In f (R) gravity, the metric, presented in the form of the multipole expansion, for the external gravitational field of a spatially compact supported source up to 1/c 3 order is provided, where c is the velocity of light in vacuum. The metric consists of General Relativity-like part and f (R) part, where the latter is the correction to the former in f (R) gravity. At the leading pole order, the metric can reduce to that for a point-like or ball-like source. For the gyroscope moving around the source without experiencing any torque, the multipole expansions of its spin's angular velocities of gravitoelectric-type precession, gravitomagnetic-type precession, f (R) precession, and Thomas precession are all derived. The first two types of precession are collectively called General Relativitylike precession, and the f (R) precession is the correction in f (R) gravity. At the leading pole order, these expansions can recover the results for the gyroscope moving around a point-like or ball-like source. If the gyroscope has a nonzero four-acceleration, its spin's total angular velocity of precession up to 1/c 3 order in f (R) gravity is the same as that in General Relativity.

The 1/r-expansion in the distance to the source is applied to the linearized f (R) gravity, and its multipole expansion in the radiation field with irreducible Cartesian tensors is presented. Then, the energy, momentum, and angular momentum in the gravitational waves are provided for linearized f (R) gravity. All of these results have two parts which are associated with the tensor part and the scalar part in the multipole expansion of linearized f (R) gravity, respectively. The former is the same as that in General Relativity, and the latter, as the correction to the result in General Relativity, is caused by the massive scalar degree of freedom, and places an important role in distinguishing GR and f (R) gravity.

The field equations of f (R, G) gravity are rewritten in the form of obvious wave equations with the stressenergy pseudotensor of the matter fields and the gravitational field as its source under the de Donder condition. The linearized field equations of f (R, G) gravity are the same as those of linearized f (R) gravity, and thus, their multipole expansions under the de Donder condition are also the same. It is also shown that the Gauss-Bonnet curvature scalar G does not contribute to the effective stress-energy tensor of gravitational waves in linearized f (R, G) gravity, though G plays an important role in the nonlinear effects in general. Further, by applying the 1/r expansion in the distance to the source to the linearized f (R, G) gravity, the energy, momentum, and angular momentum carried by gravitational waves in linearized f (R, G) gravity are provided, which shows that G, unlike the nonlinear term R 2 in the gravitational Lagrangian, does not contribute to them either.