Self-Organized Criticality in a Model of Loop Quantum Gravity

Summer Research (2008)

Continuation of the work of Tim and Sean.

Primordial Gravitational Waves with a LQG Correction Term

Senior Project

Using the Friedmann-Robertson-Walker (FRW) metric for a homogeneous and isotrpic universe, we introduce the Laplacian and use it to find the wave equation in a space time with the FRW metric. We claim that gravitational waves exist and they satisfy the wave equation for an expanding space time. We review the calculation by Dodelson to find the power spectrum from gravitational waves generated by the initial inflationary period of expansion of the universe. We investigate what gravitational waves generated during the initial expansion of the universe might look like with a quantum loop theory modification found by Yubo Lu. We describe an approach for analytical solutions to solve the equation for gravitational waves modified by loop quantum gravity effects.

Self-Organized Criticality in a Model of Loop Quantum Gravity

Summer Research (2007)

Loop quantum gravity proposes a physical theory for the smallest possible scales. A key feature of the theory is that space is not continuous, but discrete. A spin network represents this discrete space. In this representation edges labeled with color determine areas while the vertices, where the edges meet, determine volumes. We use a two-dimensional triangular spin network and evolve it in time to determine whether it exhibits “self-organized criticality”. Self-organized critical systems “evolve on their own,” without any outside tuning of parameters. Self-organized critical systems have the unique feature that there is a power law relation between the size of the effects and the frequency of their occurrence. Using a numerical simulation, we model this system and evolve it using specific rules that roughly correspond to LQG theory. We find evidence of self-organized criticality, with a power law relation F(s)=a*s-b, where F is the frequency of avalanches, s is the size, a is a constant, and b is approximately 0.9.

On Modified Dispersion Relations and the Chandrasekhar Mass Limit

Summer Research (2006,2007)
Senior Project
SUNY Downstate College of Medicine

Modified dispersion relations from effective field theory are shown to alter the Chandrasekhar mass limit. At exceptionally high densities, the modifications affect the pressure of a degenerate electron gas and can increase or decrease the mass limit, depending on the sign of the modifications. These changes to the mass limit are unlikely to be relevant for the astrophysics of white dwarf or neutron stars due to well-known dynamical instabilities that occur at lower densities.
Also, project titled, "Explanation for the Pioneer Anomaly Based on a Discrete Spacetime"

Quantum Cosmological Effects on Primordial Gravitational Waves

Summer Research (2006, 2007)
Cornell Law School

Primordial gravitational waves are oscillations of spacetime that were generated during the Big Bang. According to one approach to the theory of quantum gravity, geometry is discrete. In the very early universe, the effects of discrete geometry may be significant and so primordial gravitational waves may be affected. Two different classes of modified gravitational wave equations were derived by quantizing the inverse scale factor based on the Loop Quantum Cosmological model and the Husain-Winkler Cosmological model. The ambiguities in the parameterization were also studied. It was shown that the leading order contribution scales as a power of the ratio of the Planck lendgth to the scale factor at the end of inflation.

Galactic Heavyweights: Constraints on Hadronic Modified Dispersion Relations from UHE Cosmic Rays

Summer Research Project

In 1966, Kenneth Greisen, Vadim Kuzmin and Georgiy Zatsepin found that the upper threshold energy for charged particles from distant sources is 5 x 10^19 eV. However, current data indicates that some ultra high energy cosmic rays possess more energy than this GZK threshold. One way to solve this conundrum is through a modified, Lorentz-violating dispersion relation that involves a third order correction term. A new constraint on this correction term was derived from the kinematics of proton decay.

A No-Go Theorem for the q-Loop Algebra and Braided Preons as a Basis for the Standard Model

Summer Research Project
Williams College (undergraduate)

In the first part of the summer the consistency of the combinatorics of the Temperley-Lieb algebra and a deformation of the q-loop loop algebra was studied. It was found that no non-trivial, consistent deformation exists. In the second part, the prion model of Bilson-Thompson was investigated including problems of the model such as the modeling of mass and higher generations were explored.

Quantum Cosmological Effects on the Primordial Stochastic Gravitational Wave Background

Senior Project
Deloitte and Touche

A stochastic background of gravitational waves - spacetime oscillations -is expected to originate just after the big bang. Little is understood about the physics of universe at this very early period, which makes the gravitational background a source of great interest within the physics community. If detectable, these primordial gravity waves would reveal information about the geometry of the universe at the time of emission. Different cosmological models produce different gravitational wave spectra, and therefore it might be possible to determine which model generated these gravitational waves. Two particular models are studied, Loop Quantum Cosmology and the Husain-Winkler Cosmology. Modified wave equations, including corrections, are presented and the gravity wave energy densities for each are calculated. Although the order of the corrections differ, the result indicates that primordial gravitational waves in both cosmological models are observationally indistinguishable from those generated by the standard de Sitter model.

