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Decay of solutions to the Klein-Gordon equation on some expanding cosmological spacetimes

Sasane, Amol LU (2019) FYTM03 20192
Theoretical Particle Physics
Abstract (Swedish)
The decay of solutions to the Klein-Gordon equation is studied in two expanding cosmological spacetimes, namely (1) the de Sitter universe in flat Friedmann-Lemaitre-Robertson-Walker (FLRW) form, and (2) the cosmological region of the Reissner-Nordström-de Sitter (RNdS) model. Using energy methods, for initial data with finite higher order energies, bounds on the decay rates of the solution are obtained. Also, a previously established bound on the decay rate of the time derivative of the solution to the wave equation, in an expanding de~Sitter universe in flat FLRW form, is improved, proving Rendall's conjecture. A similar improvement is also given for the wave equation in the cosmological region of the RNdS spacetime.
Popular Abstract
One of the most profound realisations of physics of the twentieth century is that space and time should be considered as a single whole, a four-dimensional geometric object called spacetime, rather than two separate entities. Gravitation arising from matter manifests itself as curvature of spacetime, and the motion of matter is described by the straightest possible curves on this curved spacetime. Theoretically there are many possible spacetimes, all of which arise as solutions to the so-called Einstein field equations, relating matter content on the one hand with the curvature of spacetime on the other. While these are highly nonlinear difficult equations, important insights are achievable by simplifications obtained by linearising them.... (More)
One of the most profound realisations of physics of the twentieth century is that space and time should be considered as a single whole, a four-dimensional geometric object called spacetime, rather than two separate entities. Gravitation arising from matter manifests itself as curvature of spacetime, and the motion of matter is described by the straightest possible curves on this curved spacetime. Theoretically there are many possible spacetimes, all of which arise as solutions to the so-called Einstein field equations, relating matter content on the one hand with the curvature of spacetime on the other. While these are highly nonlinear difficult equations, important insights are achievable by simplifications obtained by linearising them. These simplifications take the form of the familiar wave equations one meets in everyday life, for example those used in describing ripples of water on the surface of a pond, albeit on a curved spacetime. A more general wave equation, which is satisfied by matter fields on spacetime is the so-called Klein-Gordon equation. It is a natural question to ask what the qualitative long-term behaviour is, of solutions to these wave equations. Are they bounded, do they grow, or do they decay? If they decay, at what rate do they decay? The answers to these questions have played an important role in recent theoretical developments in general relativity, for example in progress towards the resolution of the cosmic no-hair conjecture (roughly speaking saying that all solutions to the Einstein equations with a positive cosmological constant `eventually' look alike, namely they resemble the so-called de Sitter spacetime). This thesis contributes answers to the questions of obtaining exact decay rates in two expanding spacetimes for solutions to the Klein-Gordon equation and to the wave equation. (Less)
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author
Sasane, Amol LU
supervisor
organization
course
FYTM03 20192
year
type
H2 - Master's Degree (Two Years)
subject
keywords
Klein-Gordon equation, general relativity, FLRW models, Schwarzschild-de Sitter, Reissner-Nordström-de Sitter
language
English
additional info
The thesis was supervised by Jose Natario (Institute Superior Tecnico, Lisbon, Portugal) and by Johan Bijnens (Lund University).
id
8997523
alternative location
https://arxiv.org/abs/1909.01292
date added to LUP
2019-11-08 15:05:13
date last changed
2019-11-08 15:05:13
@misc{8997523,
  abstract     = {The decay of solutions to the Klein-Gordon equation is studied in two expanding cosmological spacetimes, namely (1) the de Sitter universe in flat Friedmann-Lemaitre-Robertson-Walker (FLRW) form, and (2) the cosmological region of the Reissner-Nordström-de Sitter (RNdS) model. Using energy methods, for initial data with finite higher order energies, bounds on the decay rates of the solution are obtained. Also, a previously established bound on the decay rate of the time derivative of the solution to the wave equation, in an expanding de~Sitter universe in flat FLRW form, is improved, proving Rendall's conjecture. A similar improvement is also given for the wave equation in the cosmological region of the RNdS spacetime.},
  author       = {Sasane, Amol},
  keyword      = {Klein-Gordon equation,general relativity,FLRW models,Schwarzschild-de Sitter,Reissner-Nordström-de Sitter},
  language     = {eng},
  note         = {Student Paper},
  title        = {Decay of solutions to the Klein-Gordon equation on some expanding cosmological spacetimes},
  year         = {2019},
}