Cosmology, Relativity, and Gravity Commons^™

Black Holes, Disk Structures, And Cosmological Implications In E-Dimensional Space, Subhash Kak, Menas C. Kafatos

Mathematics, Physics, and Computer Science Faculty Articles and Research

We examine a modern view of the universe that builds on achieved successes of quantum mechanics, general relativity, and information theory, bringing them together in integrated approach that is founded on the realization that space itself is e-dimensional. The global and local implications of noninteger dimensionality are examined, and how it may have increased from the value of zero to its current value is investigated. We find surprising aspects that tie to structures in the universe, black holes, and the role of observations.

What's New In Differentialgeometry Release Dg2022, Ian M. Anderson, Charles G. Torre

Tutorials on... in 1 hour or less

This Maple worksheet demonstrates the salient new features and functionalities of the 2022 release of the DifferentialGeometry software package.

Spacetime Groups, Ian M. Anderson, Charles G. Torre

Publications

A spacetime group is a connected 4-dimensional Lie group G endowed with a left invariant Lorentz metric h and such that the connected component of the isometry group of h is G itself. The Newman-Penrose formalism is used to give an algebraic classification of spacetime groups, that is, we determine a complete list of inequivalent spacetime Lie algebras, which are pairs (g,η), with g being a 4-dimensional Lie algebra and η being a Lorentzian inner product on g. A full analysis of the equivalence problem for spacetime Lie algebras is given which leads to a completely algorithmic solution to the …

How To Make Tetrads, Charles G. Torre

How to... in 10 minutes or less

This is a worksheet which demonstrates tools for creating orthonormal and null tetrads for a given spacetime.

Introduction To The Usu Library Of Solutions To The Einstein Field Equations, Ian M. Anderson, Charles G. Torre

Tutorials on... in 1 hour or less

This is a Maple worksheet providing an introduction to the USU Library of Solutions to the Einstein Field Equations. The library is part of the DifferentialGeometry software project and is a collection of symbolic data and metadata describing solutions to the Einstein equations.

Exotic Statistics For Strings In 4d Bf Theory, John C. Baez, Derek K. Wise, Alissa S. Crans

Alissa Crans

After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can …

Black Holes In The Conical Ensemble, Robert Mcnees, Daniel Grumiller

Robert A McNees IV

We consider black holes in an “unsuitable box”: a finite cavity coupled to a thermal reservoir at a temperature different than the black hole’s Hawking temperature. These black holes are described by metrics that are continuous but not differentiable due to a conical singularity at the horizon. We include them in the Euclidean path integral sum over configurations, and analyze the effect this has on black hole thermodynamics in the canonical ensemble. Black holes with a small deficit (or surplus) angle may have a smaller internal energy or larger density of states than the nearby smooth black hole, but they …

On The Existence And Uniqueness Of Static, Spherically Symmetric Stellar Models In General Relativity, Josh Michael Lipsmeyer

Masters Theses

The "Fluid Ball Conjecture" states that a static stellar model is spherically symmetric. This conjecture has been the motivation of much work since first mentioned by Kunzle and Savage in 1980. There have been many partial results( ul-Alam, Lindblom, Beig and Simon,etc) which rely heavily on arguments using the positive mass theorem and the equivalence of conformal flatness and spherical symmetry. The purpose of this paper is to outline the general problem, analyze and compare the key differences in several of the partial results, and give existence and uniqueness proofs for a particular class of equations of state which represents …

Geometrization Conditions For Perfect Fluids, Scalar Fields, And Electromagnetic Fields, Charles G. Torre, Dionisios Krongos

Charles G. Torre

Rainich-type conditions giving a spacetime “geometrization” of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and electromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equations are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Geometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and formulas for constructing the scalar field from …

Schwarzschild Spacetime And Friedmann-Lemaitre-Robertson-Walker Cosmology, Zachary Cohen

Honors Scholar Theses

The advent of General Relativity via Einstein's field equations revolutionized our understanding of gravity in our solar system and universe. The idea of General Relativity posits that gravity is entirely due to the geometry of the universe -- that is, the mass distribution throughout the universe results in the "curving" of spacetime, which gives us the physics we see on a large scale. In the framework of General Relativity, we find that the universe behaves differently than was predicted in the model of gravitation developed by Newton. We will derive the general relativistic model for a simple system near a …

