Generalization of the f(R) Gravity Field Equations in Four-Index Form

Although there have been extensive studies on f(R) gravity, Gauss-Bonnet gravity, and other modified gravity theories, this work focuses on the 4-index formulation of f(R) gravity. A 2017 study on Generalization of Einstein's Gravitational Field Equations introduced a 4-index formulation of general relativity by generalizing the Einstein-Hilbert Lagrangian, introducing new scalars constructed from the Riemann and Ricci tensors, which ultimately reduce to the Ricci scalar. Here, the Lagrangian is generalized to an arbitrary function f(R), leading to the 4-index f(R) gravitational field equations. Furthermore, this formulation is employed to derive the effective energy-momentum tensor and the modified Weyl field equations. By imposing the conservation of the total energy-momentum tensor on the 4-index field equations, we obtain Compatibility equations that are consistent with the 4-index general relativity formulation and the covariant conservation law. Lastly, maximally symmetric spacetime and power-law cosmology were investigated, where the curvature condition and the scale factor, respectively, successfully passed the consistency check against established results.