Characteristic Hypersurfaces Research Articles

Survival of the black hole’s Cauchy horizon under noncompact perturbations

We study numerically the evolution of spacetime, and in particular of a spacetime singularity, inside a black hole under a class of perturbations of noncompact support. We use a very simplified toy model of a spherical charged black hole which is perturbed nonlinearly by a self-gravitating, spherical scalar field. The latter grows logarithmically with advanced time along an outgoing characteristic hypersurface. We find that for that class of perturbations a portion of the Cauchy horizon survives as a noncentral, null singularity.

Mixed characteristic hypersurfaces of finite Cohen–Macaulay type

We define the mixed ADE singularities, which are generalizations of the ADE plane curve singularities to the case of mixed characteristic. The ADE plane curve singularities are precisely the equicharacteristic plane curve singularities of finite Cohen–Macaulay type; we show that the mixed ADE singularities also have finite Cohen–Macaulay type.

THE GOURSAT PROBLEM FOR HYPERBOLIC LINEAR THIRD ORDER EQUATIONS

The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality.

The Goursat problem for hyperbolic linear third order equations

Abstract The third order hyperbolic linear differential equation is considered in the non‐cylindrical domain of multidimensional Euclidean space. The equation operator is a composition of a differentiation operator of the first order and second order operator, which is hyperbolic with respect to the prescribed vector field. Apart from the equation, Goursat and Cauchy conditions are defined for an unknown function. Thus the boundary of the domain, where this hyperbolic equation is defined, consists of characteristic hypersurfaces, the hypersur‐faces, where Cauchy conditions are prescribed, and hypersurfaces with no conditions. For the mentioned problem the existence and uniqueness of the strong solution are proved using mollifying operators with a variable step and functional analysis methods on the base of the previously proved energy inequality.

Creation and propagation of logarithmic singularities by interaction of two piecewise smooth progressing waves

Our aim is to understand the non-conservation of the piecewise smooth regularity by a semi-linear interaction of two transverse progressing waves. Indeed, we know that this phenomenon occurs when the number of characteristic hypersurfaces passing through the locus of interaction, that is, a two-codimensional variety, is strictly inferior to the size of the considered first order strictly hyperbolic system. Thanks to the study of a significant example, we explain the obstruction to the piecewise smooth propagation by a loss of transmission property for the symbols describing the conormal singularities, which originates logarithmic singularities.

Divergence-Measure Fields and Hyperbolic Conservation Laws

We analyze a class of L 1 vector elds, called divergence-measure elds. We establish the Gauss-Green formula, the normal traces over subsets of Lipschitz boundaries, and the product rule for this class of L 1 elds. Then we apply this theory to analyzeL 1 entropy solutions of initial-boundary-value problems for hyperbolic conservation laws and to study the ways in which the solutions assume their initial and boundary data. The examples of conservation laws include multidimensional scalar equations, the system of nonlinear elasticity, and a class ofm m systems with afne characteristic hypersurfaces. The analysis inL 1 also extends toL p .

Construction de solutions ramifiées autour de deux hypersurfaces caractéristiques pour des opérateurs quasi-linéaires

Let P(x,D)u=v be a partial differential equation, where P(x,D) is a quasi-linear operator of second order with simple characteristic hypersurfaces; we get solutions u which are ramified around two simple characteristic hypersurfaces if v is ramified function and D t 1v, D t 2,v are bounded.

The Characteristic Treatment of Black Holes

The characteristic initial value problem has been successfully implemented as a robust computational algorithm (the PITT NULL CODE) to evolve 4-dimensional vacuum spacetimes. It has been applied to the calculation of gravitational waveforms emitted by black holes and to the event horizon structure in the merger of black holes. The characteristic code also has potential application to the binary black hole problem via Cauchy-characteristic matching. Because the event horizon is itself a characteristic hypersurface, it can be analyzed by characteristic techniques as a stand-alone object. We have developed an analytic conformal model of null hypersurfaces which gives new insight into the intrinsic geometry of the pairof-pants horizon found in the numerical simulation of the head-on collision of black holes and into the initially toroidal horizon found in the simulation of a collapsing, rotating cluster. Most studies of black hole formation and merger have been restricted to axisymmetry. However, axisymmetric horizons, like the Schwarzschild horizon, are non-generic. When applied to a non-axisymmetric horizon, the characteristic approach reveals substantially new features. In particular, coalescing black holes generically go through a toroidal phase before they become spherical. The conformal structure of the event horizon supplies part of the data for a simulation of the exterior space-time. This provides a new way to calculate the post-merger waveforms from a binary black hole inspiral. §1. Null cone evolution A longterm project 1) to develop the pioneering work of Bondi 2) and Penrose 3) into a computational algorithm for the characteristic initial value problem has recently culminated in a highly accurate,efficient and robust code — the PITT NULL CODE. 4) Because the evolution proceeds on a space-time foliation by null cones which are generated by the characteristic rays of the theory,this approach offers several advantages for numerical work. I will describe here the fruits of these investigations relevant to black hole physics. In null cone coordinates,Einstein’s equations reduce to propagation equations along the radial light rays,which can be integrated in hierarchical order for one variable at a time. This leads to a highly efficient characteristic marching algorithm. There is one complex evolution variable which describes the free degrees of freedom of the gravitational field and four auxiliary variables. A compactified grid,based upon Penrose’s conformal description of null infinity,removes the necessity of an artificial outgoing radiation condition and makes possible a rigorous description of geometrical quantities such as the Bondi mass and news function. The news function supplies both the true waveform and polarization of the gravitational radiation incident on a distant antenna. Furthermore the grid domain is exactly the region in which waves propagate,which is ideally efficient for the purpose of radiation studies. Since each

