Showing papers in "Classical and Quantum Gravity in 2001"

E(11) and M theory

Abstract: We argue that 11-dimensional supergravity can be described by a nonlinear realization based on the group E11. This requires a formulation of 11-dimensional supergravity in which the gravitational degrees of freedom are described by two fields which are related by duality. We show the existence of such a description of gravity.

Liouville theory revisited

Abstract: We try to develop a coherent picture of Liouville theory as a two-dimensional conformal field theory that takes into account the perspectives of the path-integral approach, bootstrap, canonical quantization and the operator approach. To do this, we need to develop further some of these approaches. This includes in particular the construction of general exponential field operators from a set of covariant chiral operators. The latter are shown to satisfy braid relations that allow one to prove the locality of the former.

New formulations of D = 10 supersymmetry and D8-O8 domain walls

Abstract: We discuss a generalized form of IIA/IIB supergravity depending on all R-R potentials C-(p) (p = 0, 1,..., 9) as the effective field theory of type IIA/IIB superstring theory. For the IIA case we explicitly break this R-R democracy to either p less than or equal to 3 or p greater than or equal to 5, which allows us to write a new bulk action that can be coupled to N = 1 supersymmetric brane actions. The case of eight-braves is studied in detail using the new bulk & brane action. The supersymmetric negative-tension branes without matter excitations can be viewed as orientifolds in the effective action. These D8-branes and O8-planes are fundamental in type I' string theory. A BPS eight-brave solution is given which satisfies the jump conditions on the wall. It implies a quantization of the mass parameter in string units. Also, we find a maximal distance between the two walls, depending on the string coupling and the mass parameter. We derive the same results via supersymmetric flow equations.

Analogue gravity from Bose-Einstein condensates

Abstract: We analyse prospects for the use of Bose–Einstein condensates as condensedmatter systems suitable for generating a generic ‘effective metric’, and for mimicking kinematic aspects of general relativity. We extend the analysis due to Garay et al (2000 Phys. Rev. Lett. 85 4643, 2001 Phys. Rev. A 63 023611). Taking a long-term view, we ask what the ultimate limits of such a system might be. To this end, we consider a very general version of the nonlinear Schr¨ odinger equation (with a 3-tensor position-dependent mass and arbitrary nonlinearity). Such equations can be used, for example, in discussing Bose–Einstein condensates in heterogeneous and highly nonlinear systems. We demonstrate that at low momenta linearized excitations of the phase of the condensate wavefunction obey a ( 3+1 )-dimensional d’Alembertian equation coupling to a ( 3+1 )-dimensional Lorentzian-signature ‘effective metric’ that is generic, and depends algebraically on the background field. Thus at low momenta this system serves as an analogue for the curved spacetime of general relativity. In contrast, at high momenta we demonstrate how one can use the eikonal approximation to extract a well controlled Bogoliubovlike dispersion relation, and (perhaps unexpectedly) recover non-relativistic Newtonian physics at high momenta. Bose–Einstein condensates appear to be an extremely promising analogue system for probing kinematic aspects of general relativity.

Gauge field theory coherent states (GCS): I. General properties

Abstract: In this paper we outline a rather general construction of diffeomorphism covariant coherent states for quantum gauge theories. By this we mean states ψ(A,E), labelled by a point (A,E) in the classical phase space, consisting of canonically conjugate pairs of connections A and electric fields E, respectively, such that: (a) they are eigenstates of a corresponding annihilation operator which is a generalization of A-iE smeared in a suitable way; (b) normal ordered polynomials of generalized annihilation and creation operators have the correct expectation value; (c) they saturate the Heisenberg uncertainty bound for the fluctuations of Â,E; and (d) they do not use any background structure for their definition, that is, they are diffeomorphism covariant. This is the first paper in a series of articles entitled `Gauge field theory coherent states (GCS)' which aims to connect non-perturbative quantum general relativity with the low-energy physics of the standard model. In particular, coherent states enable us for the first time to take into account quantum metrics which are excited everywhere in an asymptotically flat spacetime manifold as is needed for semiclassical considerations. The formalism introduced in this paper is immediately applicable also to lattice gauge theory in the presence of a (Minkowski) background structure on a possibly infinite lattice.

