Average trapped surfaces on an arbitrary initial data set.

Abstract

We study the behavior of the expansion [Theta] of an outgoing null geodesic congruence on an initial surface [Sigma]. If the intrinsic geometry (and/or topology) allows a global or local slicing of [Sigma] by two-dimensional closed surfaces, then the rate of change of [Theta] along the normal and tangential direction of each slice is derived. We find that the rate of change of [Theta] along the normal direction is governed by the initial data and also by the Gaussian, mean, and extrinsic curvature of each slice. Based on this differential equation, a formula is established that defines the total expansion along any two-dimensional slice. As it turns out, the total expansion depends upon the amount of rest mass enclosed, the amount of matter current flowing across the surface, as well as upon other geometrical characteristics of the slice. Using this formula a necessary and sufficient condition is established that guarantees whether or not a particular two-slice [ital S] is an average trapped surface. In the special case of time-symmetric initial data the criterion takes a simple and tractable form. Roughly, an average trapped surface forms when the interior rest mass [ital M] is greater than the value of an integralmore » involving only geometrical characteristics of the slice. In the special case of the spherical slices or particular slicing by axisymmetric ellipsoids, we are able to find the upper and lower bounds of the above-mentioned integral by quantities related to the size of [ital S]. Thus we are also able to establish a condition upon the amount of rest mass [ital M] needed to make [ital S] an average trapped surface.« less