arXiv subjects

Eugene Trubowitz

Publications and source records attributed to Eugene Trubowitz.

At least 19 recordsLinked to original sources

arXiv2020

Filtered expansions in general relativity II

This is the second of two papers in which we construct formal power series solutions in external parameters to the vacuum Einstein equations, implementing one bounce for the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially inhomogeneous spacetimes. Here we show that spatially inhomogeneous perturbations of spatially homogeneous elements are unobstructed. A spectral sequence for a filtered complex, and a homological contraction based on gauge-fixing, are used to do this.

math-ph↗

arXiv2020

Semiglobal non-oscillatory big bang singular spacetimes for the Einstein-scalar field system

We construct semiglobal singular spacetimes for the Einstein equations coupled to a massless scalar field. Consistent with the heuristic analysis of Belinskii, Khalatnikov, Lifshitz or BKL for this system, there are no oscillations due to the scalar field. (This is much simpler than the oscillatory BKL heuristics for the Einstein vacuum equations.) Prior results are due to Andersson and Rendall in the real analytic case, and Rodnianski and Speck in the smooth near-spatially-flat-FLRW case. Similar to Andersson and Rendall we give asymptotic data at the singularity, which we refer to as final data, but our construction is not limited to real analytic solutions. This paper is a test application of tools (a graded Lie algebra formulation of the Einstein equations and a filtration) intended for the more subtle vacuum case. We use homological algebra tools to construct a formal series solution, then symmetric hyperbolic energy estimates to construct a true solution well-approximated by truncations of the formal one. We conjecture that the image of the map from final data to initial data is an open set of anisotropic initial data.

math-ph↗

arXiv2019

Filtered expansions in general relativity I

This is the first of two papers in which we construct formal power series solutions in external parameters to the vacuum Einstein equations, implementing one bounce for the Belinskii-Khalatnikov-Lifshitz (BKL) proposal for spatially inhomogeneous spacetimes. We use a graded Lie algebra, homological framework. A dedicated filtration encodes key features of the BKL proposal, and we use it to set up an unobstructed perturbative problem.

math-ph↗

arXiv2018

The graded Lie algebra of general relativity

We construct a graded Lie algebra $\mathcal{E}$ in which the Maurer-Cartan equation is equivalent to the vacuum Einstein equations. The gauge groupoid is the groupoid of rank 4 real vector bundles with a conformal inner product, over a 4-dimensional base manifold, and the graded Lie algebra construction is a functor out of this groupoid. As usual, each Maurer-Cartan element in $\mathcal{E}^1$ yields a differential on $\mathcal{E}$. Its first homology is linearized gravity about that element. We introduce a gauge-fixing algorithm that generates, for each gauge object $G$, a contraction to a much smaller complex whose modules are the kernels of linear, symmetric hyperbolic partial differential operators. This contraction opens the way to the application of homological algebra to the analysis of the vacuum Einstein equations. We view general relativity, at least at the perturbative level, as an instance of `homological PDE' at the crossroads of algebra and analysis.

math-ph↗

arXiv2016

Power Series Representations for Complex Bosonic Effective Actions. III. Substitution and Fixed Point Equations

We have previously developed a polymer-like expansion that applies when the (effective) action in a functional integral is an analytic function of the fields being integrated. Here, we develop methods to aid the application of this technique when the method of steepest descent is used to analyze the functional integral. We develop a version of the Banach fixed point theorem that can be used to construct and control the critical fields, as analytic functions of external fields, and substitution formulae to control the change in norms that occurs when one replaces the integration fields by the sum of the critical fields and the fluctuation fields.

math-ph↗

arXiv2016

Bloch Theory for Periodic Block Spin Transformations

Block spin renormalization group is the main tool used in our program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It generates operators, like the fluctuation integral covariance, that act on some lattice but are translation invariant only with respect to a proper sublattice. This paper constructs a Bloch/Floquet framework that is appropriate for bounding such operators.

math-ph↗

arXiv2016

The Algebra of Block Spin Renormalization Group Transformations

Block spin renormalization group is the main tool used in our program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. In this paper, we discuss some of its purely algebraic aspects in an abstract setting. For example, we derive some "well known" identities like the composition rule and the relation between critical fields and background fields.

math-ph↗

arXiv2016

Operators for Parabolic Block Spin Transformations

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It is part of an analysis of the "small field" approximation to the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well. Bounds on the fluctuation integral covariance, as well as on some other linear operators, are an important ingredient in our renormalization group step analysis. These bounds are proven here.

