Influence of Λ on Blackhole Spacetimes

Tripathy, Santanu (2023) Influence of Λ on Blackhole Spacetimes. PhD thesis, Indian Institute of Science Education and Research Kolkata.

Text (PhD thesis of Santanu Tripathy (12RS048)) 12RS048.pdf - Submitted Version Restricted to Repository staff only Download (19MB)

Abstract

In this thesis, we investigate the cosmological influence on blackhole spacetimes. Our study starts with a generalization of Vaidya’s static solution with blackhole in the background of the Einstein static universe to incorporate the cosmological constant ⇤. This static solution to Einstein’s equations with the Whittaker equation of state, we referrer to as Schwarzschild-Whittaker (SW) solution. We further extend the SW solution to include the outgoing/ingoing radiation field. We use the SW solution to explore the physical phenomenon outside the event-horizon to show the influence of cosmological parameters. We start with a condition for the existence of an event horizon. We show that in general, the SW solution allows for two horizons, an inner blackhole event-horizon and an outer cosmological horizon. We also briefly investigate geodesic motion in the SW spacetime. It is straightforward to separate geodesic equations in terms of conserved quantities. We write the geodesic equation of motions and find the e↵ective radial potential. With orbital equations, we explore the periastron precession of elliptical orbits. We develop a numerical scheme by dividing an orbit into di↵erent patches using radial turning points. We integrate the orbital equation in each patch as a function of proper time and terminate the integration at the turning point. Finally, we combine all patches to obtain a complete smooth orbit. With numerically integrated orbit, we determine the periastron precession, closer to the event horizon where the e↵ects of fields are strong. We test our method with the Kepler problem, for which the orbit is a non-precession ellipse, which can be computed using the analytical methods. Next, we study the inflection of R and ⇤ in precession using the SW spacetime. Since precession is sensitive to orbital parameters such as orbital eccentricity(ϵ) and semi-major axis(a). We studied the precession of the periastron as functions of ϵ and a. Thiswill enable us to distinguish the precession parameterised by R and Λ, which represents the cosmological influence. We observe that the parameters R and ⇤ have the opposite e↵ect on precession. Next, we analyzed the evolution of a massless scalar field in the SW spacetime; these could serve as a preliminary investigation of the stability of the SW spacetime. Here, the radial equation can be written as a time-independent schr¨odinger equation with an e↵ective radial potential. we conclude that for lower values of the parameters R and l = 0, the modes or the radial modes appear to be unstable. However, the higher l modes could be stable with bound states. Next, we study the orbital resonance that corresponds to an extended object approximated up to the dipole-order term in Kerr spacetime. We start with the Mathisson-Papapetrou equations under the linear spin approximation and primarily concentrate on two particular events: first, when the orbits are nearly circular and execute a small oscillation about the equatorial plane, and second, a generic trajectory confined on the equatorial plane. While in the first case, all three fundamental frequencies, namely radial Ωr, angular Ωθ , and azimuthal Ωɸ can be commensurate with each other and give rise to the resonance phenomenon, the latter is only accompanied by the resonance between ⌦r and Ωɸ as we set θ = π/2. We provide a detailed derivation of the location of the prograde resonant orbits in either of these cases and also study the role played by the spin of the blackhole. The implications related to spin-spin interactions between the object and the blackhole are also demonstrated.

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