Browsing by Author "Bray, Hubert L"

This dissertation is based on the manuscripts \cite{BHJ, Hirsch, HKK, HirschZhang}.The paper \cite{BHJ} is joint with Simon Brendle and Florian Johne, \cite{HKK} is joint with Demetre Kazaras and Marcus Khuri, \cite{HirschZhang} is joint with Yiyue Zhang, and \cite{Hirsch} is a solo-authored.

First, we introduce \emph{$m$-intermediate curvature $\mathcal C_m$} which interpolates between Ricci ($m=1$) and scalar curvature ($m=n-1$) and prove in this context a generalized Geroch conjecture \cite{BHJ}.In particular, we show that $M^{n-m}\times \mathbb T^m$, $n\le7$, does not admit a metric with $\mathcal C_m>0$.

Next we study initial data sets $(M,g,k)$ which are used in General Relativity to describe isolated gravitational systems.We introduce \emph{spacetime harmonic functions}, i.e. functions solving the PDE $\Delta u=-\tr_gk|\nabla u|$, to give a new lower bound for the mass of $(M,g,k)$. This lower bound in particular implies the spacetime positive mass theorem \cite{HKK} including the case of equality \cite{HirschZhang}.

Finally, we discuss recent progress towards the spacetime Penrose conjecture \cite{Hirsch}.We demonstrate how the famous monotonicity formula for the Hawking energy under inverse mean curvature flow can be generalized to initial data sets. This leads to new notion of \emph{spacetime inverse mean curvature flow} which is based on double null foliations.

Further papers I wrote during my time in graduate school \cite{BHHWZ, BHKKZ, BHKKZ2, HKKZ, HKKZ2, HirschLesourd, HirschLi, HirschMiao, HMT2, HMT, HirschZhu} will not be discussed.

Einstein's General Theory of Relativity sits among the pillars of modern physics as the means by which we describe the universe across an enormous range of scales. This theory has furnished our most robust understanding of the origins of the universe, the dynamics of astronomical objects, and the fundamental structure of space and time. For all of general relativity's successes, however, a wide array of deep questions remain. Its sophisticated mathematical structure renders foundational questions surrounding the extent to which the theory is well-posed difficult to answer (and indeed, difficult to ask), and consistent systematic discrepancies between the universe's dynamics and what the theory leads us to expect given our knowledge of the structure of matter leave us puzzling over which of general relativity and particle physics is more incomplete.

This thesis seeks to explore a small cross-section of the fundamental challenges faced by general relativity through two distinct avenues. The first is an investigation of the cosmological properties of scalar field dark matter, often informed by the fact that it may arise through a minor geometric adjustment to the core structure of the theory. The novel cosmological phenomena under consideration primarily include a dark-matter dominated regime in the early universe and a modification to the standard gravitational redshift, and we generally find that (though they are not ruled out) there is little compelling evidence for either amongst the empirical probes considered herein, namely the anisotropies in the cosmic microwave background radiation as measured by the Planck collaboration and a six-year time-domain survey of spectra across many astronomical sources completed by the Anglo-Australian Telescope. The second is a reflection on both the challenge and posing of the Weak Cosmic Censorship Conjecture, the problem of whether singularities in general relativity must generically reside within black holes. We demonstrate that violating singularities are generic within a particular class of spherically symmetric spacetimes, the Vaidya spacetimes, and this reflection leads us to the development of a novel characterization of the phenomenon of black holes, utilized to formulate a more comprehensive rigorous statement of weak cosmic censorship.

In general relativity, the Riemannian Penrose inequality (RPI) provides a lower bound for the ADM mass of an asymptotically flat manifold of nonnegative scalar curvature in terms of the area of the outermost minimal surface, if one exists. In physical terms, an equivalent statement is that the total mass of an asymptotically flat spacetime admitting a time-symmetric spacelike slice is at least the mass of any black holes that are present, assuming nonnegative energy density. The main goal of this thesis is to deduce geometric lower bounds for the ADM mass of manifolds to which neither the RPI nor the famous positive mass theorem (PMT) apply. This is the case, for instance, for manifolds that contain metric singularities or have boundary components that are not minimal surfaces.

The fundamental technique is the use of conformal deformations of a given Riemannian metric to arrive at a new Riemannian manifold to which either the PMT or RPI applies. Along the way we are led to consider the geometry of certain types non-smooth metrics. We prove a result regarding the local structure of area-minimizing hypersurfaces with respect such metrics using geometric measure theory.

One application is to the theory of ``zero area singularities,'' a type of singularity that generalizes the degenerate behavior of the Schwarzschild metric of negative mass. Another application deals with constructing and understanding some new invariants of the harmonic conformal class of an asymptotically flat metric.