Spacetime as a Feynman diagram: the connection formulation

Abstract: Spin-foam models are the path-integral counterparts to loop-quantized canonical theories. Over the last few years several spin-foam models of gravity have been proposed, most of which live on finite simplicial lattice spacetime. The lattice truncates the presumably infinite set of gravitational degrees of freedom down to a finite set. Models that can accommodate an infinite set of degrees of freedom and that are independent of any background simplicial structure, or indeed any a priori spacetime topology, can be obtained from the lattice models by summing them over all lattice spacetimes. Here we show that this sum can be realized as the sum over Feynman diagrams of a quantum field theory living on a suitable group manifold, with each Feynman diagram defining a particular lattice spacetime. We give an explicit formula for the action of the field theory corresponding to any given spin-foam model in a wide class which includes several gravity models. Such a field theory was recently found for a particular gravity model. Our work generalizes this result as well as Boulatov's and Ooguri's models of three- and four-dimensional topological field theories, and ultimately the old matrix models of two-dimensional systems with dynamical topology. A first version of our result has appeared in a companion paper: here we present a new and more detailed derivation based on the connection formulation of the spin-foam models.

Vanishing theorems and string backgrounds

Abstract: We show various vanishing theorems for the cohomology groups of compact Hermitian manifolds for which the Bismut connection has a (restricted) holonomy contained in SU(n) and classify all such manifolds of dimension four. In this way we provide necessary conditions for the existence of such structures on compact Hermitian manifolds with vanishing first Chern class of non-Kahler type. Then we apply our results to solutions of the string equations and show that such solutions admit various cohomological restrictions such as, for example, that under certain natural assumptions the plurigenera vanish. We also find that under some assumptions the string equations are equivalent to the condition that a certain vector is parallel with respect to the Bismut connection.

Gauge field theory coherent states (GCS): II. Peakedness properties

Abstract: In this article we apply the methods outlined in the previous paper of this series to the particular set of states obtained by choosing the complexifier to be a Laplace operator for each edge of a graph. The corresponding coherent state transform was introduced by Hall for one edge and generalized by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann to arbitrary, finite, piecewise analytic graphs. However, both of these works were incomplete with respect to the following two issues : (a) The focus was on the unitarity of the transform and left the properties of the corresponding coherent states themselves untouched. (b) While these states depend in some sense on complexified connections, it remained unclear what the complexification was in terms of the coordinates of the underlying real phase space. In this paper we complement these results : First, we explicitly derive the com- plexification of the configuration space underlying these heat kernel coherent states and, secondly, prove that this family of states satisfies all the usual properties : i) Peakedness in the configuration, momentum and phase space (or Bargmann-Segal) representation. ii) Saturation of the unquenched Heisenberg uncertainty bound. iii) (Over)completeness. These states therefore comprise a candidate family for the semi-classical anal- ysis of canonical quantum gravity and quantum gauge theory coupled to quantum gravity. They also enable error-controlled approximations to difficult analytical cal- culations and therefore set a new starting point for numerical canonical quantum general relativity and gauge theory. The text is supplemented by an appendix which contains extensive graphics in order to give a feeling for the so far unknown peakedness properties of the states constructed.

