Understanding the area-proportionality of black hole entropy (the `Area Law')
from an underlying fundamental theory has been one of the goals of all models
of quantum gravity. A key question that one asks is: where are the degrees of
freedom giving rise to black hole entropy located? Taking the point of view
that entanglement between field degrees of freedom inside and outside the
horizon can be a source of this entropy, we show that when the field is in its
ground state, the degrees of freedom near the horizon contribute most to the
entropy, and the area law is obeyed. However, when it is in an excited state,
degrees of freedom far from the horizon contribute more significantly, and
deviations from the area law are observed. In other words, we demonstrate that
horizon degrees of freedom are responsible for the area law.Comment: 5 pages, 3 eps figures, uses Revtex4, References added, Minor changes
  to match published versio

Chandrasekhar S

Das S Shankaranarayanan S Sur S

Dasgupta A

Landau L D

Marolf D

S Shankaranarayanan

Saurya Das

Wald R M

Wheeler J

English

arXiv

Understanding the area-proportionality of black hole entropy (the `Area Law') from an underlying fundamental theory has been one of the goals of all models of quantum gravity. A key question that one asks is: where are the degrees of freedom giving rise to black hole entropy located? Taking the point of view that entanglement between field degrees of freedom inside and outside the horizon can be a source of this entropy, we show that when the field is in its ground state, the degrees of freedom near the horizon contribute most to the entropy, and the area law is obeyed. However, when it is in an excited state, degrees of freedom far from the horizon contribute more significantly, and deviations from the area law are observed. In other words, we demonstrate that horizon degrees of freedom are responsible for the area law

Das, Saurya

Shankaranarayanan, Subramaniam

Max Planck Society eDoc Server

Where are the black-hole entropy degrees of freedom ?

Das, S.

Shankaranarayanan, S.

MPG.PuRe

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Where are the black-hole entropy degrees of freedom?

