arXiv subjects

Hubert Bray

Publications and source records attributed to Hubert Bray.

9 recordsLinked to original sources

arXiv2026

Spiral structure from the interference of gravitational eigenmodes in ultralight dark matter halos

We use a variety of analytic and numerical techniques to approximate the geometry of individual dark matter halos when modeled as collections of ultralight scalar bosons (ULDM), followed by an analysis on the future directions and implications of this work. Ultimately, we aim to understand the morphology and creation of complex structures (like spirals) in galaxies from a geometric standpoint and to investigate the connection between dark matter and baryonic matter in galaxies. We find that superpositions of gravitational eigenmodes for ULDM in a fixed background potential can yield spiral density perturbations which rotate on the order of one hundred million years and appear within ten to twenty kpc of the halo center, both features comparable to realistic galaxies, and that this model provides a natural explanation for the density waves necessary to seed baryonic spirals.

astro-ph.HE↗

arXiv2025

Topological Big Bangs: Reflection, Itty-Bitty Blenders, and Eternal Trumpets

We discuss and formalize topological means by which the initial singularity might be mollified, at the level of the spacetime manifold's structure, in classical cosmological models of a homogeneous expanding universe. One construction, dubbed a "reflective" topological big bang, generalizes Schrodinger's elliptic de Sitter space and is built to be compatible with the standard Friedmann-Lemaitre-Robertson-Walker (FLRW) picture of the large-scale universe, only minimally modifying it via some nontrivial topology at an earliest "moment" in the universe's history. We establish a mathematical characterization of the admissible topological structures of reflective topological big bangs, and we discuss implications for a standard concern in cosmology, the horizon problem. We present a nonreflective example that we've christened the Itty-Bitty Blender spacetime: this spacetime and its universal cover, the Eternal Trumpet spacetime, exhibit interesting potential structures of spacetimes avoiding the Hawking and Penrose singularity theorems. While these toy models provide a proof-of-concept picture, several questions remain regarding the capacity to realize these structures under physical energy conditions.

gr-qc↗

arXiv2024

The Riemannian Penrose Inequality with Matter Density

Riemannian Penrose Inequalities are precise geometric statements that imply that the total mass of a zero second fundamental form slice of a spacetime is at least the mass contributed by the black holes, assuming that the spacetime has nonnegative matter density everywhere. In this paper, we remove this last assumption, and prove stronger statements that the total mass is at least the mass contributed by the black holes, plus a contribution coming from the matter density along the slice. We use the first author's conformal flow to achieve this, combined with Stern's harmonic level set techniques in the first case, and spinors in the second case. We then compare these new results to results previously known from Huisken-Ilmanen's inverse mean curvature flow techniques.

math.DG↗

arXiv2021

Spacetime Harmonic Functions and Applications to Mass

In the pioneering work of Stern, level sets of harmonic functions have been shown to be an effective tool in the study of scalar curvature in dimension 3. Generalizations of this idea, utilizing level sets of so called spacetime harmonic functions as well as other elliptic equations, are similarly effective in treating geometric inequalities involving the ADM mass. In this paper, we survey recent results in this context, focusing on applications of spacetime harmonic functions to the asymptotically flat and asymptotically hyperbolic versions of the spacetime positive mass theorem, and additionally introduce a new concept of total mass valid in both settings which is encoded in interpolation regions between generic initial data and model geometries. Furthermore, a novel and elementary proof of the positive mass theorem with charge is presented, and the level set approach to the Penrose inequality given by Huisken and Ilmanen is related to the current developments. Lastly, we discuss several open problems.

