Numerical evolution of plane gravitational waves
- Authors: Hakata, Jonathan
- Date: 2021-10
- Subjects: Gravitational waves , Space and time , Einstein field equations , de Sitter metric , Perturbed spacetime
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/190248 , vital:44977
- Description: Unlike electromagnetic waves, gravitational waves self interact. This interaction is non-linear and can have very interesting properties which effect the curvature of space-time. A gravitational plane wave collider, implemented in the Python package COFFEE [20] that been developed in recent years by the Otago relativity group and implements the method of lines, can be reliably used to study this self-interaction. This was shown to work well numerically as profounded by Frauendiener, Stevens and Whale in 2014 [24]. For this reason, COFFEE will be used to study these gravitational wave propagations and subsequently collisions. The Einstein field equations are formulated as a well-posed initial boundary value problem (IBVP) in the Friedrich-Nagy gauge [26] and due to the large class of boundary conditions admitted by this framework, a variety of investigations into the propagation of plane gravitational waves could be carried out. This study focuses on the propagation of plane gravitational waves in the de Sitter (dS) space-time, which is the maximally symmetric solution of the Einstein’s vacuum field equations with a positive cosmological constant λ. There is substantial cosmological evidence that our universe is asymptotically de Sitter, yet no work, analytical nor numerical, has been done on gravitational plane waves propagating on such a space-time, mainly due to the increased complexity from the non-vanishing λ. Firstly, it is found analytically that with an arbitrary cosmological constant λ and a non-vanishing energy momentum tensor, the constraints will propagate. This means that we still have a wellposed IBVP, which is nontrivial since the Friedrich-Nagy gauge has only been shown to lead to a wellposed IBVP without matter [26]. Using this system, we consider one ingoing wave propagating on said space-time in vacuum. The area of the ingoing wave profile is varied and inferences are made about the different phenomena that arise in the curvature of space-time during the evolution. It is found that there exists a critical value of the wave’s area, ac, whereby taking the area below this value the system asymptotes to its initial state, and above the system diverges, indicating the presence of a singularity. Furthermore, we define an expansion parameter H to measure how the gravitational waves influence the accelerated expansion, generalising (numerically) results of Tsamis and Woodard. , Thesis (MSc) -- Faculty of Science, Mathematics, 2021
- Full Text:
- Authors: Hakata, Jonathan
- Date: 2021-10
- Subjects: Gravitational waves , Space and time , Einstein field equations , de Sitter metric , Perturbed spacetime
- Language: English
- Type: Master's theses , text
- Identifier: http://hdl.handle.net/10962/190248 , vital:44977
- Description: Unlike electromagnetic waves, gravitational waves self interact. This interaction is non-linear and can have very interesting properties which effect the curvature of space-time. A gravitational plane wave collider, implemented in the Python package COFFEE [20] that been developed in recent years by the Otago relativity group and implements the method of lines, can be reliably used to study this self-interaction. This was shown to work well numerically as profounded by Frauendiener, Stevens and Whale in 2014 [24]. For this reason, COFFEE will be used to study these gravitational wave propagations and subsequently collisions. The Einstein field equations are formulated as a well-posed initial boundary value problem (IBVP) in the Friedrich-Nagy gauge [26] and due to the large class of boundary conditions admitted by this framework, a variety of investigations into the propagation of plane gravitational waves could be carried out. This study focuses on the propagation of plane gravitational waves in the de Sitter (dS) space-time, which is the maximally symmetric solution of the Einstein’s vacuum field equations with a positive cosmological constant λ. There is substantial cosmological evidence that our universe is asymptotically de Sitter, yet no work, analytical nor numerical, has been done on gravitational plane waves propagating on such a space-time, mainly due to the increased complexity from the non-vanishing λ. Firstly, it is found analytically that with an arbitrary cosmological constant λ and a non-vanishing energy momentum tensor, the constraints will propagate. This means that we still have a wellposed IBVP, which is nontrivial since the Friedrich-Nagy gauge has only been shown to lead to a wellposed IBVP without matter [26]. Using this system, we consider one ingoing wave propagating on said space-time in vacuum. The area of the ingoing wave profile is varied and inferences are made about the different phenomena that arise in the curvature of space-time during the evolution. It is found that there exists a critical value of the wave’s area, ac, whereby taking the area below this value the system asymptotes to its initial state, and above the system diverges, indicating the presence of a singularity. Furthermore, we define an expansion parameter H to measure how the gravitational waves influence the accelerated expansion, generalising (numerically) results of Tsamis and Woodard. , Thesis (MSc) -- Faculty of Science, Mathematics, 2021
