scholarly journals The Einstein–Klein–Gordon–AdS system for general boundary conditions

2015 ◽  
Vol 12 (02) ◽  
pp. 293-342 ◽  
Author(s):  
Gustav Holzegel ◽  
Claude M. Warnick
Keyword(s):  
Boundary Conditions ◽  
Global Dynamics ◽  
Linear Wave ◽  
Spherically Symmetric ◽  
General Boundary Conditions ◽  
De Sitter ◽  
Linear Wave Equation ◽  
Starting Point ◽  
Klein Gordon ◽  
Local Solutions

We construct unique local solutions for the spherically-symmetric Einstein–Klein–Gordon–anti-de Sitter (AdS) system subject to a large class of initial and boundary conditions including some considered in the context of the AdS-CFT correspondence. The proof relies on estimates developed for the linear wave equation by the second author and involves a careful renormalization of the dynamical variables, including a renormalization of the well-known Hawking mass. For some of the boundary conditions considered this system is expected to exhibit rich global dynamics, including the existence of hairy black holes. This paper furnishes a starting point for such global investigations.

PROPAGATION OF SINGULARITIES ON AS SPACETIMES FOR GENERAL BOUNDARY CONDITIONS AND THE HOLOGRAPHIC HADAMARD CONDITION

2020 ◽  
pp. 1-61 ◽  
Author(s):  
Oran Gannot ◽  
Michał Wrochna
Keyword(s):  
Boundary Conditions ◽  
Dirichlet Boundary ◽  
Gordon Equation ◽  
Dirichlet Boundary Conditions ◽  
General Boundary Conditions ◽  
De Sitter ◽  
Hadamard Condition ◽  
Klein Gordon ◽  
Propagation Of Singularities ◽  
Klein Gordon Equation

We consider the Klein–Gordon equation on asymptotically anti-de-Sitter spacetimes subject to Neumann or Robin (or Dirichlet) boundary conditions and prove propagation of singularities along generalized broken bicharacteristics. The result is formulated in terms of conormal regularity relative to a twisted Sobolev space. We use this to show the uniqueness, modulo regularizing terms, of parametrices with prescribed $\text{b}$ -wavefront set. Furthermore, in the context of quantum fields, we show a similar result for two-point functions satisfying a holographic Hadamard condition on the $\text{b}$ -wavefront set.

scholarly journals Asymptotic Properties of Linear Field Equations in Anti-de Sitter Space

2019 ◽  
Vol 374 (2) ◽  
pp. 1125-1178 ◽  
Author(s):  
Gustav Holzegel ◽  
Jonathan Luk ◽  
Jacques Smulevici ◽  
Claude Warnick
Keyword(s):  
Boundary Conditions ◽  
Asymptotic Properties ◽  
Uniform Boundedness ◽  
Global Dynamics ◽  
Decay Estimates ◽  
De Sitter ◽  
Field Equations ◽  
Sitter Spacetime ◽  
Loss Of Derivatives ◽  
Dissipative Boundary

Abstract We study the global dynamics of the wave equation, Maxwell’s equation and the linearized Bianchi equations on a fixed anti-de Sitter (AdS) background. Provided dissipative boundary conditions are imposed on the dynamical fields we prove uniform boundedness of the natural energy as well as both degenerate (near the AdS boundary) and non-degenerate integrated decay estimates. Remarkably, the non-degenerate estimates “lose a derivative”. We relate this loss to a trapping phenomenon near the AdS boundary, which itself originates from the properties of (approximately) gliding rays near the boundary. Using the Gaussian beam approximation we prove that non-degenerate energy decay without loss of derivatives does not hold. As a consequence of the non-degenerate integrated decay estimates, we also obtain pointwise-in-time decay estimates for the energy. Our paper provides the key estimates for a proof of the non-linear stability of the anti-de Sitter spacetime under dissipative boundary conditions. Finally, we contrast our results with the case of reflecting boundary conditions.

scholarly journals Nonstandard Finite Difference Variational Integrators for Multisymplectic PDEs

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Cuicui Liao ◽  
Xiaohua Ding
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Variational Integrators ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Variational Integrator ◽  
Discretization Schemes ◽  
Klein Gordon ◽  
Effectiveness And Efficiency ◽  
Nonstandard Finite Difference

We use the idea of nonstandard finite difference methods to derive the discrete variational integrators for multisymplectic PDEs. We obtain a nonstandard finite difference variational integrator for linear wave equation with a triangle discretization and two nonstandard finite difference variational integrators for the nonlinear Klein-Gordon equation with a triangle discretization and a square discretization, respectively. These methods are naturally multisymplectic. Their discrete multisymplectic structures are presented by the multisymplectic form formulas. The convergence of the discretization schemes is discussed. The effectiveness and efficiency of the proposed methods are verified by the numerical experiments.

scholarly journals Fundamental solutions and Hadamard states for a scalar field with arbitrary boundary conditions on an asymptotically AdS spacetimes

2021 ◽  
Vol 24 (3) ◽  
Author(s):  
Claudio Dappiaggi ◽  
Alessio Marta
Keyword(s):  
Boundary Conditions ◽  
Equations Of Motion ◽  
Fundamental Solutions ◽  
De Sitter ◽  
Arbitrary Boundary ◽  
Arbitrary Boundary Conditions ◽  
Klein Gordon ◽  
Propagation Of Singularities ◽  
The Difference ◽  
Suitable Restriction

AbstractWe consider the Klein-Gordon operator on an n-dimensional asymptotically anti-de Sitter spacetime (M,g) together with arbitrary boundary conditions encoded by a self-adjoint pseudodifferential operator on ∂M of order up to 2. Using techniques from b-calculus and a propagation of singularities theorem, we prove that there exist advanced and retarded fundamental solutions, characterizing in addition their structural and microlocal properties. We apply this result to the problem of constructing Hadamard two-point distributions. These are bi-distributions which are weak bi-solutions of the underlying equations of motion with a prescribed form of their wavefront set and whose anti-symmetric part is proportional to the difference between the advanced and the retarded fundamental solutions. In particular, under a suitable restriction of the class of admissible boundary conditions and setting to zero the mass, we prove their existence extending to the case under scrutiny a deformation argument which is typically used on globally hyperbolic spacetimes with empty boundary.

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