Non perturbative spherical gravitational waves

It is well-known that there is no spherical/topologically spherical gravitational waves in vacuum space in general relativity. We show that a deviation from general relativity leads to exact vacuum spherical gravitational waves, no matter how tiny this deviation is. We also discuss the related topics, including Vaidya-like metric in $f(R)$ gravity. We demonstrate that the existence of spherical gravitational wave is a non perturbative property for gravities. We investigate energy carried by this nonperturbative wave. We first find the wave solution from investigations of Vaidya-like metric in $f(R)$ gravity, which has only one longitude polarization. We further extend it to a metric with two transverse polarizations by directly solving the field equation.