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Solutions of Conformal Gravity with Dynamical Mass Generation in the Solar System

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Solutions of Conformal Gravity with Dynamical Mass Generation in the Solar System Solutions of Conformal Gravity with Dynamical Mass Generation in the Solar System A thesis submitted in partial fulfilment of the requirements for the Degree of Doctor of Philosophy in Physics in the University of Canterbury by Joshua N. Wood University of Canterbury 2000 11 PHYSICAl.. SCIENCES LIBRARY Qc (1~ , W87() Abstract An investigation is made into the viability of the fourth-order conformal theory of gravity with dynamical mass generation. This is done by considering the analytical behaviour of the equations of motion and using this as a guide to producing numerical solutions to these equations. A review of some criticisms of the fourth-order gravity theory is included. Numerical solutions of the equations of motion are produced for the domain interior to the source and the exterior region within the solar system, with a variety of source conditions and under different formulations. These are analysed with consideration of reasonable physical and observational requirements, based on the well established solar system gravitational effects. The possibility of extra gravitational effects to explain anomalies in current gravitational theory is investigated. These effects include the possible anomalous acceleration of the Pioneer spacecraft, and could be extended to cover galaxy-scale phenomena such as galactic rotation curves. Conclusions are then drawn about what formulations and parameter sets are viable for good representation of the physics. The effects of the dynamical mass generation on observed phenomena are discussed, with reference to the potential resolution of gravitational anomalies. 1 4 zaU1 iv Contents 1 Introduction 1 1.1 Problems for General Relativity 2 1.2 Conformal Gravity Theories .. 4 1.2.1 Existing Work in Conformal Gravity 4 1.3 Current state of General Relativity 6 1.4 Aim of the Thesis 6 1.5 Conventions ... 8 2 The Conformal Field Equations 11 2.1 Weyl Field Equations. 11 2.1.1 Metric Forms . 12 2.1.2 Existing Solutions to the Field Equations. 15 2.2 Dynamical Mass Generation and Conformal Gravity. 17 2.2.1 Conformal Hydrostatic Equilibrium .... 19 2.2.2 Interior and Exterior Quadrature Solution 22 3 Conformal Geodesic Motion 25 3.1 Dynamical Geodesic Equations ........... 25 3.1.1 Coordinate Forms of the Geodesic Equations . 28 3.2 Application to Particle Motion. 33 3.2.1 Features ........ 33 3.2.2 Usage of the Orbital Equation 34 3.2.3 Higgs-Dependant Acceleration 37 3.2.4 Application to the Possible Anomalous Acceleration of Pioneer Spacecraft, et. al. 39 3.2.5 Analytic S Field Behaviour 40 3.3 Conformal Transformations and Observable Quantities 44 v vi Contents 4 Criticisms of Conformal Gravity 47 4.1 'Solution Matching in Weyl Gravity' - Perlick & Xu 47 4.2 Wood & Nemiroff 1991 50 4.3 Other Concerns .... 51 5 Implementation of a Numerical Solution to Conformal Gravity 53 5 .1 Numerical Techniques ..... 54 5.1.1 Extrapolation Methods. 55 5.1.2 Newton-Raphson Methods 58 5.1.3 Stability Issues . 59 5.1.4 Computational Accuracy. 61 5.2 Boundary Conditions ...... 63 5.2.1 Photospheric Boundary Condition Data 64 5.2.2 Boundary Conditions at the Origin 65 5.3 Implementation ........ 67 5.3.1 Numerical Requirements 67 6 Numerical Results 73 6.1 Exterior Solutions. 73 6.2 Interior Solutions . 83 6.2.1 Full Interior Solution 86 6.2.2 Interior Solution in band S 103 6.3 Consistent Interior/Exterior Solutions. 112 6.4 Discussion of the Methods ...... 116 7 Conclusion 121 7.1 Suggestions for Further Work 125 7.2 Acknowledgments ...... 126 A Algorithms of Extrapolation Methods 127 A.1 Runge-Kutta Methods . 127 A.2 Burlisch-Stoer Methods. 128 B Algorithms of Relaxation Methods 131 C Numerical Engine Code 135 D Effects of Numerical Limitations on Solutions 173 Contents vii E Field Equation Derivation 177 F Scalar Field Conservation Identity 181 References 183 Vlll Contents Chapter 1 Introduction Theoretical physics has made great advances in this century towards explaining the nature of the fundamental interactions and particles that make up the universe we live in. The essential method in this proceeding explanation of our reality is the unification of the fundamental forces - namely, the strong and weak nuclear forces, the electromagnetic force, and the gravitational force. It is a kind of holy grail for theorists to produce a 'theory of everything' which can include all these forces within a single framework. The nature of the fundamental forces has been explored via theories of fields, which have developed in ways peculiar to the forces being described. Gravitation is essentially and intrinsically linked to the properties and contents of the space­ time in which it is acting, and so is described in terms of metric field tensor 9J-Lv by Einstein's general relativity. This field describes the distances and curvatures of space and time, leading to a beautifully symmetric theory