Extra force in $f(R)$ modified theories of gravity

Understand

The equation of motion for test particles in $f(R)$ modified theories of gravity is derived.

• By considering an explicit coupling between an arbitrary function of the scalar curvature, $R$, and the Lagrangian density of matter, it is shown that an extra force arises.
• This extra force is orthogonal to the four-velocity and the corresponding acceleration law is obtained in the weak field limit.
• Connections with MOND and with the Pioneer anomaly are further discussed.