ASYMPTOTICS OF SOLUTIONS OF THE WAVE EQUATION ON DE SITTER-SCHWARZSCHILD SPACE

RICHARD MELROSE, ANTÔNIO SÁ BARRETO, AND ANDRÁS VASY

Abstract. Solutions to the wave equation on de Sitter-Schwarzschild space with smooth initial data on a Cauchy surface are shown to decay exponentially to a constant at temporal infinity, with corresponding uniform decay on the appropriately compactified space.

1. Introduction

In this paper we describe the asymptotics of solutions to the wave equation on de Sitter-Schwarzschild space. The static model for the latter is M = ℝ_{t} × X, X = (r_{bh}, r_{dS})_{r} × 𝕊_{ω}^{2} with the Lorentzian metric

(1.1)  g = μ dt^{2} − μ^{−1} dr^{2} − r^{2} dω^{2},

where

(1.2)  μ = 1 − \frac{2m}{r} − \frac{Λr^{2}}{3}

with Λ and m suitable positive constants, 0 < 9m^{2}Λ < 1, r_{bh}, r_{dS} the two positive roots of μ and dω^{2} the standard metric on 𝕊^{2}. We also consider the compactification of X to

X̄ = [r_{bh}, r_{dS}]_{r} × 𝕊_{ω}^{2}.

Then μ is a defining function for ∂X̄ since it vanishes simply at r_{bh}, r_{dS}, i.e. 2β = \frac{dμ}{dr} ≠ 0 at r = r_{bh}, r_{dS}. Moreover, in what follows we will sometimes consider

(1.3)  α = μ^{\frac{1}{2}}

as a boundary defining function for a different compactification of X. This amounts to changing the C^{∞} structure of X̄ by adjoining α as a smooth function. We denote the new manifold by X_{\frac{1}{2}} .

The d’Alembertian with respect to (1.1) is

□ = α^{−2}(D_{t}^{2} − α^{2}r^{−2}D_{r}(r^{2}α^{2}D_{r}) − α^{2}r^{−2}Δ_{ω}), (1.4)

where Δ_{ω} is the Laplacian on 𝕊^{2}. We shall consider solutions to □u = 0 on M.

Regarding space-time as a product, up to the conformal factor α^{2}, is in fact misleading in several ways – in particular, solutions to the wave equation do not have simple asymptotic behavior on this space. Starting from the stationary description of the metric, it is natural to first compactify the time line exponentially to an interval [0, 1]_{T} . This can be done using a diffeomorphism T : ℝ → (0, 1) with derivative T′ < 0. Set

Date: 12 November, 2008.

2000 Mathematics Subject Classification. 35L05, 35P25, 83C57, 83C30.

The authors gratefully acknowledge financial support for this project, the first from the National Science Foundation under grant DMS-0408993, the second under grant DMS-0500788, and the third under grant DMS-0201092 and DMS-0801226; they are also grateful for the environment at the Mathematical Sciences Research Institute, Berkeley, where this paper was completed.

(1.5)  T_{+} = T_{λ,+} = e^{−2λt} in t > C,

with λ to be determined and let

(1.6)  T = T_{+} in t > C.

Similarly set

(1.7)  T_{−} = T_{λ,−} = e^{2λt}, T = 1 − T_{−} in t < −C.

Near infinity T depends on the free parameter λ. The boundary hypersurface T_{+} = 0 (i.e. T = 0) in

(1.8)  [0, 1]_{T} × X̄

is called here the future temporal face, T_{−} = 0 the past temporal face, while r = r_{bh} and r = r_{dS} are the black hole, resp. de Sitter, infinity, or together spatial infinity.

In fact, it turns out that we need to use different values of λ at the two ends, λ_{bh} and λ_{dS}. This is discussed in more detail in the next section. There are product decompositions near these boundaries

[0, 1]_{T} × [r_{bh}, r_{bh} + δ) × 𝕊^{2}, [0, 1]_{T} × (r_{dS} − δ, r_{dS}] × 𝕊^{2}.

If δ is so large that these overlap, the transition function is not smooth but rather is given by taking positive powers of the defining function of the future temporal face, so the resulting space should really be thought of having a polyhomogeneous conormal (but not smooth) structure in the sense of differentiability up to the temporal faces. In particular, there is no globally preferred boundary defining function for the temporal face, rather such a function is only determined up to positive powers and multiplication by positive factors. Thus, there is no fully natural ‘unit’ of decay but we consider powers of e^{−t}, resp. e^{t}, in a neighborhood of the future and past temporal faces, respectively.

It turns out that there are two resolutions of this compactified space which play a useful role in describing asymptotics. The first arises by blowing up the corners

{0} × {r_{bh}} × 𝕊^{2}, {0} × {r_{dS}} × 𝕊^{2}, {1} × {r_{bh}} × 𝕊^{2}, {1} × {r_{dS}} × 𝕊^{2},

where the blow-up is understood to be the standard spherical blow-up when locally the future temporal face is defined by T_{λ_{dS},+}, resp. T_{λ_{bh},+} at the de Sitter and black hole ends. The resulting space is denoted M̄. The lift of the temporal and spatial faces retain their names, while the new front faces are called the scattering faces. This is closely related to to the Penrose compactification, where however the temporal faces are compressed.

Thus, a neighborhood of the lift of {0} × {r_{bh}} × 𝕊^{2} is diffeomorphic to

(1.9)  [0, ϵ)_{ρ} × [r_{bh}, r_{bh} + δ) × 𝕊_{ω}^{2}, ρ = ρ_{bh,+} = T_{λ_{bh},+}/μ.

Similarly, a neighborhood of {0} × {r_{dS}} × 𝕊^{2} is diffeomorphic to

(1.10)  [0, ϵ)_{ρ} × (r_{dS} − δ, r_{dS}] × 𝕊_{ω}^{2}, ρ = ρ_{dS,+} = T_{λ_{dS},+}/μ.

If δ > 0 is large enough, these cover a neighborhood of the future temporal face tf_{+}, given by the lift of T = 0. Thus a neighborhood of the interior of tf_{+}, is polyhomogeneous-diffeomorphic to an open subset of

[0, ϵ)_{x} × (r_{bh}, r_{dS}) × 𝕊_{ω}^{2},
x = ρ_{bh,+}^{1/(2λ_{bh,+})} for r near r_{bh}, x = ρ_{dS,+}^{1/(2λ_{dS,+})} for r near r_{dS},

(1.11)

where we let the preferred defining function (up to taking positive multiples) of tf_{+} be x = e^{−t} in the interior of tf_{+}, hence x = ρ_{bh,+}^{1/(2λ_{bh,+})} at the black hole boundary of tf_{+}. This means, in particular, that a neighborhood of tf_{+} is polyhomogeneous diffeomorphic to

(1.12)  [0, ϵ)_{x} × [r_{bh}, r_{dS}] × 𝕊_{ω}^{2}.

