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Dynamical and Hamiltonian Formulation of General Relativity

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Springer Handbook of Spacetime

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Abstract

Einstein’s theory of General Relativity describes spacetime as a solution of a set of non-linear partial differential equations. These equations are initially not in the form of evolution equations and it is hence not clear how to formulate and solve initial-value problems, as would be physically highly desirable. In this contribution it will be shown how to cast Einstein’s equations into the form of a constrained Hamiltonian system. This will allow to formulate and solve initial-value problems, integrate Einstein’s equations by numerical codes, characterize dynamical degrees of freedom, and characterize isolated systems and their conserved quantities, like energy, momentum, and angular momentum. Moreover, this reformulation of General Relativity is also the starting point for various attempts to subject the gravitational field to the program of canonical quantization. The exposition given here is, to some degree, self contained. It attempts to comprehensively account for all the relevant geometric constructions, including the relevant symplectic geometry of constrained Hamiltonian systems.

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Abbreviations

ADM:

Arnowitt, Deser, Misner

GR:

general relativity

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Authors and Affiliations

  1. Institut für Theoretische Physik, Leibniz Universität Hannover, Appelstrasse 2, 30167, Hannover, Germany

    Domenico Giulini Prof.

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  1. Domenico Giulini Prof.
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Correspondence to Domenico Giulini Prof. .

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Editors and Affiliations

  1. Department of Physics, Pennsylvania State University, 16802, University Park, PA, USA

    Abhay Ashtekar

  2. Institute for Foundational Studies Hermann Minkowski, Montreal, Quebec, Canada

    Vesselin Petkov

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Giulini, D. (2014). Dynamical and Hamiltonian Formulation of General Relativity. In: Ashtekar, A., Petkov, V. (eds) Springer Handbook of Spacetime. Springer Handbooks. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-41992-8_17

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  • DOI: https://doi.org/10.1007/978-3-642-41992-8_17

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