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MULTIDIMENSIONAL FORMULATION OF THE GENERAL RELATIVISTIC EQUATIONS OF MOTION
Title
MULTIDIMENSIONAL FORMULATION OF THE GENERAL RELATIVISTIC EQUATIONS OF MOTION.
Creator
BEYL, LAWRENCE MICHAEL., Florida Atlantic University, McGuire, James B.
Abstract/Description
A 4N-dimensional formalism is developed and a corresponding space is found. The necessity of coupling the particle's proper times to one parameter is discussed. The appropriate conditions and constraints which relate this space to the ordinary 4-space are found. The transformation properties which are consistent with general 4-space transformations are determined. These transformation properties are used to determine the form of the 4x4 matrices making up the 4N-dimensional metric tensor. The...
Show moreA 4N-dimensional formalism is developed and a corresponding space is found. The necessity of coupling the particle's proper times to one parameter is discussed. The appropriate conditions and constraints which relate this space to the ordinary 4-space are found. The transformation properties which are consistent with general 4-space transformations are determined. These transformation properties are used to determine the form of the 4x4 matrices making up the 4N-dimensional metric tensor. The form of these matrices indicates they represent interactions between particles. The diagonal matrices are shown to represent gravitational interaction and the off-diagonal matrices to represent other interparticle interactions. A metric theory to cover all interparticle interactions is then proposed. The equations of motion for one particle in this 4N-dimensional space are found. These equations are then related to the motion of N interacting particles in 4- space . Finally, an approximation procedure is applied to determine the first order equations of motion.
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Date Issued
1974
PURL
http://purl.flvc.org/fcla/dt/13659
Subject Headings
General relativity (Physics), Motion
Format
Document (PDF)
General relativistic neutrino transport in static post-collapse stellar cores
Title
General relativistic neutrino transport in static post-collapse stellar cores.
Creator
DeNisco, Kenneth R., Florida Atlantic University, Bruenn, Stephen W.
Abstract/Description
We examine the effects of general relativity on neutrino transport during the critical shock reheating phase of core collapse supernovae. We derive the equations for general relativistic, static, multi-group, flux-limited diffusion (MGFLD), and use these to modify a Newtonian MGFLD code. The steady-state neutrino distributions calculated by the revised code for several post-collapse stellar cores are compared to those computed with the original Newtonian MGFLD code. The general relativistic...
Show moreWe examine the effects of general relativity on neutrino transport during the critical shock reheating phase of core collapse supernovae. We derive the equations for general relativistic, static, multi-group, flux-limited diffusion (MGFLD), and use these to modify a Newtonian MGFLD code. The steady-state neutrino distributions calculated by the revised code for several post-collapse stellar cores are compared to those computed with the original Newtonian MGFLD code. The general relativistic transport calculations display the expected reductions in neutrino rms energy and luminosity arising from redshift and curvature effects. Although the effects of general relativity on neutrino transport work against the shock revival, firm conclusions cannot be drawn at this time because of other important general relativistic effects on the structure of the core. Nevertheless, it is clear that modeling with Newtonian transport has little connection to reality.
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Date Issued
1997
PURL
http://purl.flvc.org/fcla/dt/15479
Subject Headings
General relativity (Physics), Neutrinos, Supernovae
Format
Document (PDF)
SYMMETRY CHARGES ON REDUCED PHASE SPACE AND ASYMPTOTIC FLATNESS
Title
SYMMETRY CHARGES ON REDUCED PHASE SPACE AND ASYMPTOTIC FLATNESS.
Creator
Tan, Hongwei, Han, Muxin, Florida Atlantic University, Department of Physics, Charles E. Schmidt College of Science
Abstract/Description
Though general relativity (GR) is proven to be a successful theory in describing the macroscopical nature of our universe, it still has several problems to be resolved. One of them is known as the time problem of GR. GR is a pure constraint theory, and the time evolution of the system is a gauge transformation, without carrying any physical information. One potential resolution to this issue is the relational formalism, which considers the dynamics of a material frame by coupling it to...