Supersymmetric higher-derivative actions in 10 and 11 dimensions, the associated superalgebras and their formulation in superspace

Abstract: Higher-derivative terms in the string and M-theory effective actions are strongly constrained by supersymmetry. Using a mixture of techniques, involving both string-amplitude calculations and an analysis of supersymmetry requirements, we determine the supersymmetric completion of the R4 action in 11 dimensions to second order in the fermions, in a form compact enough for explicit further calculations. Using these results, we obtain the modifications to the field transformation rules and determine the resulting field-dependent modifications to the coefficients in the supersymmetry algebra. We then make the link to the superspace formulation of the theory and discuss the mechanism by which higher-derivative interactions lead to modifications to the supertorsion constraints. For the particular interactions under discussion we find that no such modifications are induced.

Consequences on variable Λ-models from distant type Ia supernovae and compact radio sources

Abstract: We study the magnitude-redshift relation for the type?Ia supernovae data and the angular size-redshift relation for the updated compact radio sources data (from Gurvits et?al) by considering four variable ?-models: ?~S-2, ?~H2, ?~? and ?~t-2. It is found that all the variable ?-models, as well as the constant ?-Friedmann model, fit the supernovae data equally well with ?2/dof?1 and require non-zero, positive values of ? and an accelerating expansion of the universe. The estimates of the density parameter for the variable ?-models are found to be higher than those for the constant ?-Friedmann model. From the compact radio sources data, it is found, by assuming the no-evolution hypothesis, that the Gurvits et al model (Friedmann model with ? = 0) is not the best-fitting model for the constant ? case. The best-fitting Friedmann model (with constant ?) is found to be a low-density, vacuum-dominated accelerating universe. The fits of this data set to the (variable, as well as, constant ?-) models are found to be very good with ?2/dof?0.5 and require non-zero, positive values of ? with either sign of the deceleration parameter. However, for realistic values of the matter density parameter, the only interesting solutions are (a) estimated from the supernovae data: the best-fitting solutions for the flat models (including the constant ? case); (b) estimated from the radio sources data: the global best-fitting solutions for the models ?~H2 and ?~?, the best-fitting solution for the flat model with ? = {}constant and the Gurvits et al model. It is noted that, as in the case of recent cosmic microwave background analyses, the data sets seem to favour a spherical universe (k>0).

Numerical relativity: A Review

Abstract: Computer simulations are enabling researchers to investigate systems which are extremely difficult to handle analytically. In the particular case of general relativity, numerical models have proved extremely valuable for investigations of strong-field scenarios and been crucial in revealing unexpected phenomena. Considerable efforts are being spent to simulate astrophysically relevant simulations, understand different aspects of the theory and even provide insights into the search for a quantum theory of gravity. In this paper I review the present status of the field of numerical relativity, describe the techniques most commonly used and discuss open problems and (some) future prospects.

Complex geometry of conifolds and a 5-brane wrapped on a 2-sphere

Abstract: We investigate solutions of type II supergravity which have the product R 1,3 ×M 6 structure with non-compact M 6 factor and which preserve at least four supersymmetries. In particular, we consider various conifolds and the N = 1 supersymmetric ‘NS5-brane wrapped on a 2-sphere’ solution recently discussed in hep-th/0008001. In all of these cases, we explicitly construct the complex structures, and the K¨ ahler and parallel (3, 0)-forms of the corresponding M 6 . In addition, we verify that the above solutions preserve, respectively, eight and four supersymmetries of the underlying type II theory. We also demonstrate that the ordinary and fractional D3-brane (5-brane wrapped on a 2-cycle) solutions on singular, resolved and deformed conifolds, and the (S-dual of) NS5-brane wrapped on 2-sphere can be obtained as special cases from a universal ansatz for the supergravity fields, i.e. from a single one-dimensional action governing their radial evolution. We show that like the 3-branes on conifolds, the NS5brane on a 2-sphere background can be found as a solution of a first-order system following from a superpotential.

