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Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations(AMS-210)$
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Jérémie Szeftel and Sergiu Klainerman

Print publication date: 2020

Print ISBN-13: 9780691212425

Published to Princeton Scholarship Online: May 2021

DOI: 10.23943/princeton/9780691212425.001.0001

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(p.24) Chapter Two Preliminaries
Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations

Sergiu Klainerman

Jérémie Szeftel

Princeton University Press

This chapter discusses the main quantities, equations, and basic tools needed in the following chapters. It is the main reference kit providing all main null structure and null Bianchi equations, in general null frames, in the context of axially symmetric polarized spacetimes. The chapter translates the null structure and null Bianchi identities associated to an S-foliation in the reduced picture. It starts with general, Z-invariant, S-foliation, before considering the special case of geodesic foliations. The chapter then looks at perturbations of Schwarzschild and invariant quantities. It investigates how the main Ricci and curvature quantities change relative to frame transformations, i.e., linear transformations which take null frames into null frames. Finally, the chapter presents wave equations for the invariant quantities.

Keywords:   null structure equations, null Bianchi equations, null frames, axially symmetric polarized spacetimes, geodesic foliations, Schwarzschild spacetimes, invariant quantities, wave equations

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