Characteristic Initial Value Problem Research Articles

Reduction of the characteristic initial value problem to the Cauchy problem and its applications to the Einstein equations

A method is described by means of which the characteristic initial value problem can be reduced to the Cauchy problem and examples are given of how it can be used in practice. As an application it is shown that the characteristic initial value problem for the Einstein equations in vacuum or with perfect fluid source is well posed when data are given on two transversely intersecting null hypersurfaces. A new discussion is given of the freely specifiable data for this problem.

On the zeros of solutions to nonlinear hyperbolic equations

SynopsisWe consider the hyperbolic equation uxy + c(x, y, u) =f(x, y) and the wave equationWe show that, under suitable conditions, there are bounded domains in which every solution to certain problems has a zero. Characteristic initial value problems and initial boundary value problems are considered.

Qualitative behaviour of solutions of the Goursat problem for hyperbolic differential equations

Oscillation criteria and asymptotic behavior of solutions of characteristic initial-value problems have been extended to the Goursat problem.

Multiple steady states for characteristic initial value problems

The time dependent, isentropic, quasi one-dimensional equations of gas dynamics and other model equations are considered under the constraint of characteristic boundary conditions. Analysis of the time evolution shows how different initial data may lead to different steady states and how seemingly anomalous behavior of the solution may be resolved. Numerical experimentation using time consistent explicit algorithms verifies the conclusions of the analysis. The use of implicit schemes with very large time steps leads to erroneous results.

The initial value problem for colliding gravitational and hydrodynamic waves

The collision of two plane gravitational and hydrodynamic waves with parallel polarizations is studied. In the interaction region, to the future of the collision, the space-time admits two hypersurface orthogonal Killing fields, and the problem reduces to solving two (decoupled!) linear partial differential equations. The characteristic initial value problem for these equations is explicitly solved by means of Riemann’s method. In the appendices the relevant field equations are written in two different coordinate systems that have been proved useful in the studies of colliding waves, relationships among the solutions of the gravitational and the hydrodynamic equations are obtained, separable solutions involving Bessel functions are constructed, and integral identities among the Legendre functions are obtained.

Gravitational radiation and the validity of the far-zone quadrupole formula in the Newtonian limit of general relativity

We examine the gravitational radiation emitted by a sequence of spacetimes whose near-zone Newtonian limit we have previously studied. The spacetimes are defined by initial data which scale in a Newtonian fashion: the density as ${\ensuremath{\epsilon}}^{2}$, velocity as \ensuremath{\epsilon}, pressure as ${\ensuremath{\epsilon}}^{4}$, where \ensuremath{\epsilon} is the sequence parameter. We asymptotically approximate the metric at an event which, as \ensuremath{\epsilon}\ensuremath{\rightarrow}0, remains a fixed number of gravitational wavelengths distant from the system and a fixed number of wave periods to the future of the initial hypersurface. We show that the radiation behaves like that of linearized theory in a Minkowski spacetime, since the mass of the metric vanishes as \ensuremath{\epsilon}\ensuremath{\rightarrow}0. We call this Minkowskian far-zone limiting manifold FM; it is a boundary of the sequence of spacetimes, in which the radiation carries an energy flux given asymptotically by the usual far-zone quadrupole formula (the Landau-Lifshitz formula), as measured both by the Isaacson average stress-energy tensor in FM or by the Bondi flux on ${\mathit{I}}_{\mathrm{FM}}^{+}$. This proves that the quadrupole formula is an asymptotic approximation to general relativity. We study the relation between ${\mathit{I}}_{\mathrm{\ensuremath{\epsilon}}}^{+}$, the sequence of null infinities of the individual manifolds, and ${\mathit{I}}_{\mathrm{FM}}^{+}$; and we examine the gauge-invariance of FM under certain gauge transformations. We also discuss the relation of this calculation with similar ones in the frame-work of matched asymptotic expansions and others based on the characteristic initial-value problem.

Gravitational radiation from dust

A dust cloud is examined within the framework of the general relativistic characteristic initial value problem. Unique gravitational initial data are obtained by requiring that the space-time be quasi-Newtonian. Explicit calculations of metric and matter fields are presented, which include all post-Newtonian corrections necessary to discuss the major physical properties of null infinity. These results establish a curved space version of the Einstein quadrupole formula, in the form ‘‘news function equals third time derivative of transverse quadrupole moment,’’ for this system. However, these results imply that some weakened notion of asymptotic flatness is necessary for the description of quasi-Newtonian systems.

Functional characteristic initial value problems

Functional characteristic initial value problems

Axially symmetric gravitational radiation from isolated sources

Asymptotically flat, dynamic, axially symmetric space-times are considered from the point of view of the characteristic initial value problem and the expansion of solutions in powers of r−1. Imposing conditions such that the system initially be exactly static and nonradiative finally, a news function is constructed which governs the dynamics of such a space-time and which yields a finite, nonvanishing total loss of the Bondi mass. Furthermore, the Riemann tensor (to order r−3) is known explicitly for all time since an expression for the mass aspect is also determined. It is also shown that the total change in the asymptotic shear is related to the total change in the Bondi mass. Finally the implications concerning transitions between two exactly stationary states are discussed.

