The authors prove that the maximum norm of the vorticity controls the breakdown of smooth solutions of the 3-D Euler equations. In other words, if a solution of the Euler equations is initially smooth and loses its regularity at some later time, then the maximum vorticity necessarily grows without bound as the critical time approaches; equivalently, if the vorticity remains bounded, a smooth solution persists.

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Remarks on the breakdown of smooth solutions for the 3- D Euler equations

Metric affine gauge theory of gravity: Field equations, Noether identities, world spinors, and breaking of dilation invariance

LetM be a space-time whose local mass density is non-negative everywhere. Then we prove that the total mass ofM as viewed from spatial infinity (the ADM mass) must be positive unlessM is the flat Minkowski space-time. (So far we are making the reasonable assumption of the existence of a maximal spacelike hypersurface. We will treat this topic separately.) We can generalize our result to admit wormholes in the initial-data set. In fact, we show that the total mass associated with each asymptotic regime is non-negative with equality only if the space-time is flat.

https://www.math.jhu.edu/~js/Math646/schoen-yau.pm1.pdf

On the proof of the positive mass conjecture in general relativity

Reduction of symplectic manifolds with symmetry

A development of the basic theory and applications of mechanics with an emphasis on the role of symmetry. The book includes numerous specific applications, making it beneficial to physicists and engineers. Specific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering. This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises. Internet supplements containing additional material are also available.

Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems

A systematic presentation of the quasi-linear first order symmetric hyperbolic systems of Friedrichs is presented A number of sharp regularity and smoothness properties of the solutions are obtained The present paper is devoted to the case ofRn with suitable asymptotic conditions imposed As an example, we apply this theory to give new proofs of the existence and uniqueness theorems for the Einstein equations in general relativity, due to Choquet-Bruhat and Lichnerowicz These new proofs usingfirst order techniques are considerably simplier than the classical proofs based onsecond order techniques Our existence results are as sharp as had been previously known, and our uniqueness results improve by one degree of differentiability those previously existing in the literature

/pdf/the-einstein-evolution-equations-as-a-first-order-quasi-6s9u2giuw1.pdf

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I

Deformations of the scalar curvature

In this chapter we discuss some of the interrelationships between the initial value problem, the canonical formalism, linearization stability and the space of gravitational degrees of freedom. In the last decade, these topics have experienced a resurgence of interest as more advanced mathematical methods and viewpoints have begun to show the intimate relationships among these topics. At present, the literature regarding these areas of general relativity is a rapidly expanding body of knowledge. Our purpose here is to present the current state of affairs from our own point of view. We shall use geometric methods developed by the authors to establish various conn~ctions between the above-mentioned topics. The main tools we shall use to develop this material are nonlinear functional analysis, an adjoint formalism for Hamiltonian field theories, and infinite-dimensional symplectic geometry. As we shall see, these tools and the topics we shall consider are naturally related. For a more complete picture of the current state of affairs, the reader is urged to consult Choquet-Section 4.1 develops the Hamiltonian formalism for the dynamics of general relativity, usually called the ADM (Arnowitt, Deser and Misner) formalism. This is done using invariant concepts and the adjoint formalism developed by the authors. We show how to write the Einstein dynamical system explicitly in the compact form

The initial value problem and the dynamical formulation of general relativity

1. Introduction. An important problem in general relativity is the question of whether or not a solution of the linearized Einstein field equations (relative to a given background solution) actually approximates to first order a curve of exact solutions to the nonlinear equations. Here we announce that under certain geometrical conditions on the background solution the problem can be answered affirmatively; however, in certain exceptional cases the answer may be negative. In the affirmitive case we shall say that the background metric is linearization-stable. Let (4) # be a Lorentz metric (signature —, +, +, +) on a 4-manifold V. The empty space Einstein field equations of general relativity are that the Ricci tensor of (4) # vanish :

/pdf/linearization-stability-of-the-einstein-equations-51hzei7d9b.pdf

Linearization stability of the Einstein equations

In this paper the exterior Einstein equations are explored from a differential geometric point of view. Using methods of global analysis and infinite-dimensional geometry, we answer sharply the question: "In what sense are the Einstein equations, written as equations of evolution, a Lagrangian dynamical system?" By using our global methods, several aspects of the lapse function and shift vector field are clarified. The geometrical significance of the shift becomes apparent when the Einstein evolution equations are written using Lie derivatives. The evolution equations are then interpreted as evolution equations as seen by an observer in space coordinates. Using the notion of body-space transitions, we then find the relationship between solutions with different shifts by finding the flow of a time-dependent vector field. The use of body and space coordinates is shown to be somewhat analogous to the use of such coordinates in Euler's equations for a rigid body and the use of Eulerian and Lagrangian coordinates in hydrodynamics. We also explore the geometry of the lapse function, and show how one can pass from one lapse function to another by integrating ordinary differential equations. This involves integrating what we call the "intrinsic shift vector field." The essence of our method is to extend the usual configuration space [fraktur M]=Riem(M) of Riemannian metrics to [script T]×[script D]×[fraktur M], where [script T]=C[infinity](M,R) is the group of relativistic time translations and [script D]=Diff(M) is the group of spatial coordinate transformations of M. The lapse and shift then enter the dynamical picture naturally as the velocities canonically conjugate to the configuration fields (xit,etat)[is-an-element-of][script T]×[script D]. On this extended configuration space, a degenerate Lagrangian system is constructed which allows precisely for the arbitrary specification of the lapse and shift functions. We reinterpret a metric given by DeWitt for [fraktur M] as a degenerate metric on [script D]×[fraktur M]. On [script D]×[fraktur M], however, the metric is quadratic in the velocity variables. The groups [script T] and [script D] also serve as symmetry groups for our dynamical system. We establish that the associated conserved quantities are just the usual "constraint equations." A precise theorem is given for a remark of Misner that in an empty space-time we must have [script H]=0. We study the relationship between the evolution equations for the time-dependent metric gt and the Ricci flat condition of the reconstructed Lorentz metric gL. Finally, we make some remarks about a possible "superphase space" for general relativity and how our treatment on [script T]×[script D]×[fraktur M] is related to ordinary superspace and superphase space.

/pdf/the-einstein-equations-of-evolution-a-geometric-approach-uft0p2mpnr.pdf

Arthur E. Fischer

Papers

The Einstein evolution equations as a first-order quasi-linear symmetric hyperbolic system. I

Deformations of the scalar curvature

The initial value problem and the dynamical formulation of general relativity

Linearization stability of the Einstein equations

The Einstein Equations of Evolution -- A Geometric Approach