arXiv:gr-qc/0703082v2  4 Oct 2007Where are the black hole entropy degrees of freedom ?Saurya Das§ and S. Shankaranarayanan¶§ Dept. of Physics, University of Lethbridge, 4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada∗ and¶ Max-Planck-Institut fu¨r Gravitationphysik, Am Mu¨hlenberg 1, D-14476, Potsdam, Germany †Understanding the area-proportionality of black hole entropy (the ‘Area Law’) from an underlyingfundamental theory has been one of the goals of all models of quantum gravity. A key questionthat one asks is: where are the degrees of freedom giving rise to black hole entropy located? Takingthe point of view that entanglement between field degrees of freedom inside and outside the horizoncan be a source of this entropy, we show that when the field is in its ground state, the degrees offreedom near the horizon contribute most to the entropy, and the area law is obeyed. However,when it is in an excited state, degrees of freedom far from the horizon contribute more significantly,and deviations from the area law are observed. In other words, we demonstrate that horizon degreesof freedom are responsible for the area law.PACS numbers: 04.70.Dy, 03.67.Mn, 03.65.Ud,05.50.+qThe area proportionality of black hole entropy (the‘Area Law’ (AL)),SBH=AH4ℓ2P,(ℓP≡√~G/c3 = Planck length)(1)which differs from the volume proportionality of famil-iar thermodynamic systems, has been conjectured to bemore fundamental in some sense (the Holographic Hy-pothesis). Black holes are also regarded as theoretical lab-oratories for quantum gravity. Thus, candidate models ofquantum gravity, such as string theory and loop quantumgravity, have attempted to derive the ‘macroscopic’ AL(1) by counting ‘microscopic’ degrees of freedom (DOF),using the Von-Neumann/Boltzmann formula [1]:S = −Tr[ρ ln(ρ)] = lnΩ, kB = 1 (2)where ρ and Ω correspond to the density matrix and num-ber of accessible micro-states, respectively. Depending onthe approach, one either counts certain DOF on the hori-zon, or abstract DOF related to the black hole, and theredoes not appear to be a consensus about which DOF arerelevant or about their precise location [2].In this article, we attempt to answer these questionsin a more general setting, which in fact, may be relevantin any approach. We adopt the point of view that theentanglement between quantum field DOF lying insideand outside of the horizon leads to black-hole entropy. Itwas shown in Refs. [3, 4] (see also, Refs. [5, 6]) that theentanglement entropy of a massless scalar field propagat-ing in flat space-time (by tracing over a spherical regionof radius R) is proportional to the area of the sphereSent= 0.3(Ra)2a is the UV cutoff . (3)This suggests that the area law is a generic feature of en-tanglement, and acquires a physical meaning in the caseof black-holes, the latter’s horizon providing a natural‘boundary’ of the region to trace over. Note that Eq.(3)will continue to hold if the region outside the sphere istraced over instead.The relevance of Sentto SBHcan also be understoodfrom the fact that both entropies are associated with theexistence of the horizons [7]. Consider a scalar field ona background of a collapsing star. At early times, thereis no horizon, and both the entropies are zero. How-ever, once the horizon forms, SBHis non-zero, and if thescalar field DOF inside the horizon are traced over, Sentobtained from the reduced density matrix is non-zero aswell.In Refs. [3, 4], along with the fact that calculationswere done in flat space-time, a crucial assumption wasmade that the scalar field is in the Ground State (GS). InRefs. [8, 9], the current authors studied the robustness ofthe AL by relaxing the second assumption, and showedthat for Generic Coherent States (GCS) and a class ofSqueezed States (SS), the law continues to hold, whereasfor the Excited States (ES), one obtains:Sent= κ„Ra«2α, κ = O(1) (4)where α < 1, and higher the excitation, the smaller itsvalue. In this article, we attempt to provide a physicalunderstanding of this deviation from the AL, by show-ing that for ES, DOF far from the horizon contributemore significantly than for the GS. We also extend ourflat space-time analyses for any (3 + 1)−D sphericallysymmetric non-degenerate black-hole space-times [10].We begin by considering a scalar field φ(x) propagatingin a Schwarzschild space-time [17]:ds2 = −f(r)dt2 +drf(r)+ r2dΩ22, f(r) = 