A minimal no-radiation approximation to Einstein's field equations

TL;DR

This paper introduces a simplified, no-radiation approximation to Einstein's equations using the ADM formalism, resulting in elliptic equations that facilitate realistic initial data construction and astrophysical modeling without gravitational radiation effects.

Contribution

It presents a novel approximation to Einstein's equations that simplifies the equations to elliptic form, generalizes previous truncations, and accurately reproduces stationary spacetimes.

Findings

Results in coupled elliptic equations with elliptic-type structure.

Hamiltonian matches non-dissipative Einsteinian Hamiltonian up to third post-Newtonian order.

Enables realistic initial data and astrophysical modeling without gravitational radiation.

Abstract

An approximation to Einstein's field equations in Arnowitt-Deser-Misner (ADM) canonical formalism is presented which corresponds to the magneto-hydrodynamics (MHD) approximation in electrodynamics. It results in coupled elliptic equations which represent the maximum of elliptic-type structure of Einstein's theory and naturally generalizes previous conformal-flat truncations of the theory. The Hamiltonian, in this approximation, is identical with the non-dissipative part of the Einsteinian one through the third post-Newtonian order. The proposed scheme, where stationary spacetimes are exactly reproduced, should be useful to construct {\em realistic} initial data for general relativistic simulations as well as to model astrophysical scenarios, where gravitational radiation reaction can be neglected.