The physical interpretation and eventual fate of gravitational singularities in a theory surpassing classical general relativity are puzzling questions that have generated a great deal of interest among various quantum gravity approaches. In the context of loop quantum gravity (LQG), one of the major candidates for a non-perturbative background-independent quantisation of general relativity, considerable effort has been devoted to construct effective models in which these questions can be studied. In these models, classical singularities are replaced by a "bounce" induced by quantum geometry corrections. Undesirable features may arise however depending on the details of the model. In this paper, we focus on Schwarzschild black holes and propose a new effective quantum theory based on polymerisation of new canonical phase space variables inspired by those successful in loop quantum cosmology. The quantum corrected spacetime resulting from the solutions of the effective dynamics is characterised by infinitely many pairs of trapped and anti-trapped regions connected via a space-like transition surface replacing the central singularity. Quantum effects become relevant at a unique mass independent curvature scale, while they become negligible in the low curvature region near the horizon. The effective quantum metric describes also the exterior regions and asymptotically classical Schwarzschild geometry is recovered. We however find that physically acceptable solutions require us to select a certain subset of initial conditions, corresponding to a specific mass (de-)amplification after the bounce. We also sketch the corresponding quantum theory and explicitly compute the kernel of the Hamiltonian constraint operator.
We study a spherically symmetric setup consisting of a Schwarzschild metric as the background geometry in the framework of classical polymerization. This process is an extension of the polymeric representation of quantum mechanics in such a way that a transformation maps classical variables to their polymeric counterpart. We show that the usual Schwarzschild metric can be extracted from a Hamiltonian function which in turn, gets modifications due to the classical polymerization. Then, the polymer corrected Schwarzschild metric may be obtained by solving the polymer-Hamiltonian equations of mot…
In this paper, we investigate a tunneling mechanism of massless particles from the Schwarzschild black hole in the framework of polymer quantum mechanics. According to the corresponding invariant Liouville volume, we determine the tunneling rate from Schwarzschild black hole by the polymeric quantization procedure. In this regard, we show that the temperature and tunneling radiation of the black hole receive new corrections in such a way that the exact radiant spectrum is no longer precisely thermal.
An integral equation method for scalar scattering in Schwarzschild spacetime is constructed. The zeroth-order and first-order scattering phase shift is obtained.
We quantize the Schwarzschild spacetime with naked singularity using the affine coherent states quantization method. The novelty of our approach is quantization of both temporal and spatial coordinates. Quantization smears the gravitational singularity indicated by the Kretschmann invariant avoiding its localization in the configuration space. This way we resolve the singularity problem of considered spacetime at quantum level.
We consider the modified Einstein equations obtained in the framework of effective spherically symmetric polymer models inspired by Loop Quantum Gravity. When one takes into account the anomaly free point-wise holonomy quantum corrections, the modification of Einstein equations is parametrized by a function $f(x)$ of one phase space variable. We solve explicitly these equations for a static interior black hole geometry and find the effective metric describing the trapped region, inside the black hole, for any $f(x)$. This general resolution allows to take into account a standard ambiguity inhe…
The deep reason why the equations describing axial and polar perturbations of Schwarzschild black holes have the same spectrum is far from trivial. In this article, we revisit the original proof and try to make it clearer. Still focusing on uncharged and non-rotating black holes, we extend the results to spacetimes including a cosmological constant, which have so far mostly been investigated numerically from this perspective.
We present the manifestly covariant Lagrangian of a massless polarized particle, that implies all dynamic and algebraic equations as the conditions of extreme of this variational problem. The model allows for minimal interaction with a gravitational field, leading to the equations, coinciding with Maxwell equations in the geometrical optics approximation. The model allows also a wide class of non minimal interactions, which suggests an alternative way to study the electromagnetic radiation beyond the leading order of geometrical optics. As a specific example, we construct a curvature-dependent…
Comments on "Thermodynamics and/of horizons : A comparison of Schwarzschild, Rindler and de Sitter spacetimes" (ArXiv : gr-qc/0202078), by T.Padmanabhan
A gauge-invariant treatment of the monopole- ($l=0$) and dipole ($l=1$) modes in linear perturbations of the Schwarzschild background spacetime is proposed. Through this gauge-invariant treatment, we derived the solutions to the linearized Einstein equation for these modes with a generic matter field. In the vacuum case, these solutions include the Kerr parameter perturbations in the $l=1$ odd modes and the additional mass parameter perturbations of the Schwarzschild mass in the $l=0$ even modes. The linearized version of Birkhoff's theorem is also confirmed in a gauge-invariant manner. In thi…
