119247583
Mass spectrum and statistical entropy of the BTZ black hole from canonical quantum gravity
In a recent publication we developed a canonical quantization program describing the gravitational collapse of a spherical dust cloud in $2+1$ dimensions with a negative cosmological constant $\ensuremath{-}\ensuremath{\Lambda}\ensuremath{\equiv}\ensuremath{-}{l}^{\ensuremath{-}2}l0$. In this paper we address the quantization of the Banados-Teitelboim-Zanelli (BTZ) black hole. We show that the mass function describing the black hole is made of two pieces, a constant nonvanishing boundary contribution and a discrete spectrum of the form ${\ensuremath{\mu}}_{n}=\frac{\ensuremath{\hbar}}{l}(n+\frac{1}{2})$. The discrete spectrum is obtained by applying the Wheeler-DeWitt equation with a particular choice of factor ordering and interpreted as giving the energy levels of the collapsed matter shells that form the black hole. Treating a black hole microstate as a particular distribution of shells among the levels, we determine the canonical entropy of the BTZ black hole. Comparison with the Bekenstein-Hawking entropy shows that the boundary energy is related to the central charge of the Virasoro algebra that generates the asymptotic symmetry group of the three-dimensional anti-de Sitter space ${\mathrm{AdS}}_{3}$. This gives a connection between the Wheeler-DeWitt approach and the conformal field theory approach.
27984646
Quantum general relativity and Hawking radiation
In a previous paper we set up the Wheeler-DeWitt equation which describes the quantum general relativistic collapse of a spherical dust cloud. In the present paper we specialize this equation to the case of matter perturbations around a black hole, and show that, in the WKB approximation, the wave functional describes an eternal black hole in equilibrium with a thermal bath at Hawking temperature.
118510273
THE CONNECTION BETWEEN "EMERGENCE OF TIME FROM QUANTUM GRAVITY" AND "DYNAMICAL COLLAPSE OF THE WAVE FUNCTION IN QUANTUM MECHANICS"
There are various reasons to believe that quantum theory could be an emergent phenomenon. Trace dynamics is an underlying classical dynamics of noncommuting matrices, from which quantum theory and classical mechanics have been shown to emerge, in the thermodynamic approximation. However, the time that is used to describe evolution in quantum theory is an external classical time, and is in turn expected to be an emergent feature — a relic of an underlying theory of quantum gravity. In this essay we borrow ideas from trace dynamics to show that classical time is a thermodynamic approximation to an operator time in quantum gravitational physics. This prediction will be put to test by ongoing laboratory experiments attempting to construct superposed states of macroscopic objects.
119086760
Comparing quantum black holes and naked singularities
There are models of gravitational collapse in classical general relativity which admit the formation of naked singularities as well as black holes. These include fluid models as well as models with scalar fields as matter. Even if fluid models were to be regarded as unphysical in their matter content, the remaining class of models (based on scalar fields) generically admit the formation of visible regions of finite but arbitrarily high curvature. Hence it is of interest to ask, from the point of view of astrophysics, as to what a stellar collapse leading to a naked singularity (or to a visible region of very high curvature) will look like, to a far away observer. The emission of energy during such a process may be divided into three phases - (i) the classical phase, during which matter and gravity can both be treated according to the laws of classical physics, (ii) the semiclassical phase, when gravity is treated classically but matter behaves as a quantum field, and (iii) the quantum gravitational phase. In this review, we first give a summary of the status of naked singularities in classical relativity, and then report some recent results comparing the semiclassical phase of black holes with the semiclassical phase of spherical collapse leading to a naked singularity. In particular, we ask how the quantum particle creation during the collapse leading to a naked singularity compares with the Hawking radiation from a star collapsing to form a black hole. It turns out that there is a fundamental difference between the two cases. A spherical naked star emits only about one Planck energy during its semiclassical phase, and the further evolution can only be determined by the laws of quantum gravity. This contrasts with the semiclassical evaporation of a black hole.
