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C. G. Torre

Publications and source records attributed to C. G. Torre.

At least 19 recordsLinked to original sources
arXiv2022

A New Non-Inheriting Homogeneous Solution of the Einstein-Maxwell Equations with Cosmological Term

We find a new homogeneous solution to the Einstein-Maxwell equations with a cosmological term. The spacetime manifold is $R \times S^3$. The spacetime metric admits a simply transitive isometry group $G = R \times SU(2)$ of isometries and is of Petrov type I. The spacetime is geodesically complete and globally hyperbolic. The electromagnetic field is non-null and non-inheriting: it is only invariant with respect to the $SU(2)$ subgroup and is time-dependent in a stationary reference frame.

gr-qc↗
arXiv2016

Rainich Conditions in (2+1)-Dimensional Gravity

In (3 + 1) spacetime dimensions the Rainich conditions are a set of equations expressed solely in terms of the metric tensor which are equivalent to the Einstein-Maxwell equations for non-null electromagnetic fields. Here we provide the analogous conditions for (2 + 1)-dimensional gravity coupled to electromagnetism. Both the non-null and null cases are treated. The construction of these conditions is based upon reducing the problem to that of gravity coupled to a scalar field, which we have treated elsewhere. These conditions can be easily extended to other theories of (2 + 1)-dimensional gravity. For example, we apply the geometrization conditions to topologically massive gravity coupled to the electromagnetic field and obtain a family of plane-fronted wave solutions.

gr-qc↗
arXiv2015

Geometrization Conditions for Perfect Fluids, Scalar Fields, and Electromagnetic Fields

Rainich-type conditions giving a spacetime "geometrization" of matter fields in general relativity are reviewed and extended. Three types of matter are considered: perfect fluids, scalar fields, and electromagnetic fields. Necessary and sufficient conditions on a spacetime metric for it to be part of a perfect fluid solution of the Einstein equations are given. Formulas for constructing the fluid from the metric are obtained. All fluid results hold for any spacetime dimension. Geometric conditions on a metric which are necessary and sufficient for it to define a solution of the Einstein-scalar field equations and formulas for constructing the scalar field from the metric are unified and extended to arbitrary dimensions, to include a cosmological constant, and to include any self-interaction potential. Necessary and sufficient conditions on a four-dimensional spacetime metric for it to be an electrovacuum and formulas for constructing the electromagnetic field from the metric are generalized to include a cosmological constant. Both null and non-null electromagnetic fields are treated. A number of examples and applications of these results are presented.

gr-qc↗
arXiv2014

The Spacetime Geometry of a Null Electromagnetic Field

We give a set of local geometric conditions on a spacetime metric which are necessary and sufficient for it to be a null electrovacuum, that is, the metric is part of a solution to the Einstein-Maxwell equations with a null electromagnetic field. These conditions are restrictions on a null congruence canonically constructed from the spacetime metric, and can involve up to five derivatives of the metric. The null electrovacuum conditions are counterparts of the Rainich conditions, which geometrically characterize non-null electrovacua. Given a spacetime satisfying the conditions for a null electrovacuum, a straightforward procedure builds the null electromagnetic field from the metric. Null electrovacuum geometry is illustrated using some pure radiation spacetimes taken from the literature.

gr-qc↗
arXiv2011

New Symbolic Tools for Differential Geometry, Gravitation, and Field Theory

DifferentialGeometry is a Maple software package which symbolically performs fundamental operations of calculus on manifolds, differential geometry, tensor calculus, Lie algebras, Lie groups, transformation groups, jet spaces, and the variational calculus. These capabilities, combined with dramatic recent improvements in symbolic approaches to solving algebraic and differential equations, have allowed for development of powerful new tools for solving research problems in gravitation and field theory. The purpose of this paper is to describe some of these new tools and present some advanced applications involving: Killing vector fields and isometry groups, Killing tensors and other tensorial invariants, algebraic classification of curvature, and symmetry reduction of field equations.

