Minimal no-radiation approximation to Einstein’s field equations

An approximation to Einstein’s field equations in the Arnowitt-Deser-Misner canonical formalism is presented, which corresponds to the magnetohydrodynamics 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 nondissipative 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 realistic initial data for general relativistic simulations as well as to model astrophysical scenarios, where the gravitational radiation reaction can be neglected.