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

Abstract

<jats:title>Abstract</jats:title><jats:p>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 (in: The Ninth Marcel Grossmann Meeting, World Scientific Publishing Company, Singapore, 2002) 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 <jats:italic>N</jats:italic> infalling masses coming from past timelike infinity <jats:inline-formula><jats:alternatives><jats:tex-math>$$i^-$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>. Modelling gravitational radiation by scalar radiation, we then take a first step towards a <jats:italic>dynamical understanding</jats:italic> of the non-smoothness of null infinity by constructing solutions to the spherically symmetric Einstein–Scalar field equations that arise from polynomially decaying boundary data, <jats:inline-formula><jats:alternatives><jats:tex-math>$$r\phi \sim t^{-1}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mo>∼</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> as <jats:inline-formula><jats:alternatives><jats:tex-math>$$t\rightarrow -\infty $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>t</mml:mi> <mml:mo>→</mml:mo> <mml:mo>-</mml:mo> <mml:mi>∞</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, on a timelike hypersurface (to be thought of as the surface of a star) and the no incoming radiation condition, <jats:inline-formula><jats:alternatives><jats:tex-math>$$r\partial _v\phi =0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>r</mml:mi> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mi>ϕ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>, on past null infinity. We show that if the initial Hawking mass at <jats:inline-formula><jats:alternatives><jats:tex-math>$$i^-$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>i</mml:mi> <mml:mo>-</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> is nonzero, then, in accordance with the non-smoothness of <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {I}}^+$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, the asymptotic expansion of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\partial _v(r\phi )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> near <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {I}}^+$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> reads <jats:inline-formula><jats:alternatives><jats:tex-math>$$\partial _v(r\phi )=Cr^{-3}\log r+{\mathcal {O}}(r^{-3})$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>C</mml:mi> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>log</mml:mo> <mml:mi>r</mml:mi> <mml:mo>+</mml:mo> <mml:mi>O</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> for some non-vanishing constant <jats:italic>C</jats:italic>. In fact, the same logarithmic terms appear already 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 compactly supported scattering data for the linear (or coupled) wave equation on <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {I}}^-$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:mo>-</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> and on <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {H}}^-$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>H</mml:mi> </mml:mrow> <mml:mo>-</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula>, we find that the asymptotic expansion of <jats:inline-formula><jats:alternatives><jats:tex-math>$$\partial _v(r\phi )$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>v</mml:mi> </mml:msub> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>r</mml:mi> <mml:mi>ϕ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula> near <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {I}}^+$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mrow> <mml:mi>I</mml:mi> </mml:mrow> <mml:mo>+</mml:mo> </mml:msup> </mml:math></jats:alternatives></jats:inline-formula> generically contains logarithmic terms at second order, i.e. at order <jats:inline-formula><jats:alternatives><jats:tex-math>$$r^{-4}\log r$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mi>r</mml:mi> <mml:mrow> <mml:mo>-</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:msup> <mml:mo>log</mml:mo> <mml:mi>r</mml:mi> </mml:mrow> </mml:math></jats:alternatives></jats:inline-formula>.</jats:p>