Where are the degrees of freedom responsible for black hole entropy?
Considering the entanglement between quantum field degrees of
freedom inside and outside the horizon as a plausible source of
black hole entropy, we address the question: where are the
degrees of freedom that give rise to this entropy located? When
the field is in ground state, the black hole area law is obeyed
and the degrees of freedom near the horizon contribute most to the
entropy. However, for excited state, or a superposition of ground
state and excited state, power-law corrections to the area law are
obtained, and more significant contributions from the degrees of
freedom far from the horizon are shown444Based on a talk by
SD at Theory Canada III, Edmonton, Alberta, Canada on June 16,
PACS Nos.: 04.60.-m,04.62.,04.70.-s,03.65.Ud
The study of black holes (BHs) has always been a major testing arena for models of quantum gravity. The key issue has been to identify the microscopic origin of black hole entropy . The questions that naturally arise in this context are the following: (i) Why is , given by the well-known Bekenstein-Hawking relation [1, 2],
is proportional to the horizon area , as opposed to volume (usual for thermodynamic systems)? (ii) Are there corrections to this so-called ‘area law’ (AL) and if so how generic are these corrections? (iii) Can we locate the degrees of freedom (DoF) that are relevant for giving rise to the entropy?
In the attempts to address these questions there have been two distinct approaches, viz., the one that associates with fundamental DoF such as string, loop, etc. , and the other which attributes to the entanglement of quantum field DoF inside and outside the BH event horizon [4, 5, 6, 7]. In this article we adopt the second approach and consider a quantum scalar field (in a pure state) propagating in the BH space-time. Since the BH horizon provides a boundary to an outside observer, the state restricted outside the horizon is mixed and leads to a non-zero entanglement (aka Von Neumann) entropy: Tr , where is the mixed (or reduced) density matrix obtaining by tracing over the scalar DoF inside and outside the horizon.
In Ref. [4, 5] — for scalar field is in the vacuum/ground state (GS) — it was shown that of scalar fields propagating in static BH and flat space-time (the DoF being traced inside a chosen closed surface) leads to the AL. In Refs. [8, 9, 10], the robustness of the AL is examined by considering non-vacuum states. It was shown that AL continues to hold for minimum uncertainty states like generic coherent state (GCS) or a class of squeezed states (SS), but for (1-particle) excited states (ES), or for a GS-ES superposition or mixing (MS), one obtains power-law corrections to the AL. Although for large horizon area, the correction term is negligible, for small BHs the correction is significant.
To understand physically the deviation from the AL for ES/MS, we ascertain the location of the microscopic DoF that lead to in these cases [10, 11]. We find that although the DoF close to the horizon contribute most to the total entropy, the contributions from the DOF that are far from the horizon are more significant for ES/MS than for the GS. Thus, the corrections to the AL may, in a way, be attributed to the far-away DoF. We also extend the flat space-time analysis done in  to (curved) spherically symmetric static black-hole space-times.
In the next section, we first discuss the motivation for considering scalar fields for the entanglement entropy computations and then show that the scalar field Hamiltonian in a general BH space-time in Lemaître coordinates, and at a fixed Lemaître time, reduces to that in flat space-time. In sec. (3), we briefly review the procedure of obtaining the entanglement entropy and show the numerical results and estimations for the cases of GS, ES and MS. In sec. (4), we locate the scalar field degrees of freedom that are responsible for the entanglement entropy and compare the results for GS and ES/MS. We conclude with a summary and open questions in sec. (5).
In the following, we use units with and set .
2 Hamiltonian of scalar fields in black-hole space-times
Scalar fields can be motivated from the viewpoint of gravitational perturbations in static asymptotically flat spherically symmetric space-time background with metric . For a metric perturbation , the linearized form of the Einstein-Hilbert action is invariant under the infinitesimal gauge transformation . Imposing the harmonic gauge condition, i.e.,  and keeping only the first derivatives of , one finally obtains the linearized spin-2 equation 
Assuming plane-wave propagation of the metric perturbations, i.e., [ polarization tensor], in the weak-field limit, the above action reduces to the action for a massless scalar field propagating in the background metric :
Hence, by computing the entanglement entropy of the scalar fields, we obtain the entropy of the metric perturbations of the background space-time. The Hamiltonian of a scalar field propagating in a general spherically symmetric space-time background with line-element:
where are continuous, differentiable functions of and we have decomposed in terms of the real spherical harmonics , i.e., .
