TY - GEN

T1 - On the formation of trapped surfaces

AU - Klainerman, Sergiu

PY - 2012

Y1 - 2012

N2 - 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.

AB - 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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U2 - 10.1007/978-3-642-25361-4_10

DO - 10.1007/978-3-642-25361-4_10

M3 - Conference contribution

AN - SCOPUS:84883608815

SN - 9783642253607

T3 - Nonlinear Partial Differential Equations: The Abel Symposium 2010

SP - 181

EP - 206

BT - Nonlinear Partial Differential Equations

T2 - Abel Symposium 2010: Nonlinear Partial Differential Equations

Y2 - 28 September 2010 through 2 October 2010

ER -