The Zel'dovich Approximation and the Relativistic Hamilton-Jacobi Equation
arXiv:astro-ph/9403053 · doi:10.1093/mnras/271.4.1005
Abstract
Beginning with a relativistic action principle for the irrotational flow of collisionless matter, we compute higher order corrections to the Zel'dovich approximation by deriving a nonlinear Hamilton-Jacobi equation for the velocity potential. It is shown that the velocity of the field may always be derived from a potential which however may be a multi-valued function of the space-time coordinates. In the Newtonian limit, the results are nonlocal because one must solve the Newton-Poisson equation. By considering the Hamilton-Jacobi equation for general relativity, we set up gauge-invariant equations which respect causality. A spatial gradient expansion leads to simple and useful results which are local --- they require only derivatives of the initial gravitational potential.
25 pages, DAMTP R94/6, ALBERTA THY/06-94
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