Black Holes in Fourth Order Gravity

As we know, in order to fit the standard model of cosmology, which is based on General Relativity coupled to standard matter (baryons and radiation), the introduction of two dark components (matter and energy) are needed to achieve a consistent picture. To get over this problem of having ``exotic matter fields'', one of the theoretical proposals that has received a considerable amount of attention recently, is that Dark Energy has a geometrical origin. Most of the work on this idea has focused on ``fourth order gravity'', in which the standard Hilbert-Einstein action is modified with terms that are at most of order four in the metric tensor. However, it is not very well understood as to how these modifications change the standard spherically symmetric solutions of GR.  In this talk I would discuss a new covariant formalism to treat spherically symmetric spacetimes in $f(R)$ theories of gravity. Using this formalism we can determine the
conditions on the function $f$ for which the Schwarzschild metric is the only vacuum solution with vanishing Ricci scalar.  We can also show that this general framework provides a clear way of showing that the Schwarzschild solution is not a unique static spherically symmetric solution, providing some insights on how the current form of Birkhoff's theorem breaks down for these theories.