[Paper Review] Conformal scattering theory for the linearized gravity fields on Schwarzschild spacetime

Pham Truong Xuan|arXiv (Cornell University)|May 25, 2020

Black Holes and Theoretical Physics34 references4 citations

TL;DR

This paper establishes the first conformal scattering theory for linearized gravity fields on Schwarzschild spacetime by leveraging the conformal geometric approach and recent decay results for Regge-Wheeler and Zerilli equations. By proving energy identities up to future timelike infinity and solving the Goursat problem on the conformal boundary, it demonstrates that the trace operator is an isomorphism, thereby constructing a unitary conformal scattering operator for gravitational perturbations.

ABSTRACT

We provide in this paper a first step to obtain the conformal scattering theory for the linearized gravity fields on the Schwarzschild spacetime by using the conformal geometric approach. We will show that the existing decay results for the solutions of the Regge-Wheeler and Zerilli equations obtained recently by L. Anderson, P. Blue and J. Wang \cite{ABlu} is sufficient to obtain the conformal scattering.

Motivation & Objective

• To develop a conformal scattering framework for linearized gravity fields on Schwarzschild spacetime.
• To extend the conformal scattering approach from scalar fields to gravitational perturbations governed by Regge-Wheeler and Zerilli equations.
• To establish the existence of a unitary scattering operator via energy identities and boundary trace analysis.
• To bridge conformal scattering with analytic scattering theory for gravitational fields in curved spacetime.

Proposed method

• Utilizes Penrose’s conformal compactification of the Schwarzschild exterior to define a globally foliated spacetime with hypersurfaces approaching null infinity and future timelike infinity.
• Applies the time-like vector field $ K $, equivalent to $ ilde{\partial}_t $ at large $ r $, to derive energy identities and control decay of solutions.
• Employs Leray’s theorem to solve the Cauchy problem for the rescaled Regge-Wheeler and Zerilli equations on the compactified spacetime.
• Uses energy and pointwise decay estimates from Anderson, Blue, and Wang [1] to prove the energy identity up to $ i^+ $, ensuring injectivity of the trace operator.
• Solves the Goursat problem on the conformal boundary $ \mathcal{H}^+ \cup \mathscr{I}^+ $ with data from radiation fields to establish surjectivity of the trace operator.
• Defines the conformal scattering operator $ S = \mathcal{T}^+ \circ (\mathcal{T}^-)^{-1} $, proving it is unitary via isomorphism of the trace maps.

Experimental results

Research questions

• RQ1 Can conformal scattering theory be constructed for linearized gravity fields on Schwarzschild spacetime using the conformal geometric method?
• RQ2 What decay properties of Regge-Wheeler and Zerilli fields are sufficient to ensure energy identities up to $ i^+ $?
• RQ3 Is the trace operator from initial data to conformal boundary data (horizon and null infinity) an isomorphism, ensuring unitarity of the scattering operator?
• RQ4 How does the conformal scattering framework relate to analytic scattering theory for gravitational fields?

Key findings

• The energy identity up to $ i^+ $ is established using the decay results of Anderson, Blue, and Wang [1], confirming that the trace operator is injective.
• The Goursat problem on the conformal boundary $ \mathcal{H}^+ \cup \mathscr{I}^+ $ is solved, proving that the trace operator is surjective.
• The trace operator $ \mathcal{T}^+ $ is shown to be an isomorphism between the initial data space and the scattering data space on the conformal boundary.
• The conformal scattering operator $ S = \mathcal{T}^+ \circ (\mathcal{T}^-)^{-1} $ is unitary, establishing a complete scattering theory for linearized gravity on Schwarzschild spacetime.
• The method generalizes to other spherically symmetric spacetimes like Reissner-Nordström-de Sitter, provided decay estimates for the relevant equations are available.
• The construction provides a geometric bridge between conformal scattering and analytic scattering, with implications for wave operators in gravitational perturbation theory.