Global Nonlinear Stability of Schwarzschild Spacetime under Polarized Perturbations
Jérémie Szeftel, Sergiu Klainerman
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Naturwissenschaften, Medizin, Informatik, Technik / Mathematik
Essential mathematical insights into one of the most important and challenging open problems in general relativity—the stability of black holes
One of the major outstanding questions about black holes is whether they remain stable when subject to small perturbations. An affirmative answer to this question would provide strong theoretical support for the physical reality of black holes. In this book, Sergiu Klainerman and Jérémie Szeftel take a first important step toward solving the fundamental black hole stability problem in general relativity by establishing the stability of nonrotating black holes—or Schwarzschild spacetimes—under so-called polarized perturbations. This restriction ensures that the final state of evolution is itself a Schwarzschild space. Building on the remarkable advances made in the past fifteen years in establishing quantitative linear stability, Klainerman and Szeftel introduce a series of new ideas to deal with the strongly nonlinear, covariant features of the Einstein equations. Most preeminent among them is the general covariant modulation (GCM) procedure that allows them to determine the center of mass frame and the mass of the final black hole state. Essential reading for mathematicians and physicists alike, this book introduces a rich theoretical framework relevant to situations such as the full setting of the Kerr stability conjecture.
Stationary spacetime, Variable (mathematics), Penrose diagram, Tensor, Fourier series, Initial value formulation (general relativity), Monotonic function, Analytic function, Wave equation, Linear stability, Angular momentum, Cosmological constant, Curvature tensor, Einstein tensor, General relativity, Gauge theory, Scalar curvature, Linear equation, Compactification (mathematics), Lagrangian (field theory), Photon sphere, Einstein field equations, Convection–diffusion equation, Equation, Existential quantification, Topology, Ricci curvature, Space form, Schwarzschild coordinates, Support (mathematics), Coercivity, Tangent vector, Parameter, Stress–energy tensor, Causal structure, Scientific notation, Lie derivative, Geodesic, Implicit function, Renormalization, Commutator, Foliation, Second fundamental form, Pullback, Linearization, Null vector, Cauchy distribution, Hypersurface, Dimension, Boundedness, Cauchy problem, Eigenvalues and eigenvectors, Iterative method, Big O notation, Linear differential equation, Orbital stability, I0, Calculation, Estimation, Error term, Nonlinear system, Special case, Integration by parts, Initial value problem, Lyapunov stability, A priori estimate, Lorentz transformation, Transition function, Riemann curvature tensor, Derivative, Tangent space, Schwarzschild metric, Null hypersurface, Three-dimensional space (mathematics), Covariant derivative, Hodge dual, Metric tensor (general relativity), Quantity, Exterior (topology), Cauchy horizon, Partial differential equation, Pseudo-Riemannian manifold, Main effect, Mathematics, Symmetry group, Kerr metric, Coefficient, Geodesics in general relativity, Center of mass (relativistic), Curvature invariant (general relativity), Vector field, Iteration, Conjecture, Theorem, Minkowski space, Curvature, Simultaneous equations, Diffeomorphism, Coordinate system, Initial condition