Characteristic Initial Value Problem Research Articles

Improved decay estimates and C2-asymptotic stability of solutions to the Einstein-scalar field system in spherical symmetry

We investigate the asymptotic stability of solutions to the characteristic initial value problem for the Einstein (massless) scalar field system with a positive cosmological constant. We prescribe spherically symmetric initial data on a future null cone with a wider range of decaying profiles than previously considered. New estimates are then derived in order to prove that, for small data, the system has a unique global classical solution. We also show that the solution decays exponentially in (Bondi) time and that the radial decay is essentially polynomial, although containing logarithmic factors in some special cases. This improved asymptotic analysis allows us to show that, under appropriate and natural decaying conditions on the initial data, the future asymptotic solution is differentiable, up to and including spatial null-infinity, and approaches the de Sitter solution, uniformly, in a neighborhood of infinity. Moreover, we analyze the decay of derivatives of the solution up to second order showing the (uniform) [Formula: see text]-asymptotic stability of the de Sitter attractor in this setting. This corresponds to a surprisingly strong realization of the cosmic no-hair conjecture.

Covariant definition of double null data and geometric uniqueness of the characteristic initial value problem

The characteristic Cauchy problem of the Einstein field equations has been recently addressed from a completely abstract viewpoint by means of hypersurface data and, in particular, via the notion of double null data (DND). However, this definition was given in a partially gauge-fixed form. In this paper we generalize the notion of DND in a fully diffeomorphism and gauge covariant way, and show that the definition is complete by proving that no extra conditions are needed to embed the DND in some spacetime. The second aim of the paper is to show that the characteristic Cauchy problem satisfies a geometric uniqueness property. Specifically, we introduce a natural notion of isometry at the abstract level such that two DND that are isometric in this sense give rise to isometric spacetimes.

The characteristic initial value problem for the conformally invariant wave equation on a Schwarzschild background

We resume former discussions of the conformally invariant wave equation on a Schwarzschild background, with a particular focus on the behaviour of solutions near the ‘cylinder’, i.e. Friedrich’s representation of spacelike infinity. This analysis can be considered a toy model for the behaviour of the full Einstein equations and the resulting logarithmic singularities that appear to be characteristic for massive spacetimes. The investigation of the Cauchy problem for the conformally invariant wave equation (Frauendiener and Hennig 2018 Class. Quantum Grav. 35 065015) showed that solutions generically develop logarithmic singularities at infinitely many expansion orders at the cylinder, but an arbitrary finite number of these singularities can be removed by appropriately restricting the initial data prescribed at t = 0. From a physical point of view, any data at t = 0 are determined from the earlier history of the system and hence not exactly ‘free data’. Therefore, it is appropriate to ask what happens if we ‘go further back in time’ and prescribe initial data as early as possible, namely at a portion of past null infinity, and on a second past null hypersurface to complete the initial value problem. Will regular data at past null infinity automatically lead to a regular evolution up to future null infinity? Or does past regularity restrict the solutions too much, and regularity at both null infinities is mutually exclusive? Or do we still have suitable degrees of freedom for the data that can be chosen to influence regularity of the solutions to any desired degree? In order to answer these questions, we study the corresponding characteristic initial value problem. In particular, we investigate in detail the appearance of singularities at expansion orders for angular modes .

Electrodynamics under the action of null cosmic strings

A method to study electromagnetic (EM) effects generated by a straight null cosmic string moving in classical EM fields is suggested. The string is shown to induce an additional EM field which can be described as a solution to homogeneous Maxwell equations with initial data set on a null surface, the string event horizon, where the string world-sheet belongs to. The initial data ensure the required holonomy of the string space-time caused by the gravity of the string. This characteristic initial value problem is used to study interaction of plane waves with null strings and perturbations by the strings of the Coulomb fields of electric charges. It is shown that parts of an incoming EM wave crossing the string horizon from different sides of the string are refracted with respect to each other and leave behind the string a wedge-like region of interference. A null string moving near an electric charge results in two effects: it creates a self-force of the charge and induces a pulse of EM radiation traveling away from the charge in the direction close to trajectory of the string.

