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_μνT^μν)=α(T_μνT^μν)^η with α and η being constants. We have shown that EMPG with η≥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, M_ast, 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(α,η,M)=M+M_empg(α,η,M), M being the actual physical mass and M_empg being the modification due to EMPG. Accordingly, while in GR we simply have the relation M_ast=M, in EMPG we have M_ast=M+M_empg. Within the framework of EMPG, if there is information about the values of {α,η} 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 M_ast alone from astronomical observations, but M and M_empg 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/ρ_eff|≪1 (where p_eff=p+p_empg and ρ_eff=ρ+ρ_empg), whereas this condition leads to |p/ρ|≪1 in GR.