Theorems on Existence and Global Dynamics for the Einstein Equations

Rendall, Alan D.

doi:10.12942/lrr-2005-6

Cited by 60 publications

(39 citation statements)

References 319 publications

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“…The condition (4.7) thus means that we choose such a time coordinate in the perturbed FRW universes that the extrinsic curvature of the constant hypersurfaces is the same as in the unperturbed universe, i.e., it is constant along each hypersurface. Much work has been done on the existence and properties of such foliations (see [51] for the recent review and references). The gauge condition (4.7) for time slicing combined with the gauge conditions (4.3) for the spatial part of the metric will be called the Mach 1 gauge.…”

Section: B Gauge Conditions On the Time Slicingmentioning

confidence: 99%

Cosmological perturbation theory, instantaneous gauges, and local inertial frames

2007

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Linear perturbations of Friedmann-Robertson-Walker universes with any curvature and cosmological constant are studied in a general gauge without decomposition into harmonics. Desirable gauges are selected as those which embody best Mach's principle: in these gauges local inertial frames can be determined instantaneously via the perturbed Einstein field equations from the distributions of energy and momentum in the universe. The inertial frames are identified by their 'accelerations and rotations' with respect to the cosmological frames associated with the 'Machian gauges'. In closed spherical universes, integral gauge conditions are imposed to eliminate motions generated by the conformal Killing vectors. The meaning of Traschen's integral constraint vectors is thus elucidated. For all three types of FRW universes the Machian gauges admit much less residual freedom than the synchronous or generalized harmonic gauge. Mach's principle is best exhibited in the Machian gauges in closed spherical universes. Independent of any Machian motivation the general perturbation equations and discussion of gauges are useful for cosmological perturbation theory.

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Section: B Gauge Conditions On the Time Slicingmentioning

confidence: 99%

Cosmological perturbation theory, instantaneous gauges, and local inertial frames

2007

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“…This need is apparent both from the PDE point of view on general relativity, i.e., the initial value problem for the Einstein equations, and from physically relevant models. In fact, the standard local existence result for the vacuum Einstein equations [47] assumes the metric to be of Sobolev regularity (with ). Recently, the regularity of the metric has been lowered even further (e.g., [28]).…”

Section: Introductionmentioning

confidence: 99%

Lorentzian length spaces

2018

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We introduce an analogue of the theory of length spaces into the setting of Lorentzian geometry and causality theory. The rôle of the metric is taken over by the time separation function, in terms of which all basic notions are formulated. In this way, we recover many fundamental results in greater generality, while at the same time clarifying the minimal requirements for and the interdependence of the basic building blocks of the theory. A main focus of this work is the introduction of synthetic curvature bounds, akin to the theory of Alexandrov and CAT( k )-spaces, based on triangle comparison. Applications include Lorentzian manifolds with metrics of low regularity, closed cone structures, and certain approaches to quantum gravity.

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“…Provided there are no closed timelike lines, existence of suitable initial data on a spacelike surface S satisfying the initial value equations [34], together with suitable equations of state for whatever matter may be present, determines a unique solution for the Einstein equations within the future domain of dependence of S: that is, the region of spacetime such that all past timelike and null curves intersect S. Applied to the gravitational equations, the spacetime developing from the initial data is called the future Cauchy development of the data on S [5,35]. An important technical point is that one can prove existence of timelike and null geodesics from S to every point in this domain of dependence.…”

Section: Initial Data Existence and Uniqueness Theoremsmentioning

confidence: 99%

100 Years of General Relativity

Ellis¹

2015

General Relativity and Gravitation

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2015). It gives a survey of themes that have been developed during the 100 years of development of general relativity theory.This chapter aims to provide a broad historical overview of the major developments in General Relativity Theory ('GR') after the theory had been developed in its final form. It will not relate the well-documented story of the discovery of the theory by Albert Einstein, but rather will consider the spectacular growth of the subject as it developed into a mainstream branch of physics, high energy astrophysics, and cosmology. Literally hundreds of exact solutions of the full non-linear field equations are now known, despite their complexity [1]. The most important ones are the Schwarzschild and Kerr solutions, determining the geometry of the solar system and of black holes (Section 2), and the Friedmann-Lemaître-Robertson-Walker solutions, which are basic to cosmology (Section 4). Perturbations of these solutions make them the key to astrophysical applications.Rather than tracing a historical story, this chapter is structured in terms of key themes in the study and application of GR:1. The study of dynamic geometry (Section 1) through development of various technical tools, in particular the introduction of global methods, resulting in global existence and uniqueness theorems and singularity theorems.2. The study of the vacuum Schwarzschild solution and its application to the Solar system (Section 2), giving very accurate tests of general relativity, and underlying the crucial role of GR in the accuracy of useful GPSindexGPS systems.3. The understanding of gravitational collapse and the nature of Black Holes (Section 3), with major applications in astrophysics, in particular as regards quasi-stellar objects and active galactic nuclei.4. The development of cosmological models (Section 4), providing the basis for our understandings of both the origin and evolution of the universe as a whole, and of structure formation within it;5. The study of gravitational lensing and its astronomical applications, including detection of dark matter (Section 5).6. Theoretical studies of gravitational waves, in particular resulting in major developments in numerical relativity (Section 6), and with development of gravitational wave observatories that have the potential to become an essential tool in precision cosmology. This Chapter will not discuss quantum gravity, covered in Part Four. It is of course impossible to refer to all relevant literature. I have attempted to give the reader a judicious mix of path-breaking original research articles, and good review articles. Ferreira has recently discussed the historical development at greater length [2].

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Theorems on Existence and Global Dynamics for the Einstein Equations

Cited by 60 publications

References 319 publications

Cosmological perturbation theory, instantaneous gauges, and local inertial frames

Cosmological perturbation theory, instantaneous gauges, and local inertial frames

Lorentzian length spaces

100 Years of General Relativity

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