scholarly journals QUANTUM AMPLITUDES IN BLACK-HOLE EVAPORATION: COMPLEX APPROACH AND SPIN-0 AMPLITUDE

2011 ◽  
Vol 20 (02) ◽  
pp. 133-159
Author(s):  
A. N. St. J. FARLEY ◽  
P. D. D'EATH
Keyword(s):  
Black Hole ◽  
Gravitational Field ◽  
Scalar Field ◽  
Boundary Value Problem ◽  
Boundary Data ◽  
Proper Time ◽  
Boundary Value ◽  
Constrained Systems ◽  
Time Interval ◽  
Late Time

This paper is concerned with the quantum-mechanical decay of a Schwarzschild-like black hole, formed by gravitational collapse, into almost-flat space–time and weak radiation at a very late time. We evaluate quantum amplitudes (not just probabilities) for transitions from initial to final states. This quantum description shows that no information is lost in collapse to a black hole. Boundary data for the gravitational field and (in this paper) a scalar field are posed on an initial space-like hypersurface ΣI and a final surface ΣF. These asymptotically flat three-surfaces are separated by a Lorentzian proper-time interval T (typically very large), as measured at spatial infinity. The boundary-value problem is made well-posed, both classically and quantum-mechanically, by a rotation of T into the lower-half complex plane: T → |T| exp (- iθ), with 0 < θ ≤ π/2. This corresponds to Feynman's +iϵ prescription. We consider the classical boundary-value problem and calculate the second-variation classical Lorentzian action [Formula: see text] as a functional of the boundary data. Following Feynman, the Lorentzian quantum amplitude is recovered in the limit θ → 0+ from the well-defined complex-T amplitude. Dirac's canonical approach to the quantisation of constrained systems shows that, for locally supersymmetric theories of gravity, the amplitude is exactly semi-classical, namely [Formula: see text] for weak perturbations, apart from delta functionals of the supersymmetry constraints. We treat such quantum amplitudes for weak scalar-field configurations on ΣF, taking (for simplicity) the weak final gravitational field to be spherically symmetric. The treatment involves adiabatic solutions to the scalar wave equation. This considerably extends work reported in previous papers, by giving explicit expressions for the real and imaginary parts of such quantum amplitudes.

scholarly journals BOGOLIUBOV TRANSFORMATIONS IN BLACK-HOLE EVAPORATION

2007 ◽  
Vol 16 (04) ◽  
pp. 569-590 ◽  
Author(s):  
A. N. St. J. FARLEY ◽  
P. D. D'EATH
Keyword(s):  
Black Hole ◽  
Scalar Field ◽  
Probability Distribution ◽  
Boundary Value Problem ◽  
Boundary Data ◽  
Boundary Value ◽  
Spherically Symmetric ◽  
Black Hole Evaporation ◽  
Quantum Amplitude ◽  
Classical Boundary

In previous papers and letters on quantum amplitudes in black-hole evaporation, a boundary-value approach was developed for calculating (for example) the quantum amplitude to have a prescribed slightly non-spherical configuration of a massless scalar field ϕ on a final hypersurface ΣF at a very late time T, given initial almost-stationary spherically symmetric gravitational and scalar data on a space-like hypersurface ΣI at time t = 0. For definiteness, we assumed that the gravitational data are also spherically symmetric on ΣF. Such boundary data can correspond to a classical solution for the Einstein/scalar system, describing gravitational collapse from an early low-density configuration to a nearly Schwarzschild black hole. This approach provides the quantum amplitude (not just the probability) for a transition from an initial to a final state. For a real Lorentzian time-interval T, the classical boundary-value problem refers to a set of hyperbolic equations (modulo gauge), and is badly posed. Instead, the boundary-value approach of the previous letters and papers requires (following Feynman) a rotation into the complex: T → |T| exp (-iθ), for 0 < θ ≤ π/2, of the time-separation-at-infinity T. The classical boundary-value problem, for a complex solution of the coupled nonlinear classical field equations, is expected to be well-posed for 0 < θ ≤ π/2. For a locally supersymmetric Lagrangian, containing supergravity coupled to supermatter, the classical Lorentzian action S class , a functional of the boundary data (which include the complexified T), yields a quantum amplitude proportional to exp (iS class ), apart from possible loop corrections which are negligible for boundary data with frequencies below the Planck scale. Finally (still following Feynman), one computes the Lorentzian quantum amplitude by taking the limit of exp (iS class ) as θ → 0+. In the present paper, a connection is made between the above boundary-value approach and the original approach to quantum evaporation in gravitational collapse to a black hole, via Bogoliubov coefficients. This connection is developed through consideration of the radial equation obeyed by the (adiabatic) non-spherical classical perturbations. When one studies the probability distribution for configurations of the final scalar field, based on our quantum amplitudes above, one finds that this distribution can also be interpreted in terms of the Wigner quasi-probability distribution for a harmonic oscillator.

scholarly journals Harvesting correlations in Schwarzschild and collapsing shell spacetimes