If μ is replaced by α as the defining function of the boundary of X, i.e. X̄ and T_{+} are replaced by X̄_{1/2} and T_{+}^{1/2} (and analogously in the past) the resulting space is denoted M̄_{1/2}. Thus M̄_{1/2} is the square-root blow up of M̄, where the square root of the defining function of every boundary hypersurface has been appended to the smooth structure. Here tf_{+} is naturally diffeomorphic to X̄ in M̄, and to X̄_{1/2} in M̄_{1/2}. Both M̄ and M̄_{1/2} have polyhomogeneous conormal structures at tf_{+} and tf_{−}; we let the preferred defining function (up to taking positive multiples) of tf_{+} be x = e^{−t} in the interior of tf_{+}, hence x = ρ^{1/(2λ)} at ∂tf_{+}.

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Figure 1. On the left, the space-time product compactification of de Sitter-Schwarzschild space is shown (ignoring the product with 𝕊^{2}), with the time and space coordinate lines indicated by thin lines. On the right, M̄ is shown, with the time and space coordinates indicated by thin lines. These are no longer valid coordinates on M̄. Valid coordinates near the top left corner are ρ and μ.

Solutions to the wave equation, when lifted to this space have simpler asymptotics than on the product compactification, (1.8). The first indication of this is that g extends to be C^{∞} and non-degenerate, up to the scattering faces, μ = 0, away from spatial infinity and uniformly up to the temporal face; the scattering faces are characteristic with respect to the metric. We can thus extend M̄ across μ = 0 to a manifold M̃, by allowing μ to take negative values; then the scattering face becomes an interior characteristic hypersurface.

A further indication of the utility of this space can be seen from our main result which is stated in terms of

(1.13)  A_{tf_{+}}^{m}(M̄).

This consists of those functions which are C^{∞} on M̄ away from tf_{+}, and are conormal at tf_{+}, including smoothness up to the boundary of tf_{+}. Such spaces are welldefined, even though the smooth structure on M̄ is not; the conormal structure suffices. Thus, the elements of (1.13), are fixed by the condition that for any k and smooth vector fields V_{1}, …, V_{k} on M̄ which are tangent to tf_{+},

V_{1} … V_{k}v ∈ x^{m}L_{b,tf_{+}}^{2}(M̄),

where L_{b,tf_{+}}^{2}(M̄) is the L^{2}-space with respect a density ν_{b} such that xν_{b} is smooth and strictly positive on M̄. Such a density is well-defined up to a strictly positive polyhomogeneous multiple even under the operation of replacing x by a positive power, although the weight x^{m} is not. Thus, for all m ∈ ℝ,

x^{m+ϵ}C^{∞}(M̄) ⊂ A_{tf_{+}}^{m}(M̄) ⊂ x^{m}L^{∞}(M̄), ϵ > 0.

The main result on wave propagation is:

Theorem 1.1. Suppose u ∈ C^{∞}(M̄) satisfies □u = 0 for x ∈ (0, 1), then there exists a constant c and ϵ > 0 such that

u − c ∈ A_{tf_{+}}^{ϵ}(M̄) = x^{ϵ}A_{tf_{+}}^{0}(M̄).

Thus, u has an asymptotic limit, which happens to be a constant, at tf_{+}, uniformly on X̄.

While we have concerned ourselves with the behavior of the metric at the corner, in regions where ρ < C (i.e. near temporal infinity), it is worthwhile considering what happens where ρ > C, i.e. at spatial infinity. As we shall see, spatial infinity can be blown down, i.e. there is a manifold \overline{M} and a C^{∞} map β, β : M̄ → \overline{M} such that β is a diffeomorphism away from spatial infinity, and such that g lifts to a C^{∞} Lorentz b-metric on \overline{M}, with tangent (i.e. b-) behavior at the temporal face, smooth at the other faces, with respect to which the non-temporal faces are characteristic. One valid coordinate system in a neighborhood of the image of a neighborhood of the black hole end of spatial infinity, disjoint from temporal infinity, is given by exponentiated versions of Eddington-Finkelstein coordinates. In our notation, this corresponds to

s_{bh,+} = α/T_{λ_{bh},+}^{1/2} = ρ_{bh,+}^{−1/2}, s_{bh,−} = α/T_{λ_{bh},−}^{1/2} = αT_{λ_{bh},+}^{1/2} = μρ_{bh,+}^{1/2}, ω,

where as usual ω denotes coordinates on 𝕊^{2}. Here

F_{bh,+} = {s_{bh,−} = 0}

is the characteristic surface given by μ = 0 in T > 0 (i.e. the front face of the blow up of the corner), and

F_{bh,−} = {s_{bh,+} = 0}

is its negative time analogue. The change of coordinates (ρ_{bh,+}, μ) ↦ (s_{bh,+}, s_{bh,−}) is a diffeomorphism from (0, ∞) × (0, δ) onto its image, i.e. these coordinates are indeed compatible. As we show in the next section, the metric is C^{∞} and nondegenerate on \overline{M}, and the boundary faces s_{bh,+} = 0 and s_{bh,−} = 0 are characteristic. We can again extend \overline{M} to M̃, which has only two boundary faces (the two temporal ones) by allowing s_{bh,±}, and analogously s_{dS,±}, to take on negative values. Thus, \overline{M} has six boundary faces,

tf_{+}, tf_{−}, F_{bh,±}, F_{dS,±},

called the future and past temporal faces, and the future (+) and past (−) black hole and de Sitter scattering faces.

[figure: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; ^{.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; ^{.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; ^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . ^{...............................................................................}; . ^{......................................................................................................................}; . ^{...............................................................................}; M̄; ^{tf^{+}}; ^{tf^{−}}; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; t = 0; ρ; μ; ^{.}  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}; ^{.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}; ^{^{.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; ^{^{.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; M̃; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; ^{.  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; ^{. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . ^{.............................................................................................................................................................}; . ^{.............................................................................................................................................................}; \overline{M}; F_{dS,+}; F_{dS,−}; F_{bh,+}; F_{bh,−}; ^{tf^{+}}; ^{tf^{−}}; s_{bh,+}; s_{bh,−}; ρ; μ; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; t = 0; ^{.}  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}; ^{.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}; ^{^{.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; ^{^{.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; ^{^{.}  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; . ^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}; ^{^{.} . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .}; . ^{^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}^{.}}; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .; M̃]

Figure 2. On the left, M̄ is shown, while on the right its blowdown \overline{M}. The time and space coordinate lines corresponding to the product decomposition are indicated by thin lines in the interior. The temporal boundary hypersurfaces of M̄ are continued by thin lines, as are the characteristic surfaces F_{bh,±} and F_{dS,±}, to show that the Lorentz metric extends smoothly across F_{bh,±} and F_{dS,±} (but not across the temporal face!). The extended spaces are denoted by M̃ and M̃. Valid coordinates near F_{bh,+} ∩ F_{bh,−} are (apart from the spherical coordinates) s_{bh,+} and s_{bh,−}, as shown.

