scholarly journals DIFFERENTIAL FORMS AND WAVE EQUATIONS FOR GENERAL RELATIVITY

1998 ◽  
Vol 07 (06) ◽  
pp. 857-885 ◽  
Author(s):  
STEPHEN R. LAU
Keyword(s):  
General Relativity ◽  
Wave Equation ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
Evolution Equations ◽  
Differential Forms ◽  
Extrinsic Curvature ◽  
The Cauchy Problem ◽  
Hyperbolic Form ◽  
Tensor Index

In recent papers, Choquet–Bruhat and York and Abrahams, Anderson, Choquet–Bruhat, and York (we refer to both works jointly as AACY) have cast the 3 + 1 evolution equations of general relativity in gauge-covariant and causal "first-order symmetric hyperbolic form," thereby cleanly separating physical from gauge degrees of freedom in the Cauchy problem for general relativity. A key ingredient in their construction is a certain wave equation which governs the light-speed propagation of the extrinsic curvature tensor. Along a similar line, we construct a related wave equation which, as the key equation in a system, describes vacuum general relativity. Whereas the approach of AACY is based on tensor-index methods, the present formulation is written solely in the language of differential forms. Our approach starts with Sparling's tetrad-dependent differential forms, and our wave equation governs the propagation of Sparling's two-form, which in the "time-gauge" is built linearly from the "extrinsic curvature one-form." The tensor-index version of our wave equation describes the propagation of (what is essentially) the Arnowitt–Deser–Misner gravitational momentum.

WAVE AND MAXWELL'S EQUATIONS IN CARNOT GROUPS

2012 ◽  
Vol 14 (05) ◽  
pp. 1250032 ◽  
Author(s):  
BRUNO FRANCHI ◽  
MARIA CARLA TESI
Keyword(s):  
Vector Potential ◽  
Maxwell’S Equations ◽  
Wave Equations ◽  
Maxwell's Equations ◽  
Evolution Equations ◽  
Differential Forms ◽  
Higher Order ◽  
Carnot Groups ◽  
New Class ◽  
Euclidean Setting

In this paper we define Maxwell's equations in the setting of the intrinsic complex of differential forms in Carnot groups introduced by M. Rumin. It turns out that these equations are higher-order equations in the horizontal derivatives. In addition, when looking for a vector potential, we have to deal with a new class of higher-order evolution equations that replace usual wave equations of the Euclidean setting and that are no more hyperbolic. We prove equivalence of these equations with the "geometric equations" defined in the intrinsic complex, as well as existence and properties of solutions.

scholarly journals Time-Dependent Wave Equations on Graded Groups

2021 ◽  
Vol 171 (1) ◽  
Author(s):  
Michael Ruzhansky ◽  
Chiara Alba Taranto
Keyword(s):  
Wave Equation ◽  
Lie Groups ◽  
Classical Result ◽  
Wave Equations ◽  
Evolution Equations ◽  
Time Dependent ◽  
Well Posedness ◽  
Local Loss ◽  
Stratified Lie Groups ◽  
Loss Of Regularity

AbstractIn this paper we consider the wave equations for hypoelliptic homogeneous left-invariant operators on graded Lie groups with time-dependent Hölder (or more regular) non-negative propagation speeds. The examples are the time-dependent wave equation for the sub-Laplacian on the Heisenberg group or on general stratified Lie groups, or $p$ p -evolution equations for higher order operators on ${{\mathbb{R}}}^{n}$ R n or on groups, already in all these cases our results being new. We establish sharp well-posedness results in the spirit of the classical result by Colombini, De Giorgi and Spagnolo. In particular, we describe an interesting local loss of regularity phenomenon depending on the step of the group (for stratified groups) and on the order of the considered operator.

