Constructing hyperbolic systems in the Ashtekar formulation of general relativity

Details Constructing hyperbolic systems in the Ashtekar formulation of general relativity Constructing hyperbolic systems in the Ashtekar formulation of general relativity

1999-01-19

arxiv.org/abs/gr-qc/9901053v4

Int.J.Mod.Phys. D9 (2000) 13-34

Astronomy and Astrophysics

Astrophysics

General Relativity and Quantum Cosmology

15 pages, RevTeX, minor changes in Introduction. published as Int. J. Mod. Phys. D 9 (2000) 13

Scientific paper

10.1142/S0218271800000037

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

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