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 $f(T_{\mu \nu }{T}^{\mu \nu })=\alpha (T_{\mu \nu }{T}^{\mu \nu }{)}^{\eta }$ with $\alpha$ and $\eta$ being constants. We have shown that EMPG with $\eta \ge 0$ 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, $M_{\mathrm{ast}}$, the mass of an astrophysical object inferred from astronomical observations such as planetary orbits and deflection of light, corresponds to the effective mass $M_{\mathrm{eff}}(\alpha ,\eta ,M)=M+M_{\mathrm{empg}}(\alpha ,\eta ,M)$, $M$ being the actual physical mass and $M_{\mathrm{empg}}$ being the modification due to EMPG. Accordingly, while in GR we simply have the relation $M_{\mathrm{ast}}=M$, in EMPG we have $M_{\mathrm{ast}}=M+M_{\mathrm{empg}}$. 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 $M_{\mathrm{ast}}$ alone from astronomical observations, but $M$ and $M_{\mathrm{empg}}$ 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 $|p_{\mathrm{eff}}/{\rho }_{\mathrm{eff}}|\ll 1$ (where $p_{\mathrm{eff}}=p+p_{\mathrm{empg}}$ and ${\rho }_{\mathrm{eff}}=\rho +{\rho }_{\mathrm{empg}}$), whereas this condition leads to $|p/\rho |\ll 1$ in GR.

arXiv comment13 pages, no figures and tables; matches the version published in Physics of the Dark Universe

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