scholarly journals Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

2007 ◽  
Vol 60 (10) ◽  
pp. 1522-1557 ◽  
Author(s):  
Giuseppe Alì ◽  
John K. Hunter
Keyword(s):  
Wave Equations ◽  
Geometrical Optics ◽  
Einstein Equations ◽  
Variational Wave

scholarly journals POLYHOMOGENEOUS SOLUTIONS OF NONLINEAR WAVE EQUATIONS WITHOUT CORNER CONDITIONS

2006 ◽  
Vol 03 (01) ◽  
pp. 81-141 ◽  
Author(s):  
PIOTR T. CHRUŚCIEL ◽  
SZYMON ŁȨSKI
Keyword(s):  
Initial Data ◽  
Wave Equations ◽  
Space Time ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Linear Wave ◽  
Cauchy Problems ◽  
Time Dimension ◽  
Wave Maps ◽  
Corner Conditions

The study of Einstein equations leads naturally to Cauchy problems with initial data on hypersurfaces which closely resemble hyperboloids in Minkowski space-time, and with initial data with polyhomogeneous asymptotics, that is, with asymptotic expansions in terms of powers of ln r and inverse powers of r. Such expansions also arise in the conformal method for analysing wave equations in odd space-time dimension. In recent work it has been shown that for non-linear wave equations, or for wave maps, polyhomogeneous initial data lead to solutions which are also polyhomogeneous provided that an infinite hierarchy of corner conditions holds. In this paper we show that the result is true regardless of corner conditions.

Variational wave equations for relativistic few-body systems in QFT

2006 ◽  
Vol 84 (6-7) ◽  
pp. 625-632 ◽  
Author(s):  
J W Darewych
Keyword(s):  
Field Theory ◽  
Bound States ◽  
Wave Equations ◽  
Body Wave ◽  
Hamiltonian Formalism ◽  
Approximate Solutions ◽  
Quantum Field ◽  
Fermion Fields ◽  
Variational Wave ◽  
03.65 Ge

The variational method in a reformulated Hamiltonian formalism of quantum field theory is used to derive relativistic few-body wave equations for scalar and Fermion fields. Analytic and approximate solutions of some two-body bound states are presented.PACS Nos.: 03.65.Pm, 03.65.Ge, 03.70.+k, 11.10.Ef, 11.10.St, 11.15.Tk, 36.10.Dr

scholarly journals On Classical Global Solutions of Nonlinear Wave Equations with Large Data

2017 ◽  
Vol 2019 (19) ◽  
pp. 5859-5913 ◽  
Author(s):  
Shuang Miao ◽  
Long Pei ◽  
Pin Yu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Wave Equations ◽  
Global Solutions ◽  
Large Data ◽  
Einstein Equations ◽  
Nonlinear Wave Equations ◽  
Classical Solutions ◽  
Linear Wave ◽  
The Cauchy Problem

Abstract This article studies the Cauchy problem for systems of semi-linear wave equations on $\mathbb{R}^{3+1}$ with nonlinear terms satisfying the null conditions. We construct future global-in-time classical solutions with arbitrarily large initial energy. The choice of the large Cauchy initial data is inspired by Christodoulou's characteristic initial data in his work [2] on formation of black holes. The main innovation of the current work is that we discovered a relaxed energy ansatz which allows us to prove decay-in-time-estimate. Therefore, the new estimates can also be applied in studying the Cauchy problem for Einstein equations.

scholarly journals A parametrix for quantum gravity?

2016 ◽  
Vol 13 (05) ◽  
pp. 1650060
Author(s):  
Giampiero Esposito
Keyword(s):  
Quantum Gravity ◽  
Hyperbolic System ◽  
Wave Operator ◽  
Wave Equations ◽  
Inverse Operator ◽  
Einstein Equations ◽  
Poisson Brackets ◽  
Maxwell Theory ◽  
Kirchhoff Type ◽  
Nonlinear Hyperbolic System

