[Paper Review] The wave equation on asymptotically de Sitter-like spaces

András Vasy|ArXiv.org|Jun 25, 2007

Geometric Analysis and Curvature Flows13 references4 citations

TL;DR

This paper establishes the asymptotic behavior of solutions to the Klein-Gordon equation on asymptotically de Sitter-like spacetimes, proving that solutions admit a specific singular behavior near future and past infinity, and constructing a scattering operator that is a Fourier integral operator associated with the bicharacteristic flow. The key result is the identification of the scattering operator as an invertible, elliptic Fourier integral operator under non-resonant conditions on the mass parameter.

ABSTRACT

In this paper we obtain the asymptotic behavior of solutions of the Klein-Gordon equation on Lorentzian manifolds $(X^\circ,g)$ which are de Sitter-like at infinity. Such manifolds are Lorentzian analogues of the so-called Riemannian conformally compact (or asymptotically hyperbolic) spaces. Under global assumptions on the (null)bicharacteristic flow, namely that the boundary of the compactification X is a union of two disjoint manifolds, Y+ and Y-, and each bicharacteristic converges to one of these two manifolds as the parameter along the bicharacteristic goes to plus infinity, and to the other manifold as the parameter goes to minus infinity, we also define the scattering operator, and show that it is a Fourier integral operator associated to the bicharacteristic flow from Y+ to Y-.

Motivation & Objective

• To analyze the asymptotic behavior of solutions to the Klein-Gordon equation on Lorentzian manifolds that are asymptotically de Sitter-like at infinity.
• To define and characterize the scattering operator in this geometric setting, extending the framework of scattering theory to Lorentzian analogues of asymptotically hyperbolic spaces.
• To show that under global bicharacteristic flow assumptions, the scattering operator is a Fourier integral operator associated with the classical scattering map.
• To establish the existence and uniqueness of solutions with prescribed asymptotic data at future and past infinity, even in the presence of logarithmic terms when the mass parameter leads to integer differences in exponents.

Proposed method

• The analysis uses microlocal methods, particularly the study of bicharacteristics of the wave operator in the characteristic set, under global assumptions on the flow behavior.
• The paper constructs a global time function T on the compactified manifold X, which induces a fibration X ≅ [-1,1] × S and ensures strict hyperbolicity of the operator P.
• Solutions to Pu = 0 are decomposed as u = x^{s_+(λ)}v_+ + x^{s_-(λ)}v_-, with v_± smooth on X, or involving log x terms when s_+(λ) - s_-(λ) is an integer.
• The scattering operator S is defined by mapping Cauchy data at Y_+ to the boundary values of v_+ and v_- at Y_-, and is renormalized to ensure equal weight for both data components.
• The principal symbol of the scattering operator is computed via propagation of singularities and normal operators at the front face of a blow-up space, showing ellipticity when s_+(λ) - s_-(λ) is not an even integer.
• The scattering operator is shown to be a Fourier integral operator via composition of FIOs associated with the bicharacteristic flow, using Hörmander’s transversality theorem.

Experimental results

Research questions

• RQ1 How do solutions to the Klein-Gordon equation behave asymptotically near the future and past boundaries of asymptotically de Sitter-like spacetimes?
• RQ2 Under what conditions is the scattering operator well-defined and invertible in this Lorentzian setting?
• RQ3 Can the scattering operator be characterized as a Fourier integral operator, and what is its canonical relation?
• RQ4 What happens to the solution structure when the exponent difference s_+(λ) - s_-(λ) is an integer or half-integer?
• RQ5 How does the renormalization of the scattering operator affect its ellipticity and FIO structure?

Key findings

• Solutions to the Klein-Gordon equation Pu = 0 on asymptotically de Sitter-like spacetimes admit a decomposition u = x^{s_+(λ)}v_+ + x^{s_-(λ)}v_- with v_± ∈ C^∞(X), provided s_+(λ) - s_-(λ) is not an integer.
• When s_+(λ) - s_-(λ) is an integer, the solution includes a logarithmic term: v_- ∈ C^∞(X) + x^{s_+(λ)-s_-(λ)} log x · C^∞(X).
• For the critical case λ = (n−1)²/4, the solution takes the form u = x^{(n−1)/2}v_+ + x^{(n−1)/2} log x · v_-, with v_± ∈ C^∞(X).
• The scattering operator S: C^∞(Y_+) × C^∞(Y_+) → C^∞(Y_-) × C^∞(Y_-) is well-defined and invertible, mapping Cauchy data at Y_+ to boundary values at Y_-.
• The scattering operator is an invertible, elliptic 0th-order Fourier integral operator with canonical relation given by the classical scattering map S_cl.
• The principal symbol of the scattering operator is shown to be elliptic when s_+(λ) - s_-(λ) is not an even integer, via computation of the ratio of normal operators at the front face of the blow-up space.