Characteristic hypersurfaces in a relativistic superfluid theory

By discussing the Cauchy problem, we determine the covariant equation of the characteristic hypersurfaces in a relativistic superfluid theory.

The generation of semilinear singularities by a swallowtail caustic

We construct examples of bounded solutions to a semilinear system P u = f ( z, u ), z ∈ Ω ⊂ [inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="01i" /], f smooth, P = P · Id, with P a strictly hyperbolic differential operator of second order with smooth coefficients, such that for a time function t of P , the following properties are satisfied: (1) u is conormal to a smooth characteristic hypersurface Σ in t < 0 such that Σ develops a swallowtail singularity at time t = 0; (2) For t > 0, u is singular, not only at Σ, but also on the forward characteristic cone over the swallowtail tip.

CR functions vanishing on open sets (almost) complex structures and Cohen's example

This paper deals with (non-)uniqueness in the Cauchy problem, for functions annihilated by a complex vector field. (Do Lu ≡ 0 and the Cauchy data u = 0 on some non characteristic hypersurface imply u ≡ 0, in a neighborhood of the hypersurface?) Most of the results are more or less well known. But the proofs/constructions are mostly new, using very elementary considerations on (almost) complex structures. Hopefully this makes the topic more accessible, in particular to complex analysts. The failure of uniqueness in the Cauchy problem provides a striking example of non embeddable strictly pseudoconvex CR structure.

Un résultat de ramification globale

We study the non-characteristic linear ramified Cauchy problem for an operator whose coefficients are entire functions, those of the principal part being constant; we show that the solution of the Cauchy problem is ramified around the characteristic hypersurfaces in the whole space.

Cauchy–Characteristic Evolution and Waveforms

We investigate a new methodology for the computation of waves generated by isolated sources. This approach consists of a global spacetime evolution algorithm based on a Cauchy initial-value formulation in a bounded interior region and based on characteristic hypersurfaces in the exterior; we match the two schemes at their common interface. The characteristic formulation allows accurate description of radiative infinity in a compactified finite coordinate interval, so that our numerical solution extends to infinity and accurately models the free-space problem. The matching interface need not be situated far from the sources, the wavefronts may have arbitrary nonspherical geometry, and strong nonlinearity may be present in both the interior and the exterior regions. Stability and second-order convergence of the algorithms (to the exact solution of the infinite-domain problem) are established numerically in three space dimensions. The matching algorithm is compared with examples of both local and nonlocal radiation boundary conditions proposed in the literature. For linear problems, matching outperformed the local radiation conditions chosen for testing, and was about as accurate (for the same grid resolution) as the exact nonlocal conditions. However, since the computational cost of the nonlocal conditions is many times that of matching, this algorithm may be used with higher grid resolutions, yielding a significantly higher final accuracy. For strongly nonlinear problems, matching was significantly more accurate than all other methods tested. This seems to be due to the fact that currently available local and nonlocal conditions are based on linearizing the governing equations in the far field, while matching consistently takes nonlinearity into account in both interior and exterior regions.

Interaction of conormal waves with strong and weak singularities for semi-linear equations

We first define a kind of new function space, called the space of twice conormal distributions. With some estimates on these function spaces, we can prove that if there exist two characteristic hypersurfaces bearing strong and weak singularities respectively intersect transversally, then some new singularities will take place and their strength will be no more than the original weak one.