The state space and physical interpretation of self-similar spherically symmetric perfect-fluid models

Abstract: The purpose of this paper is to further investigate the solution space of self-similar spherically symmetric perfect-fluid models and gain a deeper understanding of the physical aspects of these solutions We achieve this by combining the state space description of the homothetic approach with the use of the physically interesting quantities arising in the comoving approach We focus on three types of models First, we consider models that are natural inhomogeneous generalizations of the Friedmann universe; such models are asymptotically Friedmann in their past and evolve fluctuations in the energy density at later times Secondly, we consider so-called quasi-static models This class includes models that undergo self-similar gravitational collapse and is important for studying the formation of naked singularities If naked singularities do form, they have profound implications for the predictability of general relativity as a theory Thirdly, we consider a new class of asymptotically Minkowski self-similar spacetimes, emphasizing that some of them are associated with the self-similar solutions associated with the critical behaviour observed in recent gravitational collapse calculations

Gauge field theory coherent states (GCS): III. Ehrenfest theorems

Abstract: In the preceding paper of this series of articles we established peakedness properties of a family of coherent states that were introduced by Hall for any compact gauge group and were later generalized to gauge field theory by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann. In this paper we establish the `Ehrenfest property' of these states which are labelled by a point (A,E), a connection and an electric field, in the classical phase space. By this we mean that the expectation values of the elementary operators (and of their commutators divided by i, respectively) in a coherent state labelled by the (A,E) are, to zeroth order in , given by the values of the corresponding elementary functions (and of their Poisson brackets, respectively) at the point (A,E). These results can be extended to all polynomials of elementary operators and to a certain non-polynomial function of the elementary operators associated with the volume operator of quantum general relativity. These findings are another step towards establishing that the infinitesimal quantum dynamics of quantum general relativity might, to lowest order in , indeed be given by classical general relativity.

Bayesian methods for cosmological parameter estimation from cosmic microwave background measurements

Abstract: We present a strategy for a statistically rigorous Bayesian approach to the problem of determining cosmological parameters from the results of observations of anisotropies in the cosmic microwave background. Our strategy relies on Markov chain Monte Carlo methods, specifically the Metropolis-Hastings algorithm, to perform the necessary high-dimensional integrals. We describe the Metropolis-Hastings algorithm in detail and discuss the results of our test on simulated data.

Logarithmic correction to the Bekenstein-Hawking entropy of the BTZ black hole

Abstract: We derive an exact expression for the partition function of the Euclidean BTZ black hole. Using this, we show that for a black hole with large horizon area, the correction to the Bekenstein-Hawking entropy is -3/2log area), in agreement with that for the Schwarzschild black hole obtained in the four-dimensional canonical gravity formalism and also in a Lorentzian computation of BTZ black hole entropy. We find that the correct expression for the logarithmic correction in the context of the BTZ black hole comes from the modular invariance associated with the toroidal boundary of the black hole.

Gauge field theory coherent states (GCS): IV. Infinite tensor product and thermodynamical limit

Abstract: In the canonical approach to Lorentzian quantum general relativity in four spacetime dimensions an important step forward has been made by Ashtekar, Isham and Lewandowski some eight years ago through the introduction of a Hilbert space structure, which was later proved to be a faithful representation of the canonical commutation and adjointness relations of the quantum field algebra of diffeomorphism invariant gauge field theories by Ashtekar, Lewandowski, Marolf, Mourao and Thiemann. This Hilbert space, together with its generalization due to Baez and Sawin, is appropriate for semi-classical quantum general relativity if the spacetime is spatially compact. In the spatially non-compact case, however, an extension of the Hilbert space is needed in order to approximate metrics that are macroscopically nowhere degenerate. For this purpose, in this paper we apply the theory of the infinite tensor product (ITP) of Hilbert Spaces, developed by von Neumann more than sixty years ago, to quantum general relativity. The cardinality of the number of tensor product factors can take the value of any possible Cantor aleph, making this mathematical theory well suited to our problem in which a Hilbert space is attached to each edge of an arbitrarily complicated, generally infinite graph. The new framework opens access to a new arsenal of techniques, appropriate to describe fascinating physics such as quantum topology change, semi-classical quantum gravity, effective low-energy physics etc from the universal point of view of the ITP. In particular, the study of photons and gravitons propagating on fluctuating quantum spacetimes should now be in reach.

