Null second order corrections to Casimir energy in weak gravitational field: the Schwinger's approach

We show through Schwinger's approach that in a static weak gravitational background, the Casimir Energy for a real massless scalar field obeying Dirichlet boundary conditions on rectangular plates is unaltered from it's flat space-time value, at least to second order in the gravitational parameters. The result is obtained considering an rather general class of backgrounds, adding further generality and consistency test for the previous work. The proposed result has direct consequences on earlier works in the literature that found gravity related corrections for similar setups and in the measurability of future vacuum weight experiments. We also investigate higher order corrections using a simplified toy model geometry, finding nonzero corrections in this case.