Physics Commons^™

Divergence-Free Tensor Densities In Two Dimensions, Tyler Hansen

All Graduate Theses and Dissertations, Fall 2023 to Present

In physics, a common method for exploring the way a physical system changes over time is to look at the system’s energy. Roughly speaking, the energy in these systems are either motion-based (kinetic energy, a bullet in flight) or position-based (potential energy, a rock sitting at the top of a hill). The difference between the system’s total kinetic and potential energies is quantified by an expression called the Lagrangian. Using a special procedure, this Lagrangian is massaged to produce a group of equations called the Euler-Lagrange equations; if the initial configuration of the system is provided, the solution to these …

Equivalence: A Covariantly Constant Problem In General Relativity, Jaren Hobbs

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

In studying the space-time structures described by Einstein’s theory of general relativity, it is often useful to identify particular properties referred to as geometrical invariants. These are attributes of the space-times which do not change regardless of the underlying coordinate systems used to study them. This project is part of a larger effort to catalogue space-times studied in general relativity. Specifically, computational software was used to identify structures known as covariantly constant vector fields.

Fundamental Aspects Of Black Holes, Jacob Fisher Ciafre

All Graduate Plan B and other Reports, Spring 1920 to Spring 2023

The literature study here seeks to present the foundations of black hole physics in General Relativity. The report includes a discussion of the Kerr black hole metric, black hole entropy, particle creation, the laws of black hole mechanics, and a bilinear mass formula for the Kerr-Newman black hole solution.

Blandford-Znajek Process In Vacuo And Its Holographic Dual, Ted Jacobson, Maria J. Rodriguez

All Physics Faculty Presentations

Blandford and Znajek discovered a process by which a spinning black hole can transfer rotational energy to a plasma, offering a mechanism for energy and jet emissions from quasars. Here we describe a version of this mechanism that operates with only vacuum electromagnetic fields outside the black hole. The setting, which is not astrophysically realistic, involves either a cylindrical black hole or one that lives in 2+1 spacetime dimensions, and the field is given in simple, closed form for a wide class of metrics. For asymptotically anti–de Sitter black holes in 2+1 dimensions, the holographic dual of this mechanism is …

General Relativity As A Biconformal Gauge Theory, James Thomas Wheeler

All Physics Faculty Publications

We consider the conformal group of a space of dim n=p+q, with SO(p,q) metric. The quotient of this group by its homogeneous Weyl subgroup gives a principal fiber bundle with 2n-dim base manifold and Weyl fibers. The Cartan generalization to a curved 2n-dim geometry admits an action functional linear in the curvatures. Because symmetry is maintained between the translations and the special conformal transformations in the construction, these spaces are called biconformal; this same symmetry gives biconformal spaces overlapping structures with double field theories, including manifest T-duality. We establish that biconformal geometry is …

Does The Black Hole Shadow Probe The Event Horizon Geometry?, Pedro V. P. Cunha, Carlos A. R. Herdeiro, Maria J. Rodriguez

All Physics Faculty Publications

There is an exciting prospect of obtaining the shadow of astrophysical black holes (BHs) in the near future with the Event Horizon Telescope. As a matter of principle, this justifies asking how much one can learn about the BH horizon itself from such a measurement. Since the shadow is determined by a set of special photon orbits, rather than horizon properties, it is possible that different horizon geometries yield similar shadows. One may then ask how sensitive is the shadow to details of the horizon geometry? As a case study, we consider the double Schwarzschild BH and analyze the impact …

General Relativity, 1, David Peak

General Relativity

In special relativity, events occur in the arena of space-time which may be coordinatized differently by different observers, but which is otherwise immutable. Adding gravity to relativity provides an amazing result: space-time becomes “organic,” taking its form from the matter and energy it contains. This is Einstein’s general theory of relativity and it has the capacity to tell us about the past and future of the universe. Embedded in the history book of the cosmos are several chapters on the origins of matter. As a result, relativity + gravity unites the structures of matter on the largest and smallest scales.

General Relativity, 7, David Peak

General Relativity

The expanding universe

The fact that the vast majority of galaxies have a spectral redshift can be interpreted as implying that the universe is expanding. This interpretation stems from the Doppler effect in which the relative motion of an emitter and a detector produces a frequency shift of the detected light with respect to the emitted light.

General Relativity, 8, David Peak

General Relativity

The Cosmic Microwave Background (CMB)

As previously noted, the universe is filled with microwave radiation. The frequency spectrum of this ubiquitous radiation follows a blackbody curve, as shown to the right. (http://map.gsfc.nasa.gov/media/ContentMedia/990015b.jpg) Note that photon energy (proportional to 1/wavelength) increases to the right. You might think the curve shown is the plot of a theoretical equation, but what is shown is actual measured data taken during the flight of the COBE (Cosmic Microwave Explorer) satellite/microwave observatory in 1990. The uncertainties in the measurements are about the thickness of the curve plotted. When compared with a theoretical blackbody curve the disagreement …

General Relativity, 4, David Peak

General Relativity

Orbital motion of small test masses

The starting point for analyzing free fall trajectories in the (2-space, 1-time) Schwarzschild spacetime is Equation (3) from GR 3:

General Relativity, 3, David Peak

General Relativity

Gravity as geometry: part II

Even in a region of space-time that is so small that tidal effects cannot be detected, gravity still seems to produce curvature. The argument for this point of view starts with the recognition that, for mechanical systems, it is impossible to distinguish a frame of reference with a uniform gravitational field from a uniformly accelerating frame that has no gravity. Thus, for example, in a (small) rocket ship with no windows it is not possible to determine whether the weight one reads standing on a scale at the tail of the rocket is due to …

General Relativity, 5, David Peak

General Relativity

No abstract provided.

