Black hole solutions in R2 gravity

We find static spherically symmetric solutions of scale invariant R2 gravity. The latter has been shown to be equivalent to General Relativity with a positive cosmological constant and a scalar mode. Therefore, one expects that solutions of the R2 theory will be identical to that of Einstein theory. Indeed, we find that the solutions of R2 gravity are in one-to-one correspondence with solutions of General Relativity in the case of non-vanishing Ricci scalar. However, scalar-flat R = 0 solutions are global minima of the R2 action and they cannot in general be mapped to solutions of the Einstein theory. As we will discuss, the R = 0 solutions arise in Einstein gravity as solutions in the tensionless, strong coupling limit MP → 0. As a further result, there is no corresponding Birkhoff theorem and the Schwarzschild black hole is by no means unique in this framework. In fact, R2 gravity has a rich structure of vacuum static spherically symmetric solutions partially uncovered here. We also find charged static spherically symmetric backgrounds coupled to a U(1) field. Finally, we provide the entropy and energy formulas for the R2 theory and we find that entropy and energy vanish for scalar-flat backgrounds.