Show moreThough general relativity (GR) is proven to be a successful theory in describing the macroscopical nature of our universe, it still has several problems to be resolved. One of them is known as the time problem of GR. GR is a pure constraint theory, and the time evolution of the system is a gauge transformation, without carrying any physical information. One potential resolution to this issue is the relational formalism, which considers the dynamics of a material frame by coupling it to gravity. This approach allows for constructing gauge invariant observables and subsequent quantization. One realization of the relational formalism is the Brown-Kuchaˇr formalism. In this formalism, the gravity couples Brown-Kuchaˇr dust fields, and the Brown-Kuchaˇr dust fields play the roles as a family of observers. Then, one can introduce a gauge fixing scheme to the system and construct gauge invariant observables (Dirac observables) in the reduced phase Space. The probe time of the dust plays the role as the physical time of each point of the spacetime. In this thesis, we consider the Brown-Kuchaˇr formalism in an asymptotically flat background. A set of boundary conditions for the asymptotic flatness are formulated for Dirac observables on the reduced phase space. We compute the boundary term of the physical Hamiltonian, which is identical to the ADM mass. We construct a set of the symmetry charges on the reduced phase space, which encompass both the bulk terms and the boundary terms are conserved by the physical Hamiltonian evolution. The symmetry charges generate transformations preserving the asymptotically flat boundary conditions. Under the reduced-phase space Poisson bracket, the symmetry charges form an infinite dimensional Lie algebra AG after adding a central charge. A suitable quotient of AG is analogous to the BMS algebra at spatial infinity by Henneaux and Troessaert.
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Date Issued
2024
PURL
http://purl.flvc.org/fau/fd/FA00014398
Subject Headings
General relativity (Physics), Physics, Space and time, Spacetime
Format
Document (PDF)
Binary black hole mergers
Title
Binary black hole mergers: alternatives to standard puncture initial data and the impact on gravitational waveforms.
Creator
Reifenberger, George C., Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
We study the evolution of binary black hole initial data schemes as alternatives to the standard puncture construction. These alternatives are based on post-Newtonian expansions that contain realistic gravitational waves. The first scheme is based on a second order post-Newtonian expansion in Arnowitt, Deser, and Misner transverse-traceless (ADMTT) gauge that has been re-summed to approach standard puncture data at the black holes. The other schemes are based on asymptotic matching of the 4...
Show moreWe study the evolution of binary black hole initial data schemes as alternatives to the standard puncture construction. These alternatives are based on post-Newtonian expansions that contain realistic gravitational waves. The first scheme is based on a second order post-Newtonian expansion in Arnowitt, Deser, and Misner transverse-traceless (ADMTT) gauge that has been re-summed to approach standard puncture data at the black holes. The other schemes are based on asymptotic matching of the 4-metrics of two tidally perturbed Schwarzschild solutions to post-Newtonian expansions at different orders. The alternatives are encouraging and lead to quasi-circular orbits and show gravitational radiation from the onset, as well as a reduction of spurious radiation. Deficiencies compared to punctures include more eccentricity during inspiral and larger constraint violations, since the alternative data sets only approximate solutions of Einstein's equations.
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Date Issued
2013
PURL
http://purl.flvc.org/fcla/dt/3362577
Subject Headings
Black holes (Astronomy), General relativity (Physics), Gravitational waves, Inertia (Mechanics), Mass (Physics), Particles (Nuclear physics)
Format
Document (PDF)
Spin-foam dynamics of general relativity
Title
Spin-foam dynamics of general relativity.
Creator
Chaharsough Shirazi, Atousa, Engle, Jonathan S., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
In this dissertation the dynamics of general relativity is studied via the spin-foam approach to quantum gravity. Spin-foams are a proposal to compute a transition amplitude from a triangulated space-time manifold for the evolution of quantum 3d geometry via path integral. Any path integral formulation of a quantum theory has two important parts, the measure factor and a phase part. The correct measure factor is obtained by careful canonical analysis at the continuum level. The basic...