math-ph↗
arXiv2010

Symmetric Criticality in Classical Field Theory

This is a brief overview of work done by Ian Anderson, Mark Fels, and myself on symmetry reduction of Lagrangians and Euler-Lagrange equations, a subject closely related to Palais' Principle of Symmetric Criticality. After providing a little history, I describe necessary and sufficient conditions on a group action such that reduction of a group-invariant Lagrangian by the symmetry group yields the correct symmetry-reduced Euler-Lagrange equations.

math-ph↗
arXiv2009

Symmetry Reduction of Quasi-Free States

Given a group-invariant quasi-free state on the algebra of canonical commutation relations (CCR), we show how group averaging techniques can be used to obtain a symmetry reduced CCR algebra and reduced quasi-free state. When the group is compact this method of symmetry reduction leads to standard results which can be obtained using other methods. When the group is non-compact the group averaging prescription relies upon technically favorable conditions which we delineate. As an example, we consider symmetry reduction of the usual vacuum state for a Klein-Gordon field on Minkowski spacetime by a non-compact subgroup of the Poincaré group consisting of a 1-parameter family of boosts, a 1-parameter family of spatial translations and a set of discrete translations. We show that the symmetry reduced CCR algebra and vacuum state correspond to that used by each of Berger, Husain, and Pierri for the polarized Gowdy ${\bf T}^3$ quantum gravity model.

math-ph↗
arXiv2007

Functional Time Evolution, Anomaly Potentials, and the Geometric Phase

A free quantum field in 1+1 dimensions admits unitary Schrodinger picture dynamics along any foliation of spacetime by Cauchy curves. Kuchar showed that the Schrodinger picture state vectors, viewed as functionals of spacelike embeddings, satisfy a functional Schrodinger equation in which the generators of time evolution are the scalar field energy-momentum densities with a particular normal-ordering and with a (non-unique) c-number contribution. The c-number contribution to the Schrodinger equation, called the ``anomaly potential'', is needed to make the equation integrable in light of the Schwinger terms present in the commutators of the normal-ordered energy-momentum densities. Here we give a quantum geometric interpretation of the anomaly potential. In particular, we show the anomaly potential corresponds to the expression in a gauge of the natural connection on the bundle of vacuum states over the space of embeddings of Cauchy curves into the spacetime. The holonomy of this connection is the geometric phase associated with dynamical evolution along a closed path in the space of embeddings generated by the normal-ordered energy-momentum densities. The presence of the anomaly potential in the functional Schrodinger equation provides a dynamical phase which removes this holonomy, so that there is no net phase change for quantum transport around closed loops in the space of embeddings.

hep-th↗
arXiv2006

Schrodinger representation for the polarized Gowdy model

The polarized ${\bf T}^3$ Gowdy model is, in a standard gauge, characterized by a point particle degree of freedom and a scalar field degree of freedom obeying a linear field equation on ${\bf R}\times{\bf S}^1$. The Fock representation of the scalar field has been well-studied. Here we construct the Schrodinger representation for the scalar field at a fixed value of the Gowdy time in terms of square-integrable functions on a space of distributional fields with a Gaussian probability measure. We show that ``typical'' field configurations are slightly more singular than square-integrable functions on the circle. For each time the corresponding Schrodinger representation is unitarily equivalent to the Fock representation, and hence all the Schrodinger representations are equivalent. However, the failure of unitary implementability of time evolution in this model manifests itself in the mutual singularity of the Gaussian measures at different times.

gr-qc↗
arXiv2006

Uniqueness of Solutions to the Helically Reduced Wave Equation with Sommerfeld Boundary Conditions

We consider the helical reduction of the wave equation with an arbitrary source on $(n+1)$-dimensional Minkowski space, $n\geq2$. The reduced equation is of mixed elliptic-hyperbolic type on ${\bf R}^n$. We obtain a uniqueness theorem for solutions on a domain consisting of an $n$-dimensional ball $B$ centered on the reduction of the axis of helical symmetry and satisfying ingoing or outgoing Sommerfeld conditions on $\partial B\approx S^{n-1}$. Non-linear generalizations of such boundary value problems (with $n=3$) arise in the intermediate phase of binary inspiral in general relativity.

