Einstein Constraint Equations Research Articles

Examples of cosmological spacetimes without CMC Cauchy surfaces

CMC (constant mean curvature) Cauchy surfaces play an important role in mathematical relativity as finding solutions to the vacuum Einstein constraint equations is made much simpler by assuming CMC initial data. However, Bartnik (Commun Math Phys 117(4):615–624, 1988) constructed a cosmological spacetime without a CMC Cauchy surface whose spatial topology is the connected sum of two three-dimensional tori. Similarly, Chruściel et al. (Commun Math Phys 257(1):29–42, 2005) constructed a vacuum cosmological spacetime without CMC Cauchy surfaces whose spatial topology is also the connected sum of two tori. In this article, we enlarge the known number of spatial topologies for cosmological spacetimes without CMC Cauchy surfaces by generalizing Bartnik’s construction. Specifically, we show that there are cosmological spacetimes without CMC Cauchy surfaces whose spatial topologies are the connected sum of any compact Euclidean or hyperbolic three-manifold with any another compact Euclidean or hyperbolic three-manifold. Analogous examples in higher spacetime dimensions are also possible. We work with the Tolman–Bondi class of metrics and prove gluing results for variable marginal conditions, which allows for smooth gluing of Schwarzschild to FLRW models.

Starting inflation from inhomogeneous initial conditions with momentum

We investigate the circumstances under which cosmic inflation can arise from very inhomogeneous initial conditions using numerical relativity simulations. Previous studies have not considered cases with non-zero momentum density due to technical challenges with solving the coupled Einstein constraint equations. Here we address these, introducing and comparing several different ways of constructing cosmological initial conditions with inhomogeneous scalar field and time derivative profiles. We evolve such initial conditions with large inhomogeneities in both single- and two-field inflationary models. We study cases where the initial gradient and kinetic energy are much larger than the inflationary energy scale, and black holes can form, as well as cases where the initial scalar potential energy is comparable, as in scenarios where inflation occurs at nearly Planckian densities, finding large-field inflation to be generally robust. We consider examples of initial conditions where a large scalar field velocity towards non-inflationary values would prevent inflation from occurring in the homogeneous case, finding that the addition of large gradients in the scalar field can actually dilute this effect, with the increased expansion and non-vanishing restoring force leading to inflation.

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.

Asymptotically hyperbolic Einstein constraint equations with apparent horizon boundary and the Penrose inequality for perturbations of Schwarzschild-AdS *

We prove the existence of asymptotically hyperbolic solutions to the vacuum Einstein constraint equations with a marginally outer trapped boundary of positive mean curvature, using the constant mean curvature conformal method. As an application of this result, we verify the Penrose inequality for certain perturbations of Schwarzschild Anti-de Sitter black hole initial data.

Simple numerical solutions to the Einstein constraints on various three-manifolds

Numerical solutions to the Einstein constraint equations are constructed on a selection of compact orientable three-dimensional manifolds with non-trivial topologies. A simple constant mean curvature solution and a somewhat more complicated non-constant mean curvature solution are computed on example manifolds from three of the eight Thursten geometrization classes. The constant mean curvature solutions found here are also solutions to the Yamabe problem that transforms a geometry into one with constant scalar curvature.

Future stability of self-gravitating dust balls in an expanding universe

We consider a system representing self-gravitating balls of dust in an expanding Universe. It is demonstrated that one can prescribe data for such a system at infinity and evolve it backward in time without the development of shocks or singularities. The resulting solution to the Einstein-lambda-dust equations exists for an infinite amount of time in the asymptotic region of the spacetime. Furthermore, we find that if the density is small compared to the Cosmological constant, then it is possible to construct Cosmological solutions to the Einstein constraint equations on a standard Cauchy hypersurface representing self-gravitating balls of dust. If, in addition, the density is assumed to be sufficiently small, then this initial data gives rise to a future geodesically complete solution to the Einstein-lambda-dust equations admitting a smooth conformal extension at infinity which can be regarded as a perturbation of de Sitter spacetime. The main technical tool in this analysis are Friedrichs conformal Einstein field equations for the Einstein-lambda-dust written in terms of a gauge in which the flow lines of the dust are recast as conformal geodesics.

