2.2 Linear Waves on de Sitter
2.2.2 Energy estimates
2.2.2.3 Integrated local energy decay and local redshift effect in
Proof. By (2.2.102),
d dr
f(r) exph
−α Z r
r1
g′(r′) g(r′) dr′i
≤0, (2.2.104)
which yields (2.2.103) upon integration on the interval [r1, r2].
As an immediate consequence of (2.2.101) we obtain – in view of (2.2.93) – the following estimate, which is precisely the content of Prop. 2.8.
Proposition 2.13. Let ψ be a solution of (2.2.52), then Z
Σr2
φ (
φ2 T ·ψ2
+ 1 φ2
∂ψ
∂r 2
+|∇/ ψ|2+mΛψ2 )
dµgr2 ≤
≤ Z
Σr1
φ (
φ2 T ·ψ2
+ 1 φ2
∂ψ
∂r 2
+|∇/ ψ|2+mΛψ2 )
dµgr1 (2.2.105)
for all r2 > r1 >q
3 Λ.
2.2.2.3 Integrated local energy decay and local redshift effect in the static
The current we then wish to consider is the following modification of the standard energy current associated to X:
JµX,1 =JµX +1 4 h
1 + 2
rr∗ 1−µi
∂µ
ψ2
− 1 4∂µ
h 1 + 2
rr∗ 1−µi
ψ2 (2.2.109) Note that
lim
r→√3
Λ
r∗(r)
1−µ(r)
= 0, (2.2.110)
because
r∗(r) = Z r
0
1
1−µ(r)dr= 1 2
r3 Λlog
1 + qΛ
3r 1−
qΛ 3r
. (2.2.111)
The divergence of the current (2.2.109) satisfies for any solution ψ of the wave equation (2.2.51) of vanishing spherical mean
ψ = 1 4πr2
Z
S
ψdµ
r2γ◦ = 0 (2.2.112)
the integral inequality Z
S ∇µJµX,1[ψ] dµ
r2γ◦ ≥
≥ Z
S
(1−µ) ∂ψ
∂r 2
+ 2 r2
r∗(1−µ)
r (1 +µ)ψ2+ 2Λ 3 ψ2
dµr2γ◦. (2.2.113) By Stokes theorem the energy density on the right-hand side is thus controlled in the static region by the corresponding boundary integrals of (2.2.109) which on the cosmological horizonC take the form
Z du∗
Z
S
JX, ∂
∂u∗
+ 1 4
∂ψ2
∂u∗
dµr2γ◦. (2.2.114) While the terms arising from the modification are finite on the boundary, the main con- tribution to the energy density fromT(X, ∂u∗) is of course not finite, because
f =r∗ → ∞ as r→ r3
Λ. (2.2.115)
It is for this reason that we have to introduce a cut-off in (2.2.108), which in turn neces- sitates the construction of a red-shift vectorfield.
For the multiplier (2.2.107) we will construct more generally in Section 2.2.2.4 a current for which the divergence reads
∇µJµX,1[ψ] = f(1) 1−µ
∂ψ
∂r∗ 2
+f r
∇/ ψ 2
+
−1 4
1
1−µf(3)−1
rf(2)+2µ
r2f(1)−2µ2 r3f
ψ2, (2.2.116)
where f(i)= drdi∗fi; note that for f ≥0, f(1) ≥0, and f(2) ≤0 (2.2.113) is replaced by Z
S ∇µJµX,1[ψ] dµ
r2γ◦ ≥
≥ Z
S
f(1) 1−µ
∂ψ
∂r∗ 2
+ 2f
r3(1−µ2)ψ2− 1 4
1 1−µ
d3f dr∗3ψ2
dµr2γ◦, (2.2.117) which holds as a consequence of Poincar´e’s inequality if ψ is assumed to verify (2.2.112).
As a result the error that is introduced by the cut-off is precisely of the form of the last term in (2.2.117).
Now, on very general grounds we know that in view of qΛ
3 >0 (2.2.31) the cosmological horizon exhibits a redshift effect and allows for the construction of a multiplier that gives rise to a positive current on the horizon [17]. However, while in the case of black hole horizons it suffices to provide a construction on the event horizon and then to extend the multiplier suitably into the neighborhood by a continuity argument, here an explicit redshift vectorfield is needed which extends beyond the immediate vicinity of the cosmo- logical horizon, namely into the region where the cut-off is introduced (and the last term in the above inequality becomes indefinite). In analogy to [20] we choose as a starting point for the construction of a vectorfield that captures the red-shift effect
Y = 1 +σ(1−µ)Yˆ +σ(1−µ)T (2.2.118) where
Yˆ = 2 1−µ
∂
∂v∗ , (2.2.119)
note that σ >0 is precisely the parameter that appears in the general construction [17].
