# Particle and light propagation around stellar black holes in alternative theories of gravity and implications to general relativistic tests

## Fecha

2023-01

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## Profesor Guía

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Tesis

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Universidad de Valparaíso

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## Facultad

Facultad de Ciencias

## Departamento o Escuela

Instituto de Fisica y Astronomia

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## Recolector

## Especie

## Nota general

Doctorado en Astrofísica

## Resumen

The equations of motion for particles travelling in the gravitational fields of massive objects, as formulated by the general theory of relativity, have been receiving rigor- ous attention ever since the advent of the theory. In fact, the approximate solutions to these equations, at the time, could pave the way in figuring out the trajectories of planets and light in the solar system and finally, led to some observational evidences which confirmed general relativity’s predictions (as asserted by Eddington in his fa- mous book (Eddington, 1920)). However, the more delicate the experimental tests became, the more they raised the interest in obtaining exact solutions to the equations of motion. This necessitated employing advanced mathematical methods, mainly, because of the resultant differential equations appearing in the equations of motion, which tend to calculate the arc-lengths associated with the particle trajectories. Since the early attempts by Hagihara (Hagihara, 1930) and Darwin (Darwin, 1959, 1961) in obtaining and categorizing the particle orbits in the Schwarzschild spacetime, re- searchers have been employing different approaches to the computation of the arc- lengths swept by particle trajectories in gravitating systems. These approaches are, in general, based on manipulating elliptic integrals and the resultant elliptic func- tions, covering the Jacobi and the Weierstraß elliptic functions, as the two most com- mon forms. Ever since, the elliptic and hyper-elliptic functions have received a great deal of interest in analyzing the geodesic structure of massive and mass-less particles in black hole spacetimes (Rauch & Blandford, 1994; Kraniotis & Whitehouse, 2003; Kraniotis, 2004; Beckwith & Done, 2005; Cruz et al., 2005; Kraniotis, 2005; Hackmann
& La¨ mmerzahl, 2008a,b; Bisnovatyi-Kogan & Tsupko, 2008; Kagramanova et al., 2010; Hackmann et al., 2010a,b; Kraniotis, 2011; Enolski et al., 2011; Gibbons & Vyska, 2012; Mun˜ oz, 2014a; Kraniotis, 2014; Mun˜ oz, 2014b; De Falco et al., 2016; Barlow et al., 2017; Vankov, 2017; Chatterjee et al., 2019; Uniyal et al., 2018; Jusufi et al., 2018; Ghaffarne- jad et al., 2018; Villanueva et al., 2018; Hsiao et al., 2020; Gralla & Lupsasca, 2020; Kraniotis, 2021).
In this thesis, we investigate the time-like and null geodesics that propagate in the
exterior geometry of static and rotating black holes. The first problems that are dealt with, is the derivation of the analytical solutions to the equations of motion, for both the radial and angular types of orbits. To achieve this purpose, we use the standard Lagrangian dynamics and identify the orbits in the context of the corresponding effec- tive potentials. The second objective is to apply the classical general relativistic tests (in particular in the solar system), to examine the relevant mathematical formulations for these tests that are given for each of the black holes spacetimes.
The organization of this thesis is as follows: In chapter 1, we discuss, in detail, the elliptic integrals and their solutions in terms of the Jacobian and Weierstraßian elliptic functions in all of their forms. We also bring several examples to demonstrate their ap- plicability for the relevant problems in classical physics. In chapter 2, we explore the geometrical aspects of the Lagrangian and Hamiltonian on the base manifold. This is followed by the derivation of the geodesic equation from Euler-Lagrange equations. Furthermore, to exemplify this in black hole spacetimes, we calculate the exact so- lutions to the radial and angular geodesics for the mass-less and massive particles that travel in the exterior geometry of a Schwarzschild black hole, which necessitates the exploitation of the formerly discussed elliptic functions. These trajectories are also plotted for each of the types of orbits. Moreover, we review the derivation of a modified version of the Newman-Janis algorithm to generate the stationary coun- terparts of static spacetimes. In chapter 3, we begin our studies by investigating the geodesics in a particular spacetime, derived from the fourth order Weyl conformal gravity, under certain circumstances. This static black hole spacetime, is studied in the context of the propagation of mass-less, neutral, and charged massive particles. We also apply several general relativistic tests on this black hole, by means of the de- rived analytical expressions. Finally, by employing the aforementioned algorithm, a stationary counterpart of this black hole is generated. This black hole is investigated in terms of its ergoregion, photon spheres and shadow. In chapter 4, the propaga- tion of mass-less particles in the exterior geometry of a scale-dependent BTZ black hole is discussed, together with the simulation of the possible orbits. Chapter 5 is devoted to a more complicated case, namely to a Kerr black hole immersed in a non- magnetized plasma, which produces a dielectric medium residing in a curved mani- fold. We apply an elaboration to all the previously discussed methods, and then, we employ them to the investigation of the light ray trajectories in this medium. The orbits are discussed in both planar and three-dimensional context, by solving, in- dividually, the temporal evolution of the coordinates. In chapter 6, we consider a Schwarzschild black hole associated with quintessence and cloud of strings, which is firstly calibrated in the context of standard general relativistic tests for its parame- ters. We then continue with the derivation of exact analytical solutions for null and time-like geodesics in this spacetime. We finally switch our study, in chapter 7, to the application of Carathe´ odory’s geometrothermodynamics to a static (Hayward) and a stationary (rotating scale-dependent BTZ) black hole. This discussion, although being different from those done in the previous chapters, is of great interest since it provides a new vision to the black hole thermodynamics and helps for the creation of a boost in the development of this field of study. We construct the perspective of our future studies in chapter 8.
Throughout this thesis, unless in the particular places that is adopted otherwise, we use the geometric units, in which G = c = h¯ = 1. Furthermore, in appropriate places where needed, we use the Einstein convention on summing over dummy indices, and the four-dimensional system of coordinates is adopted as xµ = (x0, x1, x2, x3 ), in which, the zero component corresponds to the time coordinate, i.e. x0 = t. All the diagrams and simulations have been generated by the software Mathematica® 12.0.

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## Palabras clave

HOYOS NEGROS (ASTRONOMIA), GRAVEDAD, TERMODINAMICA