Plenty of Nothing: Black Hole Entropy in Induced Gravity

Transcription

1 J. Astrophys. Astr. (1999) 20, Plenty of Nothing: Black Hole Entropy in Induced Gravity V. P. Frolov, Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Canada T6G 2J1 ualberta. ca D. V. Fursaev, Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Canada T6G 2J1 and Joint Institute for Nuclear Research, Bogoliubov Laboratory of Theoretical Physics, Dubna, Russia dubna. su Abstract. We demonstrate how Sakharov s idea of induced gravity allows one to explain the statistical-mechanical origin of the entropy of a black hole. According to this idea, gravity becomes dynamical as the result of quantum effects in the system of heavy constituents of the underlying theory.the black hole entropy is related to the properties of the vacuum in the induced gravity in the presence of the horizon. We obtain the Bekenstein-Hawking entropy by direct counting the states of the constituents. Key words. Black hole gravity entropy. 1. Black hole entropy problem There are physical phenomena that allow a simple description but require tremendous efforts to explain them. Without doubts the origin of black hole entropy is such a problem. A black hole of mass Μ radiates as a heated body (Hawking 1975) with temperature T H= (8πGM) 1 and has entropy (Bekenstein 1972): where A is the surface area of the black hole (A=16πG 2 M 2 ), G is Newton's constant, and c= h=1. Statistical mechanics relates entropy to the measure of disorder and qualitatively it is the logarithm of the number of microscopically different states available for given values of the macroscopical parameters. Are there internal degrees of freedom that are responsible for the Bekenstein-Hawking entropy S BH? This is the question that physicists were trying to answer for almost 25 years. What makes the problem of black hole entropy so intriguing? Before discussing more technical aspects let us make simple estimations. Consider for example a supermassive black hole of 10 9 solar mass. According to equation (1), its entropy is about This is seven orders of magnitude larger than the entropy of the other matter in the visible part of the Universe. What makes things even more complicated, a black hole is simply an empty space-time with a strong gravitational field and nothing more. Really, a black hole is plenty of nothing, or if we put this in a more physical way, the phenomenon we are dealing with is a vacuum in the strong (1) 121

2 122 V. P. Frolov & D. V. Fursaev gravitational field. This conclusion leaves us practically no other choice but to try to relate the entropy of the black hole to properties of the physical vacuum in the strong gravitational field. The black hole entropy is of the same order of magnitude as the logarithm of the number of different ways to distribute two signs + and over the cells of Planckian size on the horizon surface. This estimation suggests that a reasonable microscopical explanation of the Bekenstein-Hawking entropy must be based on the quantum gravity, this Holy Grail of the theoretical physics. The superstring theory is the best what we have and what is often considered as the modern version of quantum gravity. Recent observation that the Bekenstein-Hawking entropy can be obtained by counting of string (D-brane) states is a very interesting and important result. Still there are questions. The string calculations essentially use supersymmetry and are mainly restricted to extreme and near-extreme black holes. Moreover each model requires new calculations. And the last but not the least it remains unclear why the entropy of a black hole is universal and does not depend on the details of the theory at Planckian scales. Note that the black hole thermodynamics follows from the lowenergy gravitational theory. That is why one can expect that only a few fundamental properties of quantum gravity but not its concrete details are really important for the statistical-mechanical explanation of the black hole entropy. 2. Sakharov s induced gravity In the string theory the low-energy gravity with finite Newton's constant arises as the collective phenomenon and is the result of quantum excitations of constituents (strings) of the underlying theory. There is a certain similarity between this mechanism and Sakharov's induced gravity (Sakharov 1968). The low-energy effective action W[g] in the induced gravity is defined as a quantum average of the constituent fields propagating in a given external gravitational background g The Sakharov's basic assumption is that the gravity becomes dynamical only as the result of quantum effects of the constituent fields. The gravitons in this picture are analogous to the phonon field describing collective excitations of a crystal lattice in the low-temperature limit of the theory. Search for the statistical-mechanical origin of the black hole entropy in Sakharov's approach (Jacobson 1994) might help to understand the universality of S BH. In this paper we describe the most important features of the mechanism of generation of the Bekenstein-Hawking entropy in the induced gravity. Each particular constituent field in equation (2) gives a divergent contribution to the effective action W[g]. In the one loop approximation the divergent terms are local and of the zero order, linear and quadratic in curvature. In the induced gravity the constituents obey additional constraints, so that the divergences cancel each other. It is also assumed that some of the fields have masses comparable to the Planck mass and the constraints are chosen so that the induced cosmological constant vanishes. As a result the effective action W[g] is finite and in the low-energy limit has the form of (2)

