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

Abstract

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 $f(T_{μν}T^{μν})=α(T_{μν}T^{μν})^η$ with $α$ and $η$ being constants. We have shown that EMPG with $η\geq0$ 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, $M_{\rm ast}$, the mass of an astrophysical object inferred from astronomical observations such as planetary orbits and deflection of light, corresponds to the effective mass $M_{\rm eff}(α,η,M)=M+M_{\rm empg}(α,η,M)$, $M$ being the actual physical mass and $M_{\rm empg}$ being the modification due to EMPG. Accordingly, while in GR we simply have the relation $M_{\rm ast}=M$, in EMPG we have $M_{\rm ast}=M+M_{\rm empg}$. 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 $M_{\rm ast}$ alone from astronomical observations, but $M$ and $M_{\rm empg}$ 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 $|p_{\rm eff}/ρ_{\rm eff}|\ll1$ (where $p_{\rm eff}=p+p_{\rm empg}$ and $ρ_{\rm eff}=ρ+ρ_{\rm empg}$), whereas this condition leads to $|p/ρ|\ll1$ in GR.13 pages, no figures and tables; matches the version published in Physics of the Dark Universe