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We present a numerical study of the local stability of mean curvature flow of rotationally symmetric, complete noncompact hypersurfaces with Type-II curvature blowup. Our numerical analysis employs a novel overlap method that constructs "numerically global" (i.e., with spatial domain arbitrarily large but finite) flow solutions with initial data covering analytically distinct regions. Our numerical results show that for certain prescribed families of perturbations, there are two classes of initial data that lead to distinct behaviors under mean curvature flow. Firstly, there is a "near" class of initial data which lead to the same singular behaviour as an unperturbed solution; in particular, the curvature at the tip of the hypersurface blows up at a Type-II rate no slower than $(T-t)^{-1}$. Secondly, there is a "far" class of initial data which lead to solutions developing a local Type-I nondegenerate neckpinch under mean curvature flow. These numerical findings further suggest the existence of a "critical" class of initial data which conjecturally lead to mean curvature flow of noncompact hypersurfaces forming local Type-II degenerate neckpinches with the highest curvature blowup rate strictly slower than $(T-t)^{-1}$., Comment: 23 pages, 11 figures. Comments are welcome!

In previous work [GIKW21], we have presented evidence from numerical simulations that the Type-II singularities of mean curvature flow (MCF) of rotationally-symmetric, complete, noncompact embedded hypersurfaces constructed in [IW19, IWZ21] are stable. More precisely, it is shown in that paper that for small rotationally-symmetric perturbations of initial embeddings near the "tip", numerical simulations of MCF of such initial embeddings develop the same Type-II singularities with the same "bowl soliton" blowup behaviors in a neighborhood of the singularity. It is also shown in that work that for small rotationally-symmetric perturbations of the initial embeddings that are sufficiently far away from the tip, MCF develops Type-I "neckpinch" singularities. In this work, we again use numerical simulations to show that MCF subject to initial perturbations that are not rotationally symmetric behaves asymptotically like it does for rotationally-symmetric perturbations. In particular, if we impose sinusoidal angular dependence on the initial embeddings, we find that for perturbations near the tip, evolutions by MCF asymptotically lose their angular dependence -- becoming round -- and develop Type-II bowl soliton singularities. As well, if we impose sinusoidal angular dependence on the initial embeddings for perturbations sufficiently far from the tip, the angular dependence again disappears as Type-I neckpinch singularities develop. The numerical analysis carried out in this work is an adaptation of the "overlap" method introduced in [GIKW21] and permits angular dependence., 17 pages, 13 figures. Comments are welcome!

Gravitational wave memory and its electromagnetic analog are shown to be straightforward consequences of the wave equation. From Maxwell's equations one can derive a wave equation for the electric field, while from the Bianchi identity one can derive a wave equation for the Riemann tensor in linearized gravity. Memory in both cases is derived from the structure of the source of those wave equations., figure and references added. Discussion expanded and clarified

In this paper, a non-trivial system governed by a continuum PT-symmetric Hamiltonian is discussed. We show that this Hamiltonian is iso-spectral to the simple harmonic oscillator. We find its eigenfunctions and the path in the complex plane along which these functions form an orthonormal set. We also find the hidden symmetry operator, ${\cal C}$, for this system. All calculations are performed analytically and without approximation., Comment: Typo in equation (9) corrected

We find initial data for numerical relativity simulations of inhomogeneous cosmologies. This involves treating an exceptional case of the general relativity constraint equations. We devise analytic and numerical methods to treat this exceptional case. We apply the analytic method to the standard case of cosmology with a single scalar field. The numerical method is applied to the two-field ekpyrotic cosmology.