2020 ◽  
Vol 2020 (8) ◽  
Author(s):  
Erickson Tjoa ◽  
Robert B. Mann
Keyword(s):  
Black Hole ◽  
Scalar Field ◽  
Proper Time ◽  
Time Derivative ◽  
Vacuum State ◽  
Massless Scalar ◽  
Late Time ◽  
Massless Scalar Field ◽  
Schwarzschild Black Hole ◽  
Black Hole Spacetime

Abstract We study the harvesting of correlations by two Unruh-DeWitt static detectors from the vacuum state of a massless scalar field in a background Vaidya spacetime consisting of a collapsing null shell that forms a Schwarzschild black hole (hereafter Vaidya spacetime for brevity), and we compare the results with those associated with the three preferred vacua (Boulware, Unruh, Hartle-Hawking-Israel vacua) of the eternal Schwarzschild black hole spacetime. To do this we make use of the explicit Wightman functions for a massless scalar field available in (1+1)-dimensional models of the collapsing spacetime and Schwarzschild spacetimes, and the detectors couple to the proper time derivative of the field. First we find that, with respect to the harvesting protocol, the Unruh vacuum agrees very well with the Vaidya vacuum near the horizon even for finite-time interactions. Second, all four vacua have different capacities for creating correlations between the detectors, with the Vaidya vacuum interpolating between the Unruh vacuum near the horizon and the Boulware vacuum far from the horizon. Third, we show that the black hole horizon inhibits any correlations, not just entanglement. Finally, we show that the efficiency of the harvesting protocol depend strongly on the signalling ability of the detectors, which is highly non-trivial in presence of curvature. We provide an asymptotic analysis of the Vaidya vacuum to clarify the relationship between the Boulware/Unruh interpolation and the near/far from horizon and early/late-time limits. We demonstrate a straightforward implementation of numerical contour integration to perform all the calculations.

scholarly journals On the Solution of the Exterior Boundary Value Problem with the Aid of Series

1978 ◽  
Vol 41 ◽  
pp. 175-176
Author(s):  
M. S. Petrovskaya
Keyword(s):  
Gravitational Field ◽  
Boundary Value Problem ◽  
Small Parameter ◽  
Integral Equations ◽  
Boundary Value ◽  
Parameter Method ◽  
Small Parameter Method ◽  
Exterior Boundary ◽  
Exterior Boundary Value Problem ◽  
Molodensky Problem

AbstractThe exterior gravitational field depending on the Earth’s non-sphericity is usually determined from the analysis of satellite data or by the solution of the exterior boundary value problem. In the latter case some integral equations are solved which correlate the exterior potential with the known vector of gravity and the shape of the Earth’s surface (molodensky problem). In order to carry out the integration the small parameter method is applied. As a result, all the quantities which involve the equations should be expanded in powers of a certain small parameter, among these being the heights of the Earth’s surface points as well as the inclination α of the Earth’s physical surface. Since the angle α can be significant, especially in mountains, and in fact does not depend on any small parameter then the solution of integral equations is possible only for the Earth’s surface which is smoothed enough.

THE GREEN'S FUNCTION FOR THE BROADWELL MODEL WITH A TRANSONIC BOUNDARY

2008 ◽  
Vol 05 (02) ◽  
pp. 279-294 ◽  
Author(s):  
CHIU-YA LAN ◽  
HUEY-ER LIN ◽  
SHIH-HSIEN YU
Keyword(s):  
Boundary Value Problem ◽  
Green’S Function ◽  
Green's Function ◽  
Boundary Data ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convergent Series ◽  
Iteration Scheme ◽  
Initial Boundary Value

We study an initial boundary value problem for the Broadwell model with a transonic physical boundary. The Green's function for the initial boundary value problem is obtained by combining the estimates of the full boundary data and the Green's function for the initial value problem. The full boundary data is constructed from the imposed boundary data through an iteration scheme. The iteration scheme is designed to separate the interaction between the boundary wave and the interior wave and leads to a convergent series in the iterative boundary estimates.

scholarly journals Forced cubic Schrödinger equation with Robin boundary data: large-time asymptotics

2013 ◽  
Vol 469 (2159) ◽  
pp. 20130341 ◽  
Author(s):  
Elena I. Kaikina
Keyword(s):  
Schrödinger Equation ◽  
Boundary Value Problem ◽  
Boundary Data ◽  
Schrodinger Equation ◽  
Large Time ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Initial Boundary Value ◽  
Robin Boundary

We consider the initial-boundary-value problem for the cubic nonlinear Schrödinger equation, formulated on a half-line with inhomogeneous Robin boundary data. We study traditionally important problems of the theory of nonlinear partial differential equations, such as the global-in-time existence of solutions to the initial-boundary-value problem and the asymptotic behaviour of solutions for large time.

scholarly journals DISCUSSION ON SOME CHARACTERISTICS OF CHARGED BRANE-WORLD BLACK HOLES