The wave equation for the Lanczos potential. I

1994 ◽  
Vol 447 (1931) ◽  
pp. 557-575 ◽  
Keyword(s):  
Wave Equation ◽  
Degrees Of Freedom ◽  
Wave Equations ◽  
Conformal Curvature ◽  
Lanczos Potential ◽  
Conformal Curvature Tensor ◽  
Radiation Theory ◽  
Non Local ◽  
Spinor Wave ◽  
Gauge Conditions

The non-local part of the gravitational field in general relativity is described by the 10 component conformal curvature tensor C abcd of Weyl. For this field Lanczos found a tensor potential L abc with 16 independent components. We can make L abc have only 10 effective degrees of freedom by imposing the 6 gauge conditions L ab s :s = 0. Both fields C abcd , L abc satisfy wave equations. The wave equation satisfied by C abcd is nonlinear, even in vacuo . However, a linear spinor wave equation for the Lanczos potential has been found by Illge but no correct tensor wave equation for L abc has yet been published. Here, we derive a correct tensor wave equation for L abc and when it is simplified with the aid of some four­-dimensional identities it is equivalent to Illge’s wave equation. We also show that the nonlinear spinor wave equation of Penrose for the Weyl field can be derived from Illge’s spinor wave equation. A set of analogues of well-known results of classical electromagnetic radiation theory can now be given. We indicate how a Green’s function approach to gravitational radiation could be based on our tensor wave equation, when a global study of space-time is attempted.

A global existence theorem for the Cauchy problem of nonlinear wave equations

1988 ◽  
Vol 109 (3-4) ◽  
pp. 261-269 ◽  
Author(s):  
Jianmin Gao ◽  
Lichen Xu
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Global Existence ◽  
Nonlinear Wave ◽  
Wave Equations ◽  
Lorentz Invariance ◽  
Nonlinear Wave Equations ◽  
Homogeneous Wave ◽  
The Cauchy Problem ◽  
Global Existence Theorem

SynopsisIn this paper we consider the global existence (in time) of the Cauchy problem of the semilinear wave equation utt – Δu = F(u, Du), x ∊ Rn, t > 0. When the smooth function F(u, Du) = O((|u| + |Du|)k+1) in a small neighbourhood of the origin and the space dimension n > ½ + 2/k + (1 + (4/k)2)½/2, a unique global solution is obtained under suitable assumptions on initial data. The method used here is associated with the Lorentz invariance of the wave equation and an improved Lp–Lq decay estimate for solutions of the homogeneous wave equation. Similar results can be extended to the case of “fully nonlinear wave equations”.

Unified error analysis for nonconforming space discretizations of wave-type equations

2018 ◽  
Vol 39 (3) ◽  
pp. 1206-1245 ◽  
Author(s):  
David Hipp ◽  
Marlis Hochbruck ◽  
Christian Stohrer
Keyword(s):  
Wave Equation ◽  
Error Analysis ◽  
Hilbert Spaces ◽  
Error Bounds ◽  
Wave Equations ◽  
Evolution Equations ◽  
Convergence Rates ◽  
Linear Wave ◽  
Element Discretization ◽  
Continuous Space

Abstract This paper provides a unified error analysis for nonconforming space discretizations of linear wave equations in the time domain. We propose a framework that studies wave equations as first-order evolution equations in Hilbert spaces and their space discretizations as differential equations in finite-dimensional Hilbert spaces. A lift operator maps the semidiscrete solution from the approximation space to the continuous space. Our main results are a priori error bounds in terms of interpolation, data and conformity errors of the method. Such error bounds are the key to the systematic derivation of convergence rates for a large class of problems. To show that this approach significantly eases the proof of new convergence rates, we apply it to an isoparametric finite element discretization of the wave equation with acoustic boundary conditions in a smooth domain. Moreover, our results reproduce known convergence rates for already investigated conforming and nonconforming space discretizations in a concise and unified way. The examples discussed in this paper comprise discontinuous Galerkin discretizations of Maxwell’s equations and finite elements with mass lumping for the acoustic wave equation.

The 3+1 formalism in teleparallel and symmetric teleparallel gravity

2021 ◽  
Vol 81 (12) ◽  
Author(s):  
Salvatore Capozziello ◽  
Andrew Finch ◽  
Jackson Levi Said ◽  
Alessio Magro
Keyword(s):  
General Relativity ◽  
Degrees Of Freedom ◽  
Hamiltonian Structure ◽  
Evolution Equations ◽  
Teleparallel Gravity ◽  
Common Features ◽  
Temporal Parts ◽  
Spacetime Metric ◽  
Set Up