The following propagation result follows directly from the properties of this blowdown.

Proposition 1.2. If u satisfies □u = 0 and has C^{∞} Cauchy data on a space-like Cauchy surface Σ ⊂ M̃ ∩ {t ≥ 0}, for example Σ = {t = 0} (i.e. s_{bh,+} = s_{bh,−}), then u ∈ C^{∞}(\overline{M}^{∘}).

Combining Proposition 1.2 and Theorem 1.1, leads to the main result of this paper:

Theorem 1.3. If u satisfies □u = 0 and has C^{∞} Cauchy data on a space-like Cauchy surface Σ ⊂ M̃ ∩ {t ≥ 0} then there exists a constant c and ϵ > 0 such that

(1.14)  u − c ∈ A_{tf_{+}}^{ϵ}(\overline{M}) = x^{ϵ}A_{tf_{+}}^{0}(\overline{M})

near the future temporal face, tf_{+}.

Remark 1.4. Our methods extend further, for example to Cauchy data at t = 0 which are conormal at ∂X̄, of growth s_{bh,+}^{m}, m > −2, at the boundary. Standard hyperbolic propagation gives the same behavior at F_{bh,+} and F_{dS,+} in ρ < C, and then the resolvent estimates for the ‘spatial Laplacian’ Δ_{X} (described below) apply to yield the same asymptotic term but with convergence in the appropriate conormal space, including conormality with respect to F_{bh,+} and F_{dS,+}.

Remark 1.5. Dafermos and Rodnianski [4] have proved, by rather different methods, a similar result with an arbitrary logarithmic decay rate, i.e. an analogue of

u − c ∈ (log ρ)^{−N}A_{tf_{+}}^{0}(\overline{M})

for every N. In terms of our approach, such logarithmic convergence follows from polynomial bounds on the resolvent of Δ_{X} at the real axis, rather than in a strip for the analytic continuation; such estimates are much easier to obtain, as is explained below.

As already indicated, by looking at the appropriate compactification, one only needs to study the asymptotics near tf_{+} in M̄ (or equivalently, \overline{M}). We do this by taking the Mellin transform of the wave equation and using high-energy resolvent estimates for a ‘Laplacian’ Δ_{X} on X̄. A conjugated version of this operator is asymptotically hyperbolic, hence fits into the framework of Mazzeo and the first author [7], which in particular shows the existence of an analytic continuation for the resolvent

R(σ) = (Δ_{X} − σ^{2})^{−1}, Im σ < 0.

Here we also need high-energy estimates for R(σ).

The operator Δ_{X} has been studied by the second author and Zworski in [9], where it is shown (using the spherical symmetry to reduce to a one-dimensional problem and applying complex scaling) that the resolvent admits an analytic continuation, from the ‘physical half plane’, with only one pole, at 0, in Im σ < ϵ, for ϵ sufficiently small. Bony and Hafner in [1] extend and refine this result to derive polynomial bounds on the cutoff resolvent, χR(σ)χ, χ ∈ C_{c}^{∞}(X), as |σ| → ∞ in the strip | Im σ| < ϵ. This implies that, for initial data in C_{c}^{∞}(X), the local energy, i.e. the energy in a fixed compact set in space, decays to the energy corresponding to the 0-resonance. In our terminology this amounts to studying the behavior of the solution near a compact subset of the interior of tf_{+}. Our extension of their result is both to allow more general initial data, not necessarily of compact support, and to study the asymptotics uniformly up to the boundary at temporal infinity. This requires resolvent estimates on slightly weighted L^{2}-spaces, which were obtained by the authors in [8] together with the use of the geometric compactification M̄ (or \overline{M}). For this to succeed, it is essential that the resolvent only be applied to ‘errors’ which intersect ∂\overline{M} in the interior of F_{bh,+} and F_{dS,+}. This turns out to be a major gain since the analytic continuation of the resolvent (even arbitrarily close to the real axis) cannot be applied directly to the initial data. Thus essential use is made of the fact that once the solution has been propagated to the scattering faces, the error terms have more decay.

It is then relatively clear, as remarked above, that if one only knew polynomial growth estimates for the limiting resolvent at the real axis (rather than in a strip), one could still obtain the same asymptotics, but with error that is only superlogarithmically decaying. This observation may be of use in other settings where such polynomial bounds are relatively easy to obtain from estimates for the cutoff resolvent, as in [1], or analogous semiclassical propagation estimates at the trapped set, by pasting with well-known high energy resolvent estimates localized near infinity. This has been studied particularly by Cardoso and Vodev [3], using the method of Bruneau and Petkov [2, Section 3].

This paper is structured as follows. In Section 2 both the compactifications and the underlying geometry are discussed in more detail. The ‘spatial Laplacian’ and relevant resolvent estimates are recalled in Section 3 and in Section 4 the main result is proved using the Mellin transform.

2. Geometry

In this section the various compactifications of de Sitter-Schwarzschild space are studied after an initial examination of the simpler case of de Sitter space.

2.1. De Sitter space. We start with the extreme case of de Sitter space, corresponding to m = 0 in (1.1) and (1.2), to see what the ‘correct’ compactification of M should be. However, rather than starting from the static model, consider this as a Lorentzian symmetric space. De Sitter space is given by the hyperboloid

z_{1}^{2} + … + z_{n}^{2} = z_{n+1}^{2} + 1 in ℝ^{n+1}

equipped with the pull-back of the Minkowski metric

dz_{n+1}^{2} − dz_{1}^{2} − … − dz_{n}^{2}.

Introducing polar coordinates (R, θ) in (z_{1}, …, z_{n}), so

R = \sqrt{z_{1}^{2} + … + z_{n}^{2}} = \sqrt{1 + z_{n+1}^{2}}, θ = R^{−1}(z_{1}, …, z_{n}) ∈ 𝕊^{n−1}, τ = z_{n+1},

the hyperboloid can be identified with ℝ_{τ} × 𝕊_{θ}^{n−1} with the Lorentzian metric

\frac{dτ^{2}}{τ^{2} + 1} − (τ^{2} + 1) dθ^{2},

where dθ^{2} is the standard Riemannian metric on the sphere. For τ > 1, set x = τ^{−1}, so the metric becomes

\frac{(1 + x^{2})^{−1} dx^{2} − (1 + x^{2}) dθ^{2}}{x^{2}} .

