Semiclassical reformulation

With ˜δ^∗ defined in (8.1), we now lete>0 and take

γ1=γ and γ2=¹₂eγ. (8.2)

The constraint propagation operator is thus

e^CPg = 2δ_gG_gδ_g^∗+γ(−i_∇t∗du+(dt_∗)δ_gu+(gt_∗)u−e d(i_∇t∗u)). (8.3) We view~=γ⁻¹ as a semiclassical parameter; then,

P_~:=~²e^CPg (8.4)

is a semiclassical b-differential operator. Occasionally, we will indicate the parametere by writingPe,~. We will show that Theorem8.1follows frompurely symbolic microlocal arguments; the energy estimates we use for propagation in ther-direction beyond the horizons are rather crude, and also symbolic (albeit for differential operators) in char-acter. (We will sketch an alternative, completely microlocal symbolic argument using complex absorption in Remark8.29.) The relevant operator algebra is the semiclassical b-algebra, which is described in AppendixA.3.

In order to analyzeP_~∈Diff²_b,~(M), we first calculate the form of the second term in (8.3). We now work with the coordinatet∗ of Lemma 3.1, which simplifies computations significantly: the metricgand the dual metricGtake the form

g=µ dt²_∗−2ν dt_∗dr−c²dr²−r²/g, G=c²∂_t²_∗−2ν ∂t_∗∂r−µ ∂_r²−r⁻²G,/

(8.5)

wherec=ct_∗in the notation of Lemma 3.1, and ν=∓p

1−c²µ, ±(r−rc)>0, (8.6) withr_c=p³

3M/Λ given in Lemma3.1. See Figure8.1.

We will also make use of the boundary defining functionτ=e^−t∗ ofM. Using the identities

ν²+c²µ= 1 and 2νν⁰+c²µ⁰= 0,

one can then compute the connection coefficients and verify the following result.

0 r− rc r+

r ν

Figure 8.1. The graph of the functionνdefined in (8.6).

Lemma8.3. For w∈C^∞(M, T^∗S²) and v∈C^∞(M, TS²),we have

Lemma8.4. In the bundle decomposition (8.7),we have

−i∇t∗d=

The part of P_~corresponding to the second term in (8.3) is given by

−iL_~u:=−i∇t∗~du+(dt∗)~δgu+~(^gt∗)u−e~d(i∇t∗u)

=~







−(1+e)c²∂t∗+2ν∂r eν∂t∗+µ∂r −r⁻²/δ (1−e)c²∂r −c²∂t∗+eν∂r 0

(1−e)c²/d (e−1)ν/d −c²∂t∗+ν∂r







+~







2ν⁰+4r⁻¹ν µ⁰+2r⁻¹µ 0 0 (1+e)ν⁰+2r⁻¹ν 0

0 0 ν⁰+2r⁻¹ν





.

Thus,L_~∈Diff¹_b,~(M;^bT^∗M); the first term here is semiclassically principal, the second one subprincipal, due to the extra factor of~. We will sometimes indicate the parameter eby writingL_e,~. We denote the principal symbol ofL_e,~by

`e=σb,~(Le,~).

We decomposeL_~in the coordinates (t_∗, r, ω) as

L_~=M_t∗~D_t∗+M_r~D_r+M_ω+i~S, (8.8) with Mω capturing the /d and /δ components, and S the subprincipal term (not con-taining differentiations); the bundle endomorphismsMt∗,Mr andS of^bT^∗M have real coefficients.

Since

P_~=^CPg,~−iL_~ and ^CPg,~:=~^2CPg =~²δgGgδ_g^∗

(see (2.13)), we see directly thatP_~isnotprincipally scalar, due to the non-scalar nature ofL_~; of course, the principal part in the sense of differentiable order is scalar and equal toGId, whereGis the dual metric function.

Lemma 8.5. For e61 sufficiently close to 1, the semiclassical principal symbol of Pe,~is elliptic in^bT^∗M\(o∪(Σ∩^bS^∗M)),whereΣ=G⁻¹(0)⊂^bT^∗M is the characteristic set,and ^bS^∗M is the boundary of ^bT^∗M at fiber infinity.

Proof. At a point ζ∈^bS^∗M, the ellipticity of P_~ is equivalent to G(ζ)6=0, proving that Ell_b,~(P_~)∩^bS^∗M=^bS^∗M\Σ.

