We investigate the post-Newtonian (PN) dynamics of energy-momentum squared gravity (EMSG), with particular emphasis on the possibility of self-acceleration in $N$-body systems. A central challenge in matter-type modified gravity theories, including EMSG, is the nonvanishing divergence of the energy-momentum tensor, arising from the nonminimal interaction between the standard and modified matter fields. This feature can, in principle, influence the $N$-body dynamics. In our previous work, its effects on the external-dependent part of the motion were studied in an EMSG class known as quadratic EMSG. Here, we extend the analysis to the internal-structure-dependent contributions, namely self-acceleration. To this end, we relax the reflection-symmetric assumption adopted in E. Nazari, Phys. Rev. D 110, 064023 (2024) and derive the complete equations of motion for a self-gravitating body in an N-body system up to the first PN order. By introducing a suitable expression for the center-of-mass acceleration and employing virial identities—including one newly emerging within the quadratic-EMSG framework—it is shown that self-acceleration vanishes. Furthermore, we establish a PN integral conservation law for the total momentum, demonstrating that, as in general relativity, EMSG admits a conserved linear momentum compatible with the absence of self-acceleration. Binary pulsar experiments provide stringent bounds on self-acceleration, and our analysis shows that, within the present level of accuracy, EMSG is consistent with these constraints. Therefore, the theory remains viable in the strong-gravity regime probed by binary pulsars.

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• Alternative gravity theories
• Gravitation