31 research outputs found
On emerging scarred surfaces for the Einstein vacuum equations
This is a follow up on our previous work in which we have presented a
modified, simpler version of the remarkable recent result of Christodoulou on
the formation of trapped surfaces. In this paper we prove two related results.
First we extend the semi-global existence result, which was at the heart of our
previous work, to an optimal range. We then use it to establish the formation
of surfaces with multiple pre-scarred angular components
On the relation between mathematical and numerical relativity
The large scale binary black hole effort in numerical relativity has led to
an increasing distinction between numerical and mathematical relativity. This
note discusses this situation and gives some examples of succesful interactions
between numerical and mathematical methods is general relativity.Comment: 12 page
Second-order corrections to mean field evolution for weakly interacting Bosons. I
Inspired by the works of Rodnianski and Schlein and Wu, we derive a new
nonlinear Schr\"odinger equation that describes a second-order correction to
the usual tensor product (mean-field) approximation for the Hamiltonian
evolution of a many-particle system in Bose-Einstein condensation. We show that
our new equation, if it has solutions with appropriate smoothness and decay
properties, implies a new Fock space estimate. We also show that for an
interaction potential , where is
sufficiently small and , our program can be easily
implemented locally in time. We leave global in time issues, more singular
potentials and sophisticated estimates for a subsequent part (part II) of this
paper