We perform a careful investigation of the problem of physically realistic gravitational collapse of massive stars in $f(R)$ gravity. We show that the extra matching conditions that arise in the modified gravity impose strong constraints on the stellar structure and thermodynamic properties. In our opinion these constraints are unphysical. We prove that no homogeneous stars with nonconstant Ricci scalar can be matched smoothly with a static exterior for any nonlinear function $f(R)$. Therefore, these extra constraints make classes of physically realistic collapse scenarios in general relativity, nonadmissible in these theories. We also find an exact solution for an inhomogeneous collapsing star in the Starobinski model that obeys all the energy and matching conditions. However, we argue that such solutions are fine-tuned and unstable to matter perturbations. Possible consequences on black hole physics and the cosmic censorship conjecture are also discussed.

We develop a new covariant formalism to treat spherically symmetric spacetimes in metric f(R) theories of gravity. Using this formalism we derive the general equations for a static and spherically symmetric metric in a general f(R) gravity. These equations are used to determine the conditions for which the Schwarzschild metric is the only vacuum solution with vanishing Ricci scalar. We also show that our general framework provides a clear way of showing that the Schwarzschild solution is not a unique static spherically symmetric solution, providing some insight into how the current form of Birkhoff's theorem breaks down for these theories.

We study the gravitational collapse of a generalized Vaidya spacetime in the context of the cosmic censorship hypothesis. We develop a general mathematical framework to study the conditions on the mass function so that future directed nonspacelike geodesics can terminate at the singularity in the past. Thus our result generalizes earlier works on gravitational collapse of the combinations of Type-I and Type-II matter fields. Our analysis shows transparently that there exist classes of generalized Vaidya mass functions for which the collapse terminates with a locally naked central singularity. We calculate the strength of these singularities to show that they are strong curvature singularities and there can be no extension of spacetime through them.

We generalize the Buchdahl–Bondi limit for the case of static, spherically symmetric, relativistic compact stars immersed in Schwarzschild vacuum in the $f(R)$-theory of gravity, subject to very generic regularity, thermodynamic stability, and matching conditions. Similar to the case of general relativity, our result is model independent and remains true for any physically realistic equation of state of standard stellar matter. We show that an extramassive stable star can exist in these theories, with a surface redshift larger than 2, which is forbidden in general relativity. This result gives a novel and interesting observational test for validity or otherwise of general relativity and also provides a possible solution to the dark matter problem.

The energy conditions are studied, in the relativistic astrophysical setting, for higher dimensional Hawking-Ellis Type I and Type II matter fields. The null, weak, dominant and strong energy conditions are investigated for a higher dimensional inhomogeneous, composite fluid distribution consisting of anisotropy, shear stresses, nonvanishing viscosity as well as a null dust and null string energy density. These conditions are expressed as a system of six equations in the matter variables where the presence of the higher dimension N is explicit. The form and structure of the energy conditions is influenced by the geometry of the (N - 2)-sphere. The energy conditions for the higher dimensional Type II fluid are also generated, and it is shown that under certain restrictions the conditions for a Type I fluid are regained. All previous treatments for four dimensions are contained in our work.

In this paper we show in a covariant and gauge invariant way that in general relativity, tidal forces are actually a hidden form of gravitational waves. This must be so because gravitational effects cannot occur faster than the speed of light. Any two body gravitating system, where the bodies are orbiting around each other, may generate negligible gravitational waves, but it is via these waves that non-negligible tidal forces (causing shape distortions) act on these bodies. Although the tidal forces are caused by the electric part of the Weyl tensor, we transparently show that some small time varying magnetic part of the Weyl tensor with non zero curl must be present in the system that mediates the tidal forces via gravitational wave type effects. The outcome is a new test of whether gravitational effects propagate at the speed of light.