16490636
The final fate of spherical inhomogeneous dust collapse
We examine, in a general manner, the role played by the initial density and velocity distributions in the gravitational collapse of a spherically symmetric inhomogeneous dust cloud. Such a collapse is described by the Tolman - Bondi metric which has two free functions: the `mass function' and the `energy function', which are determined by the initial density and velocity profiles of the cloud. The collapse can end in either a black hole or a naked singularity, depending on the initial parameters characterizing these profiles. In the marginally bound case, we find that the collapse ends in a naked singularity if the leading non-vanishing derivative of the density at the centre is either the first one or the second one. If the first two derivatives are zero, and the third derivative non-zero, the singularity could either be naked or covered, depending on a quantity determined by the third derivative and the central density. If the first three derivatives are zero, the collapse ends in a black hole. In particular, the classic result of Oppenheimer and Snyder, that homogeneous dust collapse leads to a black hole, is recovered as a special case. Analogous results are found when the cloud is not marginally bound, and also for the case of a cloud starting from rest. A condition on the initial density profile is given for the singularity to be globally naked. We also show how the strength of the naked singularity depends on the density and velocity distribution. Our analysis generalizes and simplifies the earlier works of Christodoulou and Newman by dropping the assumption of evenness of density functions. It turns out that relaxing this assumption allows for a smooth transition from the naked singularity phase to the black hole phase, and also allows for the occurrence of strong curvature naked singularities.
1570088
Radiation flux and spectrum in the Vaidya collapse model
Abstract We consider the quantization of a massless scalar field, using the geometric optics approximation, in the background spacetime of a collapsing spherical self-similar Vaidya star, which forms a black hole or a naked singularity. We show that the outgoing radiation flux of the quantized scalar field diverges on the Cauchy horizon. The spectrum of the produced scalar partcles is non-thermal when the background develops a naked singularity. These results are analogous to those obtained for the scalar quantization on a self-similar dust cloud.
122321601
Scalar-tensor theories of gravitation: Foundations and prospects
A generalization of Einstein's theory is discussed in which the gravitation is described by a tensor and a scalar field. The theory is more consistent with Mach's principle and less reliant on absolute properties of space. The modification involves a violation of the “strong principle of equivalence” on which Einstein's theory is based. In the original version of this new theory, the “constant” of gravitationG is varying and particle masses are fixed. Later on another version of the theory was given in whichG is truly a constant and the particle masses vary. The two versions are related by a conformal transformation. The physical and mathematical foundations of this theory have been discussed and the field equations have been derived. The astrophysical and cosmological consequences of the theory have been elaborately reviewed.
121368055
GENERAL CLASS OF SCALAR-TENSOR THEORIES: A REVIEW
The general class of scalar-tensor theories is discussed in which, in contrast to Brans-Dicke theory, the coupling parameter ω is not a constant but a function of the scalar field ϕ. Some other generalizations of the Brans-Dicke theory have also been discussed in brief. The physical aspects of these theories have been dealt. The cosmological models in scalar-tensor theories have been discussed in detail. Finally some open problems in these theories have been pointed out.
121240575
Stationary Universe in General Relativity and Lyra's Geometry
The stationary universe solutions have been studied using the metrics of VAN Stockum (1937) and Godel (1949) in a New ST theory (Saez and Ballester, 1985), general relativity with variable cosmological term (Narlikar et al., 1991), Lyra's geometry (Sen, 1957) and Cosmic strings. It has been shown that the stationary universe has rigid rotation in ST theory. In the presence of variable cosmological term, there is non-rigid rotation in the presence of Zeldovich fluid for van Stockum metric. In Lyra's geometry the only solution is possible when the gauge function β is variable and there is Zeldovich fluid. However, in the presence of cosmic strings, there is no solution with van Stockum or Godel metrics.
117011561
On exact static spherically symmetric solutions in general relativity, Einstein-Cartan theory and higher-dimensional theories
The equations describing the hydrostatic equilibrium of a static general relativistic fluid sphere are obtained for magnetofluids, Einstein-Cartan theory and anisotropic fluids in four dimensions and perfect fluids in D (D ≥ 4) dimensions. The method due to Hojman and Santamarina and Berger et al. is used to obtain a new and larger class of solutions. Illustrative examples have been given which lead to generalization of the Schwarschild exterior solution and the relativistic models for spiral galaxies and dense stars (Hojman et al.).
124426353
Lyra's geometry and cosmology: A Review
The Lyra's geometry, which is a generalization of Weyl's geometry (removing the defect of non-integrability of length transfer) and the gravitational theory and cosmology based on this geometry have been reviewed. Further, cosmological models with both constant and time dependent displacement field have been discussed. Some of these models solve the singularity, entropy and horizon problems. Finally some possibilities of further problems and their investigations have been pointed out.