In the time-dependent Lemaître coordinates [13, 14] the line-element is given by (4) with . This line-element is related to that in the time-independent Schwarzschild coordinates by the following transformation relations :
As opposed to the Schwarzschild coordinate, the Lemaître coordinate is not singular at the horizon , and (or, ) is space(or, time)-like everywhere while (or, ) is space(or, time)-like only for . Choosing a fixed Lemaître time (), the relations (6) lead to: . Plugging this in Eq.(5) and on performing the canonical transformations: , the Hamiltonian reduces to that of a free scalar field propagating in flat space-time 
This holds for any fixed , provided the scalar field is traced over either the region or the region . Hence, evaluating the entanglement entropy of the scalar field in flat space-time corresponds to the evaluation of entropy of BH perturbations at a fixed Lemaître time.
3 Entanglement entropy of scalar fields
We discretize the scalar field Hamiltonian (7) on a radial lattice with spacing :
where are the momenta conjugate of , is the infrared cut-off. in Eq.(8) is of the form of the Hamiltonian of coupled harmonic oscillators (HOs):
where the interaction matrix is given by:
The last two terms denote nearest-neighbour interactions and originate from the derivative term in (7).
The most general eigenstate of the Hamiltonian (9) is a product of HO wave functions:
where , (), (diagonal), and are the indices of the Hermite polynomials (). The frequencies are ordered such that for .
The reduced density matrix is obtained by tracing over first of the oscillators:
It is not possible to obtain a closed form expression for for an arbitrary state (11). We resort to the following cases to compute the entropy numerically555The computations are done with a precision of and for and . using the relation Tr :
(i) Ground state (GS) with -particle wave-function: .
(ii) Excited (1-particle) state (ES) with -particle wave-function: , where are the expansion coefficients and normalization of requires ). We choose with the last columns being non-zero.
(iii) GS-ES linearly superposed (i.e. mixed) state (MS) with -particle wave-function: . Normalization of requires constants and related by . For simplicity, we choose . (See details in Ref. ).
Results : For GS, one recovers the AL — , where is the ultraviolet cutoff at the horizon (set to be ). For MS and ES, we obtain power-law corrections to the AL:
where constant of order unity and is a fractional index which depends on the excitation . As the horizon area increases, the correction term becomes negligible and asymptotically. For small BHs, however, the correction is significant. Fig. 1 shows the logarithm of entropy versus log characteristics for GS, MS and ES ( being the horizon radius), as well as the asymptotic equivalence of GS and MS/ES entropies, and the numerical fit that leads to the above result (13).
4 Location of the degrees of freedom
Let us take a closer look at the interaction matrix , Eq.(10), for the system of HOs. The last two terms which signify the nearest-neighbour (NN) interaction between the oscillators, are solely responsible for the entanglement entropy of black holes. Let us perform the following operations:
Operation I : We set NN interactions to zero (by hand) everywhere except in a ‘window’, such that the indices run from to , where . We thus restrict the thickness of the interaction region to radial lattice points, while allowing it to move rigidly across from the origin to a point outside the horizon. The variation of the percentage contribution of the total entropy for a fixed window size of lattice points, i.e., , as a function of is shown in Fig. 2 for , in each of the cases GS and MS, ES with . In all the cases when is far away from (i.e., horizon), whereas for values of very close to there are significant contributions to . For GS, peaks exactly at . For MS and ES, however, the peaks shift towards a value , and the amplitudes of the peaks also decrease as the amounts of excitation increase [11, 10]. Therefore (a) the near-horizon DoF contributes most to , and (b) the contributions from the far away DoF are more for MS and ES, than for GS.