An analogous formula holds for τ < −1, so compactifying the real line to an interval [0, 1]_{T}, with T = x = τ^{−1} for x < \frac{1}{4} (i.e. τ > 4), say, and T = 1 − |τ|^{−1}, τ < −4, gives a compactification, M̂, of de Sitter space on which the metric is conformal to a non-degenerate Lorentz metric. There is natural generalization, to asymptotically de Sitter-like spaces M̂, which are diffeomorphic to compactifications [0, 1]_{T} × Y of ℝ_{τ} × Y, where Y is a compact manifold without boundary, and M̂ is equipped with a Lorentz metric on its interior which is conformal to a Lorentz metric smooth up to the boundary. These space-times are Lorentzian analogues of the muchstudied conformally compact (Riemannian) spaces. On this class of space-times the solutions of the Klein-Gordon equation were analyzed by the third author in [10], and were shown to have simple asymptotics analogous to those for eigenfunctions on conformally compact manifolds.

Theorem. ([10, Theorem 1.1.]) Set s_{±}(λ) = \frac{n−1}{2}±\sqrt{\frac{(n−1)^{2}}{4} − λ}. If s_{+}(λ)−s_{−}(λ) ∉ ℕ, any solution u of the Cauchy problem for □ − λ with C^{∞} initial data at τ = 0 is of the form

u = x^{s_{+}(λ)}v_{+} + x^{s_{−}(λ)}v_{−}, v_{±} ∈ C^{∞}(M̂).

If s_{+}(λ)−s_{−}(λ) is an integer, the same conclusion holds if v_{−} ∈ C^{∞}(M̂) is replaced by v_{−} = C^{∞}(M̂) + x^{s_{+}(λ)−s_{−}(λ)} log x C^{∞}(M̂).

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Figure 3. On the left, the compactification of de Sitter space with the backward light cone from q_{+} = (1, 0, 0, 0) and forward light cone from q_{−} = (−1, 0, 0, 0) are shown. Ω_{+}, resp. Ω_{−}, denotes the intersection of these light cones with t > 0, resp. t < 0. On the right, the blow up of de Sitter space at q_{+} is shown. The interior of the light cone inside the front face ff_{q_{+}} can be identified with the spatial part of the static model of de Sitter space. The spatial and temporal coordinate lines for the static model are also shown.

The simple structure of the de Sitter metric (and to some extent the asymptotically de Sitter-like metrics) can be hidden by blowing up certain submanifolds of M̂. In particular, the static model of de Sitter space arises by singling out a point on 𝕊_{θ}^{n−1}, e.g. q_{0} = (1, 0, …, 0) ∈ 𝕊^{n−1} ⊂ ℝ^{n}. Note that (θ_{2}, …, θ_{n}) ∈ ℝ^{n−1} are local coordinates on 𝕊^{n−1} near q_{0}. Now consider the intersection of the backward light cone from q_{0} considered as a point q_{+} at future infinity, i.e. where T = 0, and the forward light cone from q_{0} considered as a point q_{−} at past infinity, i.e. where T = 1. These intersect the equator T = 1/2 (here τ = 0) in the same set, and together form a ‘diamond’, Ω̂, with a conic singularity at q_{+} and q_{−}. Explicitly Ω̂ is given by z_{2}^{2} + … + z_{n}^{2} ≤ 1 inside the hyperboloid. If q_{+}, q_{−} are blown up, as well as the corner ∂Ω ∩ {τ = 0}, i.e. where the light cones intersect τ = 0 in Ω̂, we obtain a manifold M̄, which can be blown down to (i.e. is a blow up of) the space-time product [0, 1] × \overline{𝔹^{n−1}}, with 𝔹^{n−1} = \{Z ∈ ℝ^{n−1} : |Z| < 1\} on which the Lorentz metric has a time-translation invariant warped product form. Namely, first considering the interior Ω of Ω̂ we introduce the global (in Ω) standard static coordinates (t, Z), given by (with the expressions involving x valid near T = 0)

\begin{aligned}𝔹^{n−1} ∋ Z & = (z_{2}, …, z_{n}) = x^{−1}\sqrt{1 + x^{2}}(θ_{2}, …, θ_{n}), \\ sinh t & = \frac{z_{n+1}}{\sqrt{z_{1}^{2} − z_{n+1}^{2}}} = (x^{2} − (1 + x^{2})(θ_{2}^{2} + … + θ_{n}^{2}))^{−1/2}, \end{aligned}

It is convenient to rewrite these as well in terms of polar coordinates in Z (valid away from Z = 0):

r = \sqrt{z_{2}^{2} + … + z_{n}^{2}} = \sqrt{1 + z_{n+1}^{2} − z_{1}^{2}} = x^{−1}\sqrt{1 + x^{2}}\sqrt{θ_{2}^{2} + … + θ_{n}^{2}},
sinh t = \frac{z_{n+1}}{\sqrt{z_{1}^{2} − z_{n+1}^{2}}} = (x^{2} − (1 + x^{2})(θ_{2}^{2} + … + θ_{n}^{2}))^{−1/2} = x^{−1}(1 − r^{2})^{−1/2},

ω = r^{−1}(z_{2}, …, z_{n}) = (θ_{2}^{2} + … + θ_{n}^{2})^{−1/2}(θ_{2}, …, θ_{n}) ∈ 𝕊^{n−2}.

In these coordinates the metric becomes

(1 − r^{2}) dt^{2} − (1 − r^{2})^{−1}dr^{2} − r^{2} dω^{2},

which is a special case of the de Sitter-Schwarzschild metrics with m = 0 and Λ = 3.

Lemma 2.1. The lift of Ω̂ to the blow up [M̂; q_{+}, q_{−}] is a C^{∞} manifold with corners, Ω̄. Moreover, [Ω̄, ∂Ω ∩ {τ = 0}] is naturally diffeomorphic to the C^{∞} manifold with corners obtained from [[0, 1]×\overline{𝔹^{n−1}}; \{0\}×∂𝔹^{n−1}; \{1\}×∂𝔹^{n−1}] by adding the square root of the defining function of the lift of \{0\} × \overline{𝔹^{n−1}} and \{1\} × \overline{𝔹^{n−1}} to the C^{∞} structure.

Remark 2.2. This lemma states that from the stationary point of view, the ‘right’ compactification near the top face arises by blowing up the corner ∂\overline{ℝ} × ∂\overline{𝔹^{n−1}}, although the resulting space is actually more complicated than needed, since the original boundary hypersurfaces of the stationary space can be blown down to obtain a subset of M̂.

