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Stability of the Schwarzschild Family of Solutions in General Relativity

Chaman Motlagh, Abolfazl | 2023

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 56379 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Safdari, Mohammad
  7. Abstract:
  8. In 1952, Choquet-Bruhat proved the well-posedness of the Cauchy problem for Einstein’s equation, and demonstrated that a given initial data on a Cauchy hypersurface in spacetime propagates forward in time as a solution to the wave equation. It took nearly half a century for mathematicians to build the necessary tools and techniques for proving the stability of the most basic solution of Einstein’s vacuum equations (EVE), namely Minkowski spacetime. In 1993, Christodoulou and Klainerman showed the global nonlinear stability of this solution using the concept of a double null gauge. The Schwarzschild solution, introduced by Karl Schwarzschild in 1916 for spherically symmetric spacetimes as an exact solution to the EVE. This one-parameter family of solutions, known as the simplest nontrivial exact solution of the EVE, has been one of the most important solutions in general relativity, which introduces significant physical objects like black holes and spacetime singularities. In 2016, Dafermos, Rodnianski, and Holzegel, in their continued studies on the behavior of wave equations on Schwarzschild backgeound, or more generally on spacetimes containing a black hole region, proved the linear stability of the Schwarzschild family of solutions using the double null gauge. In 2021, continuing this line of research jointly with Martin Taylor, they also proved the nonlinear stability of Schwarzschild. The linear stability of Schwarzschild solution was also proven in the harmonic gauge by Hang, Keller, and Wang in 2017, and in the generalized gauge by Johnson in 2019. In this thesis, we aim to describe the problem of Schwarzschild stability, present stability theorems in the double null gauge framework, and provide a proof outline for these theorems, focusing on the linear problem
  9. Keywords:
  10. General Relativity Theory ; Solutions to Differential Equation Stability ; Cauchy Problem ; Einstein Vaccum Equation ; Schwarzschild Spacetime ; Double Null Gauge

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