Analogue gravity from field theory normal modes

Abstract: We demonstrate that the emergence of a curved spacetime `effective Lorentzian geometry' is a common and generic result of linearizing a classical scalar field theory around some non-trivial background configuration. This investigation is motivated by considering the large number of `analogue models' of general relativity that have recently been developed based on condensed matter physics, and asking whether there is something more fundamental going on. Indeed, linearization of a classical field theory (that is, a field-theoretic `normal-mode analysis') results in fluctuations whose propagation is governed by a Lorentzian-signature curved spacetime `effective metric'. In the simple situation considered in this paper (a single classical scalar field), this procedure results in a unique effective metric, which is quite sufficient for simulating kinematic aspects of general relativity (up to and including Hawking radiation). Upon quantizing the linearized fluctuations around this background geometry, the one-loop effective action is guaranteed to contain a term proportional to the Einstein-Hilbert action of general relativity, suggesting that while classical physics is responsible for generating an `effective geometry', quantum physics can be argued to induce an `effective dynamics'. The situation is strongly reminiscent of, though not identical to, Sakharov's `induced-gravity' scenario, and suggests that Einstein gravity is an emergent low-energy long-distance phenomenon that is insensitive to the details of the high-energy short-distance physics. (We mean this in the same sense that hydrodynamics is a long-distance emergent phenomenon, many of whose predictions are insensitive to the short-distance cut-off implicit in molecular dynamics.)

Loop Quantum Cosmology IV: Discrete Time Evolution

Abstract: Using general features of recent quantizations of the Hamiltonian constraint in loop quantum gravity and loop quantum cosmology, a dynamical interpretation of the constraint equation as evolution equation is presented. This involves a transformation from the connection to a dreibein representation and the selection of an internal time variable. Due to the discrete nature of geometrical quantities in loop quantum gravity, time also turns out to be discrete leading to a difference rather than differential evolution equation. Furthermore, evolving observables are discussed within this framework, which enables an investigation of physical spectra of geometrical quantities. In particular, the physical volume spectrum is proven to equal the discrete kinematical volume spectrum in loop quantum cosmology.

Third post-Newtonian dynamics of compact binaries: Noetherian conserved quantities and equivalence between the harmonic-coordinate and ADM-Hamiltonian formalisms

Abstract: A Lagrangian from which one can derive the third post-Newtonian (3PN) equations of motion of compact binaries (neglecting the radiation reaction damping) is obtained. The 3PN equations of motion were computed previously by Blanchet and Faye in harmonic coordinates. The Lagrangian depends on the harmonic-coordinate positions, velocities and accelerations of the two bodies. At the 3PN order, the appearance of one undetermined physical parameter ? reflects the incompleteness of the point-mass regularization used when deriving the equations of motion. In addition the Lagrangian involves two unphysical (gauge-dependent) constants r'1 and r'2 parametrizing some logarithmic terms. The expressions of the ten Noetherian conserved quantities, associated with the invariance of the Lagrangian under the Poincar? group, are computed. By performing an infinitesimal `contact' transformation of the motion, we prove that the 3PN harmonic-coordinate Lagrangian is physically equivalent to the 3PN Arnowitt-Deser-Misner Hamiltonian obtained recently by Damour, Jaranowski and Sch?fer.

Exact solutions in multidimensional gravity with antisymmetric forms

Abstract: This short review deals with a multidimensional gravitational model containing dilatonic scalar fields and antisymmetric forms. The manifold is chosen in the form M = M 0 × M 1 × ... × M n , where M i are Einstein spaces (i ≥ 1). The sigma-model approach and exact solutions in the model are reviewed and the solutions with p-branes (e.g. solutions with harmonic functions, “cosmological”, spherically symmetric and black-brane ones) are considered.

