The Geometry of the Universe

The following sections are included:

• Foreword
• About the author
• Contents

Many ancient civilizations studied the heavens and plotted the positions of the visible planets. The belief that heavenly bodies such as planets affect humans was widespread, and both ancient Greek and Roman civilizations used the same words for some of their Gods as for the planets. This book is not concerned with the possibility that there might be some substance in these beliefs (Astrology), but rather with the nature and dynamics of the heavenly bodies themselves (Astronomy) and the ultimate nature of the entire universe (Cosmology). It is however a salient fact that belief in astrology powered much early astronomy leading to the construction of observatories such as the Ziggurats of the Babylonians and Stonehenge…

The following sections are included:

• Special Relativity
• General Relativity
• Model building
  • Einstein's "biggest blunder"
• de Sitter space
  • de Sitter space and Mach’s principle

The following sections are included:

• The biggest blunder
  • Weyl’s postulate
  • De Sitter space as a model for the universe
• The standard model
  • Two philosophical problems
• Dark matter
  • Sciama’s principle and inertial drag
  • The rotation curve
  • The fundamental relation
  • Spiral structure
• The Arp problem
• The quasar–galaxy spectrum
• Killing the angular momentum obstruction
• Embedding Mach’s principle in EGR
• Outline of Part 2

This chapter is concerned with a discussion of Mach’s principle and the restricted version that is needed for the dynamical applications (to quasars and spiral galaxies) in the rest of the book. The final form of the principle (the Weak Sciama Principle) hypothesises an inertial dragging effect from a rotating body which drops off asymptotically with k/r where k is a constant and r is distance from the centre. A reader who is happy to accept this principle can omit this chapter without loss. The precise assumption is repeated near the beginning of the next chapter.

The rotation curve of a galaxy with an equatorial plane (for example a spiral galaxy has its spiral arms lying roughly in such a plane) is the plot of tangential velocity against distance from the centre for a particle (star or similar) moving in the equatorial plane. In practice it is not possible to observe one star, but rather the general motion of all stars (or other radiating matter) in the equatorial plane. This makes the observed nature of rotation curves all the more striking. Typically the curve (of tangential velocity against distance from the centre) comprises two approximately straight lines with a short transition region. The first line passes through the origin, in other words rotation near the centre has constant angular velocity (plate-like rotation); the second is horizontal, in other words the tangential velocity is asymptotically constant, see Figure 5.4 (right) below. Furthermore, observations show that the horizontal straight line section of the rotation curve extends far outside the limits of the main visible parts of galaxies and the actual velocity is constant within less than an order of magnitude over all galaxies observed (typically between 100 and 300km/s) see Figure 5.5…

Quasars were first observed in the 1960’s. Through a telescope they appear to be stars but they exhibit strange features not shared by ordinary stars. They have spectra which often appear to be hugely redshifted and they vary irregularly with time scales that range from hours to months. Early in the study of quasars a heated controversy raged about these huge redshifts. Are they cosmological due to the expansion of the universe? or are they gravitational due to the near presence of a massive object (eg a black hole)? The cosmological explanation implies that quasars with large redshifts are extremely distant objects with truly phenomenal power outputs which are very hard to explain. By constrast the gravitational explanation allows the possibility that they are modest size objects, not too distant and with easily modelled power outputs. As can be seen from the Google definition, the cosmological explanation is the currently accepted one. This is despite some incontrovertible evidence in the form of observations of Halton Arp and others that quasars are often closely associated with galaxies with the redshift for the quasars significantly higher than that for the associated galaxies, a striking example of which was previewed in Section 3.4 and is reproduced in again Figure 6.1…

The chapter combines ideas from the last two chapters to give a complete description of the dynamics of spiral galaxies.

Previous chapters have established a new model for spiral galaxies based on Sciama’s principle. This has provided a satisfactory explanation for observed rotation curves without using “dark matter”, and accurately modelled the spiral structure. The salient features of this new model are (1) a central rotating mass (presumably a black hole) of 10¹¹ to 10¹⁴ solar masses which controls the dynamic via inertial drag effects and (2) a counter-rotating belt structure similar to the accretion disc around the (rather smaller) black holes in so-called “active” galaxies which feeds the roots of the spiral arms with pure H–He ions (with a trace of Li). The whole galaxy has a cyclical structure with matter ejected from the centre, condensing into stellar systems, moving outwards along the arms, burning out and falling back into the centre to be recycled. Thus the outward flow of gas mixes with dust and debris outside the belt and, as it flows outwards, condenses into violent star-producing regions illuminated by novae and supernova explosions. These regions sythesise the heavier elements needed for planets such as the earth to support chemically based life-forms. Solar systems containing such planets are formed further out along the arms…

This chapter discusses cosmological consequences of the model for galactic dynamics constructed in the earlier chapters and starts by considering the Big Bang theory. No abstract theory in physics has ever captured the general imagination in the way that this theory has. Even the fine detail has passed into everyday usage. Here for example is an excerpt from a review from The Guardian: The modern hunger to accord food spiritual “meaning” seems a relatively recent development: it is refreshing to note the absence of such inflated claims, for example, in the much-loved 1931 American cookbook The Joy of Cooking, by Irma Rombauer. [Description of low-key rhetoric in this book omitted] Yet since then foodist rhetoric has, like the early universe, experienced a period of rapid inflation. The foodist movement is desperate to claim other cultural domains as inherent virtues of food itself, so as not ever to have to stop thinking about stuffing its face. Food becomes not only spiritual nourishment but art, sex, ecology, history, fashion and ethics… Extracted from: The Guardian 29 Sep 2012, Review section, Steven Poole “Get stuffed”

Given such universal appreciation of the fine detail of the theory, it seems churlish to prove that it is wrong. But unfortunately this is the case.…

The following sections are included:

• Appendix A: Introduction to relativity
  • Special relativity
  • Causality
  • The speed of light and Michelson–Morley
  • Lorentz transformations
  • Time dilation and length contraction
  • Minkowski space
  • General Relativity
  • Manifolds and space-times
  • Curvature
    • Riemann curvature
    • Ricci curvature
  • Einstein’s equations
    • Einstein’s biggest blunder
    • Vacuum equations with cosmological constant
    • The Schwarzschild and de Sitter solutions
    • Black holes
    • De Sitter space
• Appendix B: De Sitter space
  • Minkowski space
  • Space-times
  • de Sitter and hyperbolic spaces
  • Projective geometry
  • Hyperbolic and de Sitter geometry
  • Transitivity of points and geodesics
  • The expansive metric
  • Time-like geodesics in Exp
  • The de Sitter metric
  • Explicit formulae
• Appendix C: Quasars: technical material
  • Bondi sphere radius and accretion rate
  • Kinetic energy, escape velocity and redshift
  • Potential and kinetic energy in Schwarzschild space-time
  • The critical radius and high redshift black holes
  • Calculations
    • Redshift in terms of medium factor and mass
    • Three types of redshift and the HLSW formula
    • Luminosity and magnitude
    • Eddington radius
    • Radiant temperature
  • Data
  • Conclusions
• Appendix D: Local stellar velocities
  • The observations
  • The explanations: Velocity variation increases with age
  • Asymmetric drift
  • Vertex deviation
• Appendix E: Optical distortion in the Hubble Ultra-Deep Field
  • Introduction
  • The face galaxy
  • Gravitational solitons
  • The companion face
  • The group of four
  • Four distorted spirals
  • Miscellanea
  • Conclusion
• Appendix F: Gamma Ray Bursts
  • Introduction
  • Geodesics in de Sitter space
  • Critique
  • Final remark

The following sections are included:

• Bibliography
• Index