3 Black hole entropy 123 the Einstein-Hilbert action where Newton's constant G is determined by the masses of the heavy constituents The dots in equation (3) indicate possible higher curvature corrections to W[g µv,] which are suppressed by the power factors of m 2 i when the curvature is small. The vacuum Einstein equations δ W/ δ g µv = 0 are equivalent to the requirement that the vacuum expectation values of the total stress-energy of the constituents vanishes The value of the Einstein-Hilbert action (equation 3) calculated on the Gibbons- Hawking instanton determines the classical free energy of the black hole, and hence gives the Bekenstein-Hawking entropy S BH. Sakharov's equality (equation 2) allows one to rewrite the free energy as the Euclidean functional integral over constituent fields on the Gibbons-Hawking instanton with periodic (for bosons) and anti-periodic (for fermions) in the Euclidean time boundary conditions. In this picture constituents are thermally excited and the Bekenstein-Hawking entropy can be expressed in terms of the statistical-mechanical entropy Here the operator Η is the total canonical Hamiltonian of all the constituents. How can an empty space (vacuum) possess thermodynamical properties? The entropy S SM arises as the result of the loss of information about states inside the black hole horizon (Bombelli et al. 1986; Frolov & Novikov 1993). In the induced gravity (entanglement) entropy (equation 5) is calculated for the heavy constituents. This solves the problems of earlier attempts to explain the Bekenstein-Hawking entropy as the entanglement entropy of physical ( light ) fields. (3) (4) (5) 3. Models 3.1 Models with non-minimal coupling of scalar constituents Consider the model of induced gravity (Frolov, Fursayev & Zelnikov 1997; Frolov & Fursaev 1997) that consists of N s scalar fields φ i with masses m s,i and N d of Dirac fermions ψj with masses m d, j Scalar fields can have non-minimal couplings and are described by actions (6) The action for Dirac fermions has the following standard form (7)

4 124 V. P. Frolov & D. V. Fursaev The corresponding quantum effective action of the model is (8) W[g] is a functional of the metric g µv of the background spacetime. The scalar and spinor actions follow immediately from equations (6) and (7), (9) (10) In general, the effective action (9) is ultraviolet divergent quantity. Certain constraints are to be imposed on the parameters of the constituents to eliminate the leading divergencies in W[g]. These conditions can be written down with the help of the following two functions constructed from the parameters of the constituents. Direct calculations show that the induced cosmological constant vanishes and the induced gravitational coupling constant G is finite if the following constraints are satisfied (11) (12) (13) The presence of the non-minimally coupled constituents is important. In this case it is possible to satisfy the constraints on the parameters ms,i, md and ξi which guarantee the cancellation of the leading ultraviolet divergencies of the induced gravitational action W[g], equation (2). The induced Newton's constant in this model is Conditions (12) can be easily satisfied if for example bosons and fermions enter in supersymmetric multiplets. In this case N s = 4N d and there exist N d multiplets. The masses of bosons and a fermion in each supermultiplet are equal one to another. Under these conditions p(z) = 0. If in addition we assume that the parameter of the non-minimal coupling ξ has the same value for scalar fields inside the multiplet, the equations (13) and (14) take the form Here x j = 3/2 _ 6 ξ j. This is a set of linear equations for ξ j which evidently has solution for N d 3. The relation between the Bekenstein-Hawking entropy S BH and the statisticalmechanical entropy S SM of the heavy constituents can be found explicitly and has the (14) (15)

