Evolution equations for a wide range of Einstein-matter systems
- PDF / 514,896 Bytes
- 36 Pages / 439.37 x 666.142 pts Page_size
- 41 Downloads / 181 Views
Evolution equations for a wide range of Einstein-matter systems M. Normann1
· J. A. Valiente Kroon1
Received: 5 June 2020 / Accepted: 10 October 2020 © The Author(s) 2020
Abstract We use an orthonormal frame approach to provide a general framework for the first order hyperbolic reduction of the Einstein equations coupled to a fairly generic class of matter models. Our analysis covers the special cases of dust and perfect fluid. We also provide a discussion of self-gravitating elastic matter. The frame is Fermi–Walker propagated and coordinates are chosen such as to satisfy the Lagrange condition. We show the propagation of the constraints of the Einstein-matter system.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . Overview of the article . . . . . . . . . . . . . . . . . . . Notation and conventions . . . . . . . . . . . . . . . . . 2 Geometric background . . . . . . . . . . . . . . . . . . . 3 The Einstein equations . . . . . . . . . . . . . . . . . . . A projection formalism . . . . . . . . . . . . . . . . . . 4 Gauge considerations . . . . . . . . . . . . . . . . . . . 5 Zero-quantities . . . . . . . . . . . . . . . . . . . . . . . 6 Evolution equations . . . . . . . . . . . . . . . . . . . . 6.1 Equations for the components of the frame . . . . . . 6.2 Evolution equations for the connection coefficients . 6.3 Evolution equations for the decomposed Z -tensor . . 6.4 Evolution equations for the decomposed Weyl tensor 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . 7 Propagation equations . . . . . . . . . . . . . . . . . . . 7.1 Propagation of divergence-free condition . . . . . . . 7.2 Propagation equations for the torsion . . . . . . . . . 7.3 Propagation equations for the geometric curvature . .
B
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . . . . . .
M. Normann [email protected] J. A. Valiente Kroon [email protected]
1
School of Mathematical Sciences Queen Mary, University of London, Mile End Road, London E1 4NS, UK 0123456789().: V,-vol
123
103
Page 2 of 36
7.4 Propagation of the N-tensor . . . . . . . . . . . . . . . . . . . 7.5 Propagation equations for the Bianchi identity . . . . . . . . . 7.6 Main theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 8 M
Data Loading...
