Minimal Extension of Einstein's Theory: The Quartic Gravity
  We study structure of solutions of the recently constructed minimal extensions of Einstein's gravity in four dimensions at the quartic curvature level. The extended higher derivative theory, just like Einstein's gravity, has only a massless spin-two graviton about its unique maximally symmetric vacuum. The extended theory does not admit the Schwarzschild or Kerr metrics as exact solutions, hence there is no issue of Schwarzschild type singularity but, approximately, outside a source, spherically symmetric metric with the correct Newtonian limit is recovered. We also show that for all Einstein space-times, square of the Riemann tensor (the Kretschmann scalar or the Gauss-Bonnet invariant) obeys a non-linear scalar Klein-Gordon equation. 
Atalay Karasu, Esin Kenar, Bayram Tekin
hep-th gr-qc