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Full-Text Articles in Physics
Gravitational Waves: Just Plane Symmetry, Charles G. Torre
Gravitational Waves: Just Plane Symmetry, Charles G. Torre
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In four spacetime dimensions gravitational plane waves (a special case of the plane-fronted waves with parallel rays) admit a 5 parameter isometry group. We generalize this group to n-dimensions and explore some special features of spacetimes admitting this isometry group. In particular, it is shown that every generally covariant rank-2 symmetric tensor field constructed from a metric with plane wave symmetry will vanish except multiples of the metric and Ricci tensors. We show that, in four spacetime dimensions, a particular enlargement of the plane wave symmetry group is enough to force the group-invariant metrics to satisfy all generally covariant vacuum …
Observables For The Polarized Gowdy Model, Charles G. Torre
Observables For The Polarized Gowdy Model, Charles G. Torre
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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.
Uniqueness Of Solutions To The Helically Reduced Wave Equation With Sommerfeld Boundary Conditions, Charles G. Torre
Uniqueness Of Solutions To The Helically Reduced Wave Equation With Sommerfeld Boundary Conditions, Charles G. Torre
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We consider the helical reduction of the wave equation with an arbitrary source on (n+1)-dimensional Minkowski space, n ≥ 2. The reduced equation is of mixed elliptic-hyperbolic type on Rn. 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 ∂B ≈ Sn−1. Nonlinear generalizations of such boundary value problems (with n = 3) arise in the intermediate phase of binary inspiral in general relativity.
Coherent State Path Integral For Linear Systems, Charles G. Torre
Coherent State Path Integral For Linear Systems, Charles G. Torre
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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 …
Cosmology, Cohomology, And Compactification, Charles G. Torre
Cosmology, Cohomology, And Compactification, Charles G. Torre
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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.
The Helically Reduced Wave Equation As A Symmetric Positive System, Charles G. Torre
The Helically Reduced Wave Equation As A Symmetric Positive System, Charles G. Torre
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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 …
Quantum Dynamics Of The Polarized Gowdy T3 Model, Charles G. Torre
Quantum Dynamics Of The Polarized Gowdy T3 Model, Charles G. Torre
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The polarized Gowdy T3 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 nonlinear constraint. The quantum Gowdy model has been defined by using a representation for φ on a Fock space 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 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 …
The Principle Of Symmetric Criticality In General Relativity, Mark E. Fels, Charles G. Torre
The Principle Of Symmetric Criticality In General Relativity, Mark E. Fels, Charles G. Torre
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We consider a version of Palais' principle of symmetric criticality (PSC) that is applicable to the Lie symmetry reduction of Lagrangian field theories. Given a group action on a space of fields, PSC asserts that for any group-invariant Lagrangian, the equations obtained by restriction of Euler–Lagrange equations to group-invariant fields are equivalent to the Euler–Lagrange equations of a canonically defined, symmetry-reduced Lagrangian. We investigate the validity of PSC for local gravitational theories built from a metric and show that there are two independent conditions which must be satisfied for PSC to be valid. One of these conditions, obtained previously in …
Group Invariant Solutions In Mathematical Physics And Differential Geometry, Ian M. Anderson, Mark E. Fels, Charles G. Torre
Group Invariant Solutions In Mathematical Physics And Differential Geometry, Ian M. Anderson, Mark E. Fels, Charles G. Torre
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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.
