Second post-Newtonian approximation of scalar-tensor theory of gravity

Deep space laser ranging missions like ASTROD I (Single-Spacecraft Astrodynamical Space Test of Relativity using Optical Devices) and ASTROD, together with astrometry missions like GAIA and LATOR will be able to test relativistic gravity to an unprecedented level of accuracy. More precisely, these missions will enable us to test relativistic gravity to 10^-7–10^-9${10}^{-7}''–''{10}^{-9}$ of the size of relativistic (post-Newtonian) effects, and will require second post-Newtonian approximation of relevant theories of gravity. The first post-Newtonian approximation is valid to 10^-6${10}^{-6}$ and the second post-Newtonian approximation is valid to 10^-12${10}^{-12}$ in terms of post-Newtonian effects in the solar system. The scalar-tensor theory is widely discussed and used in tests of relativistic gravity, especially after the interests in inflation models and in dark energy models. In the Lagrangian, intermediate-range gravity term has a similar form as cosmological term. Here we present the full second post-Newtonian approximation of the scalar-tensor theory including viable examples of intermediate-range gravity. We use Chandrasekhar’s approach to derive the metric coefficients and the equation of the hydrodynamics governing a perfect fluid in the second post-Newtonian approximation in scalar-tensor theory; all terms inclusive of O(c^-4)$\mathrm{O}(c^{-4})$ are retained consistently in the equations of motion.