Oscillation Properties of Nonlinear Hyperbolic Equations

A variety of oscillation properties are established for solutions of characteristic initial value problems for the nonlinear telegraph equation with a forcing term. Some analogous questions are considered for initial boundary value problems for the forced nonlinear wave equation. The principal tool is an averaging technique which enables one to establish such oscillation properties in terms of related ordinary differential inequalities.

The cosmological problem as initial value problem on the observer's past light cone: geometry

In the context of general relativity an approach to cosmology is suggested which rests exclusively on quantities accessible to the astronomical observer. The approach is based on the characteristic initial value problem for the Einstein field equations imposed on the observer's past light cone. the inner geometry of the past light cone enters the relations between observable quantities. Thus, it should become possible to determine both the cone geometry, as well as the distribution of cosmic matter near the cone, without adopting arbitrary assumptions. The propagation equations determine the gravitational field and the matter distribution at finite distances from the past cone. The author specifies the cone initial data for a general dust model. An explicit procedure in terms of successive integrations is given to determine the gravitational field near the past light cone.

Design of Supercritical Cascades with High Solidity

The method of complex characteristics of Garabedian and Korn was successfully used to design shockless cascades with solidities of up to one. A code was developed using this method and a new hodograph transformation of the flow onto an ellipse. This code allows the design of cascades with solidities of up to two and larger turning angles. The equations of potential flow are solved in a complex hodograph like domain by setting a characteristic initial value problem and integrating along suitable paths. The topology that the new mapping introduces permits a simpler construction of these paths of integration.

Numerical relativity. II. Numerical methods for the characteristic initial value problem and the evolution of the vacuum field equations for space‒times with two Killing vectors

This is the second of a sequence of papers on the numerical solution of the characteristic initial value problem in general relativity. Although the equations to be integrated have regular coefficients, the nonlinearity leads to the occurrence of singularities after a finite evolution time. In this paper we first discuss some novel techniques for integrating the equations right up to the singularities. The second half of the paper presents as examples the numerical evolution of the Schwarzschild and certain colliding plane wave space‒times.

Riemann’s function has an exponential bound

The Riemann function υ ( t ; x ) \upsilon (t;x) of a hyperbolic characteristic initial value problem has been much used in recent years to provide upper bounds for functions which satisfy Gronwall-type integral inequalities. This note gives a direct proof of the fact that υ \upsilon satisfies an inequality of the form υ ( t ; x ) ⩽ exp ⁡ ( ∫ t x b ( s ) d s \upsilon (t;x) \leqslant \exp (\int _t^x {b(s)ds} .

Qualitative theory for hyperbolic characteristic initial value problems

SynopsisCharacteristic initial value problems associated with hyperbolic equations of the form uxy + g(x, y)u = 0 are considered for (x, y)∈ℝ+× ℝ+. New criteria for the existence of a nodal line asymptotic to the axes are established, as are criteria for the existence of a zero beyond such a nodal line. Some numerical solutions are presented in graphical form and discussed relative to what is known about oscillation properties of such problems.

Numerical relativity. I. The characteristic initial value problem

In this, the first of a series of papers on numerical relativity, the characteristic initial value problem is posed in a form suitable for numerical integration. It can be reduced to the solution of two initial value problems for sets of ordinary differential equations (on the initial surfaces) and the solution of two initial value problems for hyperbolic systems of equations, one linear, one quasilinear. The initial data may be specified freely. Subsequent papers will develop numerical solutions of Einstein’s equations with use of this formalism.

Null spaces and ranges for the classical and spherical Radon transforms

Let R be the classical Radon transform that integrates a function over hyperplanes in R n and let SM be the transform that integrates a function over spheres containing the origin in R n . We prove continuity results for both transforms and explicitly give the null space of R for a class of square integrable functions on the exterior of a ball in R n as well as the null space of SM for square integrable functions on a ball. We show SM: L 2( R n ) → L 2( R n ) is one-one, and we characterize the range of SM on classes of smooth functions and square integrable functions by certain moment conditions. If g( x) is a Schwartz function on R n that is zero to infinite order at x = 0, we prove moment conditions sufficient for g to be in the range of SM( C ∞( R n )). We apply our results on SM to existence and uniqueness theorems for solutions to a characteristic initial value problem for the Darboux partial differential equation.

On the existence of analytic null asymptotically flat solutions of Einstein’s vacuum field equations

This paper proves the existence of analytic solutions of the asymptotic characteristic initial value problem for Einstein’s field equations for analytic data on past null infinity and on an incoming null hypersurface.

The asymptotic characteristic initial value problem for Einstein’s vacuum field equations as an initial value problem for a first-order quasilinear symmetric hyperbolic system

The asymptotic characteristic initial value problem for Einstein’s vacuum field equations where data are given on an incoming null hypersurface and on part of past null infinity is reduced to a characteristic initial value problem for a first-order quasilinear symmetric hyperbolic system of differential equations for which existence and uniqueness of solutions can be shown. It is delineated how the same method can be applied to the standard Cauchy problems for Einstein’s vacuum and conformal vacuum equations.

A characteristic initial-value problem approach to the formation of inhomogeneities in the early Universe

Starting from a small perturbation in the initial data for a Robertson-Walker universe, it is shown that, in a small region of space-time, it is possible for large fluctuations to develop in both the metric and physical variables describing the Universe. The initial data are specified on a characteristic hypersurface in a region of perfect fluid filled space-time and the method of analysis of the field equations is based on the techniques introduced by Bondi, van der Burg and Metzner (1962).