Einstein Vacuum Research Articles

The linearized Einstein equations with sources

On vacuum spacetimes of general dimension, we study the linearized Einstein vacuum equations with a spatially compactly supported and (necessarily) divergence-free source. We prove that the vanishing of appropriate charges of the source, defined in terms of Killing vector fields on the spacetime, is necessary and sufficient for solvability within the class of spatially compactly supported metric perturbations. The proof combines classical results by Moncrief with the solvability theory of the linearized constraint equations with control on supports developed by Corvino–Schoen and Chruściel–Delay.

Mode stability and shallow quasinormal modes of Kerr–de Sitter black holes away from extremality

A Kerr–de Sitter black hole is a solution (M,g_{\Lambda,\mathfrak{m},\mathfrak{a}}) of the Einstein vacuum equations with cosmological constant \Lambda>0 . It describes a black hole with mass \mathfrak{m}>0 and specific angular momentum \mathfrak{a}\in\mathbb{R} . We show that for any \varepsilon>0 there exists \delta>0 so that mode stability holds for the linear scalar wave equation \Box_{g_{\Lambda,\mathfrak{m},\mathfrak{a}}}\phi=0 when |\mathfrak{a}/\mathfrak{m}|\in[0,1-\varepsilon] and \Lambda\mathfrak{m}^{2}<\delta . In fact, we show that all quasinormal modes \sigma in any fixed half-space \operatorname{Im}\sigma>-C\sqrt{\Lambda} are equal to 0 or -i\sqrt{\Lambda/3}(n+o(1)) , n\in\mathbb{N} , as \Lambda\mathfrak{m}^{2}\searrow 0 . We give an analogous description of quasinormal modes for the Klein–Gordon equation. We regard a Kerr–de Sitter black hole with small \Lambda\mathfrak{m}^{2} as a singular perturbation either of a Kerr black hole with the same angular momentum-to-mass ratio, or of de Sitter spacetime without any black hole present. We use the mode stability of subextremal Kerr black holes, proved by Whiting and Shlapentokh-Rothman, as a black box; the quasinormal modes described by our main result are perturbations of those of de Sitter space. Our proof is based on careful uniform a priori estimates, in a variety of asymptotic regimes, for the spectral family and its de Sitter and Kerr model problems in the singular limit \Lambda\mathfrak{m}^{2}\searrow 0 .

Asymptotically de Sitter metrics from scattering data in all dimensions.

In space-time dimensions [Formula: see text], we show the existence of solutions of the Einstein vacuum equations which describe asymptotically de Sitter space-times with prescribed smooth data at the conformal boundary. This provides a short alternative proof of a special case of a result by Shlapentokh-Rothman and Rodnianski, and generalizes earlier results of Friedrich and Anderson to all dimensions. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Multi-localized time-symmetric initial data for the Einstein vacuum equations

Abstract We construct a class of time-symmetric initial data sets for the Einstein vacuum equations modeling elementary configurations of multiple almost isolated systems. Each such initial data set consists of a collection of several localized sources of gravitational radiation, and lies in a family of data sets which is closed under scaling out the distances between the systems by arbitrarily large amounts. This class contains data sets which are not asymptotically flat, but to which nonetheless a finite ADM mass can be ascribed. The construction proceeds by a gluing scheme using the Brill–Lindquist metric as a template. Such initial data are motivated in part by a desire to understand the dynamical interaction of distant systems in the context of general relativity. As a by-product of the construction, we produce families of complete, scalar-flat initial data with trivial topology and infinitely many minimal spheres, as well as families of initial data with infinitely many Einstein–Rosen bridges.

The case against smooth null infinity IV: Linearized gravity around Schwarzschild-an overview.

This paper is the fourth in a series dedicated to the mathematically rigorous asymptotic analysis of gravitational radiation under astrophysically realistic set-ups. It provides an overview of the physical ideas involved in setting up the mathematical problem, the mathematical challenges that need to be overcome once the problem is posed, as well as the main new results we will obtain in upcoming work. From the physical perspective, this includes a discussion of how post-Newtonian theory provides a prediction on the gravitational radiation emitted by [Formula: see text] infalling masses from the infinite past in the intermediate zone, i.e. up to some finite advanced time. From the mathematical perspective, we then take this prediction, together with the condition that there be no incoming radiation from [Formula: see text], as a starting point to set up a scattering problem for the linearized Einstein vacuum equations around Schwarzschild and near spacelike infinity, and we outline how to solve this scattering problem and obtain the asymptotic properties of the scattering solution near [Formula: see text] and [Formula: see text]. The full mathematical details will be presented in the sequel to this paper. This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

The Characteristic Gluing Problem for the Einstein Vacuum Equations: Linear and Nonlinear Analysis

The Characteristic Gluing Problem for the Einstein Vacuum Equations: Linear and Nonlinear Analysis

