Abstract:
General Relativity, while ultimately based on the Einstein equations,
also allows one to quantitatively study some aspects of the theory
without explicitly solving the Einstein equations. These geometrical notions
of the theory provide an insight to the nature of more general spacetimes.
In this thesis, the Raychaudhuri equation, the choice of the coordinate
system, the notions of surface gravity and of entropy, and restrictions
on negative energy densities on the form of the Quantum Interest Conjecture,
will be discussed.
First, using the Kodama vector, a geometrically preferred coordinate
system is built. With this coordinate system the usual quantities, such as
the Riemann and Einstein tensors, are calculated.
Then, the notion of surface gravity is generalized in two different ways.
The first generalization is developed considering radial ingoing and outgoing
null geodesics, in situations of spherical symmetry. The other generalized
surface gravity is a three-vector obtained from the spatial components
of the redshifted four acceleration of a suitable set of fiducial observers.
This vectorial surface gravity is then used to place a bound on
the entropy of both static and rotating horizonless objects. This bound is
obtain mostly by classical calculations, with a minimum use of quantum
field theory in curved spacetime.
Additionally, several improved versions of the Raychaudhuri equation
are developed and used in different scenarios, including a two congruence
generalization of the equation.
Ultimately semiclassical quantum general relativity is studied in the
specific form of the Quantum Inequalities, and the Quantum Interest Conjecture.
A variational proof of a version of the Quantum Interest Conjecture
in (3 + 1)–dimensional Minkowski space is provided.