Perturbations of the Dirac field in the Schwarzschild spacetime are characterized by two wave equations (for two chiralities) with effective potentials which are iso-spectral and one of which is positive definite. Therefore, the stability of a Dirac field in the Schwarzschild background is straightforward. This is not so for the Schwarzschild-de Sitter case, because potentials for both chiralities have negative gaps and the fact that the stability is not automatically guaranteed in the asymptotically de Sitter case was apparently omitted in the literature. Performing the time-domain integratio…
We study the scalar radiation emitted by a source orbiting a Schwarzschild--anti--de Sitter black hole in circular geodesics. We obtain the emitted power using quantum field theory in curved spacetimes at tree level with reflective boundary conditions at infinity. We compare our results with the ones obtained in Schwarzschild and Schwarzschild--de Sitter ($Λ> 0$) spacetimes. In the Schwarzschild-de Sitter spacetime we observe an enhancement of the emitted power associated to the lower multipole modes, when compared with the Schwarzschild case. In contrast, for the Schwarzschild-anti-de Sitter …
Equipped with new powerful and efficient methods for computing quantum expectation values in static-spherically symmetric spacetimes in arbitrary dimensions, we perform an in-depth investigation of how the quantum vacuum polarization varies with the parameters in the theory. In particular, we compute and compare the vacuum polarization for a quantum scalar field in the Schwarzschild anti-de Sitter black hole spacetime for a range of values of the field mass and field coupling constant as well as the black hole mass and number of spacetime dimensions. In addition, a new approximation for the va…
Kerr black hole immersed in test, asymptotically homogeneous magnetic field, aligned along the symmetry axis, is described by Wald's solution. We show how the static case of this solution may be generalized for nonlinear electromagnetic models via perturbative approach. Using this technique we find the lowest order correction to Wald's solution on Schwarzschild spacetime in Euler--Heisenberg and Born--Infeld theories. Finally, we discuss the problem of highly conducting star in asymptotically homogeneous magnetic field.
We present analogue models for the Schwarzschild and Reissner-Nordström (RN) spacetimes based on unidirectional hydrodynamic flows. We show that, with appropriate coordinate transformations, sound waves in a moving fluid propagate as scalar fields at the equatorial sections of Schwarzschild and RN black holes. The coordinate associated with the direction of the flow plays the role of the radial coordinate in standard Schwarzschild/RN coordinates, while the transversal spatial coordinate is related to the angular coordinate on each equatorial slice. The magnitude of the flow velocity is related…
We analyze the scalar radiation emitted by a source in geodesic circular orbits around a Schwarzschild-de Sitter black hole. We obtain the emitted power using quantum field theory in curved spacetimes framework at tree level. We compare our results with the scalar synchrotron radiation in Schwarzschild spacetime.
The retarded Green function for linear field perturbations of black hole spacetimes is notoriously difficult to calculate. One of the difficulties is due to a Dirac-$δ$ divergence that the Green function possesses when the two spacetime points are connected by a "direct" null geodesic. We present a procedure which notably aids its calculation in the case of Schwarzschild spacetime by separating this direct $δ$-divergence from the remainder of the retarded Green function. More precisely, the method consists of calculating the multipolar $\ell$-modes of the direct $δ$-divergence and subtracting …
This paper deals with quantum field theory in curved space-time using the Thermo Field Dynamics. The scalar field is coupled to the Schwarzschild space time and then thermalised. The Stefan-Boltzmann law is established at finite temperature and the entropy of the field is calculated. Then the Casimir energy and pressure are obtained at zero and finite temperature.
We compute the gravitational birefringence of light as it undergoes gravitational lensing. To this end we re-derive the Souriau-Saturnini equations in the Schwarzschild metric and solve them numerically and perturbatively. Our main result is an offset between the trajectories of the photons of opposite polarisations, which grows with time. We also find an intriguing instability of the spin component transverse to the momentum.
Linear perturbations for a one parameter family of stationary, closed Nambu-Goto strings winding around a flat torus in the five-dimensional Schwarzschild spacetime have been studied. It has been shown that this problem is solvable in the sense that frequency spectra and perturbation modes can be expressed only with arithmetic operations and radicals. It has been proven that the Nambu-Goto strings belonging to this family are always unstable, no matter how they are located at almost flat region distant from the event horizon.
Based on the perturbation expansion, we compute the noncommutative corrections to the minimal surface areas of the pure AdS spacetime and Schwarzschild-AdS black hole, where the noncommutative background is suitably constructed in terms of the Poincaré coordinate system. In particular, we find a reasonable tetrad with subtlety, which not only matches the metrics of the pure AdS spacetime and Schwarzschild-AdS black hole in the commutative case, but also makes the corrections real rather than complex in the noncommutative case. For the pure AdS spacetime, the nocommutative effect is only a loga…
The main aim of this paper is twofold. (1) Exact solutions of a scalar field in the Schwarzschild spacetime are presented. The exact wave functions of scattering states and bound-states are presented. Besides the exact solution, we also provide explicit approximate expressions for bound-state eigenvalues and scattering phase shifts. (2) By virtue of the exact solutions, we give a direct calculation for the discontinuous jump on the horizon for massive scalar fields, while in literature such a jump is obtained from an asymptotic solution by an analytic extension treatment.