math.DG↗

arXiv2019

Flatly Foliated Relativity

Flatly Foliated Relativity (FFR) is a new theory which conceptually lies between Special Relativity (SR) and General Relativity (GR), in which spacetime is foliated by flat Euclidean spaces. While GR is based on the idea that "matter curves spacetime", FFR is based on the idea that "matter curves spacetime, but not space". This idea, inspired by the observed spatial flatness of our local universe, is realized by considering the same action as used in GR, but restricting it only to metrics which are foliated by flat spatial slices. FFR can be thought of as describing gravity without gravitational waves. In FFR, a positive cosmological constant implies several interesting properties which do not follow in GR: the metric equations are elliptic on each euclidean slice, there exists a unique vacuum solution among those spherically symmetric at infinity, and there exists a geometric way to define the arrow of time. Furthermore, as gravitational waves do not exist in FFR, there are simple analogs to the positive mass theorem and Penrose-type inequalities. Importantly, given that gravitational waves have a negligible effect on the curvature of spacetime, and that the universe appears to be locally flat, FFR may be a good approximation of GR. Moreover, FFR still admits many notable features of GR including the big bang, an accelerating expansion of the universe, and the Schwarzschild spacetime. Lastly, FFR is already known to have an existence theory for some simplified cases, which provokes an interesting discussion regarding the possibility of a more general existence theory, which may be relevant to understanding existence of solutions to GR.

gr-qc↗

arXiv2019

Proof of Bishop's volume comparison theorem using singular soap bubbles

Bishop's volume comparison theorem states that a compact $n$-manifold with Ricci curvature larger than the standard $n$-sphere has less volume. While the traditional proof uses geodesic balls, we present another proof using isoperimetric hypersurfaces, also known as "soap bubbles," which minimize area for a given volume. Curiously, isoperimetric hypersurfaces can have codimension 7 singularities, an interesting challenge we are forced to overcome.

math.DG↗

arXiv2007

On The Capacity of Surfaces in Manifolds with Nonnegative Scalar Curvature

Given a surface in an asymptotically flat 3-manifold with nonnegative scalar curvature, we derive an upper bound for the capacity of the surface in terms of the area of the surface and the Willmore functional of the surface. The capacity of a surface is defined to be the energy of the harmonic function which equals 0 on the surface and goes to 1 at infinity. Even in the special case of Euclidean space, this is a new estimate. More generally, equality holds precisely for a spherically symmetric sphere in a spatial Schwarzschild 3-manifold. As applications, we obtain inequalities relating the capacity of the surface to the Hawking mass of the surface and the total mass of the asymptotically flat manifold.

math.DG↗

arXiv2006

Generalized inverse mean curvature flows in spacetime

Motivated by the conjectured Penrose inequality and by the work of Hawking, Geroch, Huisken and Ilmanen in the null and the Riemannian case, we examine necessary conditions on flows of two-surfaces in spacetime under which the Hawking quasilocal mass is monotone. We focus on a subclass of such flows which we call uniformly expanding, which can be considered for null as well as for spacelike directions. In the null case, local existence of the flow is guaranteed. In the spacelike case, the uniformly expanding condition leaves a 1-parameter freedom, but for the whole family, the embedding functions satisfy a forward-backward parabolic system for which local existence does not hold in general. Nevertheless, we have obtained a generalization of the weak (distributional) formulation of this class of flows, generalizing the corresponding step of Huisken and Ilmanen's proof of the Riemannian Penrose inequality.

gr-qc↗

arXiv2000

Curvature Estimates and the Positive Mass Theorem

The Positive Mass Theorem implies that any smooth, complete, asymptotically flat 3-manifold with non-negative scalar curvature which has zero total mass is isometric to (R^3, delta_{ij}). In this paper, we quantify this statement using spinors and prove that if a complete, asymptotically flat manifold with non-negative scalar curvature has small mass and bounded isoperimetric constant, then the manifold must be close to (R^3,delta_{ij}), in the sense that there is an upper bound for the L^2 norm of the Riemannian curvature tensor over the manifold except for a set of small measure. This curvature estimate allows us to extend the case of equality of the Positive Mass Theorem to include non-smooth manifolds with generalized non-negative scalar curvature, which we define.

math.DG↗