- •Preface
- •Acknowledgements
- •Contents
- •1.1 Introduction
- •1.2 Classical Physics Between the End of the XIX and the Dawn of the XX Century
- •1.2.1 Maxwell Equations
- •1.2.2 Luminiferous Aether and the Michelson Morley Experiment
- •1.2.3 Maxwell Equations and Lorentz Transformations
- •1.3 The Principle of Special Relativity
- •1.3.1 Minkowski Space
- •1.4 Mathematical Definition of the Lorentz Group
- •1.4.1 The Lorentz Lie Algebra and Its Generators
- •1.4.2 Retrieving Special Lorentz Transformations
- •1.5 Representations of the Lorentz Group
- •1.5.1 The Fundamental Spinor Representation
- •1.6 Lorentz Covariant Field Theories and the Little Group
- •1.8 Criticism of Special Relativity: Opening the Road to General Relativity
- •References
- •2.1 Introduction
- •2.2 Differentiable Manifolds
- •2.2.1 Homeomorphisms and the Definition of Manifolds
- •2.2.2 Functions on Manifolds
- •2.2.3 Germs of Smooth Functions
- •2.3 Tangent and Cotangent Spaces
- •2.4 Fibre Bundles
- •2.5 Tangent and Cotangent Bundles
- •2.5.1 Sections of a Bundle
- •2.5.2 The Lie Algebra of Vector Fields
- •2.5.3 The Cotangent Bundle and Differential Forms
- •2.5.4 Differential k-Forms
- •2.5.4.1 Exterior Forms
- •2.5.4.2 Exterior Differential Forms
- •2.6 Homotopy, Homology and Cohomology
- •2.6.1 Homotopy
- •2.6.2 Homology
- •2.6.3 Homology and Cohomology Groups: General Construction
- •2.6.4 Relation Between Homotopy and Homology
- •References
- •3.1 Introduction
- •3.2 A Historical Outline
- •3.2.1 Gauss Introduces Intrinsic Geometry and Curvilinear Coordinates
- •3.2.3 Parallel Transport and Connections
- •3.2.4 The Metric Connection and Tensor Calculus from Christoffel to Einstein, via Ricci and Levi Civita
- •3.2.5 Mobiles Frames from Frenet and Serret to Cartan
- •3.3 Connections on Principal Bundles: The Mathematical Definition
- •3.3.1 Mathematical Preliminaries on Lie Groups
- •3.3.1.1 Left-/Right-Invariant Vector Fields
- •3.3.1.2 Maurer-Cartan Forms on Lie Group Manifolds
- •3.3.1.3 Maurer Cartan Equations
- •3.3.2 Ehresmann Connections on a Principle Fibre Bundle
- •3.3.2.1 The Connection One-Form
- •Gauge Transformations
- •Horizontal Vector Fields and Covariant Derivatives
- •3.4 Connections on a Vector Bundle
- •3.5 An Illustrative Example of Fibre-Bundle and Connection
- •3.5.1 The Magnetic Monopole and the Hopf Fibration of S3
- •The U(1)-Connection of the Dirac Magnetic Monopole
- •3.6.1 Signatures
- •3.7 The Levi Civita Connection
- •3.7.1 Affine Connections
- •3.7.2 Curvature and Torsion of an Affine Connection
- •Torsion and Torsionless Connections
- •The Levi Civita Metric Connection
- •3.8 Geodesics
- •3.9 Geodesics in Lorentzian and Riemannian Manifolds: Two Simple Examples
- •3.9.1 The Lorentzian Example of dS2
- •3.9.1.1 Null Geodesics
- •3.9.1.2 Time-Like Geodesics
- •3.9.1.3 Space-Like Geodesics
- •References
- •4.1 Introduction
- •4.2 Keplerian Motions in Newtonian Mechanics
- •4.3 The Orbit Equations of a Massive Particle in Schwarzschild Geometry
- •4.3.1 Extrema of the Effective Potential and Circular Orbits
- •Minimum and Maximum
- •Energy of a Particle in a Circular Orbit
- •4.4 The Periastron Advance of Planets or Stars
- •4.4.1 Perturbative Treatment of the Periastron Advance
- •References
- •5.1 Introduction
- •5.2 Locally Inertial Frames and the Vielbein Formalism
- •5.2.1 The Vielbein
- •5.2.2 The Spin-Connection
- •5.2.3 The Poincaré Bundle
- •5.3 The Structure of Classical Electrodynamics and Yang-Mills Theories
