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

Abstract

AbstractIn 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 → ∞

.