We study both analytically and numerically the gravitational fields of stars in f (R) gravity theories. We derive the generalized Tolman-Oppenheimer-Volkov equations for these theories and show that in metric f (R) models the Parameterized Post-Newtonian parameter γPPN = 1/2 is a robust outcome for a large class of boundary conditions set at the center of the star. This result is also unchanged by introduction of dark matter in the Solar System. We find also a class of solutions with γPPN ≈ 1 in the metric f (R) = R − µ 4 /R model, but these solutions turn out to be unstable and decay in time. On the other hand, the Palatini version of the theory is found to satisfy the Solar System constraints. We also consider compact stars in the Palatini formalism, and show that these models are not inconsistent with polytropic equations of state. Finally, we comment on the equivalence between f (R) gravity and scalar-tensor theories and show that many interesting Palatini f (R) gravity models can not be understood as a limiting case of a Jordan-Brans-Dicke theory with ω → −3/2.

This article explores the overall geometric manner in which human beings make sense of the world around them by means of their physical theories; in particular, in what are nowadays called pregeometric pictures of Nature. In these, the pseudo-Riemannian manifold of general relativity is considered a flawed description of spacetime and it is attempted to replace it by theoretical constructs of a different character, ontologically prior to it. However, despite its claims to the contrary, pregeometry is found to surreptitiously and unavoidably fall prey to the very mode of description it endeavours to evade, as evidenced in its all-pervading geometric understanding of the world. The question remains as to the deeper reasons for this human, geometric predilection-present, as a matter of fact, in all of physics-and as to whether it might need to be superseded in order to achieve the goals that frontier theoretical physics sets itself at the dawn of a new century: a sounder comprehension of the physical meaning of empty spacetime.

We consider a Hamiltonian quantum theory of stationary spacetimes containing a Kerr-Newman black hole. The physical phase space of such spacetimes is just six-dimensional, and it is spanned by the mass M , the electric charge Q and angular momentum J of the hole, together with the corresponding canonical momenta. In this six-dimensional phase space we perform a canonical transformation such that the resulting configuration variables describe the dynamical properties of Kerr-Newman black holes in a natural manner. The classical Hamiltonian written in terms of these variables and their conjugate momenta is replaced by the corresponding self-adjoint Hamiltonian operator and an eigenvalue equation for the Arnowitt-Deser-Misner (ADM) mass of the hole, from the point of view of a distant observer at rest, is obtained. In a certain very restricted sense, this eigenvalue equation may be viewed as a sort of "Schrödinger equation of black holes". Our "Schrödinger equation" implies that the ADM mass, electric charge and angular momentum spectra of black holes are discrete, and the mass spectrum is bounded from below. Moreover, the spectrum of the quantity2 , where a is the angular momentum per unit mass of the hole, is strictly positive when an appropriate self-adjoint extension is chosen. The WKB analysis yields the result that the large eigenvalues of M , Q and a are of the form √ 2n, where n is an integer. It turns out that this result is closely related to Bekenstein's proposal on the discrete horizon area spectrum of black holes.