|
2.00 pm, Lecture Hall 803 Stability in General Relativity using Symplectic Geometry (Reference: arXiv:1902.08219) Prashant Kocherlakota TIFR Mumbai. 14-05-19 Abstract Beginning with a review of the statements of non-linear, linear and mode stability of autonomous dynamical systems in classical mechanics, in the setting of symplectic geometry, I will briefly discuss what the phase space and the Hamiltonian of general relativity are, what constitutes a dynamical system, and subsequently draw a formal analogy between the notions of stability in these two theories. The approach here will be pedagogical and geometric, and simplifies a formal understanding of the statements regarding the stability of stationary solutions of general relativity. In particular, the governing equations of motion of a Hamiltonian dynamical system are simply the flow equations of the associated symplectic Hamiltonian vector field, defined on phase space, and the non-linear stability analysis of its critical points has to do with the divergence of its flow there. Further, the linear stability of a critical point is related to the properties of the tangent flow (or linearization) of the Hamiltonian vector field. On a related note, I will also discuss how the study of the genericity of the formation of a particular spacetime is equivalent to an inquiry of how sensitive the orbits of the symplectic ADM Hamiltonian vector field are to changes in initial data. I will demonstrate this statement by conducting a somewhat restricted non-linear stability analysis of the formation of a Schwarzschild black hole, working in the usual initial value formulation of general relativity.
|
