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

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 (GR) 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 GR in the appropriate limits.

Appendix: Evaluation of the potential term

From (138), we get for the first derivative of $V(\phi )$ as

$$\begin{aligned} \frac{d V(\phi )}{d\phi }=\frac{d R(\phi )}{d\phi }f'(R(\phi ))+R(\phi )\frac{d}{d\phi }f'(R(\phi ))-\frac{d}{d\phi }f(R), \end{aligned}$$

(166)

and consequently the second derivative of $V(\phi )$ can be expressed as

$$\begin{aligned} \begin{aligned} \frac{d^2V(\phi )}{d\phi ^2}&= \frac{d^2R(\phi )}{d\phi ^2}f'(R(\phi )) +\frac{dR(\phi )}{d\phi }\frac{d}{d\phi }f'(R(\phi ))+ R(\phi )\frac{d^2}{d\phi ^2}f'(R(\phi ))\\&\quad +\frac{d}{d\phi }f'(R(\phi ))\frac{d R(\phi )}{d\phi } -\frac{d^2}{d\phi ^2}f(R(\phi )). \end{aligned} \end{aligned}$$

(167)

Now, using the following quantities which we encountered in the linearised metric f(R) section, i.e,

$$\begin{aligned} f(R)&=R^{(1)}+\frac{a_2}{2!}{R^{(1)}}^2,&f'(R)&=1+a_2 R^{(1)}, \end{aligned}$$

(168)

where $R^{(1)}$ is the linearised Ricci Scalar. In the subsequent calculation, we drop the suffix (1) and introduce at the end for sake of convenience. Hence, we get the following expression after evaluating for (167)

$$\begin{aligned} \frac{d^2V(\phi )}{d\phi ^2}= & {} \frac{d^2R(\phi )}{d\phi ^2}(1+a_2 R(\phi ))+\frac{dR(\phi )}{d\phi }\frac{d}{d\phi }(1+a_2 R(\phi ))+R(\phi )\frac{d^2}{d\phi ^2}(1+a_2 R(\phi ))\nonumber \\&+\frac{d}{d\phi }(1+a_2 R(\phi ))\frac{d R(\phi )}{d\phi }-\frac{d^2}{d\phi ^2}(R(\phi )+\frac{a_2}{2!}R(\phi )^2), \end{aligned}$$

(169)

or

$$\begin{aligned} \frac{d^2V(\phi )}{d\phi ^2}= & {} \frac{d^2R(\phi )}{d\phi ^2}+a_2 R(\phi )\frac{d^2R(\phi )}{d\phi ^2}+a_2\left( \frac{dR(\phi )}{d\phi }\right) ^2 +R(\phi )\frac{d^2R(\phi )}{d\phi ^2}+a_2\left( \frac{dR(\phi )}{d\phi }\right) ^2\nonumber \\&-\frac{d^2R(\phi )}{d\phi ^2} -a_2\left( \frac{dR(\phi )}{d\phi }\right) ^2-a_2 R(\phi )\frac{d^2R(\phi )}{d\phi ^2}, \end{aligned}$$

(170)

and hence, the expression drops to just

$$\begin{aligned} \frac{d^2V(\phi )}{d\phi ^2}=a_2\left( \frac{dR(\phi )}{d\phi }\right) ^2 +R(\phi )\frac{d^2R(\phi )}{d\phi ^2}. \end{aligned}$$

(171)

So, if we use the above result, from (141), we have the following expression

$$\begin{aligned} m_s^2=\frac{\phi _0}{3}\left( a_2\left( \frac{dR^{(1)}(\phi )}{d\phi }\right) ^2 +R^{(1)}(\phi )\frac{d^2R^{(1)}(\phi )}{d\phi ^2}\right) \bigg |_{\phi _0} \end{aligned}$$

(172)