Black Holes: Quartic Quasi-Topological Gravity and Greybody Factor

Abstract

Alternative theories of gravity are those which provide description of gravitational interaction different from the description of the usual general theory of relativity. In 1920, soon after the birth of general relativity, efforts were made to propose these modified theories of gravity. In those days, work in this direction was mainly due to the curiosity to challenge the newly presented general theory of relativity, but with time, interest and motivation to work on modified gravities waxed and waned. Overall, however, a continuous activity in this direction over the past ninety years may still be found. A part of this thesis is another such effort in the aforementioned approach. In this thesis, higher curvature gravity theories are considered which modify Einstein’s theory of gravity due to the addition of higher order curvature terms to the Einstein-Hilbert term. The primary motivation for this modification comes from the appearance of quadratic curvature terms in the low-energy effective action of string theory and is also due to black hole solutions in higher curvature gravities. Furthermore, these black hole solutions exhibit certain properties which are unavailable in the usual Einstein’s theory of gravity. Black hole emission and absorption phenomena is related to an important quantity, known as greybody factor. Due to greybody factor Hawking radiation’s spectrum deviates from the spectrum of black body radiations. Some studies of greybody factor are presented in this thesis which elaborate the characteristics of the radiations emitted from black holes. The outline of the thesis is as follows: In Chapter 1, historical background of general relativity is introduced followed by current challenges and then a brief description of higher curvature gravities. This chapter also introduces the notion of greybody factor and its importance. In Chapter 2, the quartic version of generalized cubic quasi-topological gravity is constructed. This class of theories includes Lovelock gravity and a known form of quartic quasi-topological gravity as special cases and possesses a number of remarkable properties: • In vacuum, or in the presence of suitable matter, there is a single independent field equation which is a total derivative. • At the linearized level, the equations of motion on a constant curvature background are second order, coinciding with the linearized Einstein equations up to a redefinition of Newton’s constant. Therefore, these theories propagate only the massless, transverse graviton on a constant curvature background. • While the Lovelock and quasi-topological terms are trivial in four dimensions, there exist four new generalized quasi-topological terms (the quartet) that are nontrivial, leading to interesting higher curvature theories in d ≥ 4 dimensions that appear well suited for holographic study. In Chapter 3, four dimensional black hole solutions to the theory are constructed and their properties are studied. Further, black brane solutions in general dimensions of the theory are studied. Results of this study may lead to interesting consequences for dual conformal field theories. In Chapter 4, generalized Reissner-Nordström anti-de Sitter black hole solution for generalized cubic quasi-topological gravity is constructed. Asymptotic and near-horizon solutions for this theory are found in d spacetime dimensions. A form of extended first law of black hole thermodynamics for these black holes is also presented. Critical values of volume, pressure and temperature are presented. v In Chapter 5, a general expression for the greybody factor of non-minimally coupled scalar fields in Reissner-Nordström-de Sitter spacetime in low frequency approximation is derived. Greybody factor as a characteristic of effective potential barrier is also presented. The role of cosmological constant, both in the absence as well as in the presence of non-minimal coupling, is presented. Considering the non-minimal coupling as a mass term, its effect on the greybody factor is discussed. The significance of the results are elaborated by giving formulae for differential energy rates and general absorption cross sections. The greybody factor gives insight into the spectrum of Hawking radiations. In Chapter 6, greybody factor of massless, uncharged scalar fields is worked out in the background of cylindrically symmetric spacetime, in the low-energy approximation. Two cases are discussed. In the first case, analytical expression for absorption probability is derived with the spacetime kinetically coupled with the Einstein tensor. In the second case, an analysis is performed in the absence of the coupling constant by using the wave equation, which is derived from Klein- Gordon equation. The radial part of the wave equation is solved in the form of hypergeometric function in the near-horizon region whereas in the far region, the solution is of the form of Bessel’s function. Finally considering the continuity of wave function, the two solutions in the low energy approximation are smoothly matched to get a formula for absorption probability. The last chapter concludes the thesis by summarizing the outcomes of this work.