In this study, we first establish that gravity models incorporating matter-related terms, such as fðLm Þ, fðgμν T μν Þ, and fðT μν T μν Þ, into the usual matter Lagrangian density Lm, are equivalent to general relativity (GR) with nonminimal matter interactions. Through the redefinition Lm þ f → Ltot m , these models are exactly GR, yet the usual material field T μν and its accompanying partner, namely, the modification μ μ mod field T mod μν , engage in nonminimal interactions. Specifically, ∇ T μν ¼ −Qν ¼ −∇ T μν , where Qν is the interaction kernel that governs the rate of energy transfer. Our focus narrows on the specific model of fðT μν T μν Þ, known as energy-momentum squared gravity (EMSG), where the usual material field T μν is accompanied by an energy-momentum squared field (EMSF), T emsf μν , along with a sui generis nonminimal 2 can be introduced by removing ∂g∂μνL∂gmσϵ interaction between them. We demonstrate that a particular T emsf μν (the new term emerging in models that incorporate scalars formed from T μν ), thanks to the freedom in determining the interaction kernel, but this approach compromises the Lagrangian formulation of EMSG. Additionally, we address the ambiguities regarding the perfect fluid stemming from this new term. We show the proper way of calculating this term for a perfect fluid, revealing that it is indeed nonzero, contrary to common assumption in the literature. Finally, we reexamine cosmological models within the realm of EMSG, offering new insights into the applicability and interpretation of our findings in EMSG and similar theoretical frameworks.

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