Cyclic spacetimes through singularity scattering maps. Plane-symmetric gravitational collisions

We study the plane-symmetric collision of two gravitational waves and describe the global spacetime geometry generated by this collision. We formulate the characteristic initial value problem for the Einstein equations, when Goursat data describing the incoming waves are prescribed on two null hypersurfaces. We construct a global solution representing a cyclic spacetime based on junction conditions associated with a prescribed singularity scattering map, as we call it. This amounts to a detailed analysis of the Goursat and Fuchsian initial value problems associated with singular hyperbolic equations, when junction conditions at interfaces are prescribed. We introduce a partition into monotonicity diamonds (that is, spacetime domains) and we construct the solution by concatenating domains across interfaces of timelike, null, or spacelike type.

Characteristic approach to the soliton resolution

As a toy model for understanding the soliton resolution phenomenon we consider a characteristic initial boundary value problem for the 4d equivariant Yang–Mills equation outside a ball. Our main objective is to illustrate the advantages of employing outgoing null (or asymptotically null) foliations in analyzing the relaxation processes due to the dispersal of energy by radiation. In particular, within this approach it is evident that the endstate of evolution must be non-radiative (meaning vanishing flux of energy at future null infinity). In our toy model such non-radiative configurations are given by a static solution (called the half-kink) plus an alternating chain of N decoupled kinks and antikinks. We show numerically that the configurations N = 0 (static half-kink) and N = 1 (superposition of the static half-kink and the antikink which recedes to infinity) appear as generic attractors and we determine a codimension-one borderline between their basins of attraction. The rates of convergence to these attractors are analyzed in detail.

Gauge structure of the Einstein field equations in Bondi-like coordinates

The characteristic initial (boundary) value problem has numerous applications in general relativity (GR) involving numerical studies, and is often formulated using Bondi-like coordinates. Recently it was shown that several prototype formulations of this type are only weakly hyperbolic. Presently we examine the root cause of this result. In a linear analysis we identify the gauge, constraint and physical blocks in the principal part of the Einstein field equations in such a gauge, and show that the subsystem related to the gauge variables is only weakly hyperbolic. Weak hyperbolicity of the full system follows as a consequence in many cases. We demonstrate this explicitly in specific examples, and thus argue that Bondi-like gauges result in weakly hyperbolic free evolution systems under quite general conditions. Consequently the characteristic initial (boundary) value problem of GR in these gauges is rendered ill-posed in the simplest norms one would like to employ. The possibility of finding good alternative norms, in which well-posedness is achieved, is discussed. So motivated, we present numerical convergence tests with an implementation of full GR which demonstrate the effect of weak hyperbolicity in practice.

About one characteristic initial value problem with prehistory and its investigation

About one characteristic initial value problem with prehistory and its investigation

Instability of spherical naked singularities of a scalar field under gravitational perturbations

In this paper, we initiate the study of the instability of naked singularities beyond spherical symmetry. In a series of papers, Christodoulou proved that naked singularities are not stable in the context of the spherically symmetric Einstein equations coupled with a massless scalar field. We study in this paper the next simplest case: a characteristic initial value problem of this coupled system with the initial data given on two intersecting null cones, the incoming one of which is assumed to be spherically symmetric and singular at its vertex, and the outgoing one of which has no symmetries. It is shown that, arbitrarily fixing the initial scalar field, the set of the initial conformal metrics on the outgoing null cone such that the maximal future development does not have any sequences of closed trapped surfaces approaching the singularity, is of first category in the whole space in which the shear tensors are continuous. Such a set can then be viewed as exceptional, although the exceptionality is weaker than the at least $1$ co‑dimensionality in spherical symmetry. Almost equivalently, it is also proved that, arbitrarily fixing an incoming null cone $\underline{C}_{\,\varepsilon}$ to the future of the initial incoming null cone, the set of the initial conformal metrics such that the maximal future development has at least one closed trapped surface before $\underline{C}_{\,\varepsilon}$, contains an open and dense subset of the whole space. Since the initial scalar field can be chosen such that the singularity is naked if the initial shear is set to be zero, we may say that the spherical naked singularities of a self-gravitating scalar field are not stable under gravitational perturbations. This in particular gives new families of non-spherically symmetric gravitational perturbations different from the original spherically symmetric scalar perturbations given by Christodoulou.

