Conformal Einstein Field Equations Research Articles

Conformal geodesics and the evolution of spacetimes with positive Cosmological constant.

This article provides a discussion on the construction of conformal Gaussian gauge systems to study the evolution of solutions to the Einstein field equations with positive Cosmological constant. This is done by means of a gauge based on the properties of conformal geodesics. The use of this gauge, combined with the extended conformal Einstein field equations, yields evolution equations in the form of a symmetric hyperbolic system for which standard Cauchy stability results can be employed. This strategy is used to study the global properties of de Sitter-like spacetimes with constant negative scalar curvature. It is then adapted to study the evolution of the Schwarzschild-de Sitter spacetime in the static region near the conformal boundary. This review is based on Minucci et al. 2021 Class. Quantum Grav. 38, 145026. (doi:10.1088/1361-6382/ac0356) and Minucci et al. 2023 Class. Quantum Grav. 40, 145005. (doi:10.1088/1361-6382/acdb3f). This article is part of a discussion meeting issue 'At the interface of asymptotics, conformal methods and analysis in general relativity'.

Linearised conformal Einstein field equations

The linearisation of a second-order formulation of the conformal Einstein field equations (CEFEs) in generalised harmonic gauge (GHG), with trace-free matter is derived. The linearised equations are obtained for a general background and then particularised for the study linear perturbations around a flat background—the inversion (conformal) representation of the Minkowski spacetime—and the solutions discussed. We show that the generalised Lorenz gauge (defined as the linear analogue of the GHG-gauge) propagates. Moreover, the equation for the conformal factor can be trivialised with an appropriate choice for the gauge source functions; this permits a scri-fixing strategy using gauge source functions for the linearised wave-like CEFE-GHG, which can in principle be generalised to the nonlinear case. As a particular application of the linearised equations, the far-field and compact source approximation is employed to derive quadrupole-like formulae for various conformal fields such as the perturbation of the rescaled Weyl tensor.

On the non-linear stability of the Cosmological region of the Schwarzschild-de Sitter spacetime

The non-linear stability of the sub-extremal Schwarzschild-de Sitter spacetime in the stationary region near the conformal boundary is analysed using a technique based on the extended conformal Einstein field equations and a conformal Gaussian gauge. This strategy relies on the observation that the Cosmological stationary region of this exact solution can be covered by a non-intersecting congruence of conformal geodesics. Thus, the future domain of dependence of suitable spacelike hypersurfaces in the Cosmological region of the spacetime can be expressed in terms of a conformal Gaussian gauge. A perturbative argument then allows to prove existence and stability results close to the conformal boundary and away from the asymptotic points where the Cosmological horizon intersects the conformal boundary. In particular, we show that small enough perturbations of initial data for the sub-extremal Schwarzschild-de Sitter spacetime give rise to a solution to the Einstein field equations which is regular at the conformal boundary. The analysis in this article can be regarded as a first step towards a stability argument for perturbation data on the Cosmological horizons.

The conformal Killing spinor initial data equations

We obtain necessary and sufficient conditions for an initial data set for the vacuum conformal Einstein field equations to give rise to a spacetime development in possession of a Killing spinor. The fact that the conformal Einstein field equations are used in our derivation allows for the possibility of the initial hypersurface S intersecting non-trivially with (or even being a subset of) null infinity I. For conciseness, these conditions are derived assuming that the initial hypersurface is spacelike. Hence, in particular, these conformal Killing spinor initial data equations encode necessary and sufficient conditions for the existence of a Killing spinor in the development of asymptotic initial data on spacelike components of I.

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.

The conformal Einstein field equations with massless Vlasov matter

We prove the stability of de Sitter space-time as a solution to the Einstein-Vlasov system with massless particles. The semi-global stability of Minkowski space-time is also addressed. The proof relies on conformal techniques, namely Friedrich's conformal Einstein field equations. We exploit the conformal invariance of the massless Vlasov equation on the cotangent bundle and adapt Kato's local existence theorem for symmetric hyperbolic systems to prove a long enough time of existence for solutions of the evolution system implied by the Vlasov equation and the conformal Einstein field equations.

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.

