The most general fourth order theory of Gravity at low energy

Understand

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.