Survey of two-time physics

Abstract: Two-time physics (2T) is a general reformulation of one-time physics (1T) that displays previously unnoticed hidden symmetries in 1T dynamical systems and establishes previously unknown duality-type relations among them. This may play a role in displaying the symmetries and constructing the dynamics of little understood systems, such as M-theory. 2T-physics describes various 1T dynamical systems as different d-dimensional `holographic' views of the same 2T system in d + 2 dimensions. The `holography' is due to gauge symmetries that tend to reduce the number of effective dimensions. Different 1T evolutions (i.e. different Hamiltonians) emerge from the same 2T-theory when gauge fixing is done with different embeddings of d dimensions inside d + 2 dimensions. Thus, in the 2T setting, the distinguished 1T which we call `time' is a gauge-dependent concept. The 2T-action also has a global SO(d,2) symmetry in flat spacetime, or a more general d + 2 symmetry in curved spacetime, under which all dimensions are on an equal footing. This symmetry is observable in many 1T-systems, but it remained unknown until discovered in the 2T formalism. The symmetry takes various nonlinear (hidden) forms in the 1T-systems, and it is realized in the same irreducible unitary representation (the same Casimir eigenvalues) in their quantum Hilbert spaces. 2T-physics has mainly been developed in the context of particles, including spin and supersymmetry, but some advances have also been made with strings and p-branes, and insights for M-theory have already emerged. In the case of particles, there exists a general worldline formulation with background fields, as well as a field theory formulation, both described in terms of fields that depend on d + 2 coordinates. All 1T particle interactions with Yang-Mills, gravitational and other fields are included in the d + 2 reformulation. In particular, the standard model of particle physics can be regarded as a gauge-fixed form of a 2T-theory in 4 + 2 dimensions. These facts already provide evidence for a new type of higher-dimensional unification.

New tests of Einstein's equivalence principle and Newton's inverse-square law

Abstract: This paper describes recent experimental work by the University of Washington Eot-Wash group on two different topics: a test of the strong equivalence principle and a search for sub-millimetre scale deviations of the Newtonian 1/r2 law. Our strong equivalence principle test was motivated by the resurgence of interest in `gravitational' scalar fields, which typically lead to violation of the equivalence principle for gravitational self-energy. Our sub-millimetre experiment was motivated by predictions of fundamentally new effects from `large' extra dimensions and from the dilaton and moduli scalar particles of string theory.

The first law of black brane mechanics

Abstract: We obtain ADM and Komar surface integrals for the energy density, tension and angular momentum density of stationary p-brane solutions of Einstein's equations. We use them to derive a Smarr-type formula for the energy density and thence a first law of black brane mechanics. The intensive variable conjugate to the worldspace p-volume is an 'effective' tension that equals the ADM tension for uncharged branes, but vanishes for isotropic boost-invariant charged branes.

A quintessence scalar field in Brans-Dicke theory

Abstract: It is shown that a minimally coupled scalar field in Brans-Dicke theory yields a non-decelerated expansion for the present universe for open, flat and closed Friedmann-Robertson-Walker models.

On the topology and area of higher-dimensional black holes

Abstract: Over the past decade there has been an increasing interest in the study of black holes, and related objects, in higher (and lower) dimensions, motivated to a large extent by developments in string theory. The aim of the present paper is to obtain higher-dimensional analogues of some well known results for black holes in 3 + 1 dimensions. More precisely, we obtain extensions to higher dimensions of Hawking’s black hole topology theorem for asymptotically flat (� = 0) black hole spacetimes, and Gibbons’ and Woolgar’s genus-dependent, lower entropy bound for topological black holes in asymptotically locally antide Sitter ( �< 0) spacetimes. In higher dimensions the genus is replaced by the so-called σ -constant, or Yamabe invariant, which is a fundamental topological invariant of smooth compact manifolds.

