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Title: Deformed Spheres in General Relativity and Beyond
Authors: Rahim, Rehana
Keywords: Physical Sciences
Issue Date: 2020
Publisher: Quaid-i-Azam University, Islamabad.
Abstract: Currently, Einstein’s general theory of relativity (GR) provides the best description for the phenomena called gravity. But it is not the only theory that does the job. There is the version given by Newton also. This version describes gravity as the force between the objects. Such a force depends on masses of the objects involved and also on the distance from each other. In GR, gravity is not a force. It is the curvature of the spacetime resulting due to the presence of the matter. The gravitational field as described by GR is a manifestation that space is the curved Riemannian one instead of the flat Minkowski. The gravitational field gets geometrized in GR, which is a tensor theory of the gravitational field instead of scalar one (Newtonian theory is a scalar theory). The gravitational field is represented in terms of the metric tensor of the Riemannian space, its source being the matter tensor. The components of the matter tensor source the gravitational field in an elegant way determined by the Einstein filed equations (EFEs). Continuous progress is being made in finding the solutions of the EFEs. Schwarzschild, Reissner-Nordstr¨om, Kerr and Kerr-Newman spacetimes are the simplest vacuum solutions of EFEs that describe the black holes. Reissner-Nordstr¨om is the charged generalization of the Schwarzschild solution, both being spherically symmetric. Kerr metric is the rotating generalization of the Schwarzschild metric. Introduction of the charge in the Kerr metric gives the Kerr-Newman metric. Kerr and Kerr-Newman spacetimes are axially symmetric. In the limit of mass being vanished, they reduce to Minkowski metric in spheroidal coordinates. Spheroids are the geometric objects which we can take as deformed spheres. Apart from the research and interest in GR, there has been a growing interest in alternate theories of gravity. One such theory is the Chern-Simons (CS) theory. The action of this theory consists of the usual Einstein-Hilbert term and a new parity violating fourdimensional correction. Two kinds of formulations exist in CS theory, namely dynamical and non-dynamical. Black hole solutions have been developed in both the cases. Our interest as regards to this thesis is the spacetime which has been developed in the former formulation. The solutions beyond GR can also be formulated by another method. Such method involves the model independent parameterization of the metric. The metric thus obtained must describe the black hole solution in any theory of gravity. The possible deviations from the Kerr spacetime are measured by the deviation parameters. The detailed outline of the thesis is as follows: Chapter 1 is about the preliminaries. In Chapter 2, the Misner-Sharp mass is generalized for the spheroidal geometry. MisnerSharp mass is a type of quasilocal mass that previously worked only in the spherically symmetric spacetimes. It also gives the location of the marginally outer trapped surface in such spacetimes. The Misner-Sharp mass is extended for spheroids within GR and iii the location of marginally outer trapped surface is determined in this new setting. The parameter which gives deviation from spherical geometry is kept small throughout the analysis. In quantum physics, the energy density which defines the Misner-Sharp mass (and ADM mass, named after Richard Arnowitt, Stanley Deser and Charles Misner) becomes a quantum observable and one could conjecture that the gravitational radius admits a similar description. The gravitational radius is made a quantum mechanical operator which acts on the “horizon wave function”. The horizon wave function is given by the quantum state of the source. The horizon quantum mechanics has been extended to the case of spheroidal sources at the end of the chapter. The next two chapters deal with the spacetimes in the alternate theories of gravity. Chapter 3 involves spacetime in dynamical CS theory. This spacetime is valid in slow rotation approximation and small coupling constant. The effects of the CS coupling constant on some physical phenomena e.g. quasilocal mass, particle motion and energy extraction process are studied. Johannsen and Psaltis developed a rotating deformed Kerr-like metric in an alternate theory of gravity other than GR. It is obtained by applying Newman-Janis algorithm to a deformed Schwarzschild metric. Motivated by this spacetime, a charged analogue of the Johannsen-Psaltis metric is developed in Chapter 4. Here the seed metric is taken as the Reissner-Nordstr¨om spacetime. The new metric is studied for the event and Killing horizons, the latter are also represented graphically. Lorentz violating regions are analyzed by the determinant of the charged version of the Johannsen-Psaltis metric. Analysis of the closed time-like curves are also included in this chapter. Considering the motion of a particle on the equatorial plane, we obtain its energy and angular momentum. Location of the circular photon orbits and innermost stable circular orbits are also determined.
Gov't Doc #: 20196
Appears in Collections:PhD Thesis of All Public / Private Sector Universities / DAIs.

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