Weak field and slow motion limits in energy-momentum powered gravity

We explore the weak field and slow motion limits, Newtonian and Post-Newtonian limits, of the energy-momentum powered gravity (EMPG), viz., the energy-momentum squared gravity (EMSG) of the form f (T mu nu T (mu nu)) = alpha(T mu nu T (mu nu))(eta) with a and. being constants. We have shown that EMPG with. = 0 and general relativity (GR) are not distinguishable by local tests, say, the Solar System tests; as they lead to the same gravitational potential form, PPN parameters, and geodesics for the test particles. However, within the EMPG framework, Mast, the mass of an astrophysical object inferred from astronomical observations such as planetary orbits and deflection of light, corresponds to the effective mass M-eff( alpha,eta, M) = M + M-empg( alpha,eta, M), M being the actual physical mass and Mempg being the modification due to EMPG. Accordingly, while in GR we simply have the relation Mast = M, in EMPG we have Mast = M + Mempg. Within the framework of EMPG, if there is information about the values of {alpha,eta} pair or M from other independent phenomena (from cosmological observations, structure of the astrophysical object, etc.), then in principle it is possible to infer not only Mast alone from astronomical observations, but M and Mempg separately. For a proper analysis within EMPG framework, it is necessary to describe the slow motion condition (also related to the Newtonian limit approximation) by |p(eff)/.(eff)| << 1 (where p(eff) = p+ p(empg) and rho(eff) =rho+rho empg), whereas this condition leads to |p/rho| << 1 in GR.