Show moreIn this dissertation the dynamics of general relativity is studied via the spin-foam approach to quantum gravity. Spin-foams are a proposal to compute a transition amplitude from a triangulated space-time manifold for the evolution of quantum 3d geometry via path integral. Any path integral formulation of a quantum theory has two important parts, the measure factor and a phase part. The correct measure factor is obtained by careful canonical analysis at the continuum level. The basic variables in the Plebanski-Holst formulation of gravity from which spin-foam is derived are a Lorentz connection and a Lorentz-algebra valued two-form, called the Plebanski two-form. However, in the final spin-foam sum, one usually sums over only spins and intertwiners, which label eigenstates of the Plebanski two-form alone. The spin-foam sum is therefore a discretized version of a Plebanski-Holst path integral in which only the Plebanski two-form appears, and in which the conne ction degrees of freedom have been integrated out. Calculating the measure factor for Plebanksi Holst formulation without the connection degrees of freedom is one of the aims of this dissertation. This analysis is at the continuum level and in order to be implemented in spin-foams one needs to properly discretize and quantize this measure factor. The correct phase is determined by semi-classical behavior. In asymptotic analysis of the Engle-Pereira-Rovelli-Livine spin-foam model, due to the inclusion of more than the usual gravitational sector, more than the usual Regge term appears in the asymptotics of the vertex amplitude. As a consequence, solutions to the classical equations of motion of GR fail to dominate in the semi-classical limit. One solution to this problem has been proposed in which one quantum mechanically imposes restriction to a single gravitational sector, yielding what has been called the “proper” spin-foam model. However, this revised model of quantum gravity, like any proposal for a theory of quantum gravity, must pass certain tests. In the regime of small curvature, one expects a given model of quantum gravity to reproduce the predictions of the linearized theory. As a consistency check we calculate the graviton two-point function predicted by the Lorentzian proper vertex and examine its semiclassical limit.
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Date Issued
2015
PURL
http://purl.flvc.org/fau/fd/FA00004488, http://purl.flvc.org/fau/fd/FA00004488
Subject Headings
General relativity (Physics), Gravitation, Mass (Physics), Mathematical physics, Quantum gravity, Quantum theory
Format
Document (PDF)
Binary neutron star mergeres: simulations with arbitrarily spinning stars
Title
Binary neutron star mergeres: simulations with arbitrarily spinning stars.
Creator
Tsatsin, Petr, Marronetti, Pedro, Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
The starting point of any general relativistic numerical simulation is a solution of the Hamiltonian and momentum constraints that (ideally) represents an astrophysically realistic scenario. This dissertation presents a new method to produce initial data sets for binary neutron stars with arbitrary spins and orbital eccentricities. The method only provides approximate solutions to the constraints. However, it was shown that the corresponding constraint violations subside after a few orbits,...
Show moreThe starting point of any general relativistic numerical simulation is a solution of the Hamiltonian and momentum constraints that (ideally) represents an astrophysically realistic scenario. This dissertation presents a new method to produce initial data sets for binary neutron stars with arbitrary spins and orbital eccentricities. The method only provides approximate solutions to the constraints. However, it was shown that the corresponding constraint violations subside after a few orbits, becoming comparable to those found in evolutions of standard conformally flat, helically symmetric binary initial data. This dissertation presents the first spinning neutron star binary simulations in circular orbits with a orbital eccentricity less then 0.01. The initial data sets corresponding to binaries with spins aligned, zero and anti-aligned with the orbital angular momentum were evolved in time. These simulations show the orbital “hang-up” effect previously seen in binary black holes. Additionally, they show orbital eccentricities that can be up to one order of magnitude smaller than those found in helically symmetric initial sets evolutions.