The fact that this approach gives ρ = x^{2} as the defining function of the temporal face, rather than x (hence necessitating adding the square root of ρ to the smooth structure), corresponds to the fact that, in the sense of Guillarmou [5], M̂ is actually even.

Proof. Coordinates on the blow up of M̂ near the lift of q_{+} are given by

x, Θ_{j} = \frac{\sqrt{1 + x^{2}}}{x} θ_{j}, j = 2, …, n;

these are all bounded in the region of validity of the coordinates, with the light cone given by ∑_{j=2}^{n} Θ_{j}^{2} = 1 (which is why the factor \sqrt{1 + x^{2}} was introduced), so the lift Ω̄ of Ω̂ is ∑_{j=2}^{n} Θ_{j}^{2} ≤ 1. As ∑_{j=2}^{n} Θ_{j}^{2} − 1 has a non-vanishing differential where it vanishes, this shows that Ω̄ is a C^{∞} manifold with corners. Near the light cone, ∂Ω̄ one can introduce polar coordinates in Θ, and use

x, r = (∑_{j=2}^{N} Θ_{j}^{2})^{1/2}, (ω_{2}, …, ω_{n}) = r^{−1}(Θ_{2}, …, Θ_{n}) ∈ 𝕊^{n−1}

as local coordinates. On the other hand, blowing up the corner of [0, 1]_{T} × \overline{𝔹^{n−1}}, where T = e^{−2t} for t > 4, say, which is equivalent to (sinh t)^{−2} there, gives coordinates near the lift of T = 0 :

r, ρ = (sinh t)^{−2}/(1 − r^{2}) = x^{2}, ω.

Thus, one almost has a diffeomorphism between the two coordinate charts, hence locally between the manifolds, except that in the blow up of [0, 1]_{T} × \overline{𝔹^{n−1}}, the defining function of the temporal face is the square of the defining function of the temporal face arising from the blow up of M̂. This is remedied by adding the square root of the defining function of the lift of \{0\} × \overline{𝔹^{n−1}} and \{1\} × \overline{𝔹^{n−1}}, i.e. of x^{2}, to the smooth structure, thus proving the lemma. □

It is worthwhile comparing the de Sitter space wave asymptotics, which is

u = x^{n−1}v_{+} + v_{−}, v_{+} ∈ C^{∞}(M̂), v_{−} ∈ C^{∞}(M̂) + x^{n−1}(log x)C^{∞}(M̂),

with our main result. The fact that the coefficients in the de Sitter expansion are C^{∞} on M̂ means that on M̄, the leading terms are constant. Thus, the de Sitter result implies (and is much stronger than) the statement that u decays to a constant on M̄ at an exponential rate.

2.2. Blow-up of the space-time product. We now turn to the compactification of de Sitter-Schwarzschild space. It turns out that while this cannot be embedded into a space as simple as M̂ the final setting is not much more complicated. In terms of de Sitter space, the difference is that while spatial infinity in [0, 1]_{T} × \overline{𝔹^{3}}, blown up at the corner, can always be blown down, the same is not true for temporal infinity.

In fact, the ‘black hole end’ r = r_{bh} resembles the ‘de Sitter end’ quite closely, which motivates the construction in the de Sitter-Schwarzschild setting. There is a simpler construction, depending on the choice of a constant λ > 0, which does not quite work because of some incompatibility between the two ends (which whilst very similar, are not quantitatively the same). With μ as in (1.2), compactify M by compactifying ℝ into [0, 1]_{T} as in (1.5)-(1.7), so

T_{+} = T_{λ,+} = e^{−2λt} in t > c,

and compactify (r_{bh}, r_{dS}) as [r_{bh}, r_{dS}], to obtain

[0, 1]_{T} × [r_{bh}, r_{dS}] × 𝕊^{2} = [0, 1]_{T} × X̄.

Then blowing up the corners

{0} × {r_{bh}} × 𝕊^{2}, {0} × {r_{dS}} × 𝕊^{2}

(and analogously at T = 1, i.e. T_{−} = 0), gives a space denoted M̄. Thus, a neighborhood U = U_{λ,+} of the ‘temporal face’ T_{+} = 0 is diffeomorphic to

(2.1)  [0, ϵ)_{ρ} × [r_{bh}, r_{dS}] × 𝕊_{ω}^{2}, ρ = T_{λ,+}/μ.

In the interior of the temporal face, where μ > 0, this is in turn diffeomorphic to an open subset of

(2.2)  [0, ϵ)_{T} × (r_{bh}, r_{dS}) × 𝕊_{ω}^{2}.

If the same construction is performed, but using the smooth structure on the the compactification of X given by α = μ^{1/2}, i.e. X̄_{1/2}, and

T̃_{λ,+} = T_{λ,+}^{1/2} = e^{−λt}, t > c,

then a neighborhood U of tf_{+} as above is diffeomorphic to

(2.3)  [0, ϵ)ρ̃ × [r_{bh}, r_{dS}]_{1/2} × 𝕊_{ω}^{2}, ρ̃ = T̃_{λ,+}/α,

where [r_{bh}, r_{dS}]_{1/2} denotes that α has been added to the smooth structure (or equivalently (r−r_{bh})^{1/2} and (r_{dS} −r)^{1/2} have been added to the smooth structure). The distinction between (2.1) and (2.3) is the same as between [0, 1]_{T} × X̄ and [0, 1]T̃ ×X̄_{1/2} (where T̃ is defined analogously to T), namely the square roots of the defining functions of all boundary hypersurfaces have been added to the smooth structure. We denote the resulting space by M̄_{1/2}.

The subtlety is that the de Sitter and black hole ends need different values of λ. So what we actually need is to paste together U_{λ_{bh},+}∩{r < r_{2}} and U_{λ_{dS},+}∩{r > r_{1}} for some r_{1}, r_{2}, r_{bh} < r_{1} < r_{2} < r_{dS} where λ_{bh} and λ_{dS} are chosen in a way that reflects the local geometry neat the two ends. The transition map in the overlap, where r ∈ (r_{1}, r_{2}), is given by

(ρ_{bh,+}, r, ω) ↦ (ρ_{dS,+}, r, ω), ρ_{dS,+} = μ^{λ_{dS}/λ_{bh}−1} ρ_{bh,+}^{λ_{dS}/λ_{bh}}.

