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
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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
}
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 -