Most general fourth-order theory of gravity at low energy

Abstract

The Newtonian limit of the most general fourth-order gravity is performed with the metric approach in the Jordan frame with no gauge condition. The most general theory with fourth-order differential equations is obtained by generalizing the fðRÞ term in the action with a generic function containing two other curvature invariants: the Ricci square (R R ) and the Riemann square (R R ). The spherically symmetric solutions of the metric tensor 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 quadratic Lagrangian of gravity. From the Gauss-Bonnet invariant one will have an incomplete interpretation of the solutions and the absence of a possible third characteristic length linked to the Riemann square contribution. From the analysis of metric potentials, generated by a pointlike source, one has a constraint condition on the derivatives of f with respect to scalar invariants.

The dark matter problem from f(R) gravity viewpoint

“…In order to deal with standard self-gravitating systems, any theory of gravity has to be developed to its Newtonian or post-Newtonian limit depending on the order of approximation of the theory in terms of power of c −2 , where c is the speed of light [49,60]. As we said above a R 2 -theory gives rise to Yukawa-like corrections in the Newtonian potential [61].…”

Gravitational waves in fourth order gravity

“…Both requirements are fulfilled by General Relativity and then they can be considered two possible paradigms to confront a given theory, at least in the weak field limit, with the General Relativity itself. The Newtonian limit of f (R)-gravity and f (R, R αβ R αβ , R αβγ δ R αβγ δ )-gravity can be investigated always remaining in the Jordan frame (Capozziello et al 2007Stabile 2010aStabile , 2010b while a preliminary study of the post-Minkowskian limit for the f (R) gravity is provided in Capozziello et al (2010b). Here we want to derive the post-Minkowskian limit of f (R, R αβ R αβ , R αβγ δ R αβγ δ ) gravity, that is for generic fourth-order theory of gravity, with the aim to investigate the gravitational radiation.…”

Casimir effect in extended theories of gravity

“…In order to perform our approximation we have to perturb Eqs. (2), (3) and (4) in a Minkowski background η µν [45,71]. This approximation is sometimes also referred to as an expansion in inverse powers of the speed of light.…”

“…while all other possible contributions in f are negligible [45,72,73]. By introducing the quantities…”

“…Models of fourth order gravity have been studied mainly in the Newtonian limit (weak-field and small velocity) [45,72], as well as in the Minkowskian limit [14]. In the former one finds modifications of the gravitational potential, whilst in the latter one obtains massive gravitational wave modes [15,16].…”