Birkhoff’s theorem in higher derivative theories of gravity II

As a continuation of a previous work, here, we examine the admittance of Birkhoff’s theorem in a class of higher derivative theories of gravity. This class is contained in a larger class of theories which are characterized by the property that the trace of the field equations are of second order in the metric. The action representing these theories are given by a sum of higher curvature terms. Moreover, the terms of a fixed order $k$ in the curvature are constructed by taking a complete contraction of $k$ conformal tensors. The general spherically (hyperbolic or plane) symmetric solution is then given by a static asymptotically Lifshitz black hole with the dynamical exponent equal to the spacetime dimensions. However, theories which are homogeneous in the curvature (i.e., of fixed order $k$) possess additional symmetry which manifests as an arbitrary conformal factor in the general solution. So, these theories are analyzed separately and have been further divided into two classes depending on the order and the spacetime dimensions.