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2016 | OriginalPaper | Buchkapitel

Some Evolution Problems in the Vacuum Einstein Equations

verfasst von : Junbin Li, Xi-Ping Zhu

Erschienen in: Geometry and Topology of Manifolds

Verlag: Springer Japan

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Abstract

We discuss two problems in the evolution of the vacuum Einstein equations. The first one is about the formation of trapped surface, and the second one is about the characteristic problems with initial data on complete null cones.

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Vorheriges Kapitel The Symplectic Critical Surfaces in a Kähler Surface

Nächstes Kapitel Willmore 2-Spheres in : A Survey

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See the Appendix on the discussion on the characteristic initial data.

Strictly speaking, they proved the existence of a closed marginally trapped surface, which refers to a 2-dim embedding space-like surface with the mean curvature with respect to the outgoing future null normals to the surface being zero.

We recall the spacetime Kerr metric in the Boyer-Lindquist coordinates:

$$\begin{aligned} g_{K}&=(-1+\frac{2mr}{r^2+a^2\cos ^2\theta })\mathrm{{d}} t^2-\frac{2mra\sin ^2\theta }{r^2+a^2\cos ^2\theta }\mathrm{{d}} t\mathrm{{d}}\varphi +\frac{r^2+a^2\cos ^2\theta }{r^2-2mr+a^2}\mathrm{{d}} r^2\\&\quad +(r^2+a^2\cos ^2\theta )\mathrm{{d}}\theta ^2+\sin ^2\theta (r^2+a^2+\frac{2mra^2\sin ^2\theta }{r^2+a^2\cos ^2\theta })\mathrm{{d}}\varphi ^2. \end{aligned}$$

The Schwarzschild spacetime is the Kerr spacetime with $a=0$.

The asymptotic flatness condition $\displaystyle \inf _{x_0\in \varSigma }Q(x_0)<+\infty $ holds.

This is only a heuristic form of the asymptotic behavior. See the Appendix for detail discussion.

In the case that the asymptotic behavior imposed on $C_0$ is the same as that in the work of Bieri [1], an analogue theorem was also proved in [19] by the authors after this paper was accepted.

Using the conformal vacuum Einstein equations, the length of $C_0$ becomes finite. Their method actually applies to a more general class of nonlinear wave equations assuming $C_0$ to be finite.

The work [8] of Christodoulou was actually the first example of combining both ingredients without symmetry.

The metric g does not exist at the first moment, but we should assume g exists and point out the geometric meaning of the initial data.

This means that s and $\underline{s}$ are affine parameters of the null generators of $C_0$ and $\underline{C}_0$ respectively.

By this we mean we should assign on $S_0$ a Riemannian metric $g \!\! /$, a one-form and two functions on $S_0$, which play the roles of $\zeta $, $\mathrm {tr}\chi $ and $\mathrm {tr}\underline{\chi }$ in the resulting solution.

These equations can be found in Chap. 1 of [8].

However, $\alpha $ can be computed directly as $\alpha =-\widehat{\frac{\partial }{\partial s}}\widehat{\chi }_{AB}$ where $\widehat{\frac{\partial }{\partial s}}$ is the trace-free part of the derivative.

The initial values of the equations below can be figured out by the values of $\zeta $, $\widehat{\chi }$, $\mathrm {tr}\chi $, $\widehat{\underline{\chi }}$ and $\mathrm {tr}\underline{\chi }$ on $S_0$, by another group of equations, that we will not present here.

One can see [4] for a related topic when the data is small.

There are some nonessential differences from the statement in [20] for simplicity and convenience.

$\overline{f}$ is defined to be the average of f on $S_s$.

$\mathscr {L} \! /_{e_3}$ means the restriction on $S_s$ of the Lie derivative in $e_3$ direction.

The Eq. (10) has been studied in Chap. 2 of [8]. Notice that $\widehat{\chi }$ is simply the derivative with respect to s of $\widehat{g \!\! /}$, up to a multiple $\phi ^2$.

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Titel: Some Evolution Problems in the Vacuum Einstein Equations
verfasst von: Junbin Li
Xi-Ping Zhu
Verlag: Springer Japan
Buch: Geometry and Topology of Manifolds
Print ISBN: 978-4-431-56019-7

Electronic ISBN: 978-4-431-56021-0

Copyright-Jahr: 2016
DOI: https://doi.org/10.1007/978-4-431-56021-0_11

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