scholarly journals Decay of solutions of the wave equation in expanding cosmological spacetimes

2019 ◽  
Vol 16 (01) ◽  
pp. 35-58
Author(s):  
João L. Costa ◽  
José Natário ◽  
Pedro F. C. Oliveira
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Decay Rate ◽  
Time Derivative ◽  
Iteration Scheme ◽  
Higher Order ◽  
Accelerated Expansion ◽  
De Sitter ◽  
Decay Of Solutions ◽  
Partial Energy

We study the decay of solutions of the wave equation in some expanding cosmological spacetimes, namely flat Friedmann–Lemaître–Robertson–Walker (FLRW) models and the cosmological region of the Reissner–Nordström–de Sitter (RNdS) solution. By introducing a partial energy and using an iteration scheme, we find that, for initial data with finite higher order energies, the decay rate of the time derivative is faster than previously existing estimates. For models undergoing accelerated expansion, our decay rate appears to be (almost) sharp.

scholarly journals On the initial value problem for the wave equation in Friedmann–Robertson–Walker space–times

2014 ◽  
Vol 470 (2169) ◽  
pp. 20140361 ◽  
Author(s):  
Bilal Abbasi ◽  
Walter Craig
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Three Dimensional ◽  
Propagation Speed ◽  
Cauchy Surface ◽  
Space Time ◽  
Decay Of Solutions ◽  
Dimensional Minkowski Space ◽  
Huygens Principle ◽  
Friedmann Robertson Walker

The propagator W ( t 0 , t 1 )( g , h ) for the wave equation in a given space–time takes initial data ( g ( x ), h ( x )) on a Cauchy surface {( t , x ) :  t = t 0 } and evaluates the solution ( u ( t 1 , x ),∂ t u ( t 1 , x )) at other times t 1 . The Friedmann–Robertson–Walker space–times are defined for t 0 , t 1 >0, whereas for t 0 →0, there is a metric singularity. There is a spherical means representation for the general solution of the wave equation with the Friedmann–Robertson–Walker background metric in the three spatial dimensional cases of curvature K =0 and K =−1 given by S. Klainerman and P. Sarnak. We derive from the expression of their representation three results about the wave propagator for the Cauchy problem in these space–times. First, we give an elementary proof of the sharp rate of time decay of solutions with compactly supported data. Second, we observe that the sharp Huygens principle is not satisfied by solutions, unlike in the case of three-dimensional Minkowski space–time (the usual Huygens principle of finite propagation speed is satisfied, of course). Third, we show that for 0< t 0 < t the limit, lim t 0 → 0 + W ( t 0 , t ) ( g , h ) = W ( 0 , t ) ( g ) exists, it is independent of h ( x ), and for all reasonable initial data g ( x ), it gives rise to a well-defined solution for all t >0 emanating from the space–time singularity at t =0. Under reflection t →− t , the Friedmann–Robertson–Walker metric gives a space–time metric for t <0 with a singular future at t =0, and the same solution formulae hold. We thus have constructed solutions u ( t , x ) of the wave equation in Friedmann–Robertson–Walker space–times which exist for all − ∞ < t < 0 and 0 < t < + ∞ , where in conformally regularized coordinates, these solutions are continuous through the singularity t =0 of space–time, taking on specified data u (0,⋅)= g (⋅) at the singular time.

scholarly journals Massive Dirac quasinormal modes in Schwarzschild–de Sitter black holes: Anomalous decay rate and fine structure

2021 ◽  
Vol 103 (6) ◽  
Author(s):  
Almendra Aragón ◽  
Ramón Bécar ◽  
P. A. González ◽  
Yerko Vásquez
Keyword(s):  
Black Holes ◽  
Fine Structure ◽  
Decay Rate ◽  
Quasinormal Modes ◽  
De Sitter

scholarly journals On Galilean Invariant and Energy Preserving BBM-Type Equations

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 878
Author(s):  
Alexei Cheviakov ◽  
Denys Dutykh ◽  
Aidar Assylbekuly
Keyword(s):  
Wave Equation ◽  
Conservation Laws ◽  
Numerical Simulations ◽  
Solitary Waves ◽  
Local Symmetry ◽  
Symmetry And Conservation Laws ◽  
Higher Order ◽  
Special Equation ◽  
Long Wave ◽  
Local Symmetries

We investigate a family of higher-order Benjamin–Bona–Mahony-type equations, which appeared in the course of study towards finding a Galilei-invariant, energy-preserving long wave equation. We perform local symmetry and conservation laws classification for this family of Partial Differential Equations (PDEs). The analysis reveals that this family includes a special equation which admits additional, higher-order local symmetries and conservation laws. We compute its solitary waves and simulate their collisions. The numerical simulations show that their collision is elastic, which is an indication of its S−integrability. This particular PDE turns out to be a rescaled version of the celebrated Camassa–Holm equation, which confirms its integrability.

scholarly journals Hamiltonian charges in the asymptotically de Sitter spacetimes

2021 ◽  
Vol 2021 (5) ◽  
Author(s):  
Maciej Kolanowski ◽  
Jerzy Lewandowski
Keyword(s):  
Phase Space ◽  
Initial Data ◽  
Exact Solutions ◽  
Gravitational Waves ◽  
Physical Meaning ◽  
De Sitter ◽  
Killing Vectors ◽  
Asymptotically Flat ◽  
Angular Momenta ◽  
The One

Abstract We generalize a notion of ‘conserved’ charges given by Wald and Zoupas to the asymptotically de Sitter spacetimes. Surprisingly, our construction is less ambiguous than the one encountered in the asymptotically flat context. An expansion around exact solutions possessing Killing vectors provides their physical meaning. In particular, we discuss a question of how to define energy and angular momenta of gravitational waves propagating on Kottler and Carter backgrounds. We show that obtained expressions have a correct limit as Λ → 0. We also comment on the relation between this approach and the one based on the canonical phase space of initial data at ℐ+.

scholarly journals Traversable wormholes in Einsteinian cubic gravity

2019 ◽  
Vol 35 (06) ◽  
pp. 2050017 ◽  
Author(s):  
Mohammad Reza Mehdizadeh ◽  
Amir Hadi Ziaie
Keyword(s):  
Equation Of State ◽  
Gravitational Lensing ◽  
Energy Condition ◽  
Higher Order ◽  
Energy Conditions ◽  
Geodesic Motion ◽  
De Sitter ◽  
Asymptotically Flat ◽  
Traversable Wormholes ◽  
Standard Energy

In this work, we investigate wormhole configurations described by a constant redshift function in Einstein-Cubic gravity ( ECG ). We derive analytical wormhole geometries by assuming a particular equation of state ( EoS ) and investigate the possibility that these solutions satisfy the standard energy conditions. We introduce exact asymptotically flat and anti-de Sitter (AdS) spacetimes that admit traversable wormholes. These solutions are obtained by imposing suitable values for the parameters of the theory so that the resulted geometries satisfy the weak energy condition ( WEC ) in the vicinity of the throat, due to the presence of higher-order curvature terms. Moreover, we find that AdS solutions satisfy the WEC throughout the spacetime. A description of the geodesic motion of time-like and null particles is presented for the obtained wormhole solutions. Also, using gravitational lensing effects, observational features of the wormhole structure are discussed.

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