In the overlap, where μ ≠ 0 so μ ↦ μ^{λ_{dS}/λ_{bh}−1} is smooth, this is a polyhomogeneousdiffeomorphism, in the sense that it is polyhomogeneous in the local defining functions of the lift of T = 0 (namely ρ_{bh,+} and ρ_{dS,+}). In particular, the front faces of the blow-ups have well defined boundary defining functions, namely μ, up to multiplying by a C^{∞} non-vanishing function, so we consider the resulting space a polyhomogeneous manifold with corners, where the ‘polyhomogeneous’ (as opposed to C^{∞}) faces are only the future and past temporal faces, tf_{+} and tf_{−}. There is also an analogous construction for M̄_{1/2}.

As indicated already, we also want a preferred defining function (up to taking positive multiples) x of tf_{+} in order to measure the rate of decay at the temporal faces; this should be polyhomogoneous-equivalent to the local defining functions ρ_{bh,+} and ρ_{dS,+}. We take this to be of the form x = f(r)e^{−t}, f > 0 smooth for r ∈ (r_{bh}, r_{dS}). Comparison with (1.9)-(1.10) shows that we need to take f(r) = μ^{−1/(2λ_{bh})} for r near r_{bh}, f(r) = μ^{−1/(2λ_{dS})} for r near r_{dS}, hence x = ρ_{bh,+}^{1/(2λ_{bh})} near

the black hole boundary of tf_{+}, and x = ρ_{bh,+}^{1/(2λ_{dS})} near the de Sitter boundary. Then, in particular, a neighborhood of tf_{+} is polyhomogeneous diffeomorphic to

(2.4)  [0, ϵ)_{x} × [r_{bh}, r_{dS}] × 𝕊_{ω}^{2}.

We still need to determine the values λ at the two ends. Writing T = T_{+} = T_{λ,+}, the dual metric (which is the principal symbol of the wave operator, □) has the form

G = 4α^{−2}λ^{2}T^{2}∂_{T}^{2} − α^{2}∂_{r}^{2} − r^{−2}∂_{ω}^{2}

in the original product compactification, with ∂_{r} = \frac{dμ}{dr}∂_{μ} = 2β∂_{μ}. The change of variables from r to μ is smooth and non-degenerate, i.e. 2β = dμ/dr ≠ 0 for μ close to 0, i.e. r close to r_{bh} or r_{dS}. Note that β > 0 for r near r_{bh}, β < 0 for r near r_{dS} since μ > 0 for r ∈ (r_{bh}, r_{dS}).

After blow-up, in the coordinates (ρ, μ, ω),

G = 4μ^{−1}λ^{2}ρ^{2}∂_{ρ}^{2} − 4μβ^{2}(∂_{μ} − μ^{−1}ρ∂_{ρ})^{2} − r^{−2}∂_{ω}^{2}.

Thus,

G = μ^{−1} (4λ^{2}ρ^{2}∂_{ρ}^{2} − 4β^{2}(μ∂_{μ} − ρ∂_{ρ})^{2}) − r^{−2}∂_{ω}^{2}.

If we set λ = β(r_{bh}) > 0 or λ = −β(r_{dS}) > 0 then^{1} the ρ^{2}∂_{ρ}^{2} terms cancel, so locally near r_{bh}

G = 4γρ^{2}∂_{ρ}^{2} + 8β^{2}ρ∂_{ρ}∂_{μ} − 4β^{2}μ∂_{μ}^{2} − r^{−2}∂_{ω}^{2}, γ = μ^{−1}(β(r_{bh})^{2} − β^{2}),

where γ is C^{∞} by Taylor’s theorem, and there is a similar expansion at r_{dS}. Thus the choice of λ determines the compactification M̄, and it is only at this point that the compactification has been specified. Note that this metric is a C^{∞} Lorentzian b-metric on [0, ϵ)_{ρ} × (r_{bh} − ϵ, r_{bh} + ϵ)_{r} × 𝕊_{ω}^{2} (i.e. is non-degenerate as a quadratic form on the b-cotangent bundle), in particular it is C^{∞} across μ = 0. Denoting this extension of M̄ by M̃ (which is now non-compact); g becomes a polyhomogeneous conormal Lorentz metric on M̃, smooth near μ = 0 (where there is a well-defined smooth structure). We write F for the set given by μ = 0, i.e. the boundary hypersurface of M̄ that is no longer a boundary hypersurface of M̃.

For this metric F is characteristic, and one has the standard propagation of singularities in ρ > 0. In particular, for C^{∞} initial data the solution is smooth in ρ > 0 across μ = 0. In fact, writing covectors as ξ \frac{dρ}{ρ} + ζ dμ + ∑ η_{j} dω_{j}, i.e. (ρ, μ, ω, ξ, ζ, η) are coordinates on ^{b}T^{∗}M̃, the dual metric, considered as a function on ^{b}T^{∗}M̃, is

G = 4γξ^{2} + 8β^{2}ξζ − 4β^{2}μζ^{2} − r^{−2}|η|^{2},

so the Hamilton vector field of G is

\begin{matrix}H_{G} = & 8(γξ + β^{2}ζ)ρ∂_{ρ} − 8β^{2}(μζ − ξ)∂_{μ} \\ − (4\frac{∂γ}{∂μ}ξ^{2} + 8β \frac{∂β}{∂μ}(2ξζ − μζ^{2}) − 4β^{2}ζ^{2} − \frac{∂r^{−2}}{∂μ} |η|^{2})∂_{ζ} − r^{−2}H_{(ω,η)}, \end{matrix}

with H_{(ω,η)} denoting the Hamilton vector field of the standard metric on the sphere. The conormal bundle N^{∗}{μ = 0} is μ = 0, ξ = 0, η = 0, so at this set

H_{G} = 8β^{2}ζρ∂_{ρ} + 4β^{2}ζ^{2}∂_{ζ}

and so is indeed tangent to N^{∗}{μ = 0}, and it is non-radial off the zero section (where ζ ≠ 0) as long as ρ ≠ 0.

At ∂F, i.e. at ρ = 0, however there are radial points over the conormal bundle of F. Rather than dealing with them directly, which can be done in the spirit of [6], we reduce the problem to the study of the high energy behavior of the resolvent of the spatial Laplacian (which gives more, in fact), which was performed in [8].