Operation II : We set the NN interactions to zero (by hand) everywhere except in a window whose center is fixed at , and the window thickness is varied from to , i.e., from the origin to the horizon. For GS we find that about of the total entropy is obtained within a width of just one lattice spacing, and within a width of the entire GS entropy is recovered. Thus most of the GS entropy comes from the DoF very close to the horizon and a small part has its origin deeper inside. For ES, however, the total entropy is recovered for much higher values of (than for GS) since the DoF that are away from the horizon contribute more and more as the excitation increases. Thus larger deviation from the area law may be attributed to larger contribution to the total entropy from the DoF far from the horizon. The left panels of Fig. 3 depict the variation of the percentage contribution to the total entropy, i.e., , as a function of for GS () and ES (with ). The situation is intermediate for MS (which interpolates between the GS and the ES), i.e., the total entropy is recovered for values of greater than that for GS, but less than that for ES (with same value of ). The percentage increase in entropy when the interaction region is incremented by one radial lattice point, , versus plots for GS and ES are shown in the right panels of Fig. 3. In the case of GS the inclusion of the first lattice point just inside the horizon leads to an increase from to of the total GS entropy. The next immediate points add more to this, but the contributions are lesser and lesser with inclusion of points further and further from the horizon. For ES however, inclusion of one lattice point adds , for , to the entropy, while the next immediate points contribute more than those for the GS.
We have thus shown that if the black hole entropy is looked upon as that due to the entanglement between scalar field DoF inside and outside the horizon, there are power-law corrections to the Bekenstein-Hawking area law when the field is in excited state or in a superposition of ground state and excited state. Although such corrections are negligible for semi-classical BHs, they become increasingly significant with decrease in horizon area as well as for increasing excitations. The deviation from the AL for ES and MS may be attributed to the fact that the scalar field DoF that are farther from the horizon contribute more to the total entropy in the ES/MS cases than in the case of GS. The near horizon DoF contribute most in any case, however. We have also extended the flat space-time analysis done in  to static spherically symmetric black hole space-times with non-degenerate horizons.
We conclude with some open questions related to our work: (i) Can a temperature emerge in the entanglement entropy scenario and would it be consistent with the first law of BH thermodynamics? (ii) Is ?, i.e., the second law of thermodynamics valid? (iii) Will the entanglement of scalar fields help us to understand the information loss problem? We hope to report on these in future.
SD and SSu are supported by the Natural Sciences and Engineering Research Council of Canada. SD thanks the organizers of Theory Canada III, Edmonton, AB, Canada for hospitality.
-  J. D. Bekenstein, Lett. Nuovo Cimento 4, 7371 (1972); Phys. Rev. D7, 2333 (1973); Phys. Rev. D9, 3292 (1974); Phys. Rev. D12, 3077 (1975).
-  S. W. Hawking, Nature 248, 30 (1974); Commun. Math. Phys. 43, 199 (1975).
-  A. Strominger, C. Vafa, Phys. Lett. B379, 99 (1996); A. Ashtekar et al, Phys. Rev. Lett. 80, 904 (1998); S. Carlip, ibid 88, 241301 (2002); A. Dasgupta, Class. Quant. Grav. 23, 635 (2006).
-  L. Bombelli, R. K. Koul, J. Lee, R. Sorkin, Phys. Rev. D34, 373 (1986).
-  M. Srednicki, Phys. Rev. Lett. 71, 666 (1993).
-  M. B. Plenio, Phys. Rev. Lett. 94, 060503 (2005); M. Cramer, J. Eisert, M.B. Plenio and J. Dreissig, Phys. Rev. A73, 012309 (2006).
-  R. Brustein, A. Yarom, Nucl. Phys. B709, 391 (2005); R. Brustein et al, JHEP 0601, 098 (2006).
-  M. Ahmadi, S. Das, S. Shankaranarayanan, Can. J. Phys. 84(S2), 1 (2006).
-  S. Das, S. Shankaranarayanan, Phys. Rev. D73, 121701 (2006); arxiv: gr-qc/0610022.
-  S. Das, S. Shankaranarayanan, S. Sur, arxiv: 0705.2070 (gr-qc).
-  S. Das, S. Shankaranarayanan, arxiv: gr-qc/0703082.
-  G. ’t Hooft and M. Veltman, Ann. Ins. Henri Poincaré A20, 69 (1975); N. H. Barth and S. M. Christensen, Phys. Rev. D28, 1876 (1983).
-  L. D. Landau, E. M. Lifshitz, Classical Theory of Fields, Pergamon Press, New York (1975).
-  S. Shankaranarayanan, Phys. Rev. D67, 084026 (2003).
-  K. Melnikov and M. Weinstein, Int. J. Mod. Phys. D13 1595 (2004).