The Case Against Smooth Null Infinity III: Early-Time Asymptotics for Higher $$\ell $$-Modes of Linear Waves on a Schwarzschild Background

Abstract

In this paper, we derive the early-time asymptotics for fixed-frequency solutions $$\phi _\ell $$ ϕ ℓ to the wave equation $$\Box _g \phi _\ell =0$$ □ g ϕ ℓ = 0 on a fixed Schwarzschild background ($$M>0$$ M > 0 ) arising from the no incoming radiation condition on $${\mathscr {I}}^-$$ I - and polynomially decaying data, $$r\phi _\ell \sim t^{-1}$$ r ϕ ℓ ∼ t - 1 as $$t\rightarrow -\infty $$ t → - ∞ , on either a timelike boundary of constant area radius $$r>2M$$ r > 2 M (I) or an ingoing null hypersurface (II). In case (I), we show that the asymptotic expansion of $$\partial _v(r\phi _\ell )$$ ∂ v ( r ϕ ℓ ) along outgoing null hypersurfaces near spacelike infinity $$i^0$$ i 0 contains logarithmic terms at order $$r^{-3-\ell }\log r$$ r - 3 - ℓ log r . In contrast, in case (II), we obtain that the asymptotic expansion of $$\partial _v(r\phi _\ell )$$ ∂ v ( r ϕ ℓ ) near spacelike infinity $$i^0$$ i 0 contains logarithmic terms already at order $$r^{-3}\log r$$ r - 3 log r (unless $$\ell =1$$ ℓ = 1 ). These results suggest an alternative approach to the study of late-time asymptotics near future timelike infinity $$i^+$$ i + that does not assume conformally smooth or compactly supported Cauchy data: In case (I), our results indicate a logarithmically modified Price’s law for each $$\ell $$ ℓ -mode. On the other hand, the data of case (II) lead to much stronger deviations from Price’s law. In particular, we conjecture that compactly supported scattering data on $${\mathscr {H}}^-$$ H - and $${\mathscr {I}}^-$$ I - lead to solutions that exhibit the same late-time asymptotics on $${\mathscr {I}}^+$$ I + for each $$\ell $$ ℓ : $$r\phi _\ell |_{{\mathscr {I}}^+}\sim u^{-2}$$ r ϕ ℓ | I + ∼ u - 2 as $$u\rightarrow \infty $$ u → ∞ .

Sharp Decay for Teukolsky Equation in Kerr Spacetimes

“…It should be noted that it is expected that generic physically interesting Cauchy data do not satisfy peeling properties. See, for instance, the recent works of Kehrberger [53][54][55] in which the author considered the precise structure of gravitational radiation near infinity for the scalar field on Schwarzschild.…”

“…The globally sharp v −1 τ −2 pointwise decay is first proven by Angelopoulos-Aretakis-Gajic [9,10] and the precise late-time asymptotic profile is calculated therein; Hintz [46] computed the v −1 τ −2 leading order term on both Schwarzschild and subextreme Kerr spacetimes and further obtained v −1 τ −2 −2 sharp asymptotics for ≥ modes in a compact region on Schwarzschild; Luk-Oh [64] derived sharp decay for the scalar field on a Reissner-Nordström background and used it to obtain linear instability of the Reissner-Nordström Cauchy horizon (see also their works [65,66] on a generalization to a nonlinear setting); Angelopoulos-Aretakis-Gajic based on their own earlier works and re-derived in [12] v −1 τ −2 −2 late time asymptotics for ≥ 0 modes in a finite radius region on Schwarzschild, and they further computed in [11] the asymptotic profiles of the = 0, = 1, and ≥ 2 modes in a subextreme Kerr spacetime; we [71] independently computed the global v −1 τ −2 −2 late time asymptotics for ≥ modes in a Schwarzschild spacetime. Additionally, Kehrberger [53][54][55] considered the precise structure of gravitational radiation near infinity for the scalar field on Schwarzschild.…”

Charge and antipodal matching across spatial infinity

“…The debate around the peeling vs non-peeling nature of null infinity is summarised in [51]. Here we point out that the assumption of smooth null infinity excludes the phenomenology of infalling matter from past timelike infinity [52][53][54][55][56]. We will also discuss logarithmic terms in u in section 7.…”

The case against smooth null infinity IV: Linearized gravity around Schwarzschild—an overview

“…Since asymptotic simplicity defines a class of global spacetimes, it has played a crucial role in understanding various global aspects of spacetimes, for instance related to black holes (which in most textbooks are defined using asymptotic simplicity). 2 Again, the conceptual approach of asymptotic simplicity is to understand aspects of spacetime by certain considerations concerning their asymptotic behaviour, which in turn are motivated by soft/formal considerations, that is to say, considerations that do not directly take into account the physical structure of the system under consideration.…”