Webs, Nijenhuis operators, and heavenly PDEs

In 1989 Mason and Newman (Commun. Math. Phys. 121 659–68) proved that there is a 1-1-correspondence between self-dual metrics satisfying Einstein vacuum equation (in complex case or in neutral signature) and pairs of commuting parameter depending vector fields which are divergence free with respect to some volume form. Earlier Plebański (1975 J. Math. Phys. 16 2395–402) showed instances of such vector fields depending of one function of four variables satisfying the so-called I or II Plebański heavenly PDEs. Other PDEs leading to Mason–Newman vector fields are also known in the literature: Park (1992 Int. J. Mod. Phys. A 7 1415), Husain (1994 Phys. Rev. Lett. 72 800), Grant (1993 Phys. Rev. D 48 2606–12), Schief (Phys. Lett. A 223 55–62). In this paper we discuss these matters in the context of the web theory, i.e. theory of collections of foliations on a manifold, understood from the point of view of Nijenhuis operators. In particular we show how to apply this theory for constructing new ‘heavenly’ PDEs based on different normal forms of Nijenhuis operators in 4D, which are integrable similarly to their predecessors. Relation with the Hirota dispersionless systems of PDEs and the corresponding Veronese webs, which was recently observed by Konopelchenko–Schief–Szereszewski, is established in all the cases. We also discuss some higher dimensional generalizations of the ‘heavenly’ PDEs and the existence of related vacuum Einstein metrics in 4D-case.

Exploring the solitary wave solutions of Einstein's vacuum field equation in the context of ambitious experiments and space missions

In the present work, Einstein's vacuum field equation is investigated analytically to explore the solitary wave solutions. This equation arises in mathematical physics, having meaningful applications in the general theory of relativity. This concept is crucial for numerous challenging experiments and space missions. The generalized exponential rational function and modified auxiliary equation approaches are used to obtain the exact solitary wave solution. Various types of solutions are extracted, including exponential functions, hyperbolic functions, trigonometric functions, and rational forms. Additionally, a stability analysis for the Einstein vacuum field equation is conducted. Appropriate parameters are chosen to draw 3-D and corresponding contour plots of some solutions, which clearly demonstrate the solitary wave behaviors. The obtained results support the idea that applying these approaches is the most effective strategy for resolving any nonlinear issues that may arise in science and technology.

Exterior Stability of Minkowski Space in Generalized Harmonic Gauge

We give a short proof of the existence of a small piece of null infinity for (3+1)-dimensional spacetimes evolving from asymptotically flat initial data as solutions of the Einstein vacuum equations. We introduce a modification of the standard wave coordinate gauge in which all non-physical metric degrees of freedom have strong decay at null infinity. Using a formulation of the gauge-fixed Einstein vacuum equations which implements constraint damping, we establish this strong decay regardless of the validity of the constraint equations. On a technical level, we use notions from geometric singular analysis to give a streamlined proof of semiglobal existence for the relevant quasilinear hyperbolic equation.

The seed-to-solution method for the Einstein constraints and the asymptotic localization problem

We establish the existence of a broad class of asymptotically Euclidean solutions to Einstein's constraint equations, whose asymptotic behavior at infinity is prescribed arbitrarily. The proposed seed-to-solution method encompasses vacuum as well as matter spaces, and relies on iterations based on the linearized Einstein operator and its dual. It generates a Riemannian manifold (with finitely many asymptotically Euclidean ends) from any seed data set consisting of(1) a Riemannian metric and a symmetric two-tensor and(2) a (density) field and a (momentum) vector field representing the matter content. We distinguish between tame or strongly tame seed data sets, depending whether the data provides a rough or an accurate asymptotic Ansatz at infinity. We encompass classes of metrics and matter fields with low decay (possibly with infinite ADM mass) or strong decay (with Schwarzschild behavior). Our analysis is motivated by Carlotto and Schoen's pioneering work on the localization problem for Einstein's vacuum equations. Dealing with metrics with low decay and, simultaneously, establishing estimates that include (and go beyond) harmonic decay require significantly new arguments which are developed in the present paper. We work in a weighted Lebesgue-Hölder framework adapted to the given seed data, and we analyze the nonlinear coupling between the Hamiltonian and momentum constraints. By establishing elliptic regularity estimates for the linearized Einstein operator and its dual, we uncover the novel notion of mass-momentum correctors which is related to the ADM mass-momentum of the manifold. We derive precise estimates for the difference between the seed data and the associate Einstein solution —a result that should be of interest for future numerical investigations. Furthermore, we introduce here and study the asymptotic localization problem in which we replace Carlotto-Schoen's exact localization requirement by an asymptotic condition at a super-harmonic rate. By applying our seed-to-solution method with a suitably constructed, parametrized family of seed data, we solve this problem by exhibiting mass-momentum correctors with harmonic decay.

