The Case Against Smooth Null Infinity I: Heuristics and Counter-Examples

2021/05/17 by Kehrberger, Lionor M. A. · 5 citations
#Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #General Relativity and Quantum Cosmology (gr-qc) #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph)

paper · doi:10.48550/arxiv.2105.08079

Abstract

This paper initiates a series of works dedicated to the rigorous study of the precise structure of gravitational radiation near infinity. We begin with a brief review of an argument due to Christodoulou [1] stating that Penrose's proposal of smooth conformal compactification of spacetime (or smooth null infinity) fails to accurately capture the structure of gravitational radiation emitted by N infalling masses coming from past timelike infinity i^-. Modelling gravitational radiation by scalar radiation, we then take a first step towards a rigorous, fully general relativistic understanding of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein-Scalar field equations. Our constructions are motivated by Christodoulou's argument: They arise dynamically from polynomially decaying boundary data, rϕ∼ t^-1 as t→-∞, on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, r∂_vϕ=0, on past null infinity I^-. We show that if the initial Hawking mass at i^- is non-zero, then, in accordance with the non-smoothness of \mathcal I⁺, ∂_v(rϕ) satisfies the following asymptotic expansion near I⁺ for some constant C≠ 0: ∂_v(rϕ)=Cr^-3log r+\mathcal O(r^-3). We also show that the same logarithmic terms appear in the linear theory, i.e. when considering the spherically symmetric linear wave equation on a fixed Schwarzschild background. As a corollary, we can apply our results to the scattering problem on Schwarzschild: Putting smooth, compactly supported scattering data for the wave equation on \mathcal I^- and on \mathcal H^-, we find that the asymptotic expansion of ∂_v(rϕ) near \mathcal I⁺ generically contains logarithmic terms at second order, i.e. at order r^-4log r.

Cited by

• Twisting asymptotically-flat spacetimes
• Asymptotic Simplicity and Scattering in General Relativity from Quantum Field Theory
• Controlled regularity at future null infinity from past asymptotic initial data: the wave equation
• From spatial to null infinity: Connecting initial data to peeling
• Newman-Penrose-like exact and approximate conservation laws: a covariant and conformal formulation

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