Preface

**Chapter 9 139**

**Chapter 10 161**

**Chapter 11 183**

**Chapter 12 203**

**Chapter 13 221**

**Chapter 14 235**

Stueckelberg-Horwitz-Piron Canonical Quantum Theory in General Relativity and Bekenstein- Sanders Gauge Fields for

Fast Indicators for Orbital Stability: A Survey on Lyapunov and

*by Pawel Gusin, Andy T. Augousti and Andrzej Radosz*

The Early Universe as a Source of Gravitational Waves

Periodic Solution of Nonlinear Conservative Systems *by Akuro Big-Alabo and Chinwuba Victor Ossia*

TeVeS

**II**

*by Lawrence P. Horwitz*

Reversibility Errors

*by Giorgio Turchetti and Federico Panichi*

BH M87: Beyond the Gates of Hell

Dark Matter within the Milky Way

*by Vladimir Gladyshev and Igor Fomin*

*by Aleksander Kaczmarek and Andrzej Radosz*

Some things are absolute and some things are relative. This is a fact of life. If I look at the teapot on my desk, I see that it sits to the left of my cup. If you are sitting opposite me, you will see the teapot to the right of my cup. "Left" and "right" are relative. Whether or not an object is found to the left or to the right of another depends on the observer. While this may be true, if the cup is filled to the top with coffee, all observers should approve this as actual fact, no matter where they sit. That, it'd seem, is an absolute statement, independent of who makes the observation.

The theory of relativity usually incorporates two interconnected theories by Albert Einstein: special relativity and general relativity. Einstein's special theory of relativity (special relativity) conceived in 1905, available within the paper "On the Electrodynamics of Moving Bodies", is about what's relative and what's absolute about time, space, and motion.

General relativity centers on gravitational, electromagnetic, and velocity fields, as well as functions of space and time, and density distributions that define masses and charges. Space–time is that the arena within which these fields accomplish their combined evolutions. It's therefore clear that we must first grasp the structure and geometry of space–time. Unluckily, because the velocity of light is so big, routine experience leads us to amass various false impressions about the geometry of space– time. This set of mistaken beliefs is known as Newtonian, or Galilean, space–time. The true (or truer) geometry of space–time was revealed through the improvement of Einstein's theory of special relativity. The foundation of this theory is the principle of relativity, in line with which the laws of physics are similar in all inertial reference frames. Einstein ran into the current principle by his investigation of Maxwell's equations.

Special relativity is restricted to things that are moving with regard to inertial frames of reference. That is, during a state of uniform motion with regard to each other, one cannot, by purely mechanical experiments, distinguish one from the opposite. Beginning with the behavior of light (and all other electromagnetic radiation), the theory of special relativity draws conclusions that conflict with daily knowledge, but is fully set by tests that examine subatomic particles at high speeds or measure minor changes between clocks traveling at different velocities. Special relativity discovered that the speed of light is a limit that cannot be reached by any material thing. It is the origin of the famous scientific equation E = mc<sup>2</sup> , which states that mass and energy are identical physical entities and might be changed one into the other.

Together with quantum mechanics, the theory of relativity is fundamental to modern physics.

This volume deals with extensions of special relativity, general relativity, and their applications in relation to intragalactic and extragalactic dynamics.

In addition, fundamental problems of these extensions are addressed, both classically and quantum mechanically, in Hamiltonian, Lagragian, and matrix formalisms by Richard P. Bocker and B. Roy Frieden in Chapter 2, Sikarin Yoo-Kong in Chapter 5, and Big-Alabo Akuro in Chapter 14, respectively.

**Chapter 1**

**Abstract**

calculated.

boundary conditions

**1. Introduction**

*dS*<sup>2</sup> *<sup>S</sup>*<sup>2</sup> … *<sup>S</sup>*<sup>2</sup>

[5, 39].

**1**

Quasinormal Modes of Dirac Field

in Generalized Nariai Spacetimes

The exact electrically charged solutions to the Dirac equation in higherdimensional generalized Nariai spacetimes are obtained. Using these solutions, the boundary conditions leading to quasinormal modes of the Dirac field are analyzed, and their correspondent quasinormal frequencies are analytically

**Keywords:** quasinormal modes, generalized Nariai spacetimes, Dirac field,

Quasinormal modes (QNMs) are eigenmodes of dissipative systems. For instance, if a spacetime with an event or cosmological horizon is perturbed from its equilibrium state, QNMs arise as damped oscillations with a spectrum of complex frequencies that do not depend on the details of the excitation. In fact, these frequencies depend just on the charges of the black hole, such as the mass, electric charge, and angular momentum [1, 2]. QNMs have been studied for a long time, and its interest has been renewed by the recent detection of gravitational waves, inasmuch as these are the modes that survive for a longer time when a background is perturbed and, therefore, these are the configurations that are generally measured by experiments [3–29]. Mathematically, this discrete spectrum of QNMs stems from the fact that certain boundary conditions must be imposed to the physical fields propagating in such background [30]. In this chapter, we consider a higher-dimensional generalization of the charged Nariai spacetime [31], namely,

, and investigate the dynamics of perturbations of the electrically

charged Dirac field (spin 1/2). In such a geometry, the spinorial formalism [32–34] is used to show that the Dirac equation is separable [35] and can be reduced to a Schrödinger-like equation [36] whose potential is contained in the Rosen-Morse class of integrable potentials, which has the so-called Pöschl-Teller potential as a particular case [37, 38]. Finally, the boundary conditions leading to QNMs are analyzed, and the quasinormal frequencies (QNFs) are analytically obtained

*Joás Venâncio and Carlos Batista*

Thus, extensions of special relativity are presented extensively by Richard Sauerheber in Chapter 3.

Extensions of general relativity are presented by Francis T.S. Yu in Chapter 6.

The foundations of these extensions are given special attention in the form of Lagrangian formalism by Kadiata Ba in Chapter 7, canonical formalism by Lawrence P. Horwitz in Chapter 9, and variational formalism by Giorgio Turchetti and Federico Panichi in Chapter 10.

In terms of applications, special attention is paid to the nature of light and dark matter, as well as dynamics involving exotic materials such as black holes, wormholes, and other structures involving special topologies. These topics are covered by Joás Venâcio and Carlos Batista in Chapter 1, S. D. Prijmenko and K.A. Lukin in Chapter 4, Maricel Agop et al. in Chapter 8, Giorgio Turchetti and Federico Panichi in Chapter 10, Radosz Andrzej in Chapters 11 and 12, and Fomin Igor in Chapter 13.

> **Calin Gheorghe Buzea** National Institute of Research and Development for Technical Physics, Iasi, Romania

### **Chapter 1**