2009 ◽  
Vol 24 (04) ◽  
pp. 719-739 ◽  
Author(s):  
M. KALAM ◽  
F. RAHAMAN ◽  
A. GHOSH ◽  
B. RAYCHAUDHURI
Keyword(s):  
Black Holes ◽  
Black Hole ◽  
Gravitational Field ◽  
Scalar Field ◽  
Brane World ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Effective Potentials ◽  
Test Particles ◽  
Jacobi Formalism

Several physical natures of charged brane-world black holes are investigated. Firstly, the timelike and null geodesics of the charged brane-world black holes are presented. We also analyze all the possible motions by plotting the effective potentials for various parameters for circular and radial geodesics. Secondly, we investigate the motion of test particles in the gravitational field of the charged brane-world black holes using the Hamilton–Jacobi formalism. We consider charged and uncharged test particles and examine their behavior in both static and nonstatic cases. Thirdly, the thermodynamics of the charged brane-world black holes are studied. Finally, it is shown that there is no phenomenon of superradiance for an incident massless scalar field for such a black hole.

Recovery of harmonic functions from partial boundary data respecting internal pointwise values

2017 ◽  
Vol 25 (2) ◽  
Author(s):  
Juliette Leblond ◽  
Dmitry Ponomarev
Keyword(s):  
Boundary Value Problem ◽  
Connected Domain ◽  
Harmonic Functions ◽  
Boundary Data ◽  
Boundary Value ◽  
Lipschitz Boundary ◽  
Simply Connected Domain ◽  
Simply Connected ◽  
Overdetermined Boundary Value Problem

Abstract We consider partially overdetermined boundary-value problem for Laplace PDE in a planar simply connected domain with Lipschitz boundary

scholarly journals A Cantilever Beam Problem with Small Deflections and Perturbed Boundary Data

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Ammar Khanfer ◽  
Lazhar Bougoffa
Keyword(s):  
Boundary Value Problem ◽  
Boundary Conditions ◽  
Existence Theorem ◽  
Cantilever Beam ◽  
Fourth Order ◽  
Boundary Data ◽  
Boundary Value ◽  
Integral Boundary Conditions ◽  
Beam Equation ◽  
Integral Boundary

The boundary value problem of a fourth-order beam equation u 4 = λ f x , u , u ′ , u ″ , u ′ ′ ′ , 0 ≤ x ≤ 1 is investigated. We formulate a nonclassical cantilever beam problem with perturbed ends. By determining appropriate values of λ and estimates for perturbation measurements on the boundary data, we establish an existence theorem for the problem under integral boundary conditions u 0 = u ′ 0 = ∫ 0 1 p x u x d x , u ″ 1 = u ′ ′ ′ 1 = ∫ 0 1 q x u ″ x d x , where p , q ∈ L 1 0 , 1 , and f is continuous on 0 , 1 × 0 , ∞ × 0 , ∞ × − ∞ , 0 × − ∞ , 0 .

A two-point Markov chain boundary-value problem

1993 ◽  
Vol 25 (3) ◽  
pp. 607-630 ◽  
Author(s):  
D. J. Daley ◽  
L. D. Servi
Keyword(s):  
Markov Chain ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Sample Path ◽  
Maximum Likelihood Estimates ◽  
Initial Time ◽  
Time Interval ◽  
Birth Process ◽  
Waiting Room ◽  
Time Point

The two-point Markov chain boundary-value problem discussed in this paper is a finite-time version of the quasi-stationary behaviour of Markov chains. Specifically, for a Markov chain {Xt:t = 0, 1, ·· ·}, given the time interval (0, n), the interest is in describing the chain at some intermediate time point r conditional on knowing both the behaviour of the chain at the initial time point 0 and that over the interval (0, n) it has avoided some subset B of the state space. The paper considers both ‘real time' estimates for r = n (i.e. the chain has avoided B since 0), and a posteriori estimates for r < n with at least partial knowledge of the behaviour of Xn. Algorithms to evaluate the distribution of Xr can be as small as O(n3) (and, for practical purposes, even O(n2 log n)). The estimates may be stochastically ordered, and the process (and hence, the estimates) may be spatially homogeneous in a certain sense. Maximum likelihood estimates of the sample path are furnished, but by example we note that these ML paths may differ markedly from the path consisting of the expected or average states. The scope for two-point boundary-value problems to have solutions in a Markovian setting is noted.Several examples are given, together with a discussion and examples of the analogous problem in continuous time. These examples include the basic M/G/k queue and variants that include a finite waiting room, reneging, balking, and Bernoulli feedback, a pure birth process and the Yule process. The queueing examples include Larson's (1990) ‘queue inference engine'.

scholarly journals On the stability of the scalar field in the gravitational field of a rotating (Kerr-) black hole

2010 ◽  
Author(s):  
Reinhard Horst Beyer ◽  
H. A. Morales-Tecotl ◽  
L. A. Urena-Lopez ◽  
R. Linares-Romero ◽  
H. H. Garcia-Compean
Keyword(s):  
Black Hole ◽  
Gravitational Field ◽  
Scalar Field ◽  
Kerr Black Hole ◽  
The Stability
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