Spherically symmetric black holes and affine-null metric formulation of Einstein’s equations

The definition of well-behaved coordinate charts for black hole spacetimes can be tricky, as they can lead for example to either unphysical coordinate singularities in the metric (e.g. $r=2M$ in the Schwarzschild black hole) or to an implicit dependence of the chosen coordinate to physical relevant coordinates (e.g. the dependence of the null coordinates in the Kruskal metric). Here we discuss two approaches for coordinate choices in spherical symmetry allowing us to discuss explicitly "solitary" and spherically symmetric black holes from a regular horizon to null infinity. The first approach relies on a construction of a regular null coordinate (where regular is meant as being defined from the horizon to null infinity) given an explicit solution of the Einstein-matter equations. The second approach is based on an affine-null formulation of the Einstein equations and the respective characteristic initial value problem. In particular, we present a derivation of the Reissner-Nordstr\"om black holes expressed in terms of these regular coordinates.

The conformal Einstein field equations and the local extension of future null infinity

We make use of an improved existence result for the characteristic initial value problem for the conformal Einstein equations to show that given initial data on two null hypersurfaces N⋆ and N⋆′ such that the conformal factor (but not its gradient) vanishes on a section of N⋆, one recovers a portion of null infinity. This result combined with the theory of the hyperboloidal initial value problem for the conformal Einstein field equations allows us to show the semi-global stability of the Minkowski spacetime from characteristic initial data.

Global solutions to the spherically symmetric Einstein-scalar field system with a positive cosmological constant in Bondi coordinates

We consider a characteristic initial value problem, with initial data given on a future null cone, for the Einstein (massless) scalar field system with a positive cosmological constant, in Bondi coordinates. We prove that, for small data, this system has a unique global classical solution which is causally geodesically complete to the future and decays polynomially in radius and exponentially in Bondi time, approaching the de Sitter solution.

A well-posedness result of a characteristic hyperbolic mixed problem

We study the well-posedness of a hyperbolic characteristic initial boundary value problem with Lipschitz continuous coefficients. Assuming more general boundary assumptions than those of maximally dissipativeness, we deal with a Friedrichs symmetrizable system of first order satisfying a minimal structure boundary condition, the so-called Uniform Kreiss–Lopatinskii Condition. We show that a semi-group estimate holds, leading to the proof of the $$L^{2}$$ well-posedness of the initial boundary value problem, provided that the source data of the interior is only $$L^{1}([0,T],L^{2}(\varOmega )),$$ with the aid of the paradifferential calculus.

Revisiting the characteristic initial value problem for the vacuum Einstein field equations

Using the Newman–Penrose formalism we study the characteristic initial value problem in vacuum General Relativity. We work in a gauge suggested by Stewart, and following the strategy taken in the work of Luk, demonstrate local existence of solutions in a neighbourhood of the set on which data are given. These data are given on intersecting null hypersurfaces. Existence near their intersection is achieved by combining the observation that the field equations are symmetric hyperbolic in this gauge with the results of Rendall. To obtain existence all the way along the null-hypersurfaces themselves, a bootstrap argument involving the Newman–Penrose variables is performed.

Hyperbolicity of general relativity in Bondi-like gauges

Bondi-like (single-null) characteristic formulations of general relativity are used for numerical work in both asymptotically flat and anti-de Sitter spacetimes. Well-posedness of the resulting systems of partial differential equations, however, remains an open question. The answer to this question affects accuracy, and potentially the reliability of conclusions drawn from numerical studies based on such formulations. A numerical approximation can converge to the continuum limit only for well-posed systems; for the initial value problem in the $L^2$ norm this is characterized by strong hyperbolicity. We find that, due to a shared pathological structure, the systems arising from the aforementioned formulations are however only weakly hyperbolic. We present numerical tests for toy models that demonstrate the consequence of this shortcoming in practice for the characteristic initial boundary value problem. Working with alternative norms in which our model problems may be well-posed we show that convergence may be recovered. Finally we examine well-posedness of a model for Cauchy-Characteristic-Matching in which model symmetric and weakly hyperbolic systems communicate through an interface, with the latter playing the role of GR in Bondi gauge on characteristic slices. We find that, due to the incompatibility of the norms associated with the two systems, the composite problem does not naturally admit energy estimates.