In the theory of general relativity, the Penrose conjecture claims a lower bound for the mass of a spacetime in terms of the area of an outermost horizon, if one exists. In physical terms, this conjecture is a geometric formulation of the statement that the total mass of a spacetime is at least the mass of any black holes that are present, assuming non-negative energy density. For the geometry of light-rays emanating off of a black hole horizon (called a nullcone) the Penrose conjecture can be reformulated to the so-called Null Penrose Conjecture (NPC). In this thesis, we define an explicit quasi-local mass functional that is non-decreasing along all foliations (satisfying a convexity assumption) of nullcones. We use this new functional to prove the NPC under fairly generic conditions.

In this dissertation we study scalar curvature rigidity phenomena for the upper hemisphere, and subsets thereof. In particular, we are interested in Min-Oo's conjecture that there exist no metrics on the upper hemisphere having scalar curvature greater or equal to that of the standard spherical metric, while satisfying certain natural geometric boundary conditions.

While the conjecture as originally stated has recently been disproved, there are still many interesting modications to consider. For instance, it has been shown that Min-Oo's rigidity conjecture holds on sufficiently small geodesic balls contained in the upper hemisphere, for metrics sufficiently close to the spherical metric. We show that this local rigidity phenomena can be extended to a larger class of domains in the hemisphere, in particular finding that it holds on larger geodesic balls, and on certain domains other than geodesic balls (which necessarily have more complicated boundary geometry). We discuss a possible method for finding the largest possible domain on which the local rigidity theorem is true, and give a Morse-theoretic interpretation of the problem.

Another interesting open question is whether or not such a rigidity statement holds for metrics that are not close to the spherical metric. We find that a scalar curvature rigidity theorem can be proved for metrics on sufficiently small geodesic balls in the hemisphere, provided certain additional geometric constraints are satisfied.

The question of ``What is Dark Matter?" has been a focus of cosmological research since the turn of the 20th century. Though the composition of Dark Matter is unknown, the existence of Dark Matter is crucial to the modern theory of cosmology. We focus on a theory of Dark Matter referred to as \textit{Scalar Field Wave Dark Matter} (SF$\psi$DM), which has received an increasing amount of interest from the research community since the late 2000s. SF$\psi$DM is a peculiar theory in which Dark Matter is composed of ultralight bosonic particles. As a result, SF$\psi$DM has an astronomically large deBroglie wavelength, generating complicated wave dynamics on the largest cosmological scales.

This thesis focuses on describing the status of SF$\psi$DM theory, SF$\psi$DM halos, and how SF$\psi$DM halos are affected by the wave-like features of the scalar field. In particular, we offer an analysis of galactic rotation curves and how they relate to SF$\psi$DM excited states. This analysis yields a novel model for an observed galactic trend referred to as the Baryonic Tully-Fisher Relation. Furthering this model, we formulate an eigenfunction decomposition which can be used to describe superpositions of excited states.

We examine the Einstein equation coupled to the Klein-Gordon equation for a complex-valued scalar field. These two equations together are known as the Einstein-Klein-Gordon system. In the low-field, non-relativistic limit, the Einstein-Klein-Gordon system reduces to the Poisson-Schrödinger system. We describe the simplest solutions of these systems in spherical symmetry, the spherically symmetric static states, and some scaling properties they obey. We also describe some approximate analytic solutions for these states.

The EKG system underlies a theory of wave dark matter, also known as scalar field dark matter (SFDM), boson star dark matter, and Bose-Einstein condensate (BEC) dark matter. We discuss a possible connection between the theory of wave dark matter and the baryonic Tully-Fisher relation, which is a scaling relation observed to hold for disk galaxies in the universe across many decades in mass. We show how fixing boundary conditions at the edge of the spherically symmetric static states implies Tully-Fisher-like relations for the states. We also catalog other ``scaling conditions'' one can impose on the static states and show that they do not lead to Tully-Fisher-like relations--barring one exception which is already known and which has nothing to do with the specifics of wave dark matter.

We consider complete asymptotically flat Riemannian manifolds that are the graphs of smooth functions over $\mathbb R^n$. By recognizing the scalar curvature of such manifolds as a divergence, we express the ADM mass as an integral of the product of the scalar curvature and a nonnegative potential function, thus proving the Riemannian positive mass theorem in this case. If the graph has convex horizons, we also prove the Riemannian Penrose inequality by giving a lower bound to the boundary integrals using the Aleksandrov-Fenchel inequality. We also prove the ZAS inequality for graphs in Minkowski space. Furthermore, we define a new quasi-local mass functional and show that it satisfies certain desirable properties.

In this thesis, we study how the scalar curvature relates to the ADM mass and volume of the manifold.