(There we construct Y transversal to the horizon and require∇YY =−σ(Y +T) for the extension to the vicinity of the null hypersurface). This multiplier gives rise to a positive current near the cosmological horizon, in the sense that
KY ≥0 (1−µ≪1) (2.2.120)
and controls all derivatives. However, it does not yield control on the error term unless σ is chosen very large; (note that (2.2.120) only holds for 1−µ≪1 while the error term may be of order (1−µ)−1). But we cannot add KY to KX,1 if the magnitude of Y is large, for the latter is required to control the former in the region where the positivity of (2.2.120) fails. We obtain a vectorfield with the desired properties by adding yet another term
Y =Y +κ η
r∗δ Yˆ +T
, (2.2.121)
where κ >0, δ >0 and η is a suitable cut-off function supported away from the horizon.
Let us also denote by Yc = χY where χ is a cut-off function such that χ = 0 for µ≪ 1 and χ= 1 for 1−µ≪1. We are then able to show that only a “small” contribution of the redshift effect is required in this construction.
A precise version of this statement is given in Proposition 2.20 on page 148.
We have seen that the proof of an integrated energy estimate compels the construction of a redshift vectorfield; alternatively we may say that the boundary terms on the cosmological horizon arising from a redshift vectorfield can only be controlled in conjuction with a positive current arising from a Morawetz vectorfield.
The construction which is described above and carried out in more detail in the following Section 2.2.2.4 leads us to the result:
Proposition 2.14. Let Σ0 be a spacelike hypersurface with normal n in the static region of de Sitter crossing the cosmological horizon C+ to the future of the sphere C+∩ C− (in (u, v)coordinates this is the sphere(0,0)), and denote byC0+ =J+(Σ0)∩C+ the segment of the cosmological horizon to the future ofΣ0. Then there exist a strictly timelike vectorfield N (normalized in the sense that g(T, N) is constant onC0+) and a constantC(Λ,Σ0)such that for all solutions of the homogeneous wave equation (2.2.51) we have
Z
C0+
∗JN ≤C(Λ,Σ0) Z
Σ0
Jn, n
. (2.2.122)
Inhomogeneous wave equation. In the inhomogeneous case we can rely on the same vectorfields but our identitites are based on modified currents. Namely, the current JX,1 is replaced by
JX,1;mΛ =JX,1− mΛ
2 X♭ψ2, (2.2.123)
whereX♭·Y =g(X, Y), and instead of (2.2.116) we have
∇µJµX,1;mΛ[ψ] = f(1) 1−µ
∂ψ
∂r∗ 2
+f r
∇/ ψ
2+mΛ
f rψ2 +
−1 4
1
1−µf(3)−1
rf(2)+2µ
r2f(1)−2µ2 r3f
ψ2. (2.2.124) In our analysis of the inhomogeneous case the zeroth order term steps into the role of the angular derivatives term of the homogeneous case. Indeed, while in (2.2.116) we have conluded by Poincar´e’s inequality that the last terms of each line taken together are always nonnegative after integration over the spheres, c.f. (2.2.117), we argue here that
mΛ
f
rψ2−2µ2
r3f ψ2 =h
mΛ−2µΛ 3
if
rψ2 ≥0 if mΛ≥2Λ
3 . (2.2.125)
This leads us to the following result.
Proposition 2.15. Let Σ0 and C+ be as in Prop. 2.14. Consider solutions ψ to the inhomogeneous wave equation (2.2.52) with mΛ ≥ 2Λ3. There exists a strictly timelike vectorfield N (as in Prop. 2.14) and a constantC(Λ,Σ0) such that for all solutions ψ we
have Z
C0+
n∗JN[ψ] +ψ2o
≤C(Λ,Σ0) Z
Σ0
n Jn[ψ], n
+ψ2o
. (2.2.126)
Remark 2.16 (Conformal Invariance of the wave equation). We expect that the lower bound on the mass mΛ is a shortcoming of the proof of Prop. 2.15, in particular we expect that (2.2.126) holds for all mΛ > 0. However, the value mΛ = 2Λ3 does have mathematical meaning, for in this case (2.2.52) is conformal to the wave equation on Minkowski space. More precisely, recall that we can cast the de Sitter spacetime as the subset
M=n
x∈R3+1 : hx, xi=−(x0)2+ (x1)2+ (x2)2+ (x3)2 >−4 κ
o
, (2.2.127) endowed with the metric
g = Ω2hdx, dxi= Ω2η , (2.2.128) where
Ω = 1
1 + κ4hx, xi, and κ= Λ
3 . (2.2.129)
Then (M, g) is a solution to (2.2.14), namely
Rµν = Λgµν. (2.2.130)
Lemma 2.17. Let ψ be a solution to gψ = 2κ ψ = 2Λ3 ψ, then ψ˜= Ωψ is a solution to ˜gψ˜= 0 where g˜= Ω−2g =η is the Minkowski metric.
This well known result demonstrates the significance of the specific value mΛ = 2Λ3, and allows us to read off qualitatively our main result in this special case. For a solution of the classical wave equation ηψ˜ = 0 will assume nonvanishing values on the boundary hx, xi = −4/κ. The decay of ψ is then entirely due to the conformal factor Ω, i.e. the
“expansion” of the spacetime.