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David Garfinkle

Publications and source records attributed to David Garfinkle.

At least 37 records · Page 2Linked to original sources
arXiv2022

A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup: II

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.

math.DG↗
arXiv2022

Gravitational wave memory and the wave equation

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.

gr-qc↗
arXiv2021

Dynamical attractors in contracting spacetimes dominated by kinetically coupled scalar fields

We present non-perturbative numerical relativity simulations of slowly contracting spacetimes in which the scalar field driving slow contraction is coupled to a second scalar field through an exponential non-linear sigma model-type kinetic interaction. These models are important because they can generate a nearly scale-invariant spectrum of super-Hubble density fluctuations fully consistent with cosmic microwave background observations. We show that the non-linear evolution rapidly approaches a homogeneous, isotropic and flat Friedmann- Robertson-Walker (FRW) geometry for a wide range of inhomogeneous and anisotropic initial conditions. Ultimately, we find, the kinetic coupling causes the evolution to deflect away from flat FRW and towards a novel Kasner-like stationary point, but in general this occurs on time scales that are too long to be observationally relevant.

gr-qc↗
arXiv2021

Can Thorne-$\dot{\textrm{Z}}$ytkow Objects source GW190814-type events?

The LIGO-Virgo collaboration reported in their third run the coalescence event GW190814 involving a 2.6 $M_{\odot}$ object with a 23 $M_{\odot}$ black hole. In this letter we study the conditions under which Thorne-$\dot{\textrm{Z}}$ytkow objects (T$\dot{\textrm{Z}}$Os) can be connected to that type of events. We evaluate first the rate of appearance of T$\dot{\textrm{Z}}$Os in the local Universe. Under the assumption that T$\dot{\textrm{Z}}$Os eventually become low mass gap black holes we evaluate how those black holes end up in binaries with other stellar mass black holes and compare to the reported rate for GW190814-type of events (1-23 $\textrm{Gpc}^{-3} \textrm{yr}^{-1}$). We find that T$\dot{\textrm{Z}}$Os in dense stellar clusters can not explain the LIGO-Virgo rate without a T$\dot{\textrm{Z}}$Os population in the field providing a dominant contribution. We also find that T$\dot{\textrm{Z}}$Os formed within hierarchical triple systems in the field with the third more distant star being the progenitor of a stellar mass black hole, may be able to give a rate comparable to that of GW190814-type events. In that case, future observations should discover mergers between stellar mass and low mass gap black holes, with the lower mass spanning the entire low mass gap range.

astro-ph.HE↗
arXiv2021

A numerical stability analysis of mean curvature flow of noncompact hypersurfaces with Type-II curvature blowup

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 $(T-t)^{-1}$.

math.DG↗
arXiv2020

Spike behavior in the approach to spacetime singularities

We perform numerical simulations of the approach to spacetime singularities. The simulations are done with sufficient resolution to resolve the small scale features (known as spikes) that form in this process. We find an analytical formula for the shape of the spikes and show that the spikes in the simulations are well described by this formula.

gr-qc↗
arXiv2020

Cosmological initial data for numerical relativity

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.

gr-qc↗
arXiv2020

Consistent Blandford-Znajek Expansion

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.

astro-ph.HE↗
arXiv2019

A No-Boundary Method for Numerical Relativity

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.

gr-qc↗
arXiv2019

Hair loss in parity violating gravity

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.

gr-qc↗
arXiv2019

On "the'' electric field of a uniformly accelerating charge

The problem of the electric field of a uniformly accelerating charge is a longstanding one that has led to several issues. We resolve these issues using techniques from linguistics, cognitive psychology, and the mathematics of partial differential equations.

gr-qc↗
arXiv2019

Black hole entropy as a consequence of excision

Black hole entropy is shown to be a consequence of restricting our description of physics to the exterior of black holes. This precludes the need for a statistical mechanical description of this entropy in terms of microstates.

gr-qc↗
arXiv2018

The shape of the orbit in FLRW spacetimes

The shape of the orbit of a free particle is examined in 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.

gr-qc↗