Collisional Stellar Dynamics in General Relativity: An Overview

This thesis presents recent research into analytic topics in the classical theory of General Relativity. It is a thesis in two parts. The first part features investigations into the spectrum of perturbed, rotating black holes. These include the study of near horizon perturbations, leading to a new generic frequency mode for black hole ringdown; an treatment of high frequency waves using WKB methods for Kerr black holes; and the discovery of a bifurcation of the quasinormal mode spectrum of rapidly rotating black holes. These results represent new discoveries in the field of black hole perturbation theory, and rely on additional approximations to the linearized field equations around the background black hole. The second part of this thesis presents a recently developed method for the visualization of curved spacetimes, using field lines called the tendex and vortex lines of the spacetime. The works presented here both introduce these visualization techniques, and explore them in simple situations. These include the visualization of asymptotic gravitational radiation; weak gravity situations with and without radiation; stationary black hole spacetimes; and some preliminary study into numerically simulated black hole mergers. The second part of thesis culminates in the investigation of perturbed black holes using these field line methods, which have uncovered new insights into the dynamics of curved spacetime around black holes.

Irrespective of local conditions imposed on the metric, any extendible spacetime U has a maximal extension containing no closed causal curves outside the chronological past of U. We prove this fact and interpret it as impossibility (in classical general relativity) of the time machines insofar as the latter are defined to be causality-violating regions created by human beings (as opposed to those appearing spontaneously).

Recent observations and stellar dynamics simulations suggest that thousand\nsolar mass black holes can form in compact massive young star clusters. Any\nsuch clusters in the bulge of their host galaxy will spiral to the center\nwithin a few hundred million years, where their intermediate-mass black holes\nare likely to merge eventually with the galaxy's supermassive black hole. If\nsuch mergers are common, then future space-based gravitational wave detectors\nsuch as the Laser Interferometer Space Antenna will detect them with such a\nhigh signal to noise ratio that towards the end of the inspiral the orbits will\nbe visible in a simple power density spectrum, without the need for matched\nfiltering. We discuss the astrophysics of the inspiral of clusters in the\nnuclear region of a galaxy and the subsequent merger of intermediate-mass with\nsupermassive black holes. We also examine the prospects for understanding the\nspacetime geometry of rotating black holes, based on phase connection of the\nstrong signals visible near the end of these extreme mass ratio inspirals.\n

Considerable efforts have been made and are still being made to verify the validity of the principle of the equivalence of gravitational and inertial mass, one of the cornerstones of the classical theory of general relativity. Specific attempts at quantum-mechanical formulations of gravity allow for non-Newtonian contributions which might lead to a difference in the gravitational force on matter and antimatter. While it is widely expected that the gravitational interaction is independent of the composition of bodies, this has only been tested for matter systems, but never yet for antimatter systems. By combining techniques from different fields, and relying on recent developments in the production of Positronium and ongoing work to laser-excite Positronium to Rydberg states, such a test with neutral antimatter has become feasible. The primary goal of the AEGIS experiment being built at the Antiproton Decelerator at CERN is to carry out the first direct measurement of the Earth's gravitational acceleration on antihydrogen by means of a classical Moiré deflectometer.

The possibility of a theory in which only matter is quantised, and gravity is described by the classical theory of general relativity, even at the fundamental level, is not ruled out by any theoretical argument nor by experiment. Here, I discuss implications of this possibility and show that, from the most naive approach for such a theory, a gravity-matter coupling according to the semiclassical Einstein equations, one obtains a nonlinear Schrodinger equation as a non-relativistic limit that may be used to derive experimental predictions. Possibilities of experimental tests are discussed, which seem feasible with existing technology in the not-too-far future.

We investigate the Church–Kalmar–Kreisel–Turing theses theoretical concerning (necessary) limitations of future computers and of deductive sciences, in view of recent results of classical general relativity theory. We argue that (i) there are several distinguished Church–Turing-type theses (not only one) and (ii) validity of some of these theses depend on the background physical theory we choose to use. In particular, if we choose classical general relativity theory as our background theory, then the above-mentioned limitations (predicted by these theses) become no more necessary, hence certain forms of the Church–Turing thesis cease to be valid (in general relativity). (For other choices of the background theory the answer might be different.) We also look at various “obstacles” to computing a nonrecursive function (by relying on relativistic phenomena) published in the literature and show that they can be avoided (by improving the “design” of our future computer). We also ask ourselves, how all this reflects on the arithmetical hierarchy and the analytical hierarchy of uncomputable functions.

