The most general fourth order theory of Gravity at low energy

TL;DR

This paper derives the Newtonian limit of the most general fourth order gravity theory involving multiple curvature invariants, revealing new characteristic lengths and constraints on the theory's functions.

Contribution

It generalizes fourth order gravity to include Ricci and Riemann square invariants and analyzes their Newtonian limit and spherically symmetric solutions.

Findings

Spherically symmetric solutions exhibit two Yukawa-like characteristic lengths.

At Newtonian order, all curvature invariant functions yield similar outcomes to quadratic gravity.

Gauss-Bonnet invariant clarifies solution interpretation and excludes a third characteristic length.

Abstract

The Newtonian limit of the most general fourth order gravity is performed with metric approach in the Jordan frame with no gauge condition. The most general theory with fourth order differential equations is obtained by generalizing the $f(R)$ term in the action with a generic function containing other two curvature invariants: \emph{Ricci square} ($R_{\alpha\beta}R^{\alpha\beta}$) and \emph{Riemann square} ($R_{\alpha\beta\gamma\delta}R^{\alpha\beta\gamma\delta}$). The spherically symmetric solutions of metric tensor yet present Yukawa-like spatial behavior, but now one has two characteristic lengths. At Newtonian order any function of curvature invariants gives us the same outcome like the so-called \emph{Quadratic Lagrangian} of Gravity. From Gauss - Bonnet invariant one have the complete interpretation of solutions and the absence of a possible third characteristic length linked to Riemann square contribution. From analysis of metric potentials, generated by point-like source, one has a constraint condition on the derivatives of $f$ with respect to scalar invariants.