Anisotropic Mass Effects on a Foucault Pendulum

Senior Project and Summer 2004
Yale

Ben's poster (pdf)

More than 40 years ago Cocconi and Salpeter proposed a formulation of Mach's Principle and considered how a nearby massive object, the Galactic Center, could produce a anisotropy of inertial mass. They provided a matrix for the inertial mass of an object for this case. This matrix represents the inertial mass relative to the direction the object moves. We investigated the Foucault pendulum in a mass anisotropic scenario. We found that a Foucault pendulum that has anisotropic effects would process faster than the classical Foucault pendulum. At the end of one day the anisotropic Foucault pendulum is processing at a rate equal to one half of a percent faster than the classical Foucault pendulum.

Numerical Experiments in Spin Network Dynamics

Senior Project and Summer 2004
University of Virginia

Sean's poster(ppt)

In one approach to quantum gravity, space is represented as labeled graphs or spin networks. I wrote a program for a model 2-D quantum gravity in which space is a trianglular lattice. The program simulates quantum gravity dynamics, as done by Borissov and Gupta. It is set up such that each vertex is trivalent and the numbers on each of the three edges must obey consistency conditions. The conditions on the edge values represent the restrictions imposed by angular momentum conservation or gauge invariance. In the dynamic process when a random edge gets its value changed, potentially a neighboring edge must be changed in order to restore gauge invariance. This process can propagate through many vertices. Some of these vertices may be distant. We hope to observe long range cascades of this type which may be an indication of gravity, a long range force.

Energy and Momentum in Modified Special Relativity

Summer 2004
SUNY New Paltz (Computer Science)

Special Relativity and Quantum Theory are not entirely compatible. Certain modifications of Special Relativity, were looked at in an attempt to reconcile these theories. Specifically, modifications were made to Special Relativity to include a minimum length. Beginning with modified energy- momentum transformations an attempt was made to derive spacetime transformations which incorporate the set minimum length.

Is Doubly Special Relativity Consistent with the Relativity Principle?

Phys.Rev. D69 (2004) 105016

We investigate the implications of adding a second observer independent scale to Einstein's theory of Special Relativity. To observe whether the new theories, known as Doubly Special Relativity, are consistent with the relativity principle we calculate particle process kinematics for two of these theories. We demand d that all inertial observers must agree on the occurrence of physical processes. The results of our analysis suggest that one DSR is consistent with the relativity principle while another is not.

Quantum Gravity Effects on Ultra High Energy Particles

New J. Phys. 4 (2002) 57
Ph.D. Waterloo/Perimeter 2007
Utrecht

The discrete structure of space predicted by loop quantum gravity implies modified particle dispersion relations and deviation from Lorentz symmetry. Observations of ultra high energy particles allow a phenomenological study of the quantum gravity effects and pose constraints on parameters characterizing particles' interactions with discrete space. Constraints in a three dimensional space (corresponding to parameters for hadrons, leptons and massless particles) show that current observational data is consistent with the quantum gravity model. The allowed parameter space is strongly constrained in the positive region where the particles would be able to move propagate at speeds faster than light, and the sign ambiguity for the hadronic parameter is removed altogether.

On the Statistical Mechanics of Quantum Geometry

Class. Quant. Grav. 18 (2001) 5125-5141
Class. Quant. Grav. 18 (2001) 5293-5298

This thesis presents an original derivation of the Bekenstein-Hawking formula for the entropy of a black hole within the framework of loop quantum gravity. This derivation differs from preceding ones in that it models the black hole as a grand canonical ensemble and makes use of a recently introduced quasi-local energy operator. It is shown that the statistical mechanics of the model reduces to that of a simple non-interacting gas of distinguishable particles with spin. For temperatures low in comparison with the Planck temperature and boundaries large in comparison with the Planck area, the entropy of the system is shown to be proportional to area (with a logarithmic correction), providing a simple derivation of the Bekenstein-Hawking result (for a certain choice of the Immirzi parameter). Also in this limit, the quantum geometry on the boundary forms a "condensate" in the lowest energy level (j = 1/2). Finally, we relate our description, in terms of the grand canonical ensemble, to previous geometric entropy calculations, which made use of area ensembles.

Angle and Volume Studies in Quantized Space

Class. Quant. Grav. 19 (2002) 2211-2227
Apker Award Finalist
Ph.D. Chicago 2008
Indiana

"Spatial observables" such as area, volume, and angle are given by the eigenvalues of Hermitian operators on spin network states in loop quantum gravity. We present results obtained in our investigations of the angle and volume operators, two operators which act on the vertices of spin networks. We find that the minimum observable angle is inversely proportional to the square root of the total spin of the vertex, a fairly slow decrease to zero. We also present numerical results indicating that the angle operator can reproduce the classical angle distribution. The volume operator is significantly harder to investigate analytically; however, we present analytical and numerical results indicating that the volume of a region scales as the 3/2 power of its bounding surface, which corresponds to the classical model of space.