Abstract
In a recent important breakthrough D. Christodoulou (The Formation of Black Holes in General Relativity. Monographs in Mathematics. Eur. Math. Soc., Zurich, 2009) has solved a long standing problem of General Relativity of evolutionary formation of trapped surfaces in the Einstein-vacuum space-times. He has identified an open set of regular initial conditions on a finite outgoing null hypersurface leading to a formation a trapped surface in the corresponding vacuum space-time to the future of the initial outgoing hypersurface and another incoming null hypersurface with the prescribed Minkowskian data. He also gave a version of the same result for data given on part of past null infinity. His proof is based on an inspired choice of the initial condition, an ansatz which he calls short pulse, and a complex argument of propagation of estimates, consistent with the ansatz, based, largely, on the methods used in the global stability of the Minkowski space (Christodoulou and Klainerman in The Global Nonlinear Stability of the Minkowski Space. Princeton Mathematical Series, vol. 41, 1993). Once such estimates are established in a sufficiently large region of the space-time the actual proof of the formation of a trapped surface is quite straightforward.
Christodoulou’s result has been significantly simplified and extended in my joint works with I. Rodnianski (Klainerman and Rodnianski in On the formation of trapped surfaces, Acta Math. 2011, in press) and (Klainerman and Rodnianski in Discrete Contin. Dyn. Syst. 28(3):1007–1031, 2010). In this note I will give a short survey of these results.
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Notes
- 1.
These allow one to cast the Einstein vacuum equations in the form of a system of nonlinear wave equations.
- 2.
Any past directed, in-extendable causal curve of the development intersects Σ 0.
- 3.
A proper definition of global solutions in GR requires a special discussion concerning the proper time of causal geodesics.
- 4.
Note however that the boundary of this extended space-time is not smooth, generically.
- 5.
Note that our normalization for Ω differ from that of [7].
- 6.
We could call such a region locally trapped, or a pre-scar.
- 7.
We use the short hand notation \({{\left\| \beta ,\rho ,\sigma ,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{\beta}\right\|}_{\mathcal{L}_{(sc)}^{2}(H_{u}^{(0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u})})}}={{\left\|\beta\right\|}_{\mathcal{L}_{(sc)}^{2}(H_{u}^{(0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u})})}}+{{\left\|\rho\right\|}_{\mathcal{L}_{(sc)}^{2}(H_{u}^{(0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u})})}}+{{\left\|\sigma\right\|}_{\mathcal{L}_{(sc)}^{2}(H_{u}^{(0,\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle-}$}}{u})})}}+\cdot\cdot\cdot.\).
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Klainerman, S. (2012). On the Formation of Trapped Surfaces. In: Holden, H., Karlsen, K. (eds) Nonlinear Partial Differential Equations. Abel Symposia, vol 7. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-25361-4_10
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