[Paper Review] Morawetz estimate for linearized gravity in Schwarzschild

Lars Andersson, Pieter Blue+1 more|Edinburgh Research Explorer|Aug 23, 2017

Black Holes and Theoretical Physics3 citations

TL;DR

This paper establishes an integrated local energy decay estimate (Morawetz estimate) for linearized gravity in Schwarzschild spacetime using the Regge-Wheeler and Zerilli equations. By combining a Morawetz vector field with $r^p$ hierarchy estimates, the authors prove $t^{-3/2}$ pointwise decay for both Regge-Wheeler and Zerilli variables in fixed spatial regions, extending energy decay results to the linearized Einstein equations in a Schwarzschild background.

ABSTRACT

The equations governing the perturbations of the Schwarzschild metric satisfy the Regge-Wheeler-Zerilli-Moncrief system. Applying the technique introduced in [2], we prove an integrated local energy decay estimate for both the Regge-Wheeler and Zerilli equations. In these proofs, we use some constants that are computed numerically. Furthermore, we make use of the $r^p$ hierarchy estimates [13, 32] to prove that both the Regge-Wheeler and Zerilli variables decay as $t^{-\frac{3}{2}}$ in fixed regions of $r$.

Motivation & Objective

• To establish an integrated local energy decay estimate (Morawetz estimate) for the Regge-Wheeler and Zerilli equations in Schwarzschild spacetime.
• To prove pointwise decay rates for solutions to the linearized gravity equations in the exterior region of a Schwarzschild black hole.
• To extend the $r^p$ hierarchy method to the Regge-Wheeler and Zerilli systems to derive improved decay estimates.
• To demonstrate that solutions decay as $t^{-3/2}$ in fixed spatial regions, using energy estimates and interpolation techniques.
• To provide a high-order decay estimate for weighted derivatives of the perturbations, ensuring uniform decay under regular initial data.

Proposed method

• Application of the Morawetz vector field technique to derive integrated local energy decay for the Regge-Wheeler and Zerilli equations.
• Use of $r^p$ hierarchy estimates with $p>0$ to control energy decay and derive pointwise bounds.
• Employment of the $1+1+2$ covariant perturbation formalism and double null foliation to analyze the equations in Schwarzschild spacetime.
• Use of retarded and advanced Eddington-Finkelstein coordinates to handle the geometry near the event horizon.
• Application of interpolation and pigeonhole principle arguments to upgrade $L^2$ energy decay to pointwise decay in time.
• High-order energy estimates via weighted derivatives $\mathbb{D} = \{r\partial_v, (1-\mu)^{-1}\partial_u, r\mbox{$\nabla\mkern-13.0mu/$ $}$\}$.

Experimental results

Research questions

• RQ1 Can an integrated local energy decay estimate (Morawetz estimate) be established for the Regge-Wheeler and Zerilli equations in Schwarzschild spacetime?
• RQ2 What is the optimal pointwise decay rate for solutions to the linearized gravity equations in fixed spatial regions of Schwarzschild spacetime?
• RQ3 How can the $r^p$ hierarchy method be adapted to the Regge-Wheeler and Zerilli systems to achieve improved decay estimates?
• RQ4 Can high-order decay estimates be derived for weighted derivatives of the perturbations under regular initial data?
• RQ5 What role do numerical constants in the Morawetz estimate play in the final decay rate?

Key findings

• An integrated local energy decay estimate is proven for both the Regge-Wheeler and Zerilli equations using a Morawetz vector field and numerical constants.
• Solutions to the Regge-Wheeler and Zerilli equations decay pointwise as $t^{-3/2}$ in fixed spatial regions of $r$.
• The $r^p$ hierarchy method is successfully applied to derive $t^{-3/2}$ decay, improving upon earlier $t^{-1}$ and $t^{-1+\delta}$ results.
• High-order decay estimates are established: $\sup_{m\in\mathbb{N}} r^{1/2}|\mathbb{D}^m\psi| \lesssim \frac{I}{\tau}$ and $\sup_{m\in\mathbb{N}} r^{1/2}|\partial_t \mathbb{D}^m\psi| \lesssim \frac{I}{\tau^2}$.
• Improved interior decay is obtained: $\sup_{m\in\mathbb{N}} |\mathbb{D}^m\psi| \lesssim \frac{I}{\tau^{3/2}}$ for $r < R$.
• The results are valid under high-order initial data with finite weighted energy, ensuring uniform decay for all derivatives.