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## Seminar on

**Mathematical General Relativity**

*Organizers:** *

* *S. Klainerman *(Princeton)*

#### P.G. LeFloch *(Paris)*

*A. Zeghib (Lyon)*

**Fondations des Sciences Mathématiques de Paris**

**ANR Project**

#### “Mathematical General Relativity. Analysis and geometry of spacetimes with low regularity”

**Thursday January 17, 2013**

### Laboratoire J-L Lions

### Université Pierre et Marie Curie, Paris

**Lecture room (see below)**

**Lecture room (see below)**

** . **

** 11h (Room **15-25- 104**) ** Sergiu Klainerman (Princeton) On the formation of trapped surfaces

**Abstract. **I will talk about a new result obtained in collaboration with J. Luk and I. Rodnianski in which we relax significantly Christodoulou’s main condition for the formation of trapped surfaces in vacuum.

** 14h (Room **15-25-326*)*** **Chung-Tse Arick Shao (Toronto) Null cones to infinity, curvature flux, and Bondi mass

**Abstract. **In general relativity, the Bondi mass in an asymptotically flat spacetime represents, roughly, the mass remaining in the system after some has radiated away. In this talk, we make sense of and control the Bondi mass for a single null cone in an Einstein-vacuum spacetime under minimal assumptions. In terms of regularity, we assume only small weighted curvature flux along the null cone and small data on an initial sphere of the cone. Furthermore, we make no global assumptions on the spacetime, as all our conditions deal only with the single null cone under consideration. This work is joint with S. Alexakis.

**15h30 **** (Room **15-25-326*)*** **Gustav Holzegel (Princeton) Existence of dynamical vacuum black holes

**(Room**

**Abstract. **This is joint work with Mihalis Dafermos and Igor Rodnianski. We prove the existence of a large class of non-stationary vacuum black holes whose exterior geometry asymptotes in time to a fixed Schwarzschild or Kerr metric. The spacetimes are constructed by solving a backwards scattering problem for the vacuum Einstein equations with characteristic data prescribed on the horizon and at null infinity. The data admits the full functional degrees of freedom to specify data for the Einstein equations. An essential feature of the construction is that the solutions converge to stationarity exponentially fast with their decay rate intimately related to the surface gravity of the horizon and hence to the strength of the celebrated redshift effect which, in our backwards construction, is seen as a blueshift.