1−r0r. (5)where r0 is the horizon. Transforming to Lemaˆıtre coor-dinates (τ, R) [12]:τ = t+ r0[ln(1−√r/r01 +√r/r0)+ 2√rr0](6)R = τ +2 r323√r0=⇒ rr0=[32(R − τ)r0]2/3,2the line-element (5) becomes:ds2 = −dτ 2+»32(R− τ )r0–− 23dR2+»32(R− τ )r0– 43r230dΩ2 . (7)Note that R(τ) is everywhere space-like (time-like), themetric is non-singular at r = r0 (corresponding to 3(R−τ)/2r0 = 1) and the metric is explicitly time-dependent.The Hamiltonian of a massless, minimally coupled scalarfield in the background (7) is given byH(τ) =∑lm12∫ ∞τdR[2P 2lm(τ, R)3(R− τ) +3r2(R− τ) (8)× [∂Rφlm(τ, R)]2 + 3√r0rℓ(ℓ+ 1)φ2lm(τ, R)],where Plm(τ, R) is the canonical conjugate momenta ofthe spherically reduced scalar field φlm(τ, R), such that[φˆlm(R, τ0), Pˆlm(R′, τ0)] = iδ(R−R′) and ℓ denotes angu-lar momenta. Note that, in these coordinates, the scalarfield and its Hamiltonian explicitly depend on time.Next, choosing a fixed Lemaˆıtre time (say, τ0 = 0) andperforming the following canonical transformation [16]Plm(r) =√r πlm(r) , φlm(r) =ϕlm(r)r, (9)the Hamiltonian (8) transforms to:H(0) =XlmZ ∞0dr2»π2lm(r) + r2h∂rϕlmri2+ℓ(ℓ+ 1)r2ϕ2lm(r)–(10)which is simply the Hamiltonian of a free scalar field inflat space-time! Eq.(10) holds for any fixed τ , for whichthe results of Refs. [4, 9], namely relations (3) and (4),go through, provided one traces over either the regionR ∈ [0, 23r0) or the region R ∈ [ 23r0,∞) [10]. Extensionto any non-degenerate spherically symmetric space-timesis straightforward [18].Remaining steps in the computation of Sentare:1. Discretize the Hamiltonian (10), i. e.,H =∑lm12aN∑j=1[π2lm,j +(j +12)2(ϕlm,jj− ϕlm,j+1j + 1)2+l(l + 1)j2ϕ2lm,j], (11)where πlm,j denotes the conjugate momenta of ϕlm,j , L =(N + 1)a is the box size and a is the radial lattice size.This is of the form of the Hamiltonian of N coupled HOsH =12N∑i=1p2i +12N∑i,j=1xiKijxj i, j = 1, . . . , N (12)with the interaction matrix elements Kij given by:Kij =1i2[l(l + 1) δij +94δi1δj1 +(N − 12)2δiN δjN+((i+12)2+(i− 12)2)δi,j(i6=1,N)]−[(j + 12 )2j(j + 1)]δi,j+1 −[(i+ 12 )2i(i+ 1)]δi,j−1 , (13)where the last two terms are the nearest-neighbor inter-actions.2. Choose the state of the quantum field: For GS (r =1, αi = 0), GCS (r = 1, αi = arbitrary), or a class ofSS (α = 0, r = arbitrary), the N-particle wave functionψ(x1 . . . xN ) is given by [9]ψ(x1 · · ·xN ) =∣∣∣∣ ΩπN∣∣∣∣ exp[−12∑irκ1/2Di (xi − αi)2].(14)For ES (or 1-Particle state), the N-particle wave functionψ(x1 . . . xN ) is given byψ(x1 · · ·xN) =˛˛˛˛ 2ΩπN˛˛˛˛14NXi=1aik14Di xi exp"−12Xjk12Dj x2j#(15)where aT = (a1, . . . , aN ) are the expansion coefficients(normalization requires aTa = 1). For our computa-tions, we choose aT = 1/√o(0, · · · 0, 1 · · ·1) with the lasto columns being non-zero. For details, see Ref. [9].3. Obtain the density matrix: For an arbitrary wavefunc-tion ψ(x1, . . . , xN ), the density matrix — tracing overfirst n of the N field points — is given by:ρ (t; t′) =∫ n∏i=1dxi ψ(x1, . . . , xn; t1, . . . , tN−n)× ψ⋆(x1, . . . , xn; t′1, . . . , t′N−n) (16)where: tj ≡ xn+j , t ≡ t1, . . . , tN−n, j = 1..(N −n), xT =(x1, . . . , xn; t1, . . . , tN−1) = (x1, . . . , xn; t). This step, ingeneral, cannot be evaluated analytically and requiresnumerical techniques. For GS/CS/SS, substituting (14)in the above expression and using the relation (2) givesEq. (3). For ES, on the other hand, this leads to therelation (4) [9].Now, to locate those DOF which are responsible forentropy, we take a closer look at the interaction matrix(13). As mentioned earlier, the last two terms are thenearest-neighbor interactions between the oscillators andconstitute entanglement. As expected, if these terms areset to zero (by hand), Sent vanishes. Instead, let us dothe following:(i) Set the interactions to zero (by hand) everywhere ex-cept in the ‘window’ (q−s ≤ i ≤ q+s), with s ≤ q ≤ n−s,i.e., restrict the thickness of the interaction region to2s+1 radial lattice points, while moving it rigidly across3from the origin to horizon. In Fig. (1), we have plot-ted the percentage contribution to the total entropy as afunction of q for the GS/GCS/SS (o = 0, solid thin curve)and the ES (o = 30(50), bold (light) thick curve). [Wechoose N = 300 and n = 100, 150, 200. The numericalerror in the evaluation of the entropy is less than 0.1%.]Fig. (1) shows that:• When the interaction region does not include thehorizon, the entanglement entropy is zero and it ismaximum if the horizon lies exactly in its middle.The first observation confirms that entanglementbetween DOF inside and outside the horizon con-tributes to entropy, while the second suggests thatDOF close to the horizon contribute more to theentropy compared to those far from it.