Boosted maximal hypersurfaces in Schwarzschild

Abstract: In Minkowski spacetime, there is a one-to-one correspondence between inertial observers, Lorentz boosts and entire spacelike maximal (of vanishing mean curvature) hypersurfaces, as a consequence of the renowned Cheng–Yau Bernstein-type theorem. The question of whether a similar correspondence exists in non-flat spacetimes was raised by Bartnik in 1988. In this talk, we take a first step towards addressing this problem in the Schwarzschild setting. We prove the existence of an entire, spacelike hypersurface of vanishing mean curvature that is asymptotic to suitably chosen coordinate-dependent boosts in the two asymptotically flat ends of the maximally extended Schwarzschild spacetime, for any choice of boost parameter. Analytically, this consists in finding solutions to the maximal surface equation, a geometric quasilinear elliptic PDE, with prescribed asymptotics. As a corollary, we generate a foliation of maximal hypersurfaces in exterior Schwarzschild, which corresponds to a geometric choice of time function associated to the boost parameter.

general relativity and quantum cosmologymathematical physicsanalysis of PDEsdifferential geometry

Audience: researchers in the topic

JoMaReC - Joint Online Mathematical Relativity Colloquium

Series comments: This monthly online colloquium is meant to be accessible to and informative for mathematicians and mathematical physicists with a background in General Relativity, widely interpreted to include Lorentzian Geometry, and Geometric Analysis of various Partial Differential Equations related to General Relativity.

It is aimed to present motivation and applications of particular results and/or introduce specific subfields, while refraining from too much technicalities.