Improved existence for the characteristic initial value problem with the conformal Einstein field equations

We adapt Luk’s analysis of the characteristic initial value problem in general relativity to the asymptotic characteristic problem for the conformal Einstein field equations to demonstrate the local existence of solutions in a neighbourhood of the set on which the data are given. In particular, we obtain existence of solutions along a narrow rectangle along null infinity which, in turn, corresponds to an infinite domain in the asymptotic region of the physical spacetime. This result generalises work by Kánnár on the local existence of solutions to the characteristic initial value problem by means of Rendall’s reduction strategy. In analysing the conformal Einstein equations we make use of the Newman–Penrose formalism and a gauge due to J. Stewart.

H\xf6rmander\u2019s method for the characteristic Cauchy problem and conformal scattering for a nonlinear wave equation

The purpose of this note is to prove the existence of a conformal scattering operator for the cubic defocusing wave equation on a non-stationary background. The proof essentially relies on solving the characteristic initial value problem by the method developed by Hörmander. This method consists in slowing down the propagation speed of the waves to transform a characteristic initial value problem into a standard Cauchy problem.

Multiple Killing horizons: the initial value formulation for -vacuum

In Mars et al (2018 Class. Quantum Grav. 35 155015) we have introduced the notion of ‘multiple Killing horizon’ and analyzed some of its general properties. Multiple Killing horizons are Killing horizons for two or more linearly independent Killing vectors simultaneously. In this paper we focus on the vacuum case, possibly with cosmological constant, and study the emergence of multiple Killing horizons in terms of characteristic initial value problems for two transversally intersecting null hypersurfaces. As a relevant outcome, a more general definition of near horizon geometry is put forward. This new definition avoids the use of Gaussian null coordinates associated to the corresponding degenerate Killing vector and thereby allows for inclusion of its fixed points.

Affine-null formulation of the gravitational equations: Spherical case

A new evolution algorithm for the characteristic initial value problem based upon an affine parameter rather than the areal radial coordinate used in the Bondi-Sachs formulation is applied in the spherically symmetric case to the gravitational collapse of a massless scalar field. The advantages over the Bondi-Sachs version are discussed, with particular emphasis on the application to critical collapse. Unexpected quadratures lead to a simple evolution algorithm based upon ordinary differential equations which can be integrated along the null rays. For collapse to a black hole in a Penrose compactified spacetime, these equations are regularized throughout the exterior and interior of the horizon up to the final singularity. They are implemented as a global numerical evolution code based upon the Galerkin method. New results regarding the global properties of critical collapse are presented.

Cosmic No-Hair in Spherically Symmetric Black Hole Spacetimes

We analyze in detail the geometry and dynamics of the cosmological region arising in spherically symmetric black hole solutions of the Einstein-Maxwell-scalar field system with a positive cosmological constant. More precisely, we solve, for such a system, a characteristic initial value problem with data emulating a dynamic cosmological horizon. Our assumptions are fairly weak, in that we only assume that the data approaches that of a subextremal Reissner-Nordstr\"om-de Sitter black hole, without imposing any rate of decay. We then show that the radius (of symmetry) blows up along any null ray parallel to the cosmological horizon ("near" $i^+$), in such a way that $r=+\infty$ is, in an appropriate sense, a spacelike hypersurface. We also prove a version of the Cosmic No-Hair Conjecture by showing that in the past of any causal curve reaching infinity both the metric and the Riemann curvature tensor asymptote those of a de Sitter spacetime. Finally, we discuss conditions under which all the previous results can be globalized.