The Weak Field Limit of Higher Order Gravity

TL;DR

This paper investigates the weak field limit of higher order gravity theories, revealing modifications to gravitational potential and propagations, and compares these with General Relativity to understand their astrophysical implications.

Contribution

It analyzes the weak field limit of $f(R, R_{\alpha\beta}R^{\alpha\beta})$ theories, deriving spherically symmetric solutions and identifying deviations from General Relativity.

Findings

Gravitational potential exhibits Yukawa-like modifications.

Weak field solutions show massive propagation effects.

Reduces to General Relativity when $f$ is the Einstein-Hilbert Lagrangian.

Abstract

The Higher Order Theories of Gravity - $f(R, R_{\alpha\beta}R^{\alpha\beta})$ - theory, where $R$ is the Ricci scalar, $R_{\alpha\beta}$ is the Ricci tensor and $f$ is any analytic function - have recently attracted a lot of interest as alternative candidates to explain the observed cosmic acceleration, the flatness of the rotation curves of spiral galaxies and other relevant astrophysical phenomena. It is a crucial point testing these alternative theories in the so called weak field and newtonian limit of a $f(R, R_{\alpha\beta}R^{\alpha\beta})$ - theory. With this "perturbation technique" it is possible to find spherically symmetric solutions and compare them with the ones of General Relativity. On both approaches we found a modification of General Relativity: the behaviour of gravitational potential presents a modification Yukawa - like in the newtonian case and a massive propagation in the weak field case. When the modification of the theory is removed (i.e. $f(R, R_{\alpha\beta}R^{\alpha\beta}) = R$, Hilbert - Einstein lagrangian) we find the usual outcomes of General Relativity. Also the Noether symmetries technique has been investigated to find some time independent spherically symmetric solutions.