On Certain Geometric Operators Between Sobolev Spaces of Sections of Tensor Bundles on Compact Manifolds Equipped with Rough Metrics

The study of Einstein constraint equations in general relativity naturally leads to considering Riemannian manifolds equipped with nonsmooth metrics. There are several important differential operators on Riemannian manifolds whose definitions depend on the metric: gradient, divergence, Laplacian, covariant derivative, conformal Killing operator, and vector Laplacian, among others. In this article, we study the approximation of such operators, defined using a rough metric, by the corresponding operators defined using a smooth metric. This paves the road to understanding to what extent the nice properties such operators possess, when defined with smooth metric, will transfer over to the corresponding operators defined using a nonsmooth metric. These properties are often assumed to hold when working with rough metrics, but to date the supporting literature is slim.

Einstein-Type Elliptic Systems

In this paper, we analyse semi-linear systems of partial differential equations which are motivated by the conformal formulation of the Einstein constraint equations coupled with realistic physical fields on asymptotically Euclidean (AE) manifolds. In particular, electromagnetic fields give rise to this kind of system. In this context, under suitable conditions, we prove a general existence theorem for such systems, and, in particular, under smallness assumptions on the free parameters of the problem, we prove existence of far from CMC (near CMC) Yamabe positive (Yamabe non-positive) solutions for charged dust coupled to the Einstein equations, satisfying a trapped surface condition on the boundary. As a bypass, we prove a Helmholtz decomposition on AE manifolds with boundary, which extends and clarifies previously known results.

Conformal fields and the structure of the space of solutions of the Einstein constraint equations

The drift method, introduced by the second author, provides a new formulation of the Einstein constraint equations, either in vacuum or with matter fields. The natural of the geometry underlying this method compensates for its slightly greater analytic complexity over, say, the conformal or conformal thin sandwich methods. We review this theory here and apply it to the study of solutions of the constraint equations with non-constant mean curvature. We show that this method reproduces previously known existence results obtained by other methods, and does better in one important regard. Namely, it can be applied even when the underlying metric admits conformal Killing (but not true Killing) vector fields. We also prove that the absence of true Killing fields holds generically.

Weak Continuity of the Cartan Structural System and Compensated Compactness on Semi-Riemannian Manifolds with Lower Regularity

We are concerned with the global weak continuity of the Cartan structural system -- or equivalently, the Gauss--Codazzi--Ricci system -- on semi-Riemannian manifolds with lower regularity. For this purpose, we first formulate and prove a geometric compensated compactness theorem on vector bundles over semi-Riemannian manifolds with lower regularity (Theorem 3.2), extending the classical quadratic theorem of compensated compactness. We then deduce the $L^p$ weak continuity of the Cartan structural system for $p>2$: For a family $\{\mathcal{W}_\varepsilon\}$ of connection $1$-forms on a semi-Riemannian manifold $(M,g)$, if $\{\mathcal{W}_\varepsilon\}$ is uniformly bounded in $L^p$ and satisfies the Cartan structural system, then any weak $L^p$ limit of $\{\mathcal{W}_\varepsilon\}$ is also a solution of the Cartan structural system. Moreover, it is proved that isometric immersions of semi-Riemannian manifolds into semi-Euclidean spaces can be constructed from the weak solutions of the Cartan structural system or the Gauss--Codazzi--Ricci system (Theorem 5.1), which leads to the $L^p$ weak continuity of the Gauss--Codazzi--Ricci system on semi-Riemannian manifolds. As further applications, the weak continuity of Einstein's constraint equations, general immersed hypersurfaces, and the quasilinear wave equations is also established.

Solutions to the Einstein Constraint Equations with a Small TT-Tensor and Vanishing Yamabe Invariant

In this note, we prove an existence result for the Einstein conformal constraint equations for metrics with vanishing Yamabe invariant assuming that the mean curvature satisfies an explicit near-CMC condition and that the TT-tensor is small in \(L^2\).

Asymptotic Gluing of Shear-Free Hyperboloidal Initial Data Sets

We present a procedure for asymptotic gluing of hyperboloidal initial data sets for the Einstein field equations that preserves the shear-free condition. Our construction is modeled on the gluing construction in Isenberg et al. (Ann Henri Poincare 11(5):881–927, 2010), but with significant modifications that incorporate the shear-free condition. We rely on the special Holder spaces, and the corresponding theory for elliptic operators on weakly asymptotically hyperbolic manifolds, introduced by the authors in Allen et al. (Commun Anal Geom 26(1):1–61, 2018) and applied to the Einstein constraint equations in Allen et al. (Class Quantum Grav 33(11):115015, 2016).

