A hypothesis of anti-gravity between matter and antimatter is presented that results in an alternative to the conventional Einstein field equations. Using the vacuum metric relationships identified by Schwarzschild in 1916, the radially symmetric vacuum anti-gravity metric is derived for the vacuum between a spherical core of either matter or anti-matter and an enclosing spherical shell of the other type of matter; anti-gravity prevents the shell from collapsing. Candidate black holes are hypothesized to consist of such a shell and core, and the gravitational behaviors of such a composite body are approximated. Observations of kinematics consistent with these behaviors would validate the existence of anti-gravity. (Version 4: Corrected equations 11 and 18a; added equation 18b, showing that the formulation of gRR is unchanged by the alternate boundary conditions.) (Version 3: Revised discussion before equation 16.) (Version 2: Revised 2nd paragraph section II; edits to equations 15a, 16, 18a.)