- •5.3.1 Hodge Duality
- •5.3.2 Geometrical Rewriting of the Gauge Action
- •5.3.3 Yang-Mills Theory in Vielbein Formalism
- •5.4 Soldering of the Lorentz Bundle to the Tangent Bundle
- •5.4.1 Gravitational Coupling of Spinorial Fields
- •5.5 Einstein Field Equations
- •5.6 The Action of Gravity
- •5.6.1 Torsion Equation
- •5.6.1.1 Torsionful Connections
- •The Torsion of Dirac Fields
- •Dilaton Torsion
- •5.6.2 The Einstein Equation
- •5.6.4 Examples of Stress-Energy-Tensors
- •The Stress-Energy Tensor of the Yang-Mills Field
- •The Stress-Energy Tensor of a Scalar Field
- •5.7 Weak Field Limit of Einstein Equations
- •5.7.1 Gauge Fixing
- •5.7.2 The Spin of the Graviton
- •5.8 The Bottom-Up Approach, or Gravity à la Feynmann
- •5.9 Retrieving the Schwarzschild Metric from Einstein Equations
- •References
- •6.1 Introduction and Historical Outline
- •6.2 The Stress Energy Tensor of a Perfect Fluid
- •6.3 Interior Solutions and the Stellar Equilibrium Equation
- •6.3.1 Integration of the Pressure Equation in the Case of Uniform Density
- •6.3.1.1 Solution in the Newtonian Case
- •6.3.1.2 Integration of the Relativistic Pressure Equation
- •6.3.2 The Central Pressure of a Relativistic Star
- •6.4 The Chandrasekhar Mass-Limit
- •6.4.1.1 Idealized Models of White Dwarfs and Neutron Stars
- •White Dwarfs
- •Neutron Stars
- •6.4.2 The Equilibrium Equation
- •6.4.3 Polytropes and the Chandrasekhar Mass
- •6.5 Conclusive Remarks on Stellar Equilibrium
- •References
- •7.1 Introduction
- •7.1.1 The Idea of GW Detectors
- •7.1.2 The Arecibo Radio Telescope
- •7.1.2.1 Discovery of the Crab Pulsar
- •7.1.2.2 The 1974 Discovery of the Binary System PSR1913+16
- •7.1.3 The Coalescence of Binaries and the Interferometer Detectors
- •7.2 Green Functions
- •7.2.1 The Laplace Operator and Potential Theory
- •7.2.2 The Relativistic Propagators
- •7.2.2.1 The Retarded Potential
- •7.3 Emission of Gravitational Waves
- •7.3.1 The Stress Energy 3-Form of the Gravitational Field
- •7.3.2 Energy and Momentum of a Plane Gravitational Wave
- •7.3.2.1 Calculation of the Spin Connection
- •7.3.3 Multipolar Expansion of the Perturbation
- •7.3.3.1 Multipolar Expansion
- •7.3.4 Energy Loss by Quadrupole Radiation
- •7.3.4.1 Integration on Solid Angles
- •7.4 Quadruple Radiation from the Binary Pulsar System
- •7.4.1 Keplerian Parameters of a Binary Star System
- •7.4.2 Shrinking of the Orbit and Gravitational Waves
- •7.4.2.1 Calculation of the Moment of Inertia Tensor
- •7.4.3 The Fate of the Binary System
- •7.4.4 The Double Pulsar
- •7.5 Conclusive Remarks on Gravitational Waves
- •References
- •Appendix A: Spinors and Gamma Matrix Algebra
- •A.2 The Clifford Algebra
- •A.2.1 Even Dimensions
- •A.2.2 Odd Dimensions
- •A.3 The Charge Conjugation Matrix
- •A.4 Majorana, Weyl and Majorana-Weyl Spinors
- •Appendix B: Mathematica Packages
- •B.1 Periastropack
- •Programme
- •Main Programme Periastro
- •Subroutine Perihelkep
- •Subroutine Perihelgr
- •Examples
- •B.2 Metrigravpack
- •Metric Gravity
- •Routines: Metrigrav
- •Mainmetric
- •Metricresume
- •Routine Metrigrav
- •Calculation of the Ricci Tensor of the Reissner Nordstrom Metric Using Metrigrav
- •Index
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7 Gravitational Waves and the Binary Pulsars |
Although this might seem incredible, the present sensitivity of the bar detectors is coming close to the critical sensitivity. No gravitational wave signal has so far been revealed, yet it might happen any day.
