Local Cauchy Problem

Since we are treating non-quantum fields, it seems reasonable that we consider the problem of classical dynamics, i.e. the problem of the evolution of initial data. The Einstein equations are a geometrical system, invariant under diffeo-morphisms of V, the associated isometries of g, and transformation of the sources. From the analyst's point of view they constitute, for the metric, a system of second-order quasilinear partial differential equations. However, the system is underdetermined...

Global Existence Theorems Asymptotically Euclidean Data

The existence of past and future complete asymptotically Euclidean1 vacuum Einsteinian spacetimes already preoccupied Einstein. He thought that these spacetimes being empty should coincide, in the absence of radiation coming from infinity, with the Minkowski spacetime. His physical intuition was correct, but it lacked a precise formulation of the absence of radiation coming from infinity. The positive mass theorem first formulated by Deser, with a partly heuristic proof2 by Brill and Deser,...

Qr q q2 v q

Similar use of Sobolev and Holder inequalities give the result Wq x Wq C Lr, with r > 1 if q > . The convergence in L1 of a u to au2, obvious when q > , results when < q < -, from the inequality P + qP < 1 if p > . We have thus proved that the equality u(A7 u au)pY (llj diudj u + au2)pY (2.41) holds under the given conditions on 7, a and u. It implies u 0 if AYu au 0 and a > 0,a 0. Theorem 2.7 The Poisson operator AY a on scalar functions for a metric Y on a...

The rigidity theorem for black holes

Section contributed by James Isenberg Well-known explicit black hole solutions (Schwarzschild, Kerr, Kerr-Newman) are both stationary and axisymmetric. Presuming that we restrict our attention to equilibrium and therefore stationary black holes, one may ask if it is necessary that such solutions be axisymmetric as well. Physical arguments - that a non-axisymmetric solution would radiate away any multipole moments of order higher than that of angular momentum - suggest that axisymmetry does...

Info

Since g is locally conformally flat, there are local coordinates in which the metric reads g JCn f (xV.U) (3.7) Straightforward7 calculation shows that the equations expressing that the Riemann tensor has constant curvature K reduce to (dxi)2 (dxj )2 From the first set of equations it follows that where Xi is a function of x1 alone. The second set of equations implies d2f d2f d2Xj d2Xk f f i.e. -U-A -UzA (3.11) Therefore, using the fact that a function of xj can equal a function of xk only if...

Stationary black holes

It is believed that very massive objects will generally undergo gravitational collapse and give rise to a black hole, but this black hole will settle down to a stationary state. It was conjectured already in the early 1940s by G. Darmois, without precise hypotheses, that the Schwarzschild spacetime was the only relevant one for vacuum static Einsteinian spacetimes. The uniqueness of general stationary black holes was predicted in the early 1950s by J. A. Wheeler using the picturesque phrase...

Lorentz Geometry

We give in this chapter a survey of the basic definitions of Riemannian and Lorentzian differential geometry that we will use in this book. In the first nine sections we use the simplest formulations, in local coordinates, as they are needed for the first five chapters and physical applications. The later sections contain material used in the following, more advanced, chapters. This chapter is a reminder to save the reader's time and to fix notations it is not a course in Lorentzian geometry....

Progressive Waves

We take afresh in this chapter the treatment of progressive1 waves for non-linear equations used in Sections III.12 and III.13 to construct weak gravitational and electromagnetic waves on a given electrovac Einsteinian spacetime. The fundamental improvement initiated by Leray in the construction of high-frequency waves for linear systems permits the extension to quasilinear systems2, and the appearance of new properties linked to the non-linearities, in some sense similar to shocks3. There is...

General Relativity And Einsteins Equations

The gravitational field is the only universal force noticeable by everyone everywhere and known since antiquity. It prevents massive free bodies from moving in straight lines with constant velocities with respect to inertial frames. This result is known to a child throwing a ball, which describes approximately a parabola1. It was known to Tycho Brahe, who observed the trajectories of the planets around the Sun, and to Kepler who showed that these trajectories are ellipses. These observations...

B

The Boyer-Linquist metric takes then the Kerr-Schild form ( 2mr 2 gKS -i 1--j- j dv2 + 2drdv + Ad92 + sin2 9d 2 + 4amr - dvd + 2a sin2 9drd (9.6) A 2 (r2 + a2)2 - Bo2 sin2 9 (9.7) This metric extends to a regular Lorentzian metric for all A > 0 it reduces to the Minkowski metric in retarded time and polar coordinates of R3 if m a 0, and it reduces to the Eddington-Finkelstein metric when a 0. The Kerr-Schild metric becomes singular for A r2 + a2 cos2 9 0, i.e. r 0, 9 2. This is a genuine...