Vacuum Einstein Equations Research Articles

Quantum geometrical properties of topological materials

The momentum space of topological insulators and topological superconductors is equipped with a quantum metric defined from the overlap of neighboring valence band states or quasihole states. We investigate the quantum geometrical properties of these materials within the framework of Dirac models and differential geometry. Their momentum space is found to be always a maximally symmetric space with a constant Ricci scalar, and the vacuum Einstein equation is satisfied in 3D with a finite cosmological constant. For linear Dirac models, several geometrical properties are found to be independent of the band gap, including a peculiar straight line geodesic, constant volume of the curved momentum space, and the exponential decay form of the nonlocal topological marker, indicating the peculiar yet universal quantum geometrical properties of these models.

Quantum Field Theory of Black Hole Perturbations with Backreaction: I General Framework

In a seminal work, Hawking showed that natural states for free quantum matter fields on classical spacetimes that solve the spherically symmetric vacuum Einstein equations are KMS states of non-vanishing temperature. Although Hawking’s calculation does not include the backreaction of matter on geometry, it is more than plausible that the corresponding Hawking radiation leads to black hole evaporation which is, in principle, observable. Obviously, an improvement of Hawking’s calculation including backreaction is a problem of quantum gravity. Since no commonly accepted quantum field theory of general relativity is available yet, it has been difficult to reliably derive the backreaction effect. An obvious approach is to use the black hole perturbation theory of a Schwarzschild black hole of fixed mass and to quantize those perturbations. However, it is not clear how to reconcile perturbation theory with gauge invariance beyond linear perturbations. In recent work, we proposed a new approach to this problem that applies when the physical situation has an approximate symmetry, such as homogeneity (cosmology), spherical symmetry (Schwarzschild), or axial symmetry (Kerr). The idea, which is surprisingly feasible, is to first construct the non-perturbative physical (reduced) Hamiltonian of the reduced phase space of fully gauge invariant observables and only then apply perturbation theory directly in terms of observables. The task to construct observables is then disentangled from perturbation theory, thus allowing to unambiguously develop perturbation theory to arbitrary orders. In this first paper of the series we outline and showcase this approach for spherical symmetry and second order in the perturbations for Einstein–Klein–Gordon–Maxwell theory. Details and generalizations to other matter and symmetry and higher orders will appear in subsequent companion papers.

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'.

Timelike Ricci bounds for low regularity spacetimes by optimal transport

We prove that a globally hyperbolic smooth spacetime endowed with a [Formula: see text]-Lorentzian metric whose Ricci tensor is bounded from below in all timelike directions, in a distributional sense, obeys the timelike measure-contraction property. This result includes a class of spacetimes with borderline regularity for which local existence results for the vacuum Einstein equation are known in the setting of spaces with timelike Ricci bounds in a synthetic sense. In particular, these spacetimes satisfy timelike Brunn–Minkowski, Bonnet–Myers, and Bishop–Gromov inequalities in sharp form, without any timelike nonbranching assumption. If the metric is even [Formula: see text], in fact the stronger timelike curvature-dimension condition holds. In this regularity, we also obtain uniqueness of chronological optimal couplings and chronological geodesics.

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.

Coercivity properties of the canonical energy in double null gauge on the 4-dimensional Schwarzschild exterior

In this paper, we study the canonical energy associated with solutions to the linearised vacuum Einstein equation on a stationary spacetime. The main result of this paper establishes, in the context of the 4-dimensional Schwarzschild exterior, a direct correspondence between the conservation law satisfied by the canonical energy and the conservation laws deduced by Holzegel for gravitational perturbations in double null gauge. Since the latter exhibit useful coercivity properties (leading to energy and pointwise boundedness statements) we obtain coercivity results for the canonical energy in the double null gauge as a corollary. More generally, the correspondence suggests a systematic way to uncover coercivity properties in the conservation laws for the canonical energy on Kerr.