• When the number of excited states is increased (i.e. o = 30, 50), the percentage contribution to thetotal entropy close to the horizon is less comparedto that of GS and the (bold/light thick) curves areflatter. These indicate that, for ES, there is a sig-nificant contribution from the DOF far-away fromthe horizon.90 92 94 96 98 100 102 104 106 108 110080100n = 100o=03050140 142 144 146 148 150 152 154 156 158 16007080100% entropy contributionn = 150o=03050190 192 194 196 198 200 202 204 206 208 21006080100qn = 200o=03050FIG. 1: Plot of the percentage contribution to the entropy forthe GS and ES as we move the window q−2 ≤ i < q+2 fromq = 2 to q = n, for N = 300 and n = 100, 150 and 200. Thesolid thin curve is for GS o = 0 and bold (light) thick curvefor o = 30(50).(ii) To further investigate, we now set the interactionsto zero (by hand) everywhere except in the window p ≤i ≤ n, with the horizon as its outer boundary, and varythe width of the window t ≡ n − p from 0 to n. Fort = n, the total entropy is recovered, while for t = 0,i.e. zero interaction region, entropy vanishes. In Fig.2, we have plotted the normalized GS/GCS/SS and ESentropies [Sent(t)] vs t for n = 100, 150 and 200. [Hereagain, the solid thin curve is for GS and bold (light) thickcurve for o = 30(50).] We infer the following:• For the GS/GCS/SS, the entropy reaches the GSentropy for as little as t = 3. In other words, theinteraction region encompassing DOF close to thehorizon contribute to most of the entropy for theGS/GCS/SS.• In the case of ES, the entropy reaches the ES en-tropy only for t = 15(20) [for o = 30(50)]. This in-dicates that: (a) The DOF far-away from the hori-zon have a greater contribution than that of GS and(b) As the number of excited states increases, con-tribution far-away from the horizon also increases.This leads us to one of the main conclusions ofthis article: the greater the deviation from the AL,the larger is the contribution of the DOF far-awayfrom the horizon. It can be shown that the densitymatrix for the ES is more spread out than for theGS.0 5 10 15 20 25 30 35 40 45 50406080100n = 100o=030500 5 10 15 20 25 30 35 40 45 50406080100% entropy contribution of Sent(t)n = 150o=030500 5 10 15 20 25 30 35 40 45 50406080100tn = 200o=03050FIG. 2: Plot of the percentage contribution of Sent(t) for theGS and ES. Here again, N = 300, n = 100, 150 and 200, andthe solid thin curve is for GS and bold (light) thick curve foro = 30(50).(iii) To understand this further, let us define∆pc(t) = pc(t)− pc(t− 1) where pc(t) =Sent(t)Sent× 100 ,where pc(t) is the percentage contribution to the total en-tropy by an interaction region of thickness t and ∆pc(t)is the percentage increase in entropy when the interac-tion region is incremented by one radial lattice point. Inother words, ∆pc(t) is the slope of the curves in Fig.(2). In Fig. (3), we have plotted ∆pc(t) vs (n − t) for4GS and ES. For the GS/GCS/SS, it is seen that whenthe first lattice point just inside the horizon is includedin the interaction region, the entropy increases from 0to 85% of the GS entropy. Inclusion of the next threepoints add another 9%, 3%, 1% respectively. The contri-butions to the entropy decrease rapidly and by the timethe (n/3)th point is included, entropy barely increases bya hundredth of a percent! For ES however, inclusion ofone lattice point adds 70(50)%, for o = 30(50), to theentropy, while the next few points add 9%, 4%, 3% · · ·respectively. The corresponding slopes are smaller.95 96 97 98 99 100020406080n = 100o=03050145 146 147 148 149 150020406080∆ pc(t)n = 150o=03050195 196 197 198 199 200020406080n − tn = 200o=03050FIG. 3: Plot of ∆pc(t) vs n − t for GS and ES. Here again,N = 300, n = 100, 150 and 200, and the solid thin curve isfor GS and bold (light) thick curve for o = 30(50).Figures (2) and (3) suggest the next key result of thisarticle - most of the entropy comes from the DOF closeto the horizon. However, the DOF deep inside must alsobe taken into account for the AL (3) to emerge for theGS/GCS/SS, and the law (4) to emerge for the ES. TheseDOF