A conformal approach to the stability of Einstein spaces with spatial sections of negative scalar curvature

It is shown how the extended conformal Einstein field equations and a gauge based on the properties of conformal geodesics can be used to analyse the non-linear stability of Einstein spaces with negative scalar curvature. This class of spacetimes admits a smooth conformal extension with a spacelike conformal boundary. Central to the analysis is the use of conformal Gaussian systems to obtain a hyperbolic reduction of the conformal Einstein field equations for which standard Cauchy stability results for symmetric hyperbolic systems can be employed. The use of conformal methods allows to rephrase the question of global existence of solutions to the Einstein field equations into considerations of finite existence time for the conformal evolution system.

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.

Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the tracefree matter case

Using a metric conformal formulation of the Einstein equations, we develop a construction of 4D anti-de Sitter-like spacetimes coupled to tracefree matter models. Our strategy relies on the formulation of an initial-boundary problem for a system of quasilinear wave equations for various conformal fields by exploiting the conformal and coordinate gauges. By analysing the conformal constraints we show a systematic procedure to prescribe initial and boundary data. This analysis is complemented by the propagation of the constraints, showing that a solution to the wave equations implies a solution to the Einstein field equations. In addition, we study three explicit tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang–Mills field. For each one of these we identify the basic data required to couple them to the system of wave equations. As our main result, we establish the local existence and uniqueness of solutions for the evolution system in a neighbourhood around the corner, provided compatibility conditions for the initial and boundary data are imposed up to a certain order.

Conformal wave equations for the Einstein-tracefree matter system

Inspired by a similar analysis for the vacuum conformal Einstein field equations by Paetz (Ann Henri Poincaré 16:2059, 2015), in this article we show how to construct a system of quasilinear wave equations for the geometric fields associated to the conformal Einstein field equations coupled to matter models whose energy-momentum tensor has vanishing trace. In this case, the equation of conservation for the energy-momentum tensor is conformally invariant. Our analysis includes the construction of a subsidiary evolution which allows to prove the propagation of the constraints. We discuss how the underlying structure behind these systems of equations is the set of integrability conditions satisfied by the conformal field equations. The main result of our analysis is that both the evolution and subsidiary equations for the geometric part of the conformal Einstein-tracefree matter field equations close without the need of any further assumption on the matter models other that the vanishing of the trace of the energy-momentum tensor. Our work is supplemented by an analysis of the evolution and subsidiary equations associated to three basic tracefree matter models: the conformally invariant scalar field, the Maxwell field and the Yang–Mills field. As an application we provide a global existence and stability result for de Sitter-like spacetimes. In particular, the result for the conformally invariant scalar field is new in the literature.

Construction of anti-de Sitter-like spacetimes using the metric conformal Einstein field equations: the vacuum case

We make use of the metric version of the conformal Einstein field equations to construct anti-de Sitter-like spacetimes by means of a suitably posed initial-boundary value problem. The evolution system associated to this initial-boundary value problem consists of a set of conformal wave equations for a number of conformal fields and the conformal metric. This formulation makes use of generalised wave coordinates and allows the free specification of the Ricci scalar of the conformal metric via a conformal gauge source function. We consider Dirichlet boundary conditions for the evolution equations at the conformal boundary and show that these boundary conditions can, in turn, be constructed from the 3D Lorentzian metric of the conformal boundary and a linear combination of the incoming and outgoing radiation as measured by certain components of the Weyl tensor. To show that a solution to the conformal evolution equations implies a solution to the Einstein field equations we also provide a discussion of the propagation of the constraints for this initial-boundary value problem. The existence of local solutions to the initial-boundary value problem in a neighbourhood of the corner where the initial hypersurface and the conformal boundary intersect is subject to compatibility conditions between the initial and boundary data. The construction described is amenable to numerical implementation and should allow the systematic exploration of boundary conditions.