5 form (Frolov, Fursaev & Zelnikov 1987): Black hole entropy 125 The important property of S SM is that it diverges because both fermions and bosons give positive and infinite contributions to this quantity. An additional term Q in (16) is proportional to the fluctuations of the non-minimally coupled scalar fields φ s on the horizon Σ and is the average value of the following operator (16) (17) The remarkable property of the model is that for the same values of the parameters of the constituents, that guarantee the finiteness of G, the divergences of S SM are exactly cancelled by the divergences of Q. So in the induced gravity the right-hand-side of (16) is finite and reproduces exactly the Bekenstein-Hawking expression. 3.2 Models with vector fields An important role in the above presented model is played by non-minimal coupling of the scalar constituents. Before discussing the statistical-mechanical meaning of the subtraction formula (16) we present another model (Frolov & Fursaev 1998) in which there is no such coupling. The model consists of N s minimally coupled scalar fields φ i with masses m s,i, N d spinors ψ j with masses m d,j, and N v vector fields Vk with masses m v,k. The classical actions for scalar and spinor felds are given by equations (6) and (7) with the only important difference that the non-minimal couplings of scalar fields vanish, ξ i = 0 The action of the vector field V kµ is where F kµv = µ V kv v V kµ. The corresponding quantum effective action of the model is As a result of equation of motion, a massive vector field V µ obeys the condition µ V µ = 0, which leaves only three independent components. Under quantization this condition can be realized as a constraint so that the effective action for vector fields takes the form (20) (21) where R µ v is the Ricci tensor. The functional Γ υ (m v,k )represents the effective action for a massive vector field which we will denote as A k,µ. The classical action for Α k,µ ~ (18) (19)

6 126 V. P. Frolov & D. V. Fursaev which results in (21) is The field A µ k obeys no constraints and carries an extra degree of freedom. The contribution of this unphysical degree of freedom in (21) is compensated by subtracting the action Γ s (m v,k ) of a scalar field with the mass m v,k,, see equation (20). As in the case of the model with non-minimally coupled scalar field, we require vanishing the cosmological constant and cancellation of the divergencies of the induced Newton constant. These conditions can be written down with the help of the following two functions (22) (23) It can be shown that the induced cosmological constant vanishes when The induced Newton constant G is finite if The constraints result in simple relations In particular, they show that one cannot construct the theory with finite cosmological and Newton constants from vector and spinor fields only. The Newton coupling constant is determined by the following expression Here, according to equation (26), we put Ν =Ns =Nd = Nv. From this expression it is easy to conclude that at least some of the constituents must be heavy and have mass comparable with the Planck mass m pl. For simplicity in what follows we assume that all the constituents are heavy. Let us analyze models where conditions (25) and (26) are satisfied. Equation (27) are trivially satisfied when all fields are in supersymmetric multiplets. However in such supersymmetric models p(z) = q(z) 0 (because masses of the fields in the same supermultiplet coincide) and the induced gravitational constant vanishes. A nontrivial induced gravity theory can be obtained if the supersymmetry is partly broken by splitting the masses of the fields in the supermultiplets. Let us demonstrate this by an example. Consider the model with Ν massive supermultiplets. Each multiplet consists of one scalar, one Dirac spinor and one vector (24) (25) (26) (27) (28)

7 Black hole entropy 127 field, so that the numbers of Bose and Fermi degrees of freedom coincide 1. We suggest that masses of vector and spinor fields are equal, m v,i = m d,i = m i, (here i is the number of the multiplet). The masses of the scalar partners are assumed to be m s,i (1+x i )mi, where x i ; is a dimensionless coefficient. The case when xi < <1 corresponds to slightly broken supersymmetry. For this case (29) Now equations (25), (26), and (27) take the simple form (30) (31) This is a system of linear equations which for Ν > 4 has nontrivial solutions. The calculations give (32) Here c = 1n (729/256). The Noether charge in this model arises due to the interaction of the vector fields Α kµ with curvature (see (22) and it is of the form (Frolov & Fursaev 1998): Here Ρ µυ is a projector onto a two-dimensional surface orthogonal to the bifurcation ^ ^ surface of horizons Σ, and d σ is the surface element of Σ. The average (Α iµ Α iν ) is understood as a regularized quantity. The quantity Q has a meaning of Wald's Noether charge associated with nonminimal interaction terms of the vector field. By using the Pauli-Villars regularization one finds that in the Rindler approximation (33) (34) This result allows one to show that the Bekenstein-Hawking entropy S BH in induced gravity is the difference of statistical-mechanical entropy S SM, see equation (32), and the Noether charge Q. As can be easily seen, the divergences of S SM are exactly canceled by the divergences of the charge Q, so that one gets the finite 1 Supersymmetric models with free massive scalar, spinor and vector fields are discussed,for instance, by Lopuszanski & Wolf (1981).