math-ph↗

arXiv2016

The Small Field Parabolic Flow for Bosonic Many-body Models: Part 1 - Main Results and Algebra

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It is part of an analysis of the "small field" approximation to the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well. Here we state the main result of this analysis, outline the strategy of the proof, which uses a renormalization group flow, and perform the first, algebraic, part of a renormalization group step.

math-ph↗

arXiv2016

The Small Field Parabolic Flow for Bosonic Many-body Models: Part 2 - Fluctuation Integral and Renormalization

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It is part of an analysis of the "small field" approximation to the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well. Here we complete the analysis of a renormalization group step by "evaluating" the fluctuation integral and renormalizing the chemical potential.

math-ph↗

arXiv2016

The Small Field Parabolic Flow for Bosonic Many-body Models: Part 3 - Nonperturbatively Small Errors

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It is part of an analysis of the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well. Here we provide arguments that suggest, but do not completey prove, that the difference between the "small field" approximation and the full model is nonperturbatively small.

math-ph↗

arXiv2016

The Small Field Parabolic Flow for Bosonic Many-body Models: Part 4 - Background and Critical Field Estimates

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It is part of an analysis of the "small field" approximation to the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well. Here we prove the existence of and bounds on the background and critical fields that arise from the steepest descent attack that is at the core of our renormalization group step anaylsis of these models.

math-ph↗

arXiv2016

Complex Bosonic Many-body Models: Overview of the Small Field Parabolic Flow

This paper is a contribution to a program to see symmetry breaking in a weakly interacting many Boson system on a three dimensional lattice at low temperature. It provides an overview of our analysis of the "small field" approximation to the "parabolic flow" which exhibits the formation of a "Mexican hat" potential well.

math-ph↗

arXiv2015

Filtered expansions in general relativity and one BKL-bounce

When the vacuum Einstein equations are formulated in terms of a frame, rather than a metric, can one perturb solutions with a degenerate frame into ones with a nondegenerate frame? In examples we point out that one can encounter issues already at the level of formal perturbative expansions; namely the cohomological, so-called space of obstructions is nonzero. In this paper we propose a perturbative expansion based on filtrations. We construct and prove properties of a specific filtration, intended to make mathematical sense of one BKL-bounce, a building block of a well-known but very heuristic conjecture due to Belinskii, Khalatnikov and Lifshitz (which would involve sticking together an infinite sequence of single bounces). It seems possible that now the space of obstructions is zero, but this question is left open.

gr-qc↗

arXiv2014

The graded Lie algebra of general relativity

We construct a graded Lie algebra in which a solution to the vacuum Einstein equations is any element of degree 1 whose bracket with itself is zero. Each solution generates a cochain complex, whose first cohomology is linearized gravity about that solution. We gauge-fix to get a smaller cochain complex with the same cohomologies (deformation retraction). The new complex is much smaller, it consists of the solution spaces of linear homogeneous wave equations (symmetric hyperbolic equations). The algorithm that produces these gauges and wave equations is both for linearized gravity and the full Einstein equations. The gauge groupoid is the groupoid of rank 2 complex vector bundles.

gr-qc↗

arXiv2012

Choptuik's critical spacetime exists

About twenty years ago, Choptuik studied numerically the gravitational collapse (Einstein field equations) of a massless scalar field in spherical symmetry, and found strong evidence for a universal, self-similar solution at the threshold of black hole formation. We prove rigorously the existence of a real analytic solution, that we interpret as the solution observed by Choptuik. Our construction covers an open neighborhood of the past light cone of the singularity. The proof is computer assisted. Starting from an explicit approximate solution, we show that nearby there is a true solution. The source code and a high precision data file (about 80 significant decimal digits, with rigorous error bounds) are included. We do not study perturbations.

gr-qc↗

arXiv2011

A class of gauges for the Einstein equations

A class of gauges for the Einstein vacuum equations is introduced, along with three symmetric hyperbolic systems. The first implies the local realizability of the gauge. The second is the dynamical subset of the field equations. The third is used to show that the constraints propagate. The gauges are for an orthonormal frame formalism, with first order, quadratically nonlinear equations. The unknowns are 16 frame components and 28 connection components. After gauge-fixing, a total of 33 remain.

gr-qc↗

arXiv2011

Strongly Focused Gravitational Waves

This paper contains a new proof of the formation of trapped spheres, in vacuum spacetimes, by the focusing of gravitational waves, from generic data. The first such result was obtained by Christodoulou [Chr]. We exploit the same physical mechanism, but give a logically independent construction of these spacetimes.

gr-qc↗