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Date Issued
2013
PURL
http://purl.flvc.org/fau/fd/FA0004067
Subject Headings
Astrophysics, Black holes (Astronomy), General relativity (Physics), Gravitational waves, Neutron stars, Particles (Nuclear physics)
Format
Document (PDF)
Quasi-local energy of rotating black hole spacetimes and isometric embeddings of 2-surfaces in Euclidean 3-space
Title
Quasi-local energy of rotating black hole spacetimes and isometric embeddings of 2-surfaces in Euclidean 3-space.
Creator
Ray, Shannon, Miller, Warner A., Florida Atlantic University, Charles E. Schmidt College of Science, Department of Physics
Abstract/Description
One of the most fundamental problems in classical general relativity is the measure of e↵ective mass of a pure gravitational field. The principle of equivalence prohibits a purely local measure of this mass. This thesis critically examines the most recent quasi-local measure by Wang and Yau for a maximally rotating black hole spacetime. In particular, it examines a family of spacelike 2-surfaces with constant radii in Boyer-Lindquist coordinates. There exists a critical radius r* below which,...
Show moreOne of the most fundamental problems in classical general relativity is the measure of e↵ective mass of a pure gravitational field. The principle of equivalence prohibits a purely local measure of this mass. This thesis critically examines the most recent quasi-local measure by Wang and Yau for a maximally rotating black hole spacetime. In particular, it examines a family of spacelike 2-surfaces with constant radii in Boyer-Lindquist coordinates. There exists a critical radius r* below which, the Wang and Yau quasi-local energy has yet to be explored. In this region, the results of this thesis indicate that the Wang and Yau quasi-local energy yields complex values and is essentially equivalent to the previously defined Brown and York quasi-local energy. However, an application of their quasi-local mass is suggested in a dynamical setting, which can potentially give new and meaningful measures. In supporting this thesis, the development of a novel adiabatic isometric mapping algorithm is included. Its purpose is to provide the isometric embedding of convex 2-surfaces with spherical topology into Euclidean 3-space necessary for completing the calculation of quasilocal energy in numerical relativity codes. The innovation of this algorithm is the guided adiabatic pull- back routine. This uses Ricci flow and Newtons method to give isometric embeddings of piecewise simplicial 2-manifolds, which allows the algorithm to provide accuracy of the edge lengths up to a user set tolerance.
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Date Issued
2017
PURL
http://purl.flvc.org/fau/fd/FA00004865, http://purl.flvc.org/fau/fd/FA00004865
Subject Headings
Gravitational fields., General relativity (Physics), Newton-Raphson method., Ricci flow.
Format
Document (PDF)
Towards reading line profiles of Bardeen-Petterson accretion disks
Title
Towards reading line profiles of Bardeen-Petterson accretion disks.
Creator
Vandernoot, Eric P., Florida Atlantic University, Miller, Warner A.
Abstract/Description
Images of highly idealized Bardeen-Petterson accretion disks around Kerr black holes and their generated energy spectra line profiles are created via computer simulations in this work. The line profiles are examined in the relation to the original disk parameters to demonstrate that Bardeen-Petterson disks can be the source of the spectra we observe, even of actual systems. The challenges in computer programming the simulations and methods to overcome the challenges are discussed in the paper...
Show moreImages of highly idealized Bardeen-Petterson accretion disks around Kerr black holes and their generated energy spectra line profiles are created via computer simulations in this work. The line profiles are examined in the relation to the original disk parameters to demonstrate that Bardeen-Petterson disks can be the source of the spectra we observe, even of actual systems. The challenges in computer programming the simulations and methods to overcome the challenges are discussed in the paper. Also discussed are the future work, improvements and intentions for the simulations. Finally, the discussion turns to the usage of indicators in these line profiles to predict parameters of the original disk systems.
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Date Issued
2005
PURL
http://purl.flvc.org/fcla/dt/13287
Subject Headings
Black holes (Astronomy), Accretion (Astrophysics), General relativity (Physics), Gravitation, Petterson, Jacobus Antonius,--1946-
Format
Document (PDF)