The Blandford-Znajek mechanism is the continuous extraction of energy from a rotating black hole via plasma currents flowing on magnetic field lines threading the horizon. In the discovery paper, Blandford and Znajek demonstrated the mechanism by solving the equations of force-free electrodynamics in a perturbative expansion valid at small black hole spin. Attempts to extend this perturbation analysis to higher order have encountered inconsistencies. We overcome this problem using the method of matched asymptotic expansions, taking care to resolve all of the singular surfaces (light surfaces) in the problem. Working with the monopole field configuration, we show explicitly how the inconsistencies are resolved in this framework and calculate the field configuration to one order higher than previously known. However, there is no correction to the energy extraction rate at this order. These results confirm the basic consistency of the split monopole at small spin and lay a foundation for further perturbative studies of the Blandford-Znajek mechanism., SCOPUS: ar.j, info:eu-repo/semantics/published

We find the electric field of a point charge in the presence of a higher-dimensional black hole. As the charge is lowered to the horizon, all higher multipole moments go to zero, and only the Coulomb field remains.

We propose a method for numerical relativity in which the spatial grid is finite and no outer boundary condition is needed. As a "proof of concept" we implement this method for the case of a self-gravitating, spherically symmetric scalar field.

The recent detection of gravitational waves by the LIGO/VIRGO collaboration has allowed for the first tests of Einstein's theory in the extreme gravity regime, where the gravitational interaction is simultaneously strong, non-linear and dynamical. One such test concerns the rate at which the binaries inspiral, or equivalently the rate at which the gravitational wave frequency increases, which can constrain the existence of hairy black holes. This is because black holes with scalar hair typically excite dipole radiation, which in turn leads to a faster decay rate and frequency chirping. In this paper, we present a mathematical proof that scalar hair is not sourced in vacuum, spherically symmetric spacetimes when considering extensions of Einstein's theory that break parity in gravity, focusing on dynamical Chern-Simons theory as a particular toy model. This result implies that the observational confirmation of the baldness of black holes cannot be used to constrain parity violation in gravity, unless the black holes observed are also spinning., 10 pages and 2 figures. Published in Classical and Quantum Gravity

We study the motion of meteors and calculate the speed at which they hit the Earth. The calculation is done in a rotating system where the Earth and the Sun are at rest. Our treatment is a pedagogical account of some well established celestial mechanics results, suited for presentation in an intermediate mechanics course. The impact speed of meteors in several regularly occurring meteor showers agree very well with our calculation.

Content-Based Music Retrieval (CBMR) for symbolic music aims to find all similar occurrences of a musical pattern within a larger database of symbolic music. To the best of our knowledge there does not currently exist a distributable CBMR software package integrated with a music analysis toolkit that facilitates extendability with new CBMR methods. This project presents a new MIR tool called "PatternFinder" satisfying these goals. PatternFinder is built with the computational musicology Python package music21, which provides a flexible platform capable of working with many music notation formats. To achieve polyphonic CBMR, we implement seven geometric algorithms developed at the University of Helsinki---four of which are being implemented and released publicly for the first time. The application of our MIR tool is then demonstrated through a musicological investigation of Renaissance imitation masses, which borrow melodic or contrapuntal material from a pre-existing musical work. In addition, we show Pattern-Finder's ability to find a contrapuntal pattern over a large dataset, Palestrina's 104 masses. Our investigations demonstrate the relevance of our tool for musicological research as well as its potential application for locating music within digital music libraries.

We explicitly calculate the gravitational wave memory effect for classical point particle sources in linearized gravity off of an even dimensional Minkowski background. We show that there is no memory effect in $d>4$ dimensions, in agreement with the general analysis of Hollands, Ishibashi, and Wald (2016)., 14 pages. Typos corrected, reference added, eq. (11) corrected and clarified (results unchanged), arguments in secs. IV,V revised (results unchanged)

We describe neutrino radiation in general relativity by introducing the energy–momentum tensor of a null fluid into the Einstein equations. Investigating the geometry and analysis at null infinity, we prove that a component of the null fluid enlarges the Christodoulou memory effect of gravitational-waves. The description of neutrinos in general relativity as a null fluid can be regarded as a limiting case of a more general description using the massless limit of the Einstein–Vlasov system. Gigantic neutrino bursts occur in our universe in core-collapse supernovae and in the mergers of neutron star binaries.

