The Zel'dovich Approximation and the Relativistic Hamilton-Jacobi Equation

Details The Zel'dovich Approximation and the Relativistic Hamilton-Jacobi Equation The Zel'dovich Approximation and the Relativistic Hamilton-Jacobi Equation

1994-03-23

arxiv.org/abs/astro-ph/9403053v1

Mon.Not.Roy.Astron.Soc. 271 (1994) 1005

Astronomy and Astrophysics

Astrophysics

25 pages, DAMTP R94/6, ALBERTA THY/06-94

Scientific paper

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.

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