Advances in Theoretical and Mathematical Physics

Volume 21 (2017)

Number 1

Trapped surfaces in vacuum arising dynamically from mild incoming radiation

Pages: 1 – 120



Xinliang An (Department of Mathematics, University of Toronto, Ontario, Canada)

Jonathan Luk (Department of Mathematics, Stanford University, Palo Alto, California, U.S.A.)


In this paper, we study the “minimal requirement” on the incoming radiation that guarantees a trapped surface to form in vacuum. First, we extend the region of existence in Christodoulou’s theorem on the formation of trapped surfaces and consequently show that the lower bound required to form a trapped surface can be relaxed. Second, we demonstrate that trapped surfaces form dynamically from a class of initial data which are large merely in a scaling-critical norm. This result is motivated in part by the scaling in Christodoulou’s formation of trapped surfaces theorem for the Einstein-scalar field system in spherical symmetry.

Published 6 April 2017