Isometries and the double copy

In the standard derivation of the Kerr-Schild double copy, the geodicity of the Kerr-Schild vector and the stationarity of the spacetime are presented as assumptions that are necessary for the single copy to satisfy Maxwell’s equations. However, it is well known that the vacuum Einstein equations imply that the Kerr-Schild vector is geodesic and shear-free, and that the spacetime possesses a distinguished vector field that is simultaneously a Killing vector of the full spacetime and the flat background, but need not be timelike with respect to the background metric. We show that the gauge field obtained by contracting this distinguished Killing vector with the Kerr-Schild graviton solves the vacuum Maxwell equations, and that this definition of the Kerr-Schild double copy implies the Weyl double copy when the spacetime is Petrov type D. When the Killing vector is taken to be timelike with respect to the background metric, we recover the familiar Kerr-Schild double copy, but the prescription is well defined for any vacuum Kerr-Schild spacetime and we present new examples where the Killing vector is null or spacelike. While most examples of physical interest are type D, vacuum Kerr-Schild spacetimes are generically of Petrov type II. We present a straightforward example of such a spacetime and study its double copy structure. Our results apply to real Lorentzian spacetimes as well as complex spacetimes and real spacetimes with Kleinian signature, and provide a simple correspondence between real and self-dual vacuum Kerr-Schild spacetimes. This correspondence allows us to study the double copy structure of a self-dual analog of the Kerr spacetime. We provide evidence that this spacetime may be diffeomorphic to the self-dual Taub-NUT solution.

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.

Hodge-de Rham Laplacian and geometric criteria for gravitational waves

The curvature tensor \(\hat{R}\) of a manifold is called harmonic, if it obeys the condition \(\Delta^{\text{(HR)}}\hat{R}=0\), where \(\Delta^{\text{(HR)}}=DD^{\ast} +
 D^{\ast}D\) is the Hodge–deRham Laplacian. It is proved that all solutions of the Einstein equations in vacuum, as well as all solutions of the Einstein–Cartan theory in vacuum have a harmonic curvature. The statement that only solutions of Einstein’s equations of type \(N\) (describing gravitational radiation) are harmonic is refuted.

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

All Order Gravitational Waveforms from Scattering Amplitudes.

Waveforms are classical observables associated with any radiative physical process. Using scattering amplitudes, these are usually computed in a weak-field regime to some finite order in the post-Newtonian or post-Minkowskian approximation. Here, we use strong-field amplitudes to compute the waveform produced in scattering of massive particles on gravitational plane waves, treated as exact nonlinear solutions of the vacuum Einstein equations. Notably, the waveform contains an infinite number of post-Minkowskian contributions, as well as tail effects. We also provide, and contrast with, analogous results in electromagnetism.

Naked singularities for the Einstein vacuum equations: The exterior solution

Naked singularities for the Einstein vacuum equations: The exterior solution

Perturbative Solution of the Einstein Constraints with Spin and Momentum Far Away From a Binary Source in the Bowen‐York Formalism

AbstractThe momentum and Hamiltonian constraints of vacuum Einstein equations, within the Bowen‐York formalism, for two interacting black holes in close separation, with anti‐parallel spins and anti‐parallel linear momenta is studied. An analytical solution using perturbation theory is given. The location and the shape of the apparent horizon which generically depend on all the parameters, angles, and the separation between the black holes are also computed. The solution only works for distances far away from the black holes. To gain more insight close into the black holes, one has to go to the higher orders in perturbation theory, which is a rather cumbersome process. But the solution presented here can be of some use for numerical computations as the latter should match our result for the described problem.

Nonassociative black holes in R-flux deformed phase spaces and relativistic models of Perelman thermodynamics

This paper explores new classes of black hole (BH) solutions in nonassociative and noncommutative gravity, focusing on features that generalize to higher dimensions. The theories we study are modelled on (co) tangent Lorentz bundles with a star product structure determined by R-flux deformations in string theory. For the nonassociative vacuum Einstein equations we consider both real and complex effective sources. In order to analyze the nonassociative vacuum Einstein equations we develop the anholonomic frame and connection deformation methods, which allows one to decoupled and solve these equations. The metric coefficients can depend on both space-time coordinates and energy-momentum. By imposing conditions on the integration functions and effective sources we find physically important, exact solutions: (1) 6-d Tangherlini BHs, which are star product and R-flux distorted to 8-d black ellipsoids (BEs) and BHs; (2) nonassocitative space-time and co-fiber space double BH and/or BE configurations generalizing Schwarzschild-de Sitter metrics. We also investigate the concept of Bekenstein-Hawking entropy and find it applicable only for very special classes of nonassociative BHs with conventional horizons and (anti) de Sitter configurations. Finally, we show how analogs of the relativistic Perelman W-entropy and related geometric thermodynamic variables can be defined and computed for general classes of off-diagonal solutions with nonassociative R-flux deformations.