Geometrization Conditions For Perfect Fluids, Scalar Fields, And Electromagnetic Fields, Charles G. Torre, Dionisios Krongos

Presentations and Publications

Rainich-type conditions giving a spacetime “geometrization” of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and elec- tromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equa- tions are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Ge- ometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and for- mulas for constructing …

Gaussian Cosmology: A New Model For The Accelerated Expansion Of The Universe, Brian Robert Neils Strigel

Senior Projects Fall 2015

In this paper I lay the groundwork for an alternative model for the contemporary expansion of the universe. The current model states that the universe is growing exponentially due to the vacuum of space pulling on it. This model states that the growth rate of the universe is linear in time. However in 1998, researchers suggested that the expansion rate of the universe is accelerating. This means the universe is not expanding logarithmically, not linearly. In my model, I lay the groundwork for future research and suggest that the universe develops in time in accordance to a Gaussian scale factor …

Static Interaction Of Black Holes In 1+1 Dimensions, Philip Greenwood

Physics

We consider a version of general relativity in two spacetime dimensions, and

study a solution describing two static black holes in the presence of a cosmological

constant. We first analytically find an embedding diagram to visualize the geometry

outside the black holes. We then examine how the two black holes must be

interacting to remain static. Our main result is to show how the black holes behave

effectively like two electric charges. This charge model exhibits both attraction and

repulsion, which evidently balance and moreover are localized in different regions

of space. We also begin an investigation of the black …

Perihelion Precession In General Relativity, Charles G. Torre

Charles G. Torre

This is a Maple worksheet providing a relatively quick and informal sketch of a demonstration that general relativistic corrections to the bound Kepler orbits introduce a perihelion precession. Any decent textbook will derive this result. My analysis aligns with that found in the old text "Introduction to General Relativity", by Adler, Bazin and Schiffer. The plan of the analysis is as follows. * Model the planetary orbits as geodesics in the (exterior) Schwarzschild spacetime. * Compute the geodesic equations. * Simplify them using symmetries and first integrals. * Isolate the differential equation expressing the radial coordinate as a function of …

The Spacetime Geometry Of A Null Electromagnetic Field, Charles G. Torre

Charles G. Torre

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null electrovacuum, a straightforward procedure builds the null electromagnetic field from …

The Spacetime Geometry Of A Null Electromagnetic Field, Charles G. Torre

Presentations and Publications

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null electrovacuum, a straightforward procedure builds the null electromagnetic field from …

Rainich-Type Conditions For Null Electrovacuum Spacetimes I, Charles G. Torre

Research Vignettes

In this worksheet I describe local Rainich-type conditions on a spacetime geometry which are necessary and sufficient for the existence of a solution of the Einstein-Maxwell equations with a null electromagnetic field. When it exists, the electromagnetic field is easily constructed.

Black Holes In The Conical Ensemble, Robert A. Mcnees Iv, Daniel Grumiller

Physics: Faculty Publications and Other Works

We consider black holes in an “unsuitable box”: a finite cavity coupled to a thermal reservoir at a temperature different than the black hole’s Hawking temperature. These black holes are described by metrics that are continuous but not differentiable due to a conical singularity at the horizon. We include them in the Euclidean path integral sum over configurations, and analyze the effect this has on black hole thermodynamics in the canonical ensemble. Black holes with a small deficit (or surplus) angle may have a smaller internal energy or larger density of states than the nearby smooth black hole, but they …

Exotic Statistics For Strings In 4d Bf Theory, John C. Baez, Derek K. Wise, Alissa S. Crans

Mathematics Faculty Works

After a review of exotic statistics for point particles in 3d BF theory, and especially 3d quantum gravity, we show that string-like defects in 4d BF theory obey exotic statistics governed by the 'loop braid group'. This group has a set of generators that switch two strings just as one would normally switch point particles, but also a set of generators that switch two strings by passing one through the other. The first set generates a copy of the symmetric group, while the second generates a copy of the braid group. Thanks to recent work of Xiao-Song Lin, we can …