Rishabh jha - Academia.edu (original) (raw)
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Papers by Rishabh jha
arXiv (Cornell University), Apr 6, 2022
General relativity is traditionally introduced in the covariant (Lagrangian) formalism. Just like... more General relativity is traditionally introduced in the covariant (Lagrangian) formalism. Just like in classical mechanics, there exists an equivalent description of general relativity using the Hamiltonian approach. The purpose of the present work is to give a pedagogical introduction to this formalism where we stick to classical Physics throughout. We build towards this goal by covering the mathematical prerequisites as well as the 3 + 1−decomposition of spacetime before discussing the Arnowitt-Deser-Misner (ADM) formalism (a Hamiltonian approach) of general relativity. Then we proceed to give a brief but self-contained introduction to homogeneous (but not necessarily isotropic) universes and discuss the associated Bianchi classification. We first study their dynamics in the Lagrangian formulation, followed by the Hamiltonian formulation to show the equivalence of both approaches. In order to get hold of the ADM formalism, we present a variety of examples: (i) free & massless scalar field coupled to homogeneous (in particular, Bianchi IX) universe, (ii) scalar field with a potential term coupled to Bianchi IX universe, (iii) electromagnetic field coupled to gravity in general, and (iv) electromagnetic field coupled to Bianchi IX universe. Through these examples, we also introduce the associated important concepts & the dynamics of the system in consideration using the Hamiltonian approach. We have tried to be detailed and self-contained in this work but we do expect as a prerequisite to have familiarity with the basic concepts in general relativity up to the point of deriving the FRWL cosmological solution from the Einstein field equations.
arXiv (Cornell University), Apr 6, 2022
General relativity is traditionally introduced in the covariant (Lagrangian) formalism. Just like... more General relativity is traditionally introduced in the covariant (Lagrangian) formalism. Just like in classical mechanics, there exists an equivalent description of general relativity using the Hamiltonian approach. The purpose of the present work is to give a pedagogical introduction to this formalism where we stick to classical Physics throughout. We build towards this goal by covering the mathematical prerequisites as well as the 3 + 1−decomposition of spacetime before discussing the Arnowitt-Deser-Misner (ADM) formalism (a Hamiltonian approach) of general relativity. Then we proceed to give a brief but self-contained introduction to homogeneous (but not necessarily isotropic) universes and discuss the associated Bianchi classification. We first study their dynamics in the Lagrangian formulation, followed by the Hamiltonian formulation to show the equivalence of both approaches. In order to get hold of the ADM formalism, we present a variety of examples: (i) free & massless scalar field coupled to homogeneous (in particular, Bianchi IX) universe, (ii) scalar field with a potential term coupled to Bianchi IX universe, (iii) electromagnetic field coupled to gravity in general, and (iv) electromagnetic field coupled to Bianchi IX universe. Through these examples, we also introduce the associated important concepts & the dynamics of the system in consideration using the Hamiltonian approach. We have tried to be detailed and self-contained in this work but we do expect as a prerequisite to have familiarity with the basic concepts in general relativity up to the point of deriving the FRWL cosmological solution from the Einstein field equations.