The positive mass theorem, first proven by Schoen and Yau in 1979, states that nonnegative local energy density implies nonnegative total mass. The harmonic level set technique pioneered by D. Stern [Ste19] has been used to prove a series of positive mass theorems, such as the Riemannian case [BKKS19], the spacetime case [HKK20] and the charged case. Using this novel technique, we prove the hyperbolic positive mass theorem in the spacetime setting, as well as some rigidity cases. A new interpretation of mass is introduced in this context. Then we solve the spacetime harmonic equation in the hyperbolic setting. We not only prove the positive mass theorem, but we also give a lower bound for the total mass without assuming the nonnegativity of the local energy density.

Additionally, We prove a scalar curvature volume comparison theorem, assuming some boundedness for Ricci curvature. The proof relies on the perturbation of the scalar curvature [BM11].

The central object of study of this thesis is inverse mean curvature vector flow of two-dimensional surfaces in four-dimensional spacetimes. Being a system of forward-backward parabolic PDEs, inverse mean curvature vector flow equation lacks a general existence theory. Our main contribution is proving that there exist infinitely many spacetimes, not necessarily spherically symmetric or static, that admit smooth global solutions to inverse mean curvature vector flow. Prior to our work, such solutions were only known in spherically symmetric and static spacetimes. The technique used in this thesis might be important to prove the Spacetime Penrose Conjecture, which remains open today.

Given a spacetime $(N^{4}, \gbar)$ and a spacelike hypersurface $M$. For any closed surface $\Sigma$ embedded in $M$ satisfying some natural conditions, one can ``steer'' the spacetime metric $\gbar$ such that the mean curvature vector field of $\Sigma$ becomes tangential to $M$ while keeping the induced metric on $M$. This can be used to construct more examples of smooth solutions to inverse mean curvature vector flow from smooth solutions to inverse mean curvature flow in a spacelike hypersurface.

We explore a model of dark matter called wave dark matter (also known as scalar field dark matter and boson stars) which has recently been motivated by a new geometric perspective by Bray. Wave dark matter describes dark matter as a scalar field which satisfies the Einstein-Klein-Gordon equations. These equations rely on a fundamental constant Upsilon (also known as the ``mass term'' of the Klein-Gordon equation). Specifically, in this dissertation, we study spherically symmetric wave dark matter and compare these results with observations of dwarf spheroidal galaxies as a first attempt to compare the implications of the theory of wave dark matter with actual observations of dark matter. This includes finding a first estimate of the fundamental constant Upsilon.

In the introductory Chapter 1, we present some preliminary background material to define and motivate the study of wave dark matter and describe some of the properties of dwarf spheroidal galaxies.

In Chapter 2, we present several different ways of describing a spherically symmetric spacetime and the resulting metrics. We then focus our discussion on an especially useful form of the metric of a spherically symmetric spacetime in polar-areal coordinates and its properties. In particular, we show how the metric component functions chosen are extremely compatible with notions in Newtonian mechanics. We also show the monotonicity of the Hawking mass in these coordinates. Finally, we discuss how these coordinates and the metric can be used to solve the spherically symmetric Einstein-Klein-Gordon equations.

In Chapter 3, we explore spherically symmetric solutions to the Einstein-Klein-Gordon equations, the defining equations of wave dark matter, where the scalar field is of the form f(t,r) = exp(i omega t) F(r) for some constant omega in R and complex-valued function F(r). We show that the corresponding metric is static if and only if F(r) = h(r)exp(i a) for some constant a in R and real-valued function h(r). We describe the behavior of the resulting solutions, which are called spherically symmetric static states of wave dark matter. We also describe how, in the low field limit, the parameters defining these static states are related and show that these relationships imply important properties of the static states.

In Chapter 4, we compare the wave dark matter model to observations to obtain a working value of Upsilon. Specifically, we compare the mass profiles of spherically symmetric static states of wave dark matter to the Burkert mass profiles that have been shown by Salucci et al. to predict well the velocity dispersion profiles of the eight classical dwarf spheroidal galaxies. We show that a reasonable working value for the fundamental constant in the wave dark matter model is Upsilon = 50 yr^(-1). We also show that under precise assumptions the value of Upsilon can be bounded above by 1000 yr^(-1).

In order to study non-static solutions of the spherically symmetric Einstein-Klein-Gordon equations, we need to be able to evolve these equations through time numerically. Chapter 5 is concerned with presenting the numerical scheme we will use to solve the spherically symmetric Einstein-Klein-Gordon equations in our future work. We will discuss how to appropriately implement the boundary conditions into the scheme as well as some artificial dissipation. We will also discuss the accuracy and stability of the scheme. Finally, we will present some examples that show the scheme in action.

In Chapter 6, we summarize our results. Finally, Appendix A contains a derivation of the Einstein-Klein-Gordon equations from its corresponding action.