AbstractTeleparallel and symmetric teleparallel gravity offer platforms in which gravity can be formulated in interesting geometric approaches, respectively given by torsion and nonmetricity. In this vein, general relativity can be expressed in three dynamically equivalent ways which may offer insights into the different properties of these decompositions such as their Hamiltonian structure, the efficiency of numerical analyses, as well as the classification of gravitational field degrees of freedom. In this work, we take a $$3+1$$ 3 + 1 decomposition of the teleparallel equivalent of general relativity and the symmetric teleparallel equivalent of general relativity which are both dynamically equivalent to curvature based general relativity. By splitting the spacetime metric and corresponding tetrad into their spatial and temporal parts as well as through finding the Gauss-like equations, it is possible to set up a general foundation for the different formulations of gravity. Based on these results, general 3-tetrad and 3-metric evolution equations are derived. Finally through the choice of the two respective connections, the metric $$3+1$$ 3 + 1 formulation for general relativity is recovered as well as the tetrad $$3+1$$ 3 + 1 formulation of the teleparallel equivalent of general relativity and the metric $$3+1$$ 3 + 1 formulation of symmetric teleparallel equivalent of general relativity. The approach is capable, in principle, of resolving common features of the various formulations of general relativity at a fundamental level and pointing out characteristics that extensions and alternatives to the various formulations can present.

The Hamiltonian formalism

2019 ◽  
pp. 101-108
Author(s):  
Steven Carlip
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Hamiltonian Systems ◽  
Gauge Invariance ◽  
Degrees Of Freedom ◽  
Hamiltonian Formalism ◽  
Extrinsic Curvature ◽  
Hamiltonian Form ◽  
Boundary Terms

So far, general relativity has been viewed from the four-dimensional Lagrangian perspective. This chapter introduces the (3+1)-dimensional Hamiltonian formalism, starting with the ADM form of the metric and extrinsic curvature. The Hamiltonian form of the action is served, and the nature of the constraints—and, more generally, of constraints and gauge invariance in Hamiltonian systems—is discussed. The formalism is used to count the physical degrees of freedom of the gravitational field. The chapter ends with a discussion of boundary terms and the ADM energy.

DE RHAM WAVE EQUATION FOR TENSOR VALUED p-FORMS

2003 ◽  
Vol 12 (08) ◽  
pp. 1363-1384 ◽  
Author(s):  
DONATO BINI ◽  
CHRISTIAN CHERUBINI ◽  
ROBERT T. JANTZEN ◽  
REMO RUFFINI
Keyword(s):  
General Relativity ◽  
Black Hole ◽  
Wave Equation ◽  
Covariant Derivative ◽  
Differential Forms ◽  
Black Hole Spacetimes ◽  
Metric Connection ◽  
De Rham ◽  
Integral Spin ◽  
Curvature Tensors

The de Rham Laplacian Δ (dR) for differential forms is a geometric generalization of the usual covariant Laplacian Δ, and it may be extended naturally to tensor-valued p-forms using the exterior covariant derivative associated with a metric connection. Using it the wave equation satisfied by the curvature tensors in general relativity takes its most compact form. This wave equation leads to the Teukolsky equations describing integral spin perturbations of black hole spacetimes.

Extended F-Expansion Method for Solving the modified Korteweg-de Vries (mKdV) Equation

2020 ◽  
Vol 11 (1) ◽  
pp. 93-100
Author(s):  
Vina Apriliani ◽  
Ikhsan Maulidi ◽  
Budi Azhari
Keyword(s):  
Wave Equation ◽  
Exact Solutions ◽  
Internal Waves ◽  
Water Waves ◽  
Wave Equations ◽  
Elliptic Functions ◽  
Expansion Method ◽  
Mkdv Equation ◽  
Innovative Methods ◽  
De Vries

One of the phenomenon in marine science that is often encountered is the phenomenon of water waves. Waves that occur below the surface of seawater are called internal waves. One of the mathematical models that can represent solitary internal waves is the modified Korteweg-de Vries (mKdV) equation. Many methods can be used to construct the solution of the mKdV wave equation, one of which is the extended F-expansion method. The purpose of this study is to determine the solution of the mKdV wave equation using the extended F-expansion method. The result of solving the mKdV wave equation is the exact solutions. The exact solutions of the mKdV wave equation are expressed in the Jacobi elliptic functions, trigonometric functions, and hyperbolic functions. From this research, it is expected to be able to add insight and knowledge about the implementation of the innovative methods for solving wave equations. 

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