Next, working away from fiber infinity, we note that σ_b,~(^CPg,~)=G is real, while σ_b,~(iL_~) has purely imaginary eigenvalues; this follows from (8.10) below for e=1, and either by direct computation or from Lemma8.13below for 0<e<1. We thus only need to show thatσ_b,~(L_~) is elliptic on the part of the light cone Σ∩(^bT^∗M\o) away from

fiber infinity and the zero section. Moreover`e, as a smooth section of π<sup>∗</sup>End(<sup>b</sup>T<sup>∗</sup>M) over <sup>b</sup>T<sup>∗</sup>M (with π:<sup>b</sup>T<sup>∗</sup>M!M), is homogeneous of degree 1 in the fibers of <sup>b</sup>T<sup>∗</sup>M; since Σ∩<sup>b</sup>S<sup>∗</sup>M is precompact (the non-compactness only being due to the boundaries r=r±±3εM being excluded in the definition ofM), the set ofefor which`eis elliptic on Σ∩(^bT^∗M\o) is therefore open. Thus, it suffices to prove the lemma fore=1. Writing b-covectors as

ζ=−σ dt∗+ξ dr+η, η∈T^∗S², (8.9) we have

`₁=







−2(c²σ+νξ) νσ−µξ −r⁻²i_η

0 −(c²σ+νξ) 0

0 0 −(c²σ+νξ)





. (8.10)

But−(c²σ+νξ)=G(dt_∗, ζ), which, by the timelike nature ofdt_∗, is non-zero for ζ∈Σ∩(^bT^∗M\o).

The last part of the proof has the following consequence.

Corollary 8.6. For e close to 1, the symbol σb,~(Le,~) is elliptic in the causal double cone {ζ∈^bT^∗M\o:G(ζ)>0}.

Schematically, the principal symbol ofP_1,~ is

G−i







2∇t_∗ ∗ ∗

0 ∇t_∗ 0

0 0 ∇t_∗





,

viewing∇t∗as a linear function on the fibers of^bT^∗M. Since in a conical neighborhood of the two components Σ^±of the light cone (see (3.21)) we have±∇t∗>0, the imaginary part has the required sign for real-principal-type propagation of regularity along the (rescaled) null-geodesic flow generated byH_G in the forward (resp. backward) direction within Σ⁺∩^bS^∗M (resp. Σ⁻∩^bS^∗M); see§8.2for details.

Near the zero section, one would like to think of L_1,~ as a coupled system of first-order ordinary differential operators, transporting energy along the orbits of the vector field∇t_∗. See Figure8.2.

Remark 8.7. We explain the structure ofLe,~in a bit more detail. We first note that one can check that the conclusion of Lemma 8.5 in fact holds for alle>0, but fails for e=0. Indeed, fore=0, one can easily compute the eigenvalues of`0(ζ),ζ∈^bT^∗M\oas in (8.9), to be−(c²σ+ξν) with 2-dimensional eigenspace,(¹⁷)andc²σ+ξν±p

ξ²+c²r⁻²|η|²

(¹⁷) This is independent ofeand can be computed explicitly: forη6=0 andξ6=0, the eigenspace is η^⊥⊕hν|η|²dt∗+c²|η|²dr−r²ξηi; ifη6=0 andξ=0, it isη^⊥⊕hν dt∗+c²dri; and, ifη=0, it isη^⊥=T^∗S².

τ=0 r− rc r+

Figure 8.2. The flow of the vector field∇t∗, including at the boundary at infinity.

with 1-dimensional eigenspaces, respectively; the vanishing of any one of the latter two eigenvalues impliesG=0. Thus, fore=0, L_~can roughly be thought of as transporting energy in phase space along the flow of ∇t_∗ for a corank-2 part of L_~, and along the flows of two other vector fields, whose projections toM are null, on two rank-1 parts.

As soon as 0<e<1, however, the projections to M of these two vector fields become future timelike, as we will discuss in Lemma8.15and Remark8.14below, and`eis still diagonalizable, due to Lemma 8.13below; for e=1, on the other hand, allthree vector fields coincide (up to positive scalars), with projection toM equal to a positive multiple of∇t∗, but`1 is no longer diagonalizable, as it is nilpotent with non-trivial Jordan block structure when evaluated on (dt∗)^⊥\o⊂^bT^∗M\o.

In summary, ellipticity considerations force us to usee>0, while the structure ofLe,~

is simplest fore=1, with the technical caveat of non-diagonalizability, which disappears fore<1; this is the reason for us to work withe<1 close to 1.