Gravity in warped compactifications and the holographic stress tensor

Abstract: We study gravitational aspects of brane-world scenarios. We show that the bulk Einstein equations together with the junction condition imply that the induced metric on the brane satisfies the full nonlinear Einstein equations with a specific effective stress-energy tensor. This result holds for any value of the bulk cosmological constant. The analysis is done by either placing the brane close to infinity or by considering the local geometry near the brane. In the case that the bulk spacetime is asymptotically AdS, we show that the effective stress-energy tensor is equal to the sum of the stress-energy tensor of matter localized on the brane and of the holographic stress-energy tensor appearing in the AdS/CFT duality. In addition, there are specific higher-curvature corrections to Einstein's equations. We analyse in detail the case of asymptotically flat spacetime. We obtain asymptotic solutions of Einstein's equations and show that the effective Newton constant on the brane depends on the position of the brane.

TOPICAL REVIEW: Brane worlds

Abstract: This is an introductory review of gravity on branes with an emphasis on codimension 1 models. However, for a new result it is also pointed out that the cosmological evolution of the 3-brane in the model of Dvali, Gabadadze and Porrati may follow the standard Friedmann equation. Contents: 1. Introduction 2. Conventions 3. The Lanczos-Israel matching conditions 4. The action principle with codimension 1 hypersurfaces: Need for the Gibbons-Hawking term 5. The Newtonian limit on thin branes 6. A remark on black holes in the model of Dvali, Gabadadze and Porrati 7. The cosmology of codimension 1 brane worlds

Manufacture of Gowdy spacetimes with spikes

Abstract: In numerical studies of Gowdy spacetimes evidence has been found for the development of localized features (`spikes') involving large gradients near the singularity. The rigorous mathematical results available up to now did not cover this kind of situation. In this work we show the existence of large classes of Gowdy spacetimes exhibiting features of the kind discovered numerically. These spacetimes are constructed by applying certain transformations to previously known spacetimes without spikes. It is possible to control the behaviour of the Kretschmann scalar near the singularity in detail. This curvature invariant is found to blow up in a way which is non-uniform near the spike in some cases. When this happens it demonstrates that the spike is a geometrically invariant feature and not an artefact of the choice of variables used to parametrize the metric. We also identify another class of spikes which are artefacts. The spikes produced by our method are compared with the results of numerical and heuristic analyses of the same situation.

Spectrum of charged black holes—the big fix mechanism revisited

Abstract: Following an earlier suggestion of the authors (Barvinsky A and Kunstatter G 1997 Mass spectrum for black holes in generic 2-D dilaton gravity Proc. 2nd International A D Sakharov Conference on Physics ed I M Dremin and A M Seminkhatov (Singapore: World Scientific) pp 210–15), we use some basic properties of Euclidean black hole thermodynamics and the quantum mechanics of systems with periodic phase space coordinate to derive the discrete two-parameter area spectrum of generic charged spherically symmetric black holes in any dimension. For the Reissner–Nordstrom black hole we get A/4G = π(2n + p + 1), where the integer p = 0, 1, 2,... gives the charge spectrum, with Q = ± √ p. The quantity π(2n + 1), n = 0, 1,..., gives a measure of the excess of the mass/energy over the critical minimum (i.e. extremal) value allowed for a given fixed charge Q. The classical critical bound cannot be saturated due to vacuum fluctuations of the horizon, so that generically extremal black holes do not appear in the physical spectrum. Consistency also requires the black hole charge to be an integer multiple of any fundamental elementary particle charge: Q = ±me, m = 0, 1, 2,.... As a by-product this yields a relation between the fine structure constant and integer parameters of the black hole—a kind of the Coleman big fix mechanism induced by black holes. In four dimensions, this relationship is e2/ = p/m2 and requires the fine structure constant to be a rational number. Finally, we prove that the horizon area is an adiabatic invariant, as has been conjectured previously.