Characteristic Hypersurfaces Research Articles

Characteristic surfaces and characteristic initial data for the generalized Einstein–Maxwell field equations

The characteristic hypersurfaces of the source-free generalized Einstein–Maxwell field equations are investigated. It is shown that such hypersurfaces can be null, and it appears as though they may also be spacelike or timelike. Examples of characteristic initial data on spacelike and timelike hypersurfaces are presented, and it turns out that these examples involve very intense gravitational and electromagnetic fields when the coupling constant k is small in magnitude.

Initial-value problem and the Dirac-bracket relations in null gravidynamics

Connections between the initial-value problem on characteristic hypersurfaces and null gravidynamics are analyzed in different gauges. A geometrical interpretation of the corner initial data is given in terms of the external curvature of one of the initial null hypersurfaces. The Dirac-Bracket relations between the dynamical variables of gravity are explicitly calculated in the Bondi gauge.

On characteristic hypersurfaces of submanifolds in Euclidean space

On characteristic hypersurfaces of submanifolds in Euclidean space

Local analytic continuation of holomorphic solutions of partial differential equations

We ask. When is it possible to continue analytically holomorphic solutions of partial differential equations. Using objects called cones of analytic continuation we get sufficient conditions generalizing results by J.-M. Bony and P. Schapira and by Y. Tsuno. The results are counterparts of earlier results by the author on local uniqueness in the Cauchy problem. We also give a necessary condition by constructing solutions with singularities. We think that the technique used here should also have other applications. Anyhow this result is a generalization of a result by Y. Tsuno related to simple characteristic hypersurfaces. It corresponds to the existence of local null solution when the initial hypersurface has constant multiplicity in the Cauchy problem.

Boundary value problems and Mayer-Vietoris sequence

We define formally non characteristic hypersurfaces for a complex of differential operators and we establish a Mayer-Vietoris sequence. We give applications to theorems on remouvable singularities and to systems of differential equations without solutions.

Non-uniqueness and uniqueness in the Cauchy problem at simple characteristic points

The subject of this paper is uniqueness and non-uniqueness in the Cauchy problem at simple characteristic points of the initial hypersurface. We only consider linear partial differential equations with analytic coefficients. We use the notation from the book [1] by H6rmander with the exception that D~ = t3/~xj, 1 0 in a neighbourhood of x °, and if H is of a type covered by [6, Theorem 3] then there is a neighbourhood V of x ° and a function u e C ~ ( V ) such that P ( x , D ) u = O , x ° ~ suppu C {x; ~0(x) =>0, x ~ V}." When we want to construct characteristic hypersurfaces that are not hyperplanes we have to rely on integration of Hamilton's equations. So we restrict ourselves to operators with real principal part and to simple characteristic points on the initial hypersurface. The three points of departure of this paper is the method to prove non-uniqueness by Malgrange [3], the uniqueness theorem [2, Theorem 8.1] by H6rmander, see Theorem 3 below, and the technique of proving uniqueness taken from Persson [5]. The part played by Theorem 3 is a passive one leaving to us to study the case when a priori the bicharacteristic curve does not leave the support of the solution. In [9] Zachmanoglou proves the following theorem.

Cauchy problems with characteristic initial hypersurface

Cauchy problems with characteristic initial hypersurface

On the initial-value problems with data on a characteristic hypersurface

On the initial-value problems with data on a characteristic hypersurface

Mathematical Methods in Wave Propagation Part I—Linear Wave Front Analysis

This paper presents a brief summary of some of the mathematical techniques that are used in wave front analysis as applied to linear hyperbolic partial differential equations. After an introductory review of the method of classification of partial differential equations, and the identification of wave propagation phenomena, the effects of dispersion on a wave profile are outlined. Thereafter the idea of a characteristic hypersurface is introduced and a little of its differential geometry is examined. Finally, all ideas are drawn together to derive the transport equation which governs the propagation of discontinuities along rays. The appearance of singularities in the solution as a result of focusing effects emerges naturally from a study of the transport equation.

Wave Equations, Multipole Solutions, and Characteristic Hypersurfaces for Massless Particles

In this paper, the canonical wave equation and auxiliary conditions for a massless particle with arbitrary integral or half-integral spin s are derived from a symmetric spinor formulation. The wave equation is solved for the multipole solutions using the appropriate parity operator for each spin. Finally, the characteristic hypersurfaces for the wave equation are derived and found to agree with that of the photon. The special cases of spins one, three-halves, and two are detailed in conclusion.

Characteristic Hypersurfaces in General Relativity. I

Lightlike (null) hypersurfaces are treated by means of an intrinsic Ricci rotation coefficient technique. This provides an effective way of dealing with the various types of geometry on a null hypersurface. The formalism is used to examine inner affinities, differential invariants, local features such as asymptotic and shear directions and geodesic lines, and to give a short description of null hypersurfaces in flat space-time. Applications to gravitational radiation theory and cosmology are briefly mentioned.

On the Characteristic Initial Value Problem in Gravitational Theory

The problem of determining a solution of the Einstein field equations for the gravitational field from data set on a pair of intersecting characteristic (that is, null) hypersurfaces and on their intersection Σ is considered. It is shown that by giving the conformal inner metric of each hypersurface, the inner geometry of Σ, the two mean extrinsic curvatures of Σ, and one additional extrinsic quantity for Σ, one completely determines a solution in a space-time region. In a suitable coordinate system giving the above data amounts to giving four functions of three variables, four functions of two variables, and one function of one variable; all these can be given without any constraints. In a given space time with given Σ the functions are almost, but not entirely, known functions of their arguments. All the derivations and discussions are applicable only in a sufficiently small four-dimensional region.