Group Invariant Solutions Without Transversality, Ian M. Anderson, Mark E. Fels, Charles G. Torre
Group Invariant Solutions Without Transversality, Ian M. Anderson, Mark E. Fels, Charles G. Torre
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We present a generalization of Lie's method for finding the group invariant solutions to a system of partial differential equations. Our generalization relaxes the standard transversality assumption and encompasses the common situation where the reduced differential equations for the group invariant solutions involve both fewer dependent and independent variables. The theoretical basis for our method is provided by a general existence theorem for the invariant sections, both local and global, of a bundle on which a finite dimensional Lie group acts. A simple and natural extension of our characterization of invariant sections leads to an intrinsic characterization of the reduced …
Functional Evolution Of Free Quantum Fields, Charles G. Torre, Madhavan Varadarajan
Functional Evolution Of Free Quantum Fields, Charles G. Torre, Madhavan Varadarajan
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We consider the problem of evolving a quantum field between any two (in general, curved) Cauchy surfaces. Classically, this dynamical evolution is represented by a canonical transformation on the phase space for the field theory. We show that this canonical transformation cannot, in general, be unitarily implemented on the Fock space for free quantum fields on flat spacetimes of dimension greater than 2. We do this by considering time evolution of a free Klein-Gordon field on a flat spacetime (with toroidal Cauchy surfaces) starting from a flat initial surface and ending on a generic final surface. The associated Bogolubov transformation …
Midisuperspace Models Of Canonical Quantum Gravity, Charles G. Torre
Midisuperspace Models Of Canonical Quantum Gravity, Charles G. Torre
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A midisuperspace model is a field theory obtained by symmetry reduction of a parent gravitational theory. Such models have proven useful for exploring the classical and quantum dynamics of the gravitational field. I present three recent classes of results pertinent to canonical quantization of vacuum general relativity in the context of midisuperspace models. (1) I give necessary and sufficient conditions such that a given symmetry reduction can be performed at the level of the Lagrangian or Hamiltonian.(2) I discuss the Hamiltonian formulation of models based upon cylindrical and toroidal symmetry. In particular, I explain how these models can be identified …
Quantum Fields At Any Time, Charles G. Torre, Madhavan Varadarajan
Quantum Fields At Any Time, Charles G. Torre, Madhavan Varadarajan
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The canonical quantum theory of a free field using arbitrary foliations of a flat two-dimensional spacetime is investigated. It is shown that dynamical evolution along arbitrary spacelike foliations is unitarily implemented on the same Fock space as that associated with inertial foliations. It follows that the Schrodinger picture exists for arbitrary foliations as a unitary image of the Heisenberg picture for the theory. An explicit construction of the Schrodinger picture image of the Heisenberg Fock space states is provided. The results presented here can be interpreted in terms of a Dirac constraint quantization of parametrized field theory. In particular, it …
Spinors, Jets, And The Einstein Equations, Charles G. Torre
Spinors, Jets, And The Einstein Equations, Charles G. Torre
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Many important features of a field theory, e.g., conserved currents, symplectic structures, energy-momentum tensors, etc., arise as tensors locally constructed from the fields and their derivatives. Such tensors are naturally defined as geometric objects on the jet space of solutions to the field equations. Modern results from the calculus on jet bundles can be combined with a powerful spinor parametrization of the jet space of Einstein metrics to unravel basic features of the Einstein equations. These techniques have been applied to computation of generalized symmetries and “characteristic cohomology” of the Einstein equations, and lead to results such as a proof …
Asymptotic Conservation Laws In Field Theory, Ian M. Anderson, Charles G. Torre
Asymptotic Conservation Laws In Field Theory, Ian M. Anderson, Charles G. Torre
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A new, general, field theoretic approach to the derivation of asymptotic conservation laws is presented. In this approach asymptotic conservation laws are constructed directly from the field equations according to a universal prescription which does not rely upon the existence of Noether identities or any Lagrangian or Hamiltonian formalisms. The resulting general expressions of the conservation laws enjoy important invariance properties and synthesize all known asymptotic conservation laws, such as the Arnowitt-Deser-Misner energy in general relativity.
Internal Time Formalism For Spacetimes With Two Killing Vectors, Joseph D. Romano, Charles G. Torre
Internal Time Formalism For Spacetimes With Two Killing Vectors, Joseph D. Romano, Charles G. Torre
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The Hamiltonian structure of spacetimes with two commuting Killing vector fields is analyzed for the purpose of addressing the various problems of time that arise in canonical gravity. Two specific models are considered: (i) cylindrically symmetric spacetimes, and (ii) toroidally symmetric spacetimes, which respectively involve open and closed universe boundary conditions. For each model canonical variables which can be used to identify points of space and instants of time, {\it i.e.}, internally defined spacetime coordinates, are identified. To do this it is necessary to extend the usual ADM phase space by a finite number of degrees of freedom. Canonical transformations …
Some Remarks On Gravitational Analogs Of Magnetic Charge, Charles G. Torre
Some Remarks On Gravitational Analogs Of Magnetic Charge, Charles G. Torre
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Existing mathematical results are applied to the problem of classifying closed p-forms which are locally constructed from Lorentzian metrics on an n-dimensional orientable manifold M(0
Gravitational Observables And Local Symmetries, Charles G. Torre
Gravitational Observables And Local Symmetries, Charles G. Torre
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Using a recent classification of local symmetries of the vacuum Einstein equations, it is shown that there can be no observables for the vacuum gravitational field (in a closed universe) built as spatial integrals of local functions of Cauchy data and their derivatives.
Symmetries Of The Einstein Equations, Charles G. Torre, Ian M. Anderson
Symmetries Of The Einstein Equations, Charles G. Torre, Ian M. Anderson
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We classify all generalized symmetries of the vacuum Einstein equations in four spacetime dimensions. They consist of constant scalings of the metric, and of the infinitesimal action of generalized spacetime diffeomorphisms. Our results rule out a large class of possible ‘‘observables’’ for the gravitational field, and suggest that the vacuum Einstein equations are not integrable.