Black hole solutions in R 2 gravity
Alex Kehagias (Physics Division, National Technical University of Athens, 15780 Zografou Campus, Athens, Greece, Department of Theoretical Physics and Center for Astroparticle Physics (CAP), 24 quai E. Ansermet, CH-1211, Geneva 4, Switzerland); Costas Kounnas (Laboratoire de Physique Théorique, Ecole Normale Supérieure, 24 rue Lhomond, 75231, Paris cedex 05, France); Dieter Lüst (Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805, München, Germany, Arnold Sommerfeld Center for Theoretical Physics, LMU, Theresienstr. 37, 80333, München, Germany, CERN, PH-TH Division, 1211, Geneva 23, Switzerland); Antonio Riotto (Department of Theoretical Physics and Center for Astroparticle Physics (CAP), 24 quai E. Ansermet, CH-1211, Geneva 4, Switzerland)
We find static spherically symmetric solutions of scale invariant R 2 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 R 2 theory will be identical to that of Einstein theory. Indeed, we find that the solutions of R 2 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 R 2 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 M P → 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, R 2 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 R 2 theory and we find that entropy and energy vanish for scalar-flat backgrounds.
