Rough Solutions of Einstein Vacuum Equations in CMCSH Gauge

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

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TWISTOR THEORY, INTEGRABILITY & THE EINSTEIN VACUUM EQUATIONS

The Ninth Marcel Grossmann Meeting ◽

10.1142/9789812777386_0073 ◽

2002 ◽

pp. 807-809

Author(s):

L.J. MASON

Keyword(s):

Twistor Theory ◽

Einstein Vacuum Equations ◽

Einstein Vacuum

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THE CHARACTERISTIC PROBLEM FOR THE EINSTEIN VACUUM EQUATIONS

Asymptotic Methods in Nonlinear Wave Phenomena ◽

10.1142/9789812708908_0014 ◽

2007 ◽

Author(s):

Francesco Nicolò

Keyword(s):

Characteristic Problem ◽

Einstein Vacuum Equations ◽

Einstein Vacuum

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CURVATURE BLOW UP ON A DENSE SUBSET OF THE SINGULARITY IN T3-GOWDY

Journal of Hyperbolic Differential Equations ◽

10.1142/s021989160500052x ◽

2005 ◽

Vol 02 (02) ◽

pp. 547-564 ◽

Cited By ~ 7

Author(s):

HANS RINGSTRÖM

Keyword(s):

Initial Data ◽

Blow Up ◽

Dense Subset ◽

Cosmic Censorship ◽

Einstein Vacuum Equations ◽

Strong Cosmic Censorship ◽

Einstein Vacuum ◽

Original Argument ◽

Gowdy Spacetimes ◽

Generic Set

This paper is concerned with the Einstein vacuum equations under the additional assumption of T3-Gowdy symmetry. We prove that there is a generic set of initial data such that the corresponding solutions exhibit curvature blow up on a dense subset of the singularity. By generic, we mean a countable intersection of open sets (i.e. a Gδ set) which is also dense. Furthermore, the set of initial data is given the C∞ topology. This result was presented at a conference in Miami 2004. Recently, we have obtained a stronger result, but the argument to prove it is different and much longer. Therefore, we here wish to present the original argument. Finally, combining the results presented here with a paper by Chruściel and Lake, one obtains strong cosmic censorship for T3-Gowdy spacetimes.

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Rough solutions of the Einstein-vacuum equations

Annals of Mathematics ◽

10.4007/annals.2005.161.1143 ◽

2005 ◽

Vol 161 (3) ◽

pp. 1143-1193 ◽

Cited By ~ 25

Author(s):

Sergiu Klainerman ◽

Igor Rodnianski

Keyword(s):

Einstein Vacuum Equations ◽

Einstein Vacuum

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GLOBAL CHARACTERISTIC PROBLEM FOR EINSTEIN VACUUM EQUATIONS WITH SMALL INITIAL DATA: (I) THE INITIAL DATA CONSTRAINTS

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891605000439 ◽

2005 ◽

Vol 02 (01) ◽

pp. 201-277 ◽

Cited By ~ 9

Author(s):

GIULIO CACIOTTA ◽

FRANCESCO NICOLÒ

Keyword(s):

Initial Data ◽

Existence Result ◽

Subsequent Paper ◽

Small Initial Data ◽

Asymptotic Decay ◽

Characteristic Problem ◽

Einstein Vacuum Equations ◽

Einstein Vacuum ◽

Data Constraints ◽

Global Existence Result

We show how to prescribe the initial data of a characteristic problem satisfying the constraints, the smallness, the regularity and the asymptotic decay suitable to prove a global existence result. In this paper, the first of two, we show in detail the construction of the initial data and give a sketch of the existence result. This proof, which mimicks the analogous one for the non-characteristic problem in [19], will be the content of a subsequent paper.

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