math-ph↗
arXiv2005

Observables for the polarized Gowdy model

We give an explicit characterization of all functions on the phase space for the polarized Gowdy 3-torus spacetimes which have weakly vanishing Poisson brackets with the Hamiltonian and momentum constraint functions.

gr-qc↗
arXiv2005

On the Coherent State Path Integral for Linear Systems

We present a computation of the coherent state path integral for a generic linear system using ``functional methods'' (as opposed to discrete time approaches). The Gaussian phase space path integral is formally given by a determinant built from a first-order differential operator with coherent state boundary conditions. We show how this determinant can be expressed in terms of the symplectic transformation generated by the (in general, time-dependent) quadratic Hamiltonian for the system. We briefly discuss the conditions under which the coherent state path integral for a linear system actually exists. A necessary -- but not sufficient -- condition for existence of the path integral is that the symplectic transformation generated by the Hamiltonian is (unitarily) implementable on the Fock space for the system.

quant-ph↗
arXiv2004

Cosmology, cohomology, and compactification

Ashtekar and Samuel have shown that Bianchi cosmological models with compact spatial sections must be of Bianchi class A. Motivated by general results on the symmetry reduction of variational principles, we show how to extend the Ashtekar-Samuel results to the setting of weakly locally homogeneous spaces as defined, e.g., by Singer and Thurston. In particular, it is shown that any m-dimensional homogeneous space G/K admitting a G-invariant volume form will allow a compact discrete quotient only if the Lie algebra cohomology of G relative to K is non-vanishing at degree m.

gr-qc↗
arXiv2003

The Helically-Reduced Wave Equation as a Symmetric-Positive System

Motivated by the partial differential equations of mixed type that arise in the reduction of the Einstein equations by a helical Killing vector field, we consider a boundary value problem for the helically-reduced wave equation with an arbitrary source in 2+1 dimensional Minkowski spacetime. The reduced equation is a second-order partial differential equation which is elliptic inside a disk and hyperbolic outside the disk. We show that the reduced equation can be cast into symmetric-positive form. Using results from the theory of symmetric-positive differential equations, we show that this form of the helically-reduced wave equation admits unique, strong solutions for a class of boundary conditions which include Sommerfeld conditions at the outer boundary.

math-ph↗
arXiv2002

Quantum Dynamics of the Polarized Gowdy Model

The polarized Gowdy ${\bf T}^3$ vacuum spacetimes are characterized, modulo gauge, by a ``point particle'' degree of freedom and a function $ϕ$ that satisfies a linear field equation and a non-linear constraint. The quantum Gowdy model has been defined by using a representation for $ϕ$ on a Fock space $\cal F$. Using this quantum model, it has recently been shown that the dynamical evolution determined by the linear field equation for $ϕ$ is not unitarily implemented on $\cal F$. In this paper: (1) We derive the classical and quantum model using the ``covariant phase space'' formalism. (2) We show that time evolution is not unitarily implemented even on the physical Hilbert space of states ${\cal H} \subset {\cal F}$ defined by the quantum constraint. (3) We show that the spatially smeared canonical coordinates and momenta as well as the time-dependent Hamiltonian for $ϕ$ are well-defined, self-adjoint operators for all time, admitting the usual probability interpretation despite the lack of unitary dynamics.

gr-qc↗
arXiv2001

Group Invariant Solutions in Mathematical Physics and Differential Geometry

This is a brief overview of our work on the theory of group invariant solutions to differential equations. The motivations and applications of this work stem from problems in differential geometry and relativistic field theory. The key feature in our theory is that we allow for non-transverse symmetry group actions, which are very common in applications.

math-ph↗
arXiv1999

Gravitational Waves: Just Plane Symmetry

We present some remarkable properties of the symmetry group for gravitational plane waves. Our main observation is that metrics with plane wave symmetry satisfy every system of generally covariant vacuum field equations except the Einstein equations. The proof uses the homothety admitted by metrics with plane wave symmetry and the scaling behavior of generally covariant field equations. We also discuss a mini-superspace description of spacetimes with plane wave symmetry.

gr-qc↗