8 128 V. P. Frolov & D. V. Fursaev expression This result coincides with (16) for the model with non-minimally coupled scalar fields. 4. Statistical-mechanical interpretation of black hole entropy in induced gravity What is the origin of the compensation mechanism in equations (16) and (35) and what is the statistical-mechanical meaning of the subtraction in this relation? The answer to ^ both questions is in the properties of the operator Q of the Noether charge (Wald 1993). In statistical-mechanical computations we consider fields localized in the black ^ hole exterior. The charge Q determines the difference between the energy Ε in the external region B, defined by means of the stress-energy tensor and the canonical ^ energy Η in the same region Here ζ µ is the timelike Killing vector field. This formula is valid for both models (for its proof, see Frolov & Fursaev (1997; 1998). ^ ^ Two energies, Ε and H, play essentially different roles. The canonical energy is the ^ value of the Hamiltonian Η which is the generator of translations of the system along ζ µ and which enters definition (5) of the statistical-mechanical entropy S SM. The energy Ε is the contribution of the constituents to the black hole mass. The number v(e) E of microscopically different physical states of the constituents in the energy interval E near Ε= 0 determines the degeneracy of the black hole mass spectrum. Since the Killing vector ζ µ vanishes at the bifurcation surface Σ of the Killing horizons, the Hamiltonian Η is degenerate. One can add to the system an arbitrary number of soft modes, i.e. modes with zero frequencies, without changing the canonical energy. On the other hand, only soft modes contribute to the average of the Noether charge Q. According to equation (36), this removes the degeneracy of the energy E. As a result the infinite number of thermal states of the constituents reduces to the finite number of physical states of the black hole and S SM reduces to the Bekenstein- Hawking value. The corresponding number density v(e=0) of physical states is exp S BH (Frolov & Fursaev 1997). There is a similarity between this mechanism and gauge theories, soft modes playing the role of the pure gauge degrees of freedom. 5. Discussion Let us note that the concrete models of the induced gravity may differ from the ones considered here, and may contain, for example, finite or infinite number of fields of higher spins. However our consideration indicates that it is quite plausible that the same mechanism of black-hole entropy generation still works Our discussion can be summarized as follows. The entropy S BH of a black hole in induced gravity is the logarithm of the number of different states of the constituents obeying the condition Τ ^ µν = 0. The statistical-mechanical explanation of the entropy (35) (36)

9 Black hole entropy 129 becomes possible only when one appeals to a more deep underlying theory in which vacuum is equipped with an additional fine structure. The vacuum in the induced gravity is a ground state of heavy constituents. Light particles are just low energy collective excitations over this vacuum. The black hole entropy S BH acquires its statistical-mechanical meaning because of this additional fine structure of the vacuum of the underlying theory. Two important elements that make the statistical-mechanical picture self-consistent and universal are a representation of gravity as an induced phenomenon and the ultraviolet finiteness of the induced gravitational couplings. These are main lessons the induced gravity teaches us. It should be emphasized, that the induced gravity approach does not pretend to explain the black hole entropy from the first principles of the fundamental theory of quantum gravity (such as the string theory) but it allows one to demonstrate the universality of the entropy and its independence of the concrete details of such theory. It gives us a hint that only a few quite general properties of the fundamental theory are really required for the statistical-mechanical explanation of the black hole entropy. This work was partially supported by the Natural Sciences and Engineering Research Council of Canada. One of the authors (V.F.) is grateful to the Killam Trust for its financial support. References Bekenstein, J. D. 1972, Nuov. Cim. Lett, 4, 737 (1972); 1973, Phys. Rev. D, 7, Bombelli, L., Koul, R., Lee, J., Sorkin, R. 1986, Phys. Rev. D, 34, 373. Frolov, V. P., Fursaev, D. V. 1997, Phys. Rev. D, 56, Frolov, V. P., Fursaev, D. V. 1998, Phys. Rev. D. 58, Frolov, V. P., Fursaev, D. V., Zelnikov, A. I. 1997, Nucl. Phys. B, 486, 339. Frolov, V., Novikov, I. 1993, Phys. Rev. D, 48, Hawking, S. W., 1975, Comm. Math. Phys., 43, 199. Jacobson, Τ. 1994, Black Hole Entropy and Induced Gravity, preprintgr-qc/ Lopuszanski, J. Τ., Wolf, Μ. 1981, Nucl. Phys. B, 184, 133 (1981). Sakharov, A. D. 1968, Sov. Phys. Doklady, 12, 1040; 1976, Theor. Math. Phys., 23, 435. Wald, R. M. 1993, Phys. Rev. D, 48, R3427.