  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.