General Relativity, 6, David Peak

General Relativity

Modern cosmography

The “normal” matter in the universe—i.e., stuff made of protons, neutrons, and electrons— consists, approximately, of lumps floating in a dilute fog. The lumps are galaxies, clusters of 10⁷ to 10¹¹ stars bound together by gravity. In the currently observable universe, it is estimated that there are roughly 10¹¹ galaxies. The dilute fog is primarily neutral atomic hydrogen gas with some helium-4 mixed in (making up a total of 98% or more of the fog); there are also very small fractions of ²H (deuterium), ³He , and ⁷Li . The ratio …

General Relativity, 2, David Peak

General Relativity

Newton’s law of gravitostatics is incompatible with special relativity. To see this, suppose at time t in frame O m₁ is at x₁(t) and m₂ is at x₂(t). Newton’s gravitational force law says F_{1on 2}(t) = Gm₁m₂ / [x₂(t) − x₁(t)]² and relativistic dynamics says dp₂ /dt = F_{1on 2}. Transforming to another frame Oʹ moving relative to O leads to dp′₂/dt′ = F′_{1on 2}. But what is F′_{1on 2} ? If x₁(t) and …

General Relativity, 9, David Peak

General Relativity

Cosmic development

As discussed in GR 8, the cosmic scale factor a in the FLWR s-t obeys the Friedmann equation

Wormholes: Gates To The Stars?, Samuel Shreeve

Physics Capstone Projects

One of the most consistently fascinating results of Albert Einstein’s theory of general relativity is the prediction of wormholes – astronomical objects which are, among other things, capable of serving as a connection between two distant regions of space. The simplest class of wormholes are Schwarzschild wormholes – wormholes that behave as non-rotating, non-charged black holes, except that the event horizon serves as a connection to another wormhole elsewhere, instead of a point of no return.

This research presentation analyzes the attributes that make a Schwarzschild wormhole unsuitable for human travel, and examines the conditions that would have to hold …

The Equivalence Problem: Einstein­-Maxwell Solutions, Rebecca Whitney

Physics Capstone Projects

The “Equivalence Problem” is part of the Digital Einstein Project. The goal of this project is to create a digital and interactive library of all known solutions to the Einstein field equations in general relativity. The “Equivalence Problem” involves determining when two solutions are physically equivalent. This requires calculating physical and geometric features to characterize each solution independently of any coordinate system. One of the principal features used to characterize the solutions is the degree of symmetry or the isometry group of the space­-time metric. We have focused on the solutions to the Einstein­-Maxwell field equations and compared the isometry …

The Schwarzschild Solution And Timelike Geodesics, Matthew Ross

Physics Capstone Projects

General Relativity is the standard theory of the gravitational interaction. It allows us to cal- culate the motions and interactions of particles in a non-Euclidean space-time. This presentation will present the derivation of the Schwarzschild metric tensor field by finding a solution of the Einstein Equation for a non-rotating, static vacuum. A general form of the metric for a static, spherically symmetric spacetime will be used to calculate the Riemann curvature tensor and sub- sequently the Ricci tensor and Ricci scalar which will then be used to find a vaccum solution to the Einstein Equation. Once the solutions of the …

The Schwarzschild Solution And Timelike Geodesics, Matthew B. Ross

Physics Capstone Projects

General Relativity is a major area of study in physics. It allows us to calculate the motion and interaction of particles in a non-Euclidean space-time. This presentation will examine the process of finding the Schwarzschild metric tensor field by finding a solution of the Einstein Equation for a non-rotating spherical mass. A general form of the Schwarzschild metric tenor field will be used to calculate the Riemann curvature tensor and subsequently the Ricci tensor and Ricci scalar which will be used to find a vacuum solution to the Einstein Equation. Once the solutions of the Einstein Equation are found, they …

Time And Dark Matter From The Conformal Symmetries Of Euclidean Space, Jeffrey S. Hazboun, James Thomas Wheeler

All Physics Faculty Publications

The quotient of the conformal group of Euclidean 4-space by its Weyl subgroup results in a geometry possessing many of the properties of relativistic phase space, including both a natural symplectic form and non-degenerate Killing metric. We show that the general solution posesses orthogonal Lagrangian submanifolds, with the induced metric and the spin connection on the submanifolds necessarily Lorentzian, despite the Euclidean starting pont. By examining the structure equations of the biconformal space in an orthonormal frame adapted to its phase space properties, we also find that two new tensor fields exist in this geometry, not present in Riemannian geometry. …

A Systematic Construction Of Curved Phase Space: A Gravitational Gauge Theory With Symplectic Form, Jeffrey Hazboun, James Thomas Wheeler

Presentations

No abstract provided.

Brst Structure Of General Relativity In Terms Of New Variables, Charles G. Torre

All Physics Faculty Publications

The structure of the Poisson-brackets algebra of constraints of general relativity is reexamined using the recently introduced spinorial variables. Three different combinations of constraints are analyzed and their relative merits are discussed. In each case we construct the corresponding expression of the Becchi-Rouet-Stora-Tyutin charge. These expressions provide a point of departure for a nonperturbative quantization scheme for general relativity.