## Scalar perturbations of Schwarzschild black holes in modified gravity.

##### Abstract

This thesis is concerned with the physics related to scalar perturbations in the
Schwarzschild geometry that arise in modifed gravity theories. It has already been
shown that the gravitational waves emitted from a Schwarzschild black hole in f(R)
gravity have no signatures on the modification of gravity from General Relativity, as the
Regge-Wheeler equation remains invariant. In this thesis we consider the perturbations
of the Ricci scalar in a vacuum Schwarzschild spacetime, which is unique to higher order
theories of gravity and is absent in General Relativity. We show that the equations that
govern these perturbations can be reduced to a Volterra integral equation. We explicitly
calculate the reflection coefficients for the Ricci scalar perturbations, when they are
scattered by the black hole potential barrier. Our analysis shows that a larger fraction
of these Ricci scalar waves are reflected compared to the gravitational waves. This may
provide a novel observational signature for fourth order gravity. We also show that
higher order curvature corrections to General Relativity, in the strong gravity regime
on scales of the order of the near horizon, produce a rapidly oscillating and infalling
Ricci scalar fireball just outside the horizon. These
fluctuations behave like an infalling
extra massive scalar field that can generate the ringdown modes of gravitational waves
having the same natural frequency as those that are generated by black hole mergers.
Our analysis provides a viable classical or semi-classical explanation for the echoes in
the ringdown modes without invoking the existence of any exotic structures at the
horizon.