In the second part of this thesis, the global study of solutions to the linear wave equation on the expansion of de Sitter and Schwarzschild de Sitter spacetimes is started. The fact that a global solution to the linear wave equation on expanding black hole spacetimes has a limit on the future.
Overview
Statement of the Theorems
Outside the black hole, the metric g takes the classical form in (t, r) coordinates. 2m and we will for that reason in Section 1.2 introduce the global geometry of (Mnm, g) using a double zero foliation, from which we derive an alternative double zero coordinate system for the outside of the black hole.
Overview of the Proof
In section 1.4.3 a more general construction of a current is given using a commutation with the angular momentum operators. The pointwise decay of Theorem 2 then follows from Theorem 1 and the (refined) integrated local energy decay estimates of Section 1.4.4 by a simple interpolation argument given in Section 1.6.
Global causal geometry of the higher dimensional Schwarzschild solution . 20
The analogous calculation of the Hessian equations in (n+ 1) dimensions in the presence of matter shows that this is precisely fulfilled for. In this section, we prove a manifestation of the local redshift effect in the Schwarzschild geometry of Section 1.2 in the framework of multiplier vector fields.
Integrated Local Energy Decay
Radial multiplier vectorfields
Although it is possible to find simple functions f ≥ 0 to asymptotically guarantee the positivity of KX,1, this is not possible in the entire domain of outer communication; this is a consequence of trapping, which emerges more concretely as the indeterminacy of sign in (1.4.21) for the photon sphere r = n−2√. In the following, our strategy will consist of proving the non-negativity of KX,1 not point by point, but by using Poincar´e inequalities after integration over the spheres (the group orbits of SO(n)).
High angular frequencies
We are now in a position to prove a non-negativity property of the terms appearing in (1.4.21) which we will denote by 0KX,1,. This estimate of the zero-order term φ2 is sufficient to obtain an estimate for all derivatives using a commutation with the vector fieldT; see Proof of Prop.
Low angular frequencies and commutation
Note also that, in view of (1.4.77), the positivity property (1.4.74) is "easily" satisfied for large values of n, indicating that there may be another simplified proof in higher dimensions. Here we obtain the time derivatives with the auxiliary current Kaux =∇µJµaux; Jaux=JXaux,0; Xaux = faux.
Boundary terms
This is sufficient to control the remaining derivatives; for the auxiliary current (B.10) gives. 1.4.90) Using Cauchy's inequality for the first term, that is. 1.4.93). Then especially from Cor. 1.4.97) But in a second step we will show that there actually exists a constant C(n) such that. For boundary integrals on t-const hypersurfaces, we will use (ii) of the following Lemma.
The following lemma will be applied to the boundary terms of the J(α)-current on the zero hypersurfaces in the regionr≤r0. We conclude the statement of the theorem by discussing the two regimes in Step 1 and Step 2 above. However, this estimate does not include the zero-order term, which we addressed separately in Prop.
The Decay Argument
Uniform Boundedness
Let Σ be a (spherically symmetric) space-like hypersurface in M, Σ′ = Σ∩ {r ≤R} and N the output null hypersurface derived from ∂Σ′; moreover, let. Let φ be the solution of the wave equation (1.1.1) with initial data on Σ0, then there exists a constant C(Σ0) such that By analogy, we can continue to estimate the energy of the local observer [17]; from the energy identity for N on the domain R(τ′, τ) = ∪τ′≤τ≤τΣττ follows.
1.9, namely the redshift effect, KN is bounded from below by (JN, n) near the horizon and from above by (JT, n) away from the horizon; since also the course of the foliation of R is bounded from above and below, we conclude that there are only constants 0< b < B depending on Σ and N thus.
Energy decay
Note that the powers ofr appearing in the bulk term are 1 less than those in the boundary terms. In a first step the decay of the solutions at future zero-infinity will be derived from the weighted energy inequality, and in a second step the continuation to the event horizon will be derived from the redshift effect. 1.5.20.
Improved interior decay of the first order energy
We will use this weighted energy inequality for χ to proceed in a hierarchy of four steps. Note the increase in powers of r compared to the boundary term appearing in Prop 1.32; furthermore, the inequality still holds if we add the integral of the non-generating energy onR′′jPττ2j+1′′′′′.
The last two terms on the right-hand side of (1.5.64) actually decay at almost the same rate as the first; for the first note here that we could have used Prop.
Digression: Conformal energy decay
While KZ,1 does not vanish identically, the following lemma shows that at least near the horizon and at infinity it has a sign. It is still necessary to show the non-negativity of the coefficient on |∇/ φ|2 in the asymptotic range. The following proposition shows that the current JZ,1 gives rise to non-negative edge terms of ont-const surfaces with the desired weights.
The last fact related to the Morawetz vector field is that the bulk term in a t-const plate with the J(α) current (see Cor. 1.29) can be controlled in a finite annular region. We deduce from (1.5.80) in conjunction with Lemma 1.30 — where we have already addressed the boundary terms of the J(α) current on t-const hypersurfaces — that. The spacelike hypersurfaces relative to which we expect the local observer's energy to decay end at future zero infinity (and on the horizon to the future of the bifurcation sphere).
The estimate of the space-time integral (1.5.87) near the horizon is the starting point of the following dove argument.
