## Details

Original language | English |
---|---|

Title of host publication | Springer Handbooks |

Subtitle of host publication | Spacetime |

Editors | Abhay Ashtekar, Vesselin Petkov |

Publisher | Springer Science and Business Media Deutschland GmbH |

Pages | 323-362 |

Number of pages | 40 |

ISBN (electronic) | 978-3-642-41992-8 |

ISBN (print) | 978-3-642-41991-1 |

Publication status | Published - Dec 2014 |

## Publication series

Name | Springer Handbooks |
---|---|

ISSN (Print) | 2522-8692 |

ISSN (electronic) | 2522-8706 |

## 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.

## Keywords

- Constrained Hamiltonian Systems, Constraint Vector, DeWitt Metric, Momentum Map, Weingarten Map

## ASJC Scopus subject areas

## Cite this

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**Dynamical and Hamiltonian Formulation of General Relativity.**/ Giulini, Domenico.

Springer Handbooks: Spacetime. ed. / Abhay Ashtekar; Vesselin Petkov. Springer Science and Business Media Deutschland GmbH, 2014. p. 323-362 (Springer Handbooks).

Research output: Chapter in book/report/conference proceeding › Contribution to book/anthology › Research › peer review

*Springer Handbooks: Spacetime.*Springer Handbooks, Springer Science and Business Media Deutschland GmbH, pp. 323-362. https://doi.org/10.48550/arXiv.1505.01403, https://doi.org/10.1007/978-3-642-41992-8_17

*Springer Handbooks: Spacetime*(pp. 323-362). (Springer Handbooks). Springer Science and Business Media Deutschland GmbH. https://doi.org/10.48550/arXiv.1505.01403, https://doi.org/10.1007/978-3-642-41992-8_17

}

TY - CHAP

T1 - Dynamical and Hamiltonian Formulation of General Relativity

AU - Giulini, Domenico

PY - 2014/12

Y1 - 2014/12

N2 - 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.

AB - 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.

KW - Constrained Hamiltonian Systems

KW - Constraint Vector

KW - DeWitt Metric

KW - Momentum Map

KW - Weingarten Map

UR - http://www.scopus.com/inward/record.url?scp=85134036910&partnerID=8YFLogxK

U2 - 10.48550/arXiv.1505.01403

DO - 10.48550/arXiv.1505.01403

M3 - Contribution to book/anthology

AN - SCOPUS:85134036910

SN - 978-3-642-41991-1

T3 - Springer Handbooks

SP - 323

EP - 362

BT - Springer Handbooks

A2 - Ashtekar, Abhay

A2 - Petkov, Vesselin

PB - Springer Science and Business Media Deutschland GmbH

ER -