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Cosmological structure formation using spectral methods
- Authors: Funcke, Michelle
- Date: 2016
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/2969 , vital:20348
- Description: Numerical simulations are becoming an increasingly important tool for understanding the growth and development of structure in the universe. Common practice is to discretize the space-time using physical variables. The discreteness is embodied by considering the dynamical variables as fields on a fixed spatial and time resolution, or by constructing the matter fields by a large number of particles which interact gravitationally (N-body methods). Recognizing that the physical quantities of interest are related to the spectrum of perturbations, we propose an alternate discretization in the frequency domain, using standard spectral methods. This approach is further aided by periodic boundary conditions which allows a straightforward decomposition of variables in a Fourier basis. Fixed resources require a high-frequency cut-off which lead to aliasing effects in non-linear equations, such as the ones considered here. This thesis describes the implementation of a 3D cosmological model based on Newtonian hydrodynamic equations in an expanding background. Initial data is constructed as a spectrum of perturbations, and evolved in the frequency domain using a pseudo-spectral evolution scheme and an explicit Runge-Kutta time integrator. The code is found to converge for both linear and non-linear evolutions, and the convergence rate is determined. The correct growth rates expected from analytical calculations are recovered in the linear case. In the non-linear model, we observe close correspondence with linear growth and are able to monitor the growth on features associated with the non-linearity. High-frequency aliasing effects were evident in the non-linear evolutions, leading to a study of two potential resolutions to this problem: a boxcar filter which adheres to“Orszag’s two thirds rule” and an exponential window function, the exponential filter suggested by Hou and Li [1], and a shifted version of the exponential filter suggested, which has the potential to alleviate high frequency- ripples resulting from the Gibbs’ phenomenon. We found that the filters were somewhat successful at reducing aliasing effects but that the Gibbs’ phenomenon could not be entirely removed by the choice of filters.
- Full Text:
- Authors: Funcke, Michelle
- Date: 2016
- Language: English
- Type: Thesis , Masters , MSc
- Identifier: http://hdl.handle.net/10962/2969 , vital:20348
- Description: Numerical simulations are becoming an increasingly important tool for understanding the growth and development of structure in the universe. Common practice is to discretize the space-time using physical variables. The discreteness is embodied by considering the dynamical variables as fields on a fixed spatial and time resolution, or by constructing the matter fields by a large number of particles which interact gravitationally (N-body methods). Recognizing that the physical quantities of interest are related to the spectrum of perturbations, we propose an alternate discretization in the frequency domain, using standard spectral methods. This approach is further aided by periodic boundary conditions which allows a straightforward decomposition of variables in a Fourier basis. Fixed resources require a high-frequency cut-off which lead to aliasing effects in non-linear equations, such as the ones considered here. This thesis describes the implementation of a 3D cosmological model based on Newtonian hydrodynamic equations in an expanding background. Initial data is constructed as a spectrum of perturbations, and evolved in the frequency domain using a pseudo-spectral evolution scheme and an explicit Runge-Kutta time integrator. The code is found to converge for both linear and non-linear evolutions, and the convergence rate is determined. The correct growth rates expected from analytical calculations are recovered in the linear case. In the non-linear model, we observe close correspondence with linear growth and are able to monitor the growth on features associated with the non-linearity. High-frequency aliasing effects were evident in the non-linear evolutions, leading to a study of two potential resolutions to this problem: a boxcar filter which adheres to“Orszag’s two thirds rule” and an exponential window function, the exponential filter suggested by Hou and Li [1], and a shifted version of the exponential filter suggested, which has the potential to alleviate high frequency- ripples resulting from the Gibbs’ phenomenon. We found that the filters were somewhat successful at reducing aliasing effects but that the Gibbs’ phenomenon could not be entirely removed by the choice of filters.
- Full Text:
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