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In the three-dimensional pure Einstein gravity, the geometries of the vacuumspace-times are always trivial, and gravitational waves (gravitons) arestrictly forbidden. For the first time, we find a vacuum circularly symmetricblack hole with nontrivial geometries in $f(R)$ gravity theory, in which a truesingularity appears. In this frame with nontrivial geometry, a perturbativegravitational wave does exist. Beyond the perturbative level, we make aconstructive proof of the existence of a gravitational wave in $f(R)$ gravity,where the Birkhoff-like theorem becomes invalid. We find two classes of exactsolutions of circularly symmetric pure gravitational wave radiation andabsorption.

Black holes and gravitational waves in three-dimensional f(R) gravity