The Zel'dovich approximation and the relativistic Hamilton-Jacobi equation

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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 non-linear 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 multivalued function of the space–time coordinates. In the Newtonian limit, the results are non-local 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 that are local – they require only derivatives of the initial gravitational potential.
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