affect the horizon DOF via the nearest-neighborinteractions in (13).Our work clearly demonstrates the close relationshipbetween the AL and the horizon DOF, and that whenthe latter become less important, the entropy scales as apower of area less than unity. This can be understoodfrom the following heuristic picture: taking the point ofview that SBHis proportional to N , the number of DOFon the horizon, and that for the GS/CS/SS there is oneDOF per Planck-area, such that N ∝ AH , the AL follows(This is known as the it-from-bit picture [14]). For EShowever, since this number is seen to get diminished, itwill be given by another function N = f(A) < A. Now,since current results are expected to be valid when A≫ 1(in Planck units) (such that backreaction effects on thebackground can be neglected), it is quite plausible thatf(A) ∼ Aα, α < 1, just as we obtain. Note that theabove reasoning continues to hold when the outside of thehorizon is traced over. Another way of understanding ourresult is as follows: all interactions being of the nearest-neighbor type [cf. Eq.(13)], the degrees of freedom deepinside the horizon influence the ones near the horizon(and hence contribute to the entropy) only indirectly, i.e.via the intermediate degrees of freedom. Evidently, theirinfluence diminishes with their increasing distance fromthe horizon. When they are excited however, they havemore energy to spare, resulting in an increased overalleffect. Further investigations with superpositions of GSand ES are expected to shed more light on this [15].We would like to thank A. Dasgupta, A. Roy, S. Sen-gupta and S. Sur for discussions. This work was sup-ported in part by the Natural Sciences and EngineeringResearch Council of Canada and the Perimeter Institutefor Theoretical Physics.∗ Electronic address: saurya.das@uleth.ca† Electronic address: shanki@aei.mpg.de[1] 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) [arXiv:gr-qc/0505017].[2] See, for instance, Sec. (6.2) in R. M. Wald, Living Rev.Rel. 4, 6 (2001).[3] L. Bombelli et al, Phys. Rev. D34, 373 (1986).[4] M. Srednicki, Phys. Rev. Lett. 71, 666 (1993).[5] M.B. Plenio et al, Phys. Rev. Lett. 94, 060503 (2005);M. Cramer et al, Phys. Rev. A73, 012309 (2006).[6] R. Brustein, A. Yarom, Nucl. Phys. B709, 391 (2005);D. Marolf, A. Yarom, JHEP 0601 141 (2006).[7] T. Jacobsonet al, Int. J. Theor. Phys. 44, 1807 (2005).[8] M. Ahmadi, S. Das, S. Shankaranarayanan, Can. J. Phys.84(S2), 1 (2006) [arXiv:hep-th/0507228].[9] S. Das, S. Shankaranarayanan, Phys.Rev. D73, 121701(2006) [gr-qc/0511066].[10] Our analysis differs from that ofS. Mukohyama et al Phys. Rev. D 58, 064001 (1998).There, the authors divide the exterior region r ≥ r0 intotwo by introducing an hypothetical surface and obtainthe Sentof the hypothetical horizon. On the contrary, weconsider the complete r ≥ r0 region and obtain Sent forthe event-horizon of the black-hole.[11] S. Chandrasekhar, The mathematical theory of black-holes, Oxford University Press (1998).[12] L. D. Landau and E. M. Lifshitz, Classical Theory ofFields, Course of Theoretical Physics, Volume 2 (Perg-amon Press, New York, 1975).[13] T. Jacobson et al, Phys. Rev. D 61, 024017 (2000).[14] J. Wheeler, Sakharov Memorial Lecture on Physics, Vol-ume 2, Ed. L. Keldysh and V. Feinberg, Nova (1992).[15] S. Das, S. Shankaranarayanan, S. Sur, arXiv:gr-qc/0705.2070.[16] K. Melnikov, M. Weinstein, Int. J. Mod. Phys. D13(2004) 1595 (arXiv:hep-th/02052231).[17] The perturbations of a (3 + 1)-dimensional static spher-ically symmetric black holes result in two kinds – axial5and polar – of gravitational perturbations. The equationof motion of the axial perturbations is same as that of atest scalar field propagating in the black-hole background[11, 15]. Hence, by computing Sentof the test scalar fields,we obtain entropy of black-hole metric perturbations.[18] Ideally, one would like to fix the vacuum state at someLemaitre time τ = 0 and study the evolution of themodes at late time leading to Hawking effect. Such ananalysis is nontrivial for the (3+1)−D black-holes. Evenin the case of (1+1)−D black-holes, this issue has beenaddressed with limited success [13]. In the rest of thearticle, we will not consider these effects.

Where are the black hole entropy degrees of freedom ?

Where are the black hole entropy degrees of freedom ?

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