the Immirzi parameter is a topological coupling in cosmological situations [12]). In particular there is an the gravity first order action not affecting the classical expected contribution to black hole entropy from the en- equations of motion (like the θ parameter of Yang-Mills tanglement entropy of quantum fields across the black theories [8, 9]). Therefore, in contrast with expression hole horizon [13]. In standard QFTs the result is (1), one would expect to find γG only in quantum correc- tions of a leading term matching exactly the Bekenstein- A Hawking entropy (as the Immirzi parameter plays an Sent = λ + corrections. (4) GN ~ important role in the quantum theory where geometric operators have discrete spectra in the fundamental area with λ left undetermined due to UV divergences and 2 ~ ℓLQG = γG ). other ambiguities such as the species problem (concretely, −2 One logical possibility [10] is to interpret equation (2) λ = λ0GN ~ǫ for λ0 a regularization dependent dimen- as a renormalization condition fixing the relationship be- sionless constant and ǫ a UV cut-off length). tween GN and G. If that were the case, then the numer- In the analysis of [11] equation (3) follows from the ical value γ0 would have to have a clear interpretation in assumption that the degeneracy contribution to the area 2 spectrum coming from vacuum fluctuations of non geo- I. WHY SURFACE DEGREES OF FREEDOM? metric degrees of freedom is given by the exponential of qual- (4) with an undetermined λ (this justifies the term In this section we will discuss a common feature be- itative 1 used above ). In the perturbative quantum grav- tween the entanglement approach to black hole entropy ity framework there are indications that the ambiguities and the statistical mechanical approach followed in loop encoded in the value of λ disappear if the gravitational quantum gravity: that the relevant correlations are the degrees of freedom are correctly taken into account [18]. UV correlations across the BH horizon in the first case; In the non perturbative framework, such possibility is re- that the relevant microstates are local excitation of the inforced in a completely independent way by the results horizon geometry and matter on the horizon in the sec- of [11] where consistency fixes λ = 1/4 up to quantum ond case. In both cases a separation of scales is necessary corrections. to isolate the relevant physics behind the notion of BH However, as just mentioned, in the argument that leads entropy. Once this ingredient is incorporated to the anal- to (3) one assumes that one can interpret the (exponen- ysis we will show that both approaches actually compute tial of the) entanglement entropy of quantum fields across the same thing: namely, the Boltzmann-Gibbs entropy the horizon as a measure of the degeneracy of the area of the ensemble of surface microstates. Therefore, from spectrum of the horizon (including all the UV degrees the appropriate perspective, BHs are standard thermo- of freedom). This assumption necessitates a suitable re- dynamical objects where the horizon plays the role of the lationship between entanglement and statistical entropy. statistical mechanical system. Exploring and establishing the degree to which such rela- tionship holds in the LQG treatment of BHs is one of the aims of this work. Some aspects concerning this question A. Long range correlation in entanglement entropy have been considered in previous literature (for a recent discussion see [19]). Entanglement of apparent horizons in toy theories have been considered in [20], regions of Let us start with some discussion of the entanglement spin network states have been studied in [21], and it is approach. In particular we want to point out the influ- an important part of the argument of [17]. ence of global features of the quantum state considered when trying to compute the entanglement entropy asso- Finally, we would like to point out in this introduction ciated to the presence of a horizon. that there is a well know relationship between entangle- The Rindler wedge with its family of uniformly ac- ment entropy and Wald’s entropy [22] (for a review see celerated observers and associated Rindler horizon cap- [23]). The present analysis deals with a similar ques- tures some of the physical aspects of black hole systems tion but at the more fundamental Planckian level where in their infinite area limit. Entanglement entropy of the a statistical mechanical account of the thermodynamical Minkowski vacuum for the wedge is ill defined for the properties of BHs becomes available (including the fully UV problems mentioned above or exactly zero if one uses quantum treatment of gravitational contributions). the regularization prescription that is consistent with the In Section I we will argue that the relevant degrees of semiclassical Einstein’s equations compatible with the flat background (i.e. for which Tµν = 0 [24]). We will freedom both for entanglement and statistical mechanical h i approaches to BH entropy are to be found on the local come back to this important point in Section I B. How- Planckian vicinity of the BH horizon. In making this ever, the changes in the entanglement entropy (relative case, we will revisit the results of [18], and gain new in- entropy [25]) under perturbations of the reduced density sights by discussing their generalisation to eternal static matrix ρ = ρ0 + δρ turn out to be well defined [18] and black holes in Section I A. In Section II we will briefly give review the quantum description of the BH horizon quan- tum degrees of freedom in LQG. With this framework δSent = Tr[(ρ0 + δρ) log(ρ + δρ)] − at hand we will establish the sense in which entangle- µ ν = 2π δ Tµν χ dΣ ment and statistical entropy are equivalent in quantum ZΣ h i geometry in Section III. δA = (5) 4GN ~ 1 This type of holographic degeneracy appears naturally when con- where χ is the corresponding boost killing vector field sidering the analytic continuation of the pure geometric degen- (to which accelerated observers are tangent), Σ is an ar- eracy from real the Immirzi parameter γ to γ →±i [14, 15].