Scaling up the extrinsic curvature in asymptotically flat gravitational initial data: Generating trapped surfaces

TL;DR

This paper explores how scaling the extrinsic curvature in asymptotically flat initial data affects the geometry, revealing that increasing extrinsic curvature can lead to trapped surfaces despite decreasing physical extrinsic curvature.

Contribution

It demonstrates that multiplying the TT tensor by a large constant causes the conformal factor to increase, leading to trapped surfaces, which is a non-intuitive result in initial data construction.

Findings

Conformal factor increases monotonically with scaling.

Physical extrinsic curvature decreases as TT tensor is scaled.

Trapped surfaces appear in the initial data when extrinsic curvature is scaled up.

Abstract

The existence of the initial value constraints means that specifying initial data for the Einstein equations is non-trivial. The standard method of constructing initial data in the asymptotically flat case is to choose an asymptotically flat 3-metric and a transverse-tracefree (TT) tensor on it. One can find a conformal transformation that maps these data into solutions of the constraints. In particular, the TT tensor becomes the extrinsic curvature of the 3-slice. We wish to understand how the physical solution changes as the free data is changed. In this paper we investigate an especially simple change: we multiply the TT tensor by a large constant. One might assume that this corresponds to pumping up the extrinsic curvature in the physical initial data. Unexpectedly, we show that, while the conformal factor monotonically increases, the physical extrinsic curvature decreases. The increase in the conformal factor however means that the physical volume increases in such a way that the ADM mass become unboundedly large. In turn, the blow-up of the mass combined with the control we have on the extrinsic curvature allows us to show that trapped surfaces, i.e., surfaces that are simultaneously future and past trapped, appear in the physical initial data.