# Gravitoelectromagnetism in metric f(R) and Brans-Dicke theories with a potential

## TL;DR

This paper develops a gravitoelectromagnetism formalism within metric f(R) and Brans-Dicke theories, revealing how scalar potentials influence deviations from General Relativity and analyzing gravitational time delay effects.

## Contribution

It introduces a formalism for gravitoelectromagnetism in metric f(R) and Brans-Dicke theories, deriving the Lorentz force law and gravitational time delay, and explores the effects of scalar potentials.

## Key findings

- Deviations from GR depend on the scalar potential's absolute value.
- f(R) effects are significant only at short distances or below the Compton wavelength.
- Gravitational time delay is primarily due to the Ricci scalar, with minimal influence from the scalar degree of freedom.

## Abstract

A Gravitoelectromagnetism formalism in the context of metric f(R) theory is presented and the analogue Lorentz force law is derived. Some interesting results such as the dependence of the deviation from General Relativity on the absolute value of the scalar potential are found, it is also found that the f(R) effects are only relevant at a shorter distance or when the distance is much less than the compton wavelength, and that the effects are attractive in nature. An investigation of gravitational time delay in the context of metric f(R) is also presented showing that the Ricci scalar alone is responsible for the time delay effect which seems to suggest that the extra scalar degree of freedom associated to f(R) does not provide any modification. Also, to generalise our results, the Lorentz force law and gravitational time delay in the case of Brans-Dicke theories with a potential are derived; it is shown that the results are consistent with those obtained in the case of metric f(R) and General Relativity in the appropriate limits.