Perturbations of the Asymptotic Region of the Schwarzschild\u2013de Sitter Spacetime

The conformal structure of the Schwarzschild–de Sitter spacetime is analysed using the extended conformal Einstein field equations. To this end, initial data for an asymptotic initial value problem for the Schwarzschild–de Sitter spacetime are obtained. This initial data allow to understand the singular behaviour of the conformal structure at the asymptotic points where the horizons of the Schwarzschild–de Sitter spacetime meet the conformal boundary. Using the insights gained from the analysis of the Schwarzschild–de Sitter spacetime in a conformal Gaussian gauge, we consider nonlinear perturbations close to the Schwarzschild–de Sitter spacetime in the asymptotic region. We show that small enough perturbations of asymptotic initial data for the Schwarzschild–de Sitter spacetime give rise to a solution to the Einstein field equations which exists to the future and has an asymptotic structure similar to that of the Schwarzschild–de Sitter spacetime.

Spinorial wave equations and stability of the Milne spacetime

The spinorial versions of the conformal vacuum Einstein field equations are used to construct a system of quasilinear wave equations for the various conformal fields. As a part of the analysis we also show how to construct a subsidiary system of wave equations for the zero quantities associated to the various conformal field equations. This subsidiary system is used, in turn, to show that under suitable assumptions on the initial data a solution to the wave equations for the conformal fields implies a solution to the actual conformal Einstein field equations. The use of spinors allows for a more unified deduction of the required wave equations and the analysis of the subsidiary equations than similar approaches based on the metric conformal field equations. As an application of our construction we study the nonlinear stability of the Milne Universe. It is shown that sufficiently small perturbations of initial hyperboloidal data for the Milne Universe gives rise to a solution to the Einstein field equations which exists towards the future and has an asymptotic structure similar to that of the Milne Universe.

On the conformal structure of the extremal Reissner–Nordström spacetime

We analyse various conformal properties of the extremal Reissner–Nordström spacetime. In particular, we obtain conformal representations of the neighbourhoods of spatial infinity, time-like infinity and the cylindrical end—the so-called cylinders at spatial infinity and at the horizon, respectively—which are regular with respect to the conformal Einstein field equations and their associated initial data sets. We discuss possible implications of these constructions for the propagation of test fields and non-linear perturbations of the gravitational field close to the horizon.

The extended conformal Einstein field equations with matter: The Einstein–Maxwell field

A discussion is given of the conformal Einstein field equations coupled with matter whose energy–momentum tensor is trace-free. These resulting equations are expressed in terms of a generic Weyl connection. The article shows how in the presence of matter it is possible to construct a conformal gauge which allows to know a priori the location of the conformal boundary. In vacuum this gauge reduces to the so-called conformal Gaussian gauge. These ideas are applied to obtain (i) a new proof of the stability of Einstein–Maxwell de Sitter-like spacetimes; (ii) a proof of the semi-global stability of purely radiative Einstein–Maxwell spacetimes.

Conformal extensions for stationary spacetimes

The construction of the cylinder at spatial infinity for stationary spacetimes is considered. Using a specific conformal gauge and frame, it is shown that the tensorial fields associated with the conformal Einstein field equations admit expansions in a neighbourhood of the cylinder at spatial infinity which are analytic with respect to some suitable time, radial and angular coordinates. It is then shown that the essentials of the construction are independent of the choice of conformal gauge. As a consequence, one finds that the construction of the cylinder at spatial infinity and the regular finite initial value problem for stationary initial data sets are, in a precise sense, as regular as they could be.

Asymptotic properties of the development of conformally flat data near spatial infinity

The analysis of the relation between Bondi-type systems (NP-gauge) and a gauge used in the analysis of the structure of spatial infinity (F-gauge) which was carried out by Friedrich and Kánnár (2000 J. Math Phys. 41 2195) is retaken and applied to the development of a suitable class of conformally flat initial data sets with non-vanishing second fundamental form. The calculations presented depend on a certain assumption about the existence and regularity of the solutions to the conformal Einstein field equations close to null and spatial infinity. As a result of the calculations the Newman–Penrose constants of both future and past null infinity are calculated in terms of initial data and are shown to be equal. It is also shown that the asymptotic shear goes to zero as one approaches spatial infinity along the generators of null infinity so that it is possible to select, in a canonical fashion, the Poincaré group out of the BMS group. An expansion—again in terms of initial data quantities—of the Bondi mass close to spatial infinity is calculated. This expansion shows that if the existence and regularity assumptions hold, the Bondi mass approaches the ADM mass. A discussion of possible conditions on the initial data which would render a peeling development is presented.