Initial data in general relativity described by expansion, conformal deformation and drift

The conformal method is a technique for finding Cauchy data in general relativity solving the Einstein constraint equations, and its parameters include a conformal class, a conformal momentum (as measured by a densitized lapse), and a mean curvature. Although the conformal method is successful in generating constant mean curvature (CMC) solutions of the constraint equations, it is unknown how well it applies in the non-CMC setting, and there have been indications that it encounters difficulties there. We are therefore motivated to investigate alternative generalizations of the CMC conformal method. Introducing a densitized lapse into the ADM Lagrangian, we find that solutions of the momentum constraint can be described in terms of three parameters. The first is conformal momentum as it appears in the standard conformal method. The second is volumetric momentum, which appears as an explicit parameter in the CMC conformal method, but not in the non-CMC formulation. We have called the third parameter drift momentum, and it is the conjugate momentum to infinitesimal motions in superspace that preserve conformal class and volume form up to independent diffeomorphisms. This decomposition of solutions of the momentum constraint leads to extensions of the CMC conformal method where conformal and volumetric momenta both appear as parameters. There is more than one way to treat drift momentum, in part because of an interesting duality that emerges, and we identify three candidates for incorporating drift into a variation of the conformal method.

Reduced thin-sandwich equations on manifolds Euclidean at infinity and on closed manifolds: Existence and multiplicity

The reduced thin-sandwich equations (RTSE) appear within Wheeler's thin-sandwich approach towards the Einstein constraint equations (ECE) of general relativity. It is known that these equations cannot be well-posed in general, but, on closed manifolds, sufficient conditions for well-posedness have been established. In particular, it has been shown that the RTSE are well posed in a neighbourhood of umbilical solutions of the constraint equations without conformal Killing fields. In this paper we will analyse such set of equations on manifolds euclidean at infinity in a neighbourhood of asymptotically euclidean (AE) solutions of the ECE. The main conclusion in this direction is that on AE-manifolds admitting a Yamabe positive metric, the solutions of the RTSE parametrize an open subset in the space of solutions of the ECE. Also, we show that in the case of closed manifolds, these equations are well-posed around umbilical solutions of the ECE admitting Killing fields and present some relevant examples. Finally, it will be shown that in the set of umbilical solutions of the vacuum ECE on closed manifolds, the RTSE are generically well-posed.

On the conformal method for the Einstein constraint equations

In this work, we use the global analysis and degree-theoretic methods introduced by Smale to study the existence and multiplicity of solutions of the vacuum Einstein constraint equations given by the conformal method of Lichnerowicz-Choquet-Bruhat-York. In particular this approach gives a new proof of the existence result of Maxwell and Holst-Nagy-Tsogtgerel. We also relate the method to the limit equation of Dahl-Gicquaud-Humbert and the non-existence result of Nguyen.

Lorentzian Einstein metrics with prescribed conformal infinity

We prove a local well-posedness theorem for the $(n+1)$-dimensional Einstein equations in Lorentzian signature, with initial data $(\widetilde{g},K)$ whose asymptotic geometry at infinity is similar to that anti-de Sitter (AdS) space, and compatible boundary data $\widehat{g}$ prescribed at the time-like conformal boundary of space-time. More precisely, we consider an $n$-dimensional asymptotically hyperbolic Riemannian manifold $(M, \widetilde{g})$ such that the conformally rescaled metric $x^2 \widetilde{g}$ (with $x$ a boundary defining function) extends to the closure $\overline{M}$ of $M$ as a metric of class $C^{n-1} (\overline{M})$ which is also poly-homogeneous of class $C^p_{\mathrm{polyhom}} (\overline{M})$. Likewise we assume that the conformally rescaled symmetric $(0, 2)$-tensor $x^ 2 K$ extends to $\overline{M}$ as a tensor field of class $C^{n-1} (\overline{M})$ which is polyhomogeneous of class $C^{p-1}_{\mathrm{polyhom}} (\overline{M})$. We assume that the initial data $(\widetilde{g}, K)$ satisfy the Einstein constraint equations and also that the boundary datum is of class $C^p$ on $\partial M \times (-T_0, T_0)$ and satisfies a set of natural compatibility conditions with the initial data. We then prove that there exists an integer $r_n$, depending only on the dimension $n$, such that if $p \geqslant 2q + r_n$, with $q$ a positive integer, then there is $T \gt 0$, depending only on the norms of the initial and boundary data, such that the Einstein equations (1.1) has a unique (up to a diffeomorphism) solution $g$ on $(-T, T) \times M$ with the above initial and boundary data, which is such that $x^2 g \in C^{n-1} ((-T, T) \times \overline{M}) \; \cap \; C^q_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$. Furthermore, if $x^2 \widetilde{g} , x^2 K$ are polyhomogeneous of class $C^{\infty}$ and $\widehat{g}$ is in $C^{\infty} ((-T_0, T_0) \times \partial \overline{M})$, then $x^2 g$ is in $C^{\infty}_{\mathrm{polyhom}} ((-T, T) \times \overline{M})$.