I argue that black hole area should be thought of as an ``entropy-like quantity'' rather than an entropy. This precludes the need for a statistical mechanical description of black hole thermodynamics in terms of microstates.

We examine gravitational wave memory in the case where sources and detector are in an expanding cosmology. For simplicity, we treat the case where the cosmology is de Sitter spacetime, and discuss the possibility of generalizing our results to the case of a more realistic cosmology. We find results very similar to those of gravitational wave memory in an asymptotically flat spacetime, but with the magnitude of the effect multiplied by a redshift factor.

Though the main applications of computer simulations in relativity are to astrophysical systems such as black holes and neutron stars, nonetheless there are important applications of numerical methods to the investigation of general relativity as a fundamental theory of the nature of space and time. This paper gives an overview of some of these applications. In particular we cover (i) investigations of the properties of spacetime singularities such as those that occur in the interior of black holes and in big bang cosmology. (ii) investigations of critical behavior at the threshold of black hole formation in gravitational collapse. (iii) investigations inspired by string theory, in particular analogs of black holes in more than 4 spacetime dimensions and gravitational collapse in spacetimes with a negative cosmological constant., Review paper submitted to Reports on Progress in Physics

A simple estimate is given of gravitational wave memory for the inspiral and merger of a binary black hole system. Here the memory is proportional to the total energy radiated and has a simple angular dependence. This estimate might be helpful as a consistency check for numerical relativity memory waveforms., New section and new references added

The classical approach to spacetime singularities leads to a simplified dynamics in which spatial derivatives become unimportant compared to time derivatives, and thus each spatial point essentially becomes uncoupled from its neighbors. This uncoupled dynamics leads to sharp features (called "spikes") as follows: particular spatial points follow an exceptional dynamical path that differs from that of their neighbors, with the consequence that, in the neighborhood of these exceptional points, the spatial profile becomes ever more sharp. Spikes are consequences of the BKL-type oscillatory evolution towards generic singularities of spacetime. Do spikes persist when the spacetime dynamics is treated using quantum mechanics? To address this question, we treat a Hamiltonian system that describes the dynamics of the approach to the singularity and consider how to quantize that system. We argue that this particular system is best treated using an affine quantization approach (rather than the more familiar methods of canonical quantization) and we set up the formalism needed for this treatment. Our investigation, based on this affine approach, shows the nonexistence of quantum spikes., 28 pages, 2 figures, matches published version in Phys. Rev. D {\bf 95}, 024014 (2017)

In 1960 Eugene Wigner published an article titled “The Unreasonable Effectiveness of Mathematics in the Natural Sciences.”

The shape of the orbit of a free particle is examined in Friedmann-Lemaitre-Robertson-Walker (FLRW) cosmologies. The spatial projection of the orbit is time-independent and has a simple geometric description. We relate this description to the expression in terms of standard FLRW coordinates.

We perform numerical simulations of gravitational collapse in Einstein-aether theory. We find that under certain conditions, the collapse results in the temporary formation of a white hole horizon.

The Planck mass is often assumed to play a role only at the extremely high energy scales where quantum gravity becomes important. However, this mass plays a role in any physical system that involves gravity, quantum mechanics, and relativity. We examine the role of the Planck mass in determining the maximum mass of white dwarf stars.

Computational simulations of the electrodynamics of cardiac fibrillation yield a great deal of useful data and provide a framework for theoretical explanations of heart behavior. Extending the application of these mathematical models to defibrillation studies requires that a simulation should sustain fibrillation without defibrillation intervention. In accordance with the critical mass hypothesis, the simulated tissue should be of a large enough size. The choice of biperiodic boundary conditions sustains fibrillation for a longer duration than no-flux boundary conditions for a given area, and so is commonly invoked. Here, we show how this leads to a boundary condition artifact that may complicate the analysis of defibrillation efficacy; we implement an alternative coordinate scheme that utilizes spherical shell topology and mitigates singularities in the Laplacian found with the usual spherical curvilinear coordinate system.