THE UNIQUENESS AND CONTINUOUS DEPENDENCE FOR CHARACTERISTIC CAUCHY PROBLEM OF PARTIAL DIFFERENTIAL EQUATIONS

In this paper, we investigate the connection between the uniqueness and the continuous dependence for the Cauchy problem of linear partial differential equations with everywhere characteristic initial hypersurface. The local Carleman's estimate used by S. Alinhac and M. S. Baouendi to prove the uniqueness for the characteristic Cauchy problem is improved to the global estimate with a specific peaking weight factor. Using this estimate, we can prove similarly the uniqueness of the characteristic Cauchy problem. In particular, we derive also the continuous dependence for the solution of this problem in the function class that a certain norm is bounded by the aid of the same inference of proving the global estimate.

Conormal and piecewise smooth solutions to quasilinear wave equations

In this paper, we show first that if a solution u u of the equation P 2 ( t , x , u , D u , D ) u = f ( t , x , u , D u ) {P_2}(t,x,u,Du,D)u = f(t,x,u,Du) , where P 2 ( t , x , u , D u , D ) {P_2}(t,x,u,Du,D) is a second order strictly hyperbolic quasilinear operator, is conormal with respect to a single characteristic hypersurface Σ \Sigma of P 2 {P_2} in the past and Σ \Sigma is smooth in the past, then Σ \Sigma is smooth and u u is conormal with respect to Σ \Sigma for all time. Second, let Σ 0 {\Sigma _0} and Σ 1 {\Sigma _1} be characteristic hypersurfaces of P 2 {P_2} which intersect transversally and let Γ = Σ 0 ∩ Σ 1 \Gamma = {\Sigma _0} \cap {\Sigma _1} . If Σ 0 {\Sigma _0} and Σ 1 {\Sigma _1} are smooth in the past and u u is conormal with repect to { Σ 0 , Σ 1 } \{ {\Sigma _0},{\Sigma _1}\} in the past, then Γ \Gamma is smooth, and u u is conormal with respect to { Σ 0 , Σ 1 } \{ {\Sigma _0},{\Sigma _1}\} locally in time outside of Γ \Gamma , even though Σ 0 {\Sigma _0} and Σ 1 {\Sigma _1} are no longer necessarily smooth across Γ \Gamma . Finally, we show that if u ( 0 , x ) u(0,x) and ∂ t u ( 0 , x ) {\partial _t}u(0,x) are in an appropriate Sobolev space and are piecewise smooth outside of Γ \Gamma , then u u is piecewise smooth locally in time outside of Σ 0 ∪ Σ 1 {\Sigma _0} \cup {\Sigma _1} .

Interactions of conormal waves for fully semilinear wave equations

An existence theorem is proved for the continuation form of the Cauchy problem Pu = ƒ(z, u(z), ▽u(z)), where P is a second order strictly hyperbolic differential operator, with initial data strongly conormal to three characteristic hypersurfaces intersecting at 0 and the light cone for P over 0. We assume that the hypersurfaces are either pairwisely transversal or two of them are tangent of finite order along the line of intersection. By reduction to a model case, conormal energy estimates are obtained for the linear equation with zero initial data. The proof of the theorem follows from an extension lemma and a contraction mapping argument.

Effective determinism in a classical field theory with spacelike characteristics

For certain classical field theories, it appears as if spacelike characteristic hypersurfaces can occur in the solutions. In principle, this means that acausal propagation is allowed, and the Cauchy problem breaks down. To investigate if indeed this happens, a class of Kasner-like solutions of a generalized Einstein–Maxwell theory is examined. It is found that the spacelike characteristics almost never form, and when they do, they cause no trouble.

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).

Uniqueness for the Characteristic Cauchy Problem and Strong Unique Continuation for Higher Order Partial Differential Inequalities

Introduction. This paper presents new results on the uniqueness for the Cauchy problem with respect to an everywhere characteristic hypersurface, and applications to strong unique continuation for partial differential inequalities of elliptic or hyperbolic type. The main theorem (Theorem 2) states uniqueness for mth order operators of the form P(t, 0, tat, ao), which are polynomials in tat, whose coefficients are pseudodifferential operators in 0 (depending smoothly on t) on a compact manifold M. Note that for non compact M, uniqueness does not hold in general [4], [9]. Our assumptions and results are parallel to those of Calderon in the classical local uniqueness theorem for the noncharacteristic Cauchy problem ([11], see also Nirenberg [16]), the operator tat replacing at. The proof of Theorem 2 combines pseudodifferential calculus on M with results on differential inequalities in Hilbert space, given in Theorem 1. Convexity and uniqueness results for similar abstract inequalities have been obtained by Agmon and Nirenberg [2], and also by Baiocchi