In the 60s, DeWitt discovered that the advanced and retarded Green functions of the wave operator on metric perturbations in the de Donder gauge make it possible to define classical Poisson brackets on the space of functionals that are invariant under the action of the full diffeomorphism group of spacetime. He therefore tried to exploit this property to define invariant commutators for the quantized gravitational field, but the operator counterpart of such classical Poisson brackets turned out to be a hard task. On the other hand, in the mathematical literature, it is by now clear that, rather than inverting exactly an hyperbolic (or elliptic) operator, it is more convenient to build a quasi-inverse, i.e. an inverse operator up to an operator of lower order which plays the role of regularizing operator. This approximate inverse, the parametrix, which is, strictly, a distribution, makes it possible to solve inhomogeneous hyperbolic (or elliptic) equations. We here suggest that such a construction might be exploited in canonical quantum gravity provided one understands what is the counterpart of classical smoothing operators in the quantization procedure. We begin with the simplest case, i.e. fundamental solution and parametrix for the linear, scalar wave operator; the next step are tensor wave equations, again for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear Einstein equations are studied, relying upon the well-established Choquet-Bruhat construction, according to which the fifth derivatives of solutions of a nonlinear hyperbolic system solve a linear hyperbolic system. The latter is solved by means of Kirchhoff-type formulas, while the former fifth-order equations can be solved by means of well-established parametrix techniques for elliptic operators. But then the metric components that solve the vacuum Einstein equations can be obtained by convolution of such a parametrix with Kirchhoff-type formulas. Some basic functional equations for the parametrix are also obtained, that help in studying classical and quantum version of the Jacobi identity.

scholarly journals Geometrical optics for scalar, electromagnetic and gravitational waves on curved spacetime

2018 ◽  
Vol 27 (11) ◽  
pp. 1843010 ◽  
Author(s):  
Sam R. Dolan
Keyword(s):  
Gravitational Waves ◽  
Cross Section ◽  
Principal Part ◽  
Wave Equations ◽  
Conjugate Point ◽  
Geometrical Optics ◽  
Transport Equations ◽  
Speed Of Light ◽  
Curved Spacetime ◽  
Null Tetrad

The geometrical-optics expansion reduces the problem of solving wave equations to one of the solving transport equations along rays. Here, we consider scalar, electromagnetic and gravitational waves propagating on a curved spacetime in general relativity. We show that each is governed by a wave equation with the same principal part. It follows that: each wave propagates at the speed of light along rays (null generators of hypersurfaces of constant phase); the square of the wave amplitude varies in inverse proportion to the cross-section of the beam; and the polarization is parallel-propagated along the ray (the Skrotskii/Rytov effect). We show that the optical scalars for a beam, and various Newman–Penrose scalars describing a parallel-propagated null tetrad, can be found by solving transport equations in a second-order formulation. Unlike the Sachs equations, this formulation makes it straightforward to find such scalars beyond the first conjugate point of a congruence, where neighboring rays cross, and the scalars diverge. We discuss differential precession across the beam which leads to a modified phase in the geometrical-optics expansion.

scholarly journals Variational wave equations of two fermions interacting via scalar, pseudoscalar, vector, pseudovector and tensor fields

Open Physics ◽  
2005 ◽  
Vol 3 (4) ◽  
Author(s):  
Askold Duviryak ◽  
Jurij Darewych
Keyword(s):  
Wave Equations ◽  
Body Wave ◽  
Vacuum State ◽  
Nonlocal Interaction ◽  
Tensor Fields ◽  
Tensor Coupling ◽  
Coordinate Representation ◽  
Interaction Terms ◽  
Breit Equation ◽  
Variational Wave

AbstractWe consider a method for deriving relativistic two-body wave equations for fermions in the coordinate representation. The Lagrangian of the theory is reformulated by eliminating the mediating fields by means of covariant Green's functions. Then, the nonlocal interaction terms in the Lagrangian are reduced to local expressions which take into account retardation effects approximately. We construct the Hamiltonian and two-fermion states of the quantized theory, employing an unconventional “empty” vacuum state, and derive relativistic two-fermion wave equations. These equations are a generalization of the Breit equation for systems with scalar, pseudoscalar, vector, pseudovector and tensor coupling.

Sign in / Sign up

Export Citation Format

Share Document