While the bar technology was developed, new impetus to the quest for gravitational waves was brought by the indirect evidence of their existence revealed by a sensational astrophysical discovery of 1974. This latter was made possible by the progresses of radio-astronomy and the construction of giant radio-telescopes as that of Arecibo.
7.1.2 The Arecibo Radio Telescope
The construction of the Arecibo telescope (see Fig. 7.2), located in United State territory in the Caribbean island of Puerto Rico, was initiated by Professor William E. Gordon of Cornell University, who originally intended to use it for the study of Earth’s ionosphere. Later, through contact with other agencies, including the Air Force Cambridge Research Laboratory, the original project was somewhat modified, also with the essential contribution of the two brothers Doundoulakis, chief engineers of the General Bronze Corporation in Garden City, New York. These latter devised and designed the cable suspension system which is the structure finally realized.
Construction began in the summer of 1960, with the official opening on November 1st, 1963. As the primary dish is spherical, its focus is along a line rather than at a single point, as would be the case for a parabolic reflector; thus complicated line feeds had to be used to carry out observations. The telescope has undergone significant upgrades, the first one in 1974, when a high precision surface was added for the current reflector. A Gregorian reflector system was installed in 1997, incorporating secondary and tertiary reflectors to focus radio waves at a single point. This allowed the installation of a suite of receivers, covering the whole 1–10 GHz range, that could be easily moved onto the focal point, giving Arecibo a new flexibility.
Fig. 7.2 The giant Arecibo radio Telescope in Puerto Rico
7.1 Introduction |
277 |
Fig. 7.3 The Crab Nebula, the remnant of the 1054 supernova. At the center there is the Crab Pulsar which, detected in 1968, gave the first solid evidence for the existence of neutron stars in our Universe
This long standing and sophisticated machine, which during the years of the cold war was also used for military purposes, notably for the localization of soviet radar installations, is responsible for a few fundamental scientific discoveries that have substantially upgraded human knowledge in the fields of Astrophysics and General Relativity.
7.1.2.1 Discovery of the Crab Pulsar
Four years after its coming into operation, Arecibo enabled Lovelace and other researchers to provide the first solid evidence for the existence of neutron stars in our Universe. This came through the detection of the radio signal emitted with a frequency of 33 milliseconds by the Crab Pulsar PSR0531+21. This compact star is at the center of the Crab Nebula, that is located at a distance of about 6,500 light years from Earth and has a diameter of about 11 light years (Fig. 7.3). Discovered in 1731 by the English doctor and amateur astronomer John Bevis,5 the Crab Nebula is just the remnant of the supernova SN 1054, one of the brightest in history, recorded by Chinese and Arab astronomers and also by the monks of St. Gallen monastery. The supernova 1054 was visible during day-time for 23 days and for 653 days in night-time. The detection of the Crab Pulsar at the center of the Crab Nebula gave final confirmation to the theory that the core of a gravitational collapsing star settles down to the equilibrium state of a neutron star, if its mass is inferior to the relevant Chandrasekhar limit. The supernova II explosion is just generated by the dramatic bouncing of the infalling matter on the incredibly hard crust of the newly formed neutron star.
5John Bevis, (1693–1771) besides the discovery of the Crab Nebula is known for his observations of the occultation of Venus by Mercury and for his studies on the eclipses of Jupiter’s moons.
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7 Gravitational Waves and the Binary Pulsars |
Fig. 7.4 The 1993 Nobel laureates R.A. Hulse and J.H. Taylor for their 1974 discovery of the binary pulsar system PSR 1913+16
7.1.2.2 The 1974 Discovery of the Binary System PSR1913+16
The 1993 Nobel Prize for Physics was awarded jointly to Joseph Hooton Taylor and Russell Alan Hulse (see Fig. 7.4) with the following motivation for the discovery of a new type of pulsar, a discovery that has opened up new possibilities for the study of gravitation. The motivation referred to their 1974 discovery of the binary pulsar system known as PSR1913+16. The first detection and subsequent thirtyyear long monitoring of the system were performed by means of the Arecibo Radio Telescope [5–7].
Born 29th March 1941 in Philadelphia in a family with Quaker roots, Taylor was Distinguished Professor of Physics at the University of Princeton from 1980 to his retirement in 2006. Previously he held a chair