#### Vol. 8, No. 7, 2015

 Recent Issues
 The Journal About the Journal Editorial Board Subscriptions Editors’ Interests Scientific Advantages Submission Guidelines Submission Form Editorial Login Ethics Statement ISSN: 1948-206X (e-only) ISSN: 2157-5045 (print) Author Index To Appear Other MSP Journals
Quantitative decay rates for dispersive solutions to the Einstein-scalar field system in spherical symmetry

### Jonathan Luk and Sung-Jin Oh

Vol. 8 (2015), No. 7, 1603–1674
##### Abstract

We study the future causally geodesically complete solutions of the spherically symmetric Einstein-scalar field system. Under the a priori assumption that the scalar field $\varphi$ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that $\varphi$ and its derivatives decay polynomially. Moreover, we show that the decay rates are sharp. In particular, we obtain sharp quantitative decay for the class of global solutions with small BV norms constructed by Christodoulou. As a consequence of our results, for every future causally geodesically complete solution with sufficiently regular initial data, we show the dichotomy that either the sharp power law tail holds or that the spacetime blows up at infinity in the sense that some scale invariant spacetime norms blow up.

##### Keywords
Einstein-scalar field system, spherical symmetry, quantitative decay rate
Primary: 35Q76