Pointwise bounds
As in the proof of Lemma 1.42, integrating from infinity and Cauchy's inequality we obtain that. It is the purpose of the work presented in this chapter to initiate the global study of linear waves on cosmological spacetimes. The Schwarzschild de Sitter family exhibits a region of spacetime bounded in the past by two cosmological horizons and in the future by a space-like hypersurface of unbounded area.
More precisely, we establish uniform energy estimates for general solutions to the linear wave equation in the expanding regions of de Sitter and Schwarzschild de Sitter spacetimes, which extend with a stable redshift mechanism to a global estimate. Let Σ ⊂ J−(Σ+) be a spacelike hypersurface in Σ+'s past such that Σ+ is in the dependence domain of Σ and such that Σ crosses the cosmological horizon and the event horizon to the future of the bifurcation spheres (see Figure 2.1). The expanding region is bounded in the past by the cosmological horizons ¯C+ ∪ C+ and in the future by the space-like hypersurface Σ+.
However, the reader is advised to first familiarize himself with our treatment of linear waves on de Sitter spacetimes in section 2.2, where many of the ideas for our approach come from and their application is seen more clearly.
Linear Waves on de Sitter
Global geometry of de Sitter
- Static region
- Cosmological horizon
We take the view of de Sitter as a member of the Schwarzschild-de Sitter family with m = 0. It is a key insight that based on these equations the mass function m is defined by 1− 2m. It is worth noting that this region S is "small": it is the intersection of the past of a point with the future of a point.
1; as discussed below, it turns out that this region of Misi is actually the past of a sphere and in this sense "big". We can use the radial function as a function of time in R, because in the expanding region the radius of the area is increasing towards the future. This is a convenient framework for decomposing Einstein's equations and the wave equation with respect to leaves (Σr), as discussed in Appendix C.1.
Together with the vector fields Ω(i) (discussed in Appendix C.2), this implies that the tangent space to Σr is spanned by vector fields that generate isometries of spacetime.
Energy estimates
- Energy estimates and global redshift in the expanding region133
- Integrated local energy decay and local redshift effect in
- Proof of integrated local energy decay in the static region 144
- Pointwise estimates on the timelike future boundary
Then there exists a strictly time-like vector field N (normalized in the sense that g(T, N) is constant on C0+) and a constant C(Λ,Σ0), such that for all solutions of the homogeneous wave equation (2.2.51) have. We expect that the lower bound on the mass mΛ is a deficiency in the proof of Prop. For a solution of the classical wave equation, ηψ˜ = 0 will assume non-vanishing values on the boundary hx, xi = −4/κ.
There exists a time-like vector fieldN and a Conly constant depending on Λ, so that for all solutions ψ of the wave equation (2.2.51) holds. As discussed in Section 2.2.2.3, our analysis of the inhomogeneous wave equation is based on the same vector fields but modified fluxes. There exists a time-like vector fieldN and a Conly constant depending on Λ, so that for all solutions ψ of the inhomogeneous wave equation (2.2.52) holds.
Let Σ be a spatial hypersurface with normal n crossing the cosmological horizon C+ towards the future of the sphere C+∩ C−.
Linear Waves on Schwarzschild de Sitter
Geometry of the expanding region of Schwarzschild de Sitter
Here it is useful to recall our discussion of the global de Sitter geometry in Section 1.2. Different manifold diagrams (M, g) are obtained in explicit form for different choices of centering (2.3.23) and corresponding choices of f, g in (2.3.24). In the following, we will briefly discuss the double zero coordinate system that spans the cosmological horizons and extends into neighboring regions, especially into the future expanding region.
Moreover, the future time-like boundary r = ∞ is identified with the space-like hyperbola uv= 1. 2.3.36b) the two components of the past boundary of the expanding region r > rC. The metric g on the diagram, which covers the region rH < r < ∞ and extends over the cosmological horizon r=rC, thus has the double zero coordinates (u, v) form. We must think of (2.3.41) as the decomposition of the metric g in the expanding region uv >0 with respect to the level sets of the time function r.
The vector field (2.3.42) extends to a global vector field that characterizes the cosmological horizon as a Death Horizon with positive surface gravity.
Energy estimates
- Energy estimate in the expanding region
- Redshift vectorfield on the cosmological horizon
In this section, we construct a vector field that captures the local redshift effect of the cosmological horizon. However, it remains to show that T+Y is time-like near the cosmological horizon. However, the discussion that follows has implications for the study of the wave equation on perturbations of Minkowski space.
In this Appendix we give as an exercise a proof of "improved" internal decay of the first order energy for the wave equation on 3+1 dimensional Minkowski space. More precisely, given a solution φ of the wave equation φ = 0 with finite initial higher order energy D <∞, we show that. In the following we use the shorthand notation Z. A.27) Sketch of Proof (of the refinement of assumption A.4).
The proof of this fact is in the same way as Assumptions A.4, A.6 and not the focus of our interest here, but we include the proof of a simplified version of Lemma A.10 without the refinement in finite regions; the difference with the proof given here amounts to keeping track of the boundary terms in the Hardy inequalities, (cf. Sketch of the proof of Assumption A.6).
Formulas
Rational functions
Radial functions
Boundary Integrals and Hardy Inequalities
Coercivity inequality on the sphere