Geometric optics approximation for the Einstein vacuum equations

Geometric optics approximation for the Einstein vacuum equations

Naked singularities for the Einstein vacuum equations: The exterior solution

Naked singularities for the Einstein vacuum equations: The exterior solution

Twisted Self-Similarity and the Einstein Vacuum Equations

Twisted Self-Similarity and the Einstein Vacuum Equations

Double null data and the characteristic problem in general relativity

General hypersurfaces of any causal character can be studied abstractly using the hypersurface data formalism. In the null case, we write down all tangential components of the ambient Ricci tensor in terms of the abstract data. Using this formalism, we formulate and solve in a completely abstract way the characteristic Cauchy problem of the Einstein vacuum field equations. The initial data is detached from any spacetime notion, and it is fully diffeomorphism and gauge covariant. The results of this paper put the characteristic problem on a similar footing as the standard Cauchy problem in General Relativity.

Quadratic Theory of Gravity with a Scalar Field and Type I Shapovalov Wave Spacetimes

For the quadratic theory of gravity with a scalar field, exact solutions are found for gravitational-wave models in Shapovalov I-type spacetimes, which do not arise in models of the general theory of relativity. The theory of gravity under consideration can effectively describe the early stages of the universe. Type I Shapovalov spaces are the most general forms of gravitational-wave Shapovalov spacetimes, whose metrics in privileged coordinate systems depend on three variables, including the wave variable. For Einstein vacuum spacetimes, these wave models degenerate into simpler types. The exact models of gravitational waves in the quadratic theory of gravity can be used to test the realism of such theories of gravity.

Einstein vacuum equations with 𝕌(1) symmetry in an elliptic gauge: Local well-posedness and blow-up criterium

In this paper, we are interested in the Einstein vacuum equations on a Lorentzian manifold displaying [Formula: see text] symmetry. We identify some freely prescribable initial data, solve the constraint equations and prove the existence of a unique and local in time solution at the [Formula: see text] level. In addition, we prove a blow-up criterium at the [Formula: see text] level. By doing so, we improve a result of Huneau and Luk in [Einstein equations under polarized [Formula: see text] symmetry in an elliptic gauge, Commun. Math. Phys. 361(3) (2018) 873–949] on a similar system, and our main motivation is to provide a framework adapted to the study of high-frequency solutions to the Einstein vacuum equations done in a forthcoming paper by Huneau and Luk. As a consequence we work in an elliptic gauge, particularly adapted to the handling of high-frequency solutions, which have large high-order norms.

The Spacelike-Characteristic Cauchy Problem of General Relativity in Low Regularity

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface $$\Sigma \simeq \overline{B_1} \subset {{\mathbb {R}}}^3$$ and the outgoing null hypersurface $${{\mathcal {H}}}$$ emanating from $${\partial }\Sigma $$ , we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in $$L^2$$ . The proof uses the bounded $$L^2$$ curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.

A scale-critical trapped surface formation criterion for the Einstein-Maxwell system

In this study we show that, from arbitrarily dispersed initial data, both the focusing of electromagnetic fields and the concentration of gravitational waves could lead to the formation of trapped surfaces. We establish a scale–critical semi–global existence result from past null infinity for the Einstein–Maxwell system by assigning the signature for decay rates to both geometric quantities and Maxwell components. This result generalizes an approach of the first author in [4] on studying the Einstein vacuum equations to the physical Einstein-Maxwell system by employing additional elliptic estimates and geometric renormalizations and it also extends a result of Yu [18] and serves as the first scale-critical result for Einstein's equations coupled with matter.

A new look at black holes via thermal dimensions and the complex coordinates/ temperature vectors correspondence

The action of active diffeomorphisms (diffs) [Formula: see text] on the Schwarzschild metric leads to metrics which are also static spherically symmetric solutions of the Einstein vacuum field equations. It is shown how in a [Formula: see text] case it allows to introduce a deformation of the manifold such that [Formula: see text], and [Formula: see text] corresponding, respectively, to the spacelike singularity and horizon of the Schwarzschild metric. In doing so, one ends up with a spherical [Formula: see text] surrounding the singularity at [Formula: see text]. In order to explore the “interior” region of this void, we introduce complex radial coordinates whose imaginary components have a direct link to the inverse Hawking temperature, and which furnish a path that provides access to interior region. In addition, we show that the black hole entropy [Formula: see text] (in Planck units) is equal to the [Formula: see text] of a rectangular strip in the [Formula: see text] radial-coordinate plane associated to this path. The gist of the physical interpretation behind this construction is that there is an emergence of thermal dimensions which unfolds as one plunges into the interior void region via the use of complex coordinates, and whose imaginary components capture the span of the thermal dimensions. Namely, the filling of the void leads to an [Formula: see text] internal/ thermal dimension via the imaginary part [Formula: see text] of the complex radial variable [Formula: see text].

Construction of GCM Spheres in Perturbations of Kerr

This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for \(|a|\ll m\). The paper builds on the strategy laid out in [6] in the context of the nonlinear stability of Schwarzschild for axially symmetric polarized perturbations. In fact the central idea of [6] was the introduction and construction of generally covariant modulated (GCM) spheres on which specific geometric quantities take Schwarzschildian values. This was made possible by taking into account the full general covariance of the Einstein vacuum equations. The goal of this, and its companion paper [7], is to get rid of the symmetry restriction in the construction of GCM spheres in [6] and thus remove an essential obstruction in extending the result to a full stability proof of the Kerr family.