Recent numerical simulations of coalescing binary neutron stars conducted by Wilson, Mathews and Marronetti (WMM) show a rising central energy density of the stars as the orbital separation shrinks, i.e. the stars are individually crushed as they near coalescence. They claim this ``star-crushing'' effect is partially due to a non-linear, first post-Newtonian order enhancement of the self-gravity of each star caused by the presence of the other star. We present a concrete calculation which shows, within general relativity, first post-Newtonian order interactions with the other star leave the central energy density unchanged as the orbital radius shrinks. The results presented here are in sharp disagreement with the WMM claim. However, alternative gravitational theories, such as Brans-Dicke theory, can exhibit a small crushing effect in the binary constituents as they near coalescence. We show that the absence of the star-crushing effect at first post-Newtonian order is related to adherence to the strong equivalence principle.

RevTex, submitted to PRL, 4 pages