Space of initial data for self-similar Schwarzschild solutions of the Einstein equations

The Einstein constraint equations describe the space of initial data for the evolution equations, dictating how space should curve within spacetime. Under certain assumptions, the constraints reduce to a scalar quasilinear parabolic equation on the sphere with various singularities, and nonlinearity being the prescribed scalar curvature of space. We focus on self-similar Schwarzschild solutions. Those describe, for example, the initial data of black holes. We construct the space of initial data for such solutions, and show that the event horizon is related with global attractors of such parabolic equations. Lastly, some properties of those attractors and its solutions are explored.

Non-CMC Solutions of the Einstein Constraint Equations on Compact Manifolds with Apparent Horizon Boundaries

In this article we continue our effort to do a systematic development of the solution theory for conformal formulations of the Einstein constraint equations on compact manifolds with boundary. By building in a natural way on our recent work in Holst and Tsogtgerel (2013), and Holst, Nagy, and Tsogtgerel (2008, 2009), and also on the work of Maxwell (2004, 2005, 2009) and Dain (2004), under reasonable assumptions on the data we prove existence of both near- and far-from-constant mean curvature solutions for a class of Robin boundary conditions commonly used in the literature for modeling black holes, with a third existence result for constant mean curvature (CMC) appearing as a special case. Dain and Maxwell addressed initial data engineering for space-times that evolve to contain black holes, determining solutions to the conformal formulation on an asymptotically Euclidean manifold in the CMC setting, with interior boundary conditions representing excised interior black hole regions. Holst and Tsogtgerel compiled the interior boundary results covered by Dain and Maxwell, and then developed general interior conditions to model the apparent horizon boundary conditions of Dain and Maxwell for compact manifolds with boundary, and subsequently proved existence of solutions to the Lichnerowicz equation on compact manifolds with such boundary conditions. This paper picks up where Holst and Tsogtgerel left off, addressing the general non-CMC case for compact manifolds with boundary. As in our previous articles, our focus here is again on low regularity data and on the interaction between different types of boundary conditions. While our work here serves primarily to extend the solution theory for the compact with boundary case, we also develop several technical tools that have potential for use with the asymptotically Euclidean case.

Smoothly Compactifiable Shear-Free Hyperboloidal Data is Dense in the Physical Topology

We show that any polyhomogeneous asymptotically hyperbolic constant-mean-curvature solution to the vacuum Einstein constraint equations can be approximated, arbitrarily closely in Holder norms determined by the physical metric, by shear-free smoothly conformally compact vacuum initial data.

Existence and blowup results for asymptotically Euclidean initial data sets generated by the conformal method

For each set of (freely chosen) seed data, the conformal method reduces the Einstein constraint equations to a system of elliptic equations, the conformal constraint equations. We prove an admissibility criterion, based on a (conformal) prescribed scalar curvature problem, which provides a necessary condition on the seed data for the conformal constraint equations to (possibly) admit a solution. We then consider sets of asymptotically Euclidean (AE) seed data for which solutions of the conformal constraint equations exist, and examine the blowup properties of these solutions as the seed data sets approach sets for which no solutions exist. We also prove that there are AE seed data sets which include a Yamabe nonpositive metric and lead to solutions of the conformal constraints. These data sets allow the mean curvature function to have zeroes.