2.3. Blow down of spatial infinity. We now discuss the manifold with corners \overline{M}, in which spatial infinity is blown down. A valid coordinate system near the image of the black hole end of spatial infinity, disjoint from temporal infinity, is given by

s_{bh,+} = α/T_{λ_{bh},+}^{1/2} = ρ_{bh,+}^{−1/2}, s_{bh,−} = α/T_{λ_{bh},−}^{1/2} = αT_{λ_{bh},+}^{1/2} = μρ_{bh,+}^{1/2}, ω,

where as usual ω denotes coordinates on 𝕊^{2}. In these coordinates the dual metric is

G = γ(s_{bh,+}^{2}∂_{s_{bh,+}}^{2} + s_{bh,−}^{2}∂_{s_{bh,−}}^{2}) − 2(β(r_{bh})^{2} + β^{2})∂_{s_{bh,+}}∂_{s_{bh,−}} − r^{−2}∂_{ω}^{2},

and the boundary faces s_{bh,+} = 0 and s_{bh,−} = 0 are characteristic. We can also extend G to a smooth non-degenerate Lorentz metric on M̃^{∘}. Recall that M̃ contains \overline{M} as a closed domain with corners, namely locally in M̃ we simply allow the four boundary defining functions s_{bh,±}, s_{dS,±}, to assume negative values (but we do not extend \overline{M} across the temporal faces). This calculation shows the following important fact:

1In terms of the ‘spatial Laplacian’, Δ_{X}, described in Section 3, −β(r_{bh})^{2} and −β(r_{dS})^{2} are the asymptotic curvatures, hence they are natural quantities from that point of view as well.

Lemma 2.3. The d’Alembertian □ ∈ Diff^{2}(M̃^{∘}), and indeed □ ∈ Diff_{b}^{2}(M̃). Moreover the scattering surfaces F_{bh,±} = {s_{bh,∓} = 0} and F_{dS,±} = {s_{dS,∓} = 0} are characteristic.

Proposition 1.2 is an immediate corollary of this lemma and standard hyperbolic propagation.

3. Resolvent estimates

Next consider the ‘spatial Laplacian’, resolvent estimates for which constitute one of the key ingredients in our analysis. From (1.4) with T = e^{−t}, it follows that

□ = α^{−2} ((T D_{T})^{2} − α^{2}r^{−2}D_{r}α^{2}r^{2}D_{r} − α^{2}r^{−2}Δ_{ω}) .

Recall that not precisely T, but rather T^{λ_{bh}} and T^{λ_{dS}}, were used above to construct the compactification.

By definition the spatial ‘Laplacian’ is

Δ_{X} = α^{2}r^{−2}D_{r}α^{2}r^{2}D_{r} + α^{2}r^{−2}Δ_{ω}

Near α = 0, where α can be used as a valid coordinate in place of r,

Δ_{X} = βr^{−2}αD_{α}βr^{2}αD_{α} + α^{2}r^{−2}Δ_{ω} ∈ Diff_{0}^{2}(X̄_{1/2}).

This is not the Laplacian of a Riemannian metric on X; however it is very similar to one. It is of the form d^{∗}d with respect to the inner product on one-forms given by the fiber inner product with respect to the ‘spatial part’ H = α^{2}∂_{r}^{2} + r^{−2}∂_{ω}^{2} of G and density on X given by dh = α^{−2}r^{2} dr dω. In what follows we will also view Δ_{X} is a 0-operator on X̄_{1/2}, self-adjoint on

L^{2}(X, |dh|), |dh| = α^{−2}r^{2} |dr| |dω| = α^{−1}|β|^{−1}r^{2} |dα| |dω|,

and we will use the techniques of [7] to study its resolvent. It is also useful to introduce the operator

L = αΔ_{X}α^{−1},

which is self-adjoint on

L^{2}(X, α^{−2} |dh|) = αL^{2}(X, dh), α^{−2}|dh| = α^{−3}|β|^{−1}r^{2} |dα| |dω|.

Thus, this space is L_{0}^{2}(X) as a Banach space, up to equivalence of norms.

Let α̃ = α^{1/λ_{bh}} ∈ C^{∞}(X), α̃ > 0, for r near r_{bh} or α̃ = α^{1/λ_{dS}} for r near r_{dS}. The normal operators N_{0,bh}(L), N_{0,dS}(L) of L in Diff_{0}^{2}(X̄_{1/2}) at r = r_{bh}, resp. r = r_{dS}, are

N_{0,bh}(L) = λ_{bh}^{2}N_{0,bh}(Δ_{ℍ^{3}}), N_{0,dS}(L) = λ_{dS}^{2}N_{0,dS}(Δ_{ℍ^{3}}),

where Δ_{ℍ^{3}} is the hyperbolic Laplacian, explaining the usefulness of this conjugation. In particular, it follows immediately from [7] (with improvements from [5]) that the resolvent

R(σ) = (L − σ^{2})^{−1}, on L_{0}^{2}(X̄_{1/2}), Im σ < 0,

continues meromorphically to a strip | Im σ| < ϵ as an operator between weighted L^{2}-spaces (as well as other spaces):

R(σ) : α̃^{δ}L_{0}^{2}(X̄_{1/2}) → α̃^{−δ}L_{0}^{2}(X̄_{1/2}), δ > ϵ;

we keep denoting the analytic continuation by R(σ). Thus,

R(σ) = (Δ_{X} − σ^{2})^{−1} = α^{−1}R(σ)α on L^{2}(X, |dh|), Im σ < 0,

continues meromorphically to a strip | Im σ| < ϵ

R(σ) : α̃^{δ}L^{2}(X, |dh|) → α̃^{−δ}L^{2}(X, |dh|), δ > ϵ.

The result we need is proved in [8], giving polynomial bounds on the resolvent in a strip around the real axis.

Proposition 3.1. If ϵ > 0 is sufficiently small the only pole of the analytic continuation of the resolvent R(σ) in Im σ < ϵ is σ = 0, which is simple, with residue given by a constant γ and for each k and δ > ϵ there exist m > 0 and C > 0 and M such that

(3.1)  ∥α̃^{−iσ}R(σ)∥_{L(α^{−1}α̃^{δ}H_{0}^{m}(X̄_{1/2}),C^{k}(X̄))} ≤ C|σ|^{M},

for |σ| > 1, Im σ < ϵ.

4. Asymptotics for solutions of the wave equation

We now proceed to study the asymptotics of solutions of the wave equation at tf_{+}.

Suppose u is a solution of the wave equation, □u = 0, and u is smooth on M̃^{∘}. Energy estimates show that u is necessarily tempered, in the sense that u ∈ ρ^{−s}L^{2}(M̄) for some s > 0. This is shown directly below.

Let ϕ = ϕ_{0}(ρ) ∈ C^{∞}(M̄) be a cutoff function, with ρ ∈ C_{c}^{∞}([0, 1)), identically 1 near 0. If u satisfies □u = 0 and u is smooth in μ (i.e. across the side faces), then ϕu is smooth in μ, and

□(ϕu) = [□, ϕ]u = f

where f is also smooth in μ, and vanishes in a neighborhood of the temporal face – since [□, ϕ] ∈ Diff_{b}^{1}(M̃), by Lemma 2.3, is supported away from the temporal face. Moreover, v = ϕu is the unique solution of □v = f in M̄^{∘} with v = 0 for ρ sufficiently large.

