Decay of solutions to the KleinGordon equation on some expanding cosmological spacetimes
(2019) FYTM03 20192Theoretical Particle Physics
 Abstract (Swedish)
 The decay of solutions to the KleinGordon equation is studied in two expanding cosmological spacetimes, namely (1) the de Sitter universe in flat FriedmannLemaitreRobertsonWalker (FLRW) form, and (2) the cosmological region of the ReissnerNordströmde 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 fourdimensional 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 socalled 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 fourdimensional 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 socalled 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 socalled KleinGordon equation. It is a natural question to ask what the qualitative longterm 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 nohair conjecture (roughly speaking saying that all solutions to the Einstein equations with a positive cosmological constant `eventually' look alike, namely they resemble the socalled de Sitter spacetime). This thesis contributes answers to the questions of obtaining exact decay rates in two expanding spacetimes for solutions to the KleinGordon equation and to the wave equation. (Less)
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/studentpapers/record/8997523
 author
 Sasane, Amol ^{LU}
 supervisor

 Johan Bijnens ^{LU}
 organization
 course
 FYTM03 20192
 year
 2019
 type
 H2  Master's Degree (Two Years)
 subject
 keywords
 KleinGordon equation, general relativity, FLRW models, Schwarzschildde Sitter, ReissnerNordströmde 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
 20191108 15:05:13
 date last changed
 20191108 15:05:13
@misc{8997523, abstract = {The decay of solutions to the KleinGordon equation is studied in two expanding cosmological spacetimes, namely (1) the de Sitter universe in flat FriedmannLemaitreRobertsonWalker (FLRW) form, and (2) the cosmological region of the ReissnerNordströmde 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 = {KleinGordon equation,general relativity,FLRW models,Schwarzschildde Sitter,ReissnerNordströmde Sitter}, language = {eng}, note = {Student Paper}, title = {Decay of solutions to the KleinGordon equation on some expanding cosmological spacetimes}, year = {2019}, }