in which the mass and energy content of the universe defines how space is curved, which in turn defines how the mass and energy will move. Developed in 1916, the General Theory of Relativity [1] has been spectacularly successful in describing the gravitational phenomena in the solar system, to the limits of definition of our observations. Some time after the development of general relativity (GR), quantum field theory (QFT) began to be considered as a good description of the fundamental particles. The theory has considerable initial difficulties, such as the possibility of particles with infinite amounts of energy. These were overcome with the advent of renormalisable theories, which through a redefinition of the idea of what a particle is, allow such infinities to be neatly removed from the theory. Particles are now being treated in a more dynamic way, as being the excitations of a field through­ out space-time, rather than being kinematic constructs. Concepts of symmetry invariance, when applied to quantum field theory, allowed the development of the theory of dynamical mass generation by Weinberg, Glashow and Salaam, which 1 2 Chapter 1. Introduction unified the weak nuclear and electromagnetic forces. This theory of dynamical mass generation by 'spontaneously broken sym­ metry' (SBS) was a revolutionary concept and took some time to be accepted (see Weinberg's book, 'Dreams of a final theory' [2]), but with the experimental discovery of the Wand Z particles at the CERN facility in Switzerland, the theory gained strong credence. It has since been expanded to include the strong nuclear force in the same theoretical framework (quantum chromodynamics, QCD ), and is so accepted today that this is referred to as the 'standard model'. However, the final unification of the four forces remains elusive. The nature of the formulation of general relativity considers matter kinematically, not dynam­ ically. A space-time containing dynamically generated matter is, to Einstein's field equations, essentially a space-time containing nothing. Despite both general relativity and quantum field theory being generally accepted, efforts towards final unification are often in essentially different theories such as superspace or string theories. 1.1 Problems for General Relativity Despite it's wide acceptance and excellent observational verification in solar sys­ tem tests, the general theory of relativity is not without it's own foils. Possibly the most obvious difficulty is the 'missing mass' problem, in it's various forms. This is perhaps most easily described in terms of galaxy dynamics. By measuring the Doppler shift of the frequency of light emitted by a galaxy that is near edge-on to our line of sight here on Earth, the relative velocity of the different parts of such a galaxy can be determined. It is found that the rotational velocity of the stars in the spiral arms of galaxies increase in a way which is not in agreement with the predictions of general relativity. Given the apparent success of GR on local scales, such as within the solar system, the usually presented argument is that the extra gravitational attraction required is provided by 'dark matter' of some kind which is not observed - i.e. there is something out there which we cannot see or detect which is responsible for fixing up the anomalous results. For reasons to do with stability of the rotation of the galaxy, the average spiral galaxy requires about nine times as much dark matter as luminous, observed matter, distributed in a spherical 'halo' around the spiral. The exact composition of the dark matter is speculative, ranging from unknown, exotic fundamental particles that we are currently unaware of, to small stellar 1.1. Problems for General Relativity 3 bodies similar to the familiar galactic bodies, but of very low luminosity and profusely distributed. Another serious question for general relativity is the expansion of the uni­ verse. This has always been a slightly nebulous issue, as the end result of the expansion depends on the density of the universe, something not yet determined. Currently, with only considering the observed luminous sources, the expansion of the universe will continue indefinitely as there is not enough matter to reverse the expansion through gravitational attraction. As this has been regarded as an undesirable destiny, mechanisms to retard the expansion have been devised. The most famous is the cosmological constant A, an 'ad hoc' term originally added to Einstein's field equations as a 'negative pressure' to augment the gravitational contraction of the universe. The exact value of A is arguably the least specified constant in any theory, as arguments from gravitational theories can be made for it to be zero, whilst from particle theory arguments it could be required to be of order 10 60 or larger. Recent developments have only exacerbated this problem. Measurements have been made of very distant high redshift type 1a supernovae, the furthest away and thus most ancient of the objects for which we can calculate distances ([3, 4]).
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