Now we wish to take the Mellin transform in T = e^{−t}, for functions supported in a neighborhood of the temporal face, namely in U = {ρ < 1}. Such a neighborhood is equipped with a fibration M̄ → X, extending the fibration M → X in the interior, and there is a natural density |dt| = \frac{|dT |}{T} on the fibers. In coordinates (ρ̃, α, ω) = (T^{λ_{bh}}/α, α, ω) valid near the temporal face boundary at r = r_{bh} this density takes the form \frac{|dρ̃|}{λ_{bh}ρ̃}. So, the Mellin transform can be taken with respect to this fibration and density. Thus the map v ↦ v̂ from functions supported near the temporal face to functions on Ω × X, Ω ⊂ ℂ,

v̂(σ, z) = ∫ T^{iσ}v(T, z) \frac{|dT |}{T}.

If v is polynomially bounded in T, supported in T ≥ 0, with values in a function space H in z, this transform gives an analytic function in a lower half plane (depending on the order of growth of v) with values in H.

In fact, writing the integral in coordinates valid near the boundary of the temporal face, ρ̃ = T^{λ_{bh}}/α,

v̂(σ, α, ω) = α^{iσ/λ_{bh}}λ_{bh}^{−1} ∫ ρ̃^{iσ/λ_{bh}}v(ρ̃, α, ω) \frac{|dρ̃|}{ρ̃}.

The integral is then the Mellin transform of v with respect to ρ̃ evaluated at σ/λ_{bh}. Thus, if v is smooth on M̃^{∘}, supported in {ρ̃ ∈ I}, I ⊂ (0, 1) compact, i.e. near but not at the temporal face, then v̂ is in fact analytic in ℂ with values in functions of the form α̃^{iσ}C^{∞}, with C^{∞} seminorms all bounded by C_{k}⟨σ⟩^{−k}, k arbitrary. If v is just supported in ρ̃ < 1 and is conormal on M̃, then v̂ is analytic in a lower half plane with values in functions of the form α̃^{iσ}C^{∞}.

Assuming for the moment that ϕu is polynomially bounded in T, □(ϕu) = f becomes N̂_{b}(□)ϕ̂u = f̂, where

N̂_{b}(□) = α^{−2}(σ^{2} − Δ_{X}) = α^{−2} (σ^{2} − α^{2}r^{−2}D_{r}α^{2}r^{2}D_{r} − α^{2}r^{−2}Δ_{ω}) ,

so

(σ^{2} − Δ_{X}) ϕ̂u = α^{2}f̂.

If ϕu is polynomially bounded in T, then both f̂ and ϕ̂u are analytic in Im σ < −C, and as f is compactly supported in ρ̃, f̂ is in entire analytic with values in functions of the form α̃^{iσ}C^{∞}, with C^{∞} seminorms all bounded by C_{k}⟨σ⟩^{−k}, k arbitrary. Thus,

ϕ̂u = R(σ)(α^{2}f̂), Im σ < −C,

and we recover ϕu by taking the inverse Mellin transform.

We now return to arbitrary (not a priori polynomially bounded) u, f = □(ϕu), as above. Thus, f̂ is analytic in all of ℂ, with values in functions of the form α̃^{iσ}C^{∞}, with C^{∞} seminorms all bounded by C_{k}⟨σ⟩^{−k}, k arbitrary. Now note that the inclusion

α^{1+s}L^{∞}(X) ↪ L_{0}^{2}(X̄_{1/2})

is continuous for every s > 0. Thus, the inclusion

α̃^{iσ−δ}α^{3}C^{∞}(X̄_{1/2}) ↪ H_{0}^{m}(X̄_{1/2})

is continuous if

λ(ϵ + δ) < 2, λ = max(λ_{bh}, λ_{dS}),

and Im σ < ϵ, which is to say

α̃^{iσ}α^{2}C^{∞}(X̄_{1/2}) ↪ α^{−1}α̃^{δ}H_{0}^{m}(X̄_{1/2})

is continuous.

In particular, then

∥α^{2}f̂∥_{α^{−1}α̃^{δ}H_{0}^{m}(X̄_{1/2})} ≤ C_{k}⟨σ⟩^{−k}

for all k in Im σ < ϵ < δ (with new constants), 0 < ϵ < δ sufficiently small. Proposition 3.1 shows that, for ϵ > 0 sufficiently small and for all N and k,

(4.1)  ∥α̃^{−iσ}R(σ)(α^{2}f̂)∥_{C^{N} (X̄)} ≤ C_{k}|σ|^{−k}, Im σ < ϵ.

The inverse Mellin transform of w = R(σ)(α^{2}f̂) is

w̌(T, z) = (2π)^{−1} ∫ T^{−iσ}w(σ, z) dσ.

Thus,

w̌(ρ̃, α, ω) = (2π)^{−1} ∫ ρ̃^{−iσ/λ_{bh}}α^{−iσ/λ_{bh}}w(σ, α, ω) dσ.

In view of (4.1) (in particular the analyticity of w̌ in the lower half plane with the stated estimates), w = 0 for T < 0 (as can be seen directly by shifting the contour to Im σ = −C, using the off spectrum resolvent estimate ∥R(σ)∥_{L(L^{2}(X,|dh|))} ≤ | Im(σ^{2})|^{−1} and letting C → +∞). Since the unique solution of □v = f, v supported in T ≥ 0, is ϕu, it follows that w̌ = ϕu.

Shifting the contour for the inverse Mellin transform for w to Im σ = ϵ gives a residue term at 0, and shows that

ρ̃^{−ϵ}(ϕu − v) ∈ L^{2}([0, δ)ρ̃; C^{∞}(X̄)),

where v arises from the residue at 0, hence is a constant function. Note that (at the cost of changing ϵ) this is equivalent to the analogous statement with ρ replaced by ρ̃. The derivatives with respect t satisfy similar estimates. Hence, the same estimates hold for the conormal derivatives with respect to ρ̃ (or equivalently ρ). We thus deduce the leading part of the asymptotics of u at the future temporal face, tf_{+}.

This completes the proof of Theorem 1.1. The main result, Theorem 1.3, follows from the combination of this result with Proposition 1.2.

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge MA 02139-4307, U.S.A.

E-mail address: rbm@math.mit.edu

Department of Mathematics, Purdue University, West Lafayette, IN 47907-1395, U.S.A.

E-mail address: sabarre@math.purdue.edu

Department of Mathematics, Stanford University, Stanford, CA 94305-2125, U.S.A.

E-mail address: andras@math.stanford.edu