Vol. 8, No. 7, 2015

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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

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 ϕ scatters locally in the scale-invariant bounded-variation (BV) norm, we prove that ϕ 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.

Einstein-scalar field system, spherical symmetry, quantitative decay rate
Mathematical Subject Classification 2010
Primary: 35Q76
Received: 4 May 2014
Revised: 16 April 2015
Accepted: 24 June 2015
Published: 18 September 2015
Jonathan Luk
Department of Pure Mathematics and Mathematical Statistics
University of Cambridge
Centre for Mathematical Sciences
Wilberforce Road
United Kingdom
Sung-Jin Oh
Department of Mathematics
UC Berkeley
970 Evans Hall
Berkeley, CA 94720
United States