The metric condition and the symmetry of affine connexions are both a priori assumed by General Relativity without much justification. To pro~ide a basis for this, we present a new approach to spacetime structure, taking as probes elementary particles of spin 1/2, baryons and leptons including neutrinos which define 'neutrino null cone'. Thus our ap· proach is much finer than the usual ones using classical particles and quite opposite to spinor formulation of General Relativity as it stands. Assuming the most general form of metric spinor and spinor connexion which includes two gauge fields, we mathematically deduce (1) the metric tensor and (2) the semimetric condition. By virtue of the Eotvos· type null experiments, we find that fermion-number gauge field interacts with matter too weakly to be detected. Furthermore, it it shown that scale gauge field neither couples to the elementary particles, nor the electromagnetic field. We are thus led to the Riemann geometry with torsion, first introduced by Cartan independently of spin concept. There is no evidence whatsoever to exclude torsion. We conclude, therefore, that torsion must seriously be taken into account, in particular when and where classical General Relativity seems to break down.

We compute the next-to-leading order term in the scattering waveform of uncharged black holes in classical general relativity and of half-BPS black holes in mathcal{N} = 8 supergravity. We propose criteria, generalizing explicit calculations at next-to-leading order, for determining the terms in amplitudes that contribute to local observables. For general relativity, we construct the relevant classical integrand through generalized unitarity in two distinct ways, (1) in a heavy-particle effective theory and (2) in general relativity minimally-coupled to scalar fields. With a suitable prescription for the matter propagator in the former, we find agreement between the two methods, thus demonstrating the absence of interference of quantum and classically-singular contributions. The classical mathcal{N} = 8 integrand for massive scalar fields is constructed through dimensional reduction of the known five-point one-loop integrand. Our calculation exhibits novel features compared to conservative calculations and inclusive observables, such as the appearance of master integrals with intersecting matter lines and the appearance of a classical infrared divergence whose absence from classical observables requires a suitable definition of the retarded time.

The central conceptual idea of the contemporary theory of general relativity--or geometrodynamics--is the identification of matter with the structure of space-time. No entities foreign to space-time, like masses, charges, or independent fields are needed, and physics thus becomes identical with the geometry of space-time. This idea implies a philosophical description of the universe that is monistic and organic, characterized by an all-encompassing interdependence of events. Moreover, it is an idea with deep roots in the history of philosophy. For these reasons, the author of this important study strives to clarify these philosophical and historical issues before proceeding to the details of the physical theory of geometrodynamics.Graves develops a general philosophical framework of realism, asserting that scientific theories have ontological import in determining the real structure of the world, and in particular that models play a central role as ontological hypotheses. He introduces two factors, the empiricist error and the logical errors that he feels have contributed to past misinterpretations of the nature of reality that were based on general relativity theory.The historic continuities that join the metaphysics of Plato and Descartes with the monistic, organic nature of geometrodynamics are contrasted with the latter's radical break with the Newtonian or atomistic worldview which is predicated on a basic separation between matter and space. The Newtonian independence and distinctness of objects is at the polar extreme from their Einsteinian interdependence and continuity.In discussing the physical principles of relativity, the author begins with a review of Einstein's original formulation (classical general relativity) and the significance of its basic assumptions and equations. He then presents the remarkable recent developments in geometrodynamics, which allow the program of identifying matter with space-time to be carried further than even Einstein suspected possible. The surprising discovery that electromagnetism can be incorporated into geometrodynamics without modifying Einstein's original equations appears to be formally correct, but reliance on multiply-connected topologies (wormholes) to represent charge raises various unresolved questions.Graves concludes that the present language of physics, like that of everyday life, is based on concepts of independence and separation, and that a wholly new language may be needed to describe the world in terms of geometrodynamics, in which space-time appears as the only substance, with curvatures as its attributes, and in which objects have no absolute individuality, distinctness, or location.

The big bang and the Schwarzschild singularities are space-like. They are generally regarded as the "final frontiers" at which space-time ends and general relativity breaks down. We review the status of such space-like singularities from three increasingly more general perspectives. They are provided by (i) A reformulation of classical general relativity motivated by the Belinskii, Khalatnikov, Lifshitz conjecture on the behavior of the gravitational field near space-like singularities; (ii) The use of test quantum fields to probe the nature of these singularities; and, (iii) An analysis of the fate of these singularities in loop quantum gravity due to quantum geometry effects. At all three levels singularities turn out to be less menacing than one might a priori expect from classical general relativity. Our goal is to present an overview of the emerging conceptual picture and suggest lines for further work. In line with the \emph{Introduction to Current Research} theme, we have made an attempt to make it easily accessible to all researchers in gravitational physics.

In this brief review, we present the results of the fractional differential approach in cosmology in the context of the exact models of cosmological accelerated expansion obtained by several authors to date. Most of these studies are devoted to the problem of introducing fractional derivatives or fractional integrals into the classical General Relativity (GR). There are several observational and theoretical motivations to investigate the modified or alternative theories of GR. Among other things, we cover General Relativity modified by a phenomenological approach dealing with fractional calculus. At the same time, a sufficiently large number of exact solutions of the cosmological equations modified by this approach were obtained. Some of these models may be especially relevant in the light of solving the problem of late accelerated expansion of the Universe. These studies are largely motivated by rapid progress in the field of observational cosmology that now allows, for the first time, precision tests of fundamental physics on the scale of the observable Universe. The purpose of this review is to provide a reference tool for researchers and students in cosmology and gravitational physics, as well as a self-contained, comprehensive and up-to-date introduction to the subject as a whole.

The physical meaning of the Planck length (ell _{P}) is investigated in the framework of the unconstrained synchronous variational formulation of classical general relativity (GR). This theoretical setting permits the establishment of manifestly-covariant Lagrangian and Hamiltonian theories for the Einstein field equations of the continuum gravitational field. It is shown that such a formulation is distinguished by the existence of a novel variational contribution expressed by an infinite series summation of suitable 4-scalar terms in which the coupling coefficients are even powers of the Planck length. However, the requirement of realization of a classical GR Hamiltonian theory places stringent constraints on the admissible Planck-length power terms to be retained. In fact, excluding the trivial gauge constant, it is proved that only the Oleft( ell _{P}^{0}right) contribution of the series is ultimately permitted, namely the unique one which is independent of ell _{P}. Therefore, the Planck length is effectively not allowed to appear at the classical level for consistency with the Hamiltonian principle. This places important consequences on the mathematical establishment of the corresponding canonical quantum gravity theory, which is then found to be correct through Oleft( ell _{P}^{2}right) . Additional implications concern the physical significance of related quantum momenta and their meaning in the semi-classical limit, as well as the role of the Planck length in the same quantum-gravity realm.

Zero-point fluctuations are a universal consequence of quantum theory. Vacuum fluctuations of electromagnetic field have provided crucial evidence and guidance for QED as a successful quantum field theory with a defining gauge symmetry through the Lamb shift, Casimir effect, and spontaneous emission. In an accelerated frame, the thermalisation of the zero-point electromagnetic field gives rise to the Unruh effect linked to the Hawking effect of a black hole via the equivalence principle. This principle is the basis of general covariance, the symmetry of general relativity as the classical theory of gravity. If quantum gravity exists, the quantum vacuum fluctuations of the gravitational field should also lead to the quantum decoherence and dissertation of general forms of energy and matter. Here we present a novel theoretical effect involving the spontaneous emission of soft gravitons by photons as they bend around a heavy mass and discuss its observational prospects. Our analytic and numerical investigations suggest that the gravitational bending of starlight predicted by classical general relativity should also be accompanied by the emission of gravitational waves. This in turn redshifts the light causing a loss of its energy somewhat analogous to the bremsstrahlung of electrons by a heavier charged particle. It is suggested that this new effect may be important for a combined astronomical source of intense gravity and high-frequency radiation such as X-ray binaries and that the proposed LISA mission may be potentially sensitive to the resulting sub-Hz stochastic gravitational waves.

Tunneling through bridges: Bohmian non-locality from higher-derivative gravity

Wormhole solutions in classical general relativity are unstable and hence non traversable. Morris and Thorne discovered a traversable wormhole solution that required the energy momentum tensor of matter sources to violate various energy conditions and are out of the purview of the standard model of particle physics. The search for traversable wormhole solutions in modified theories of gravity has been of significant interest in the decades after Morris and Thorne first published their results as such violations may be avoided in such theories. This work comprehensively reviews traversable wormhole solutions in higher dimensional extensions of general relativity that satisfy the various energy conditions with an in depth look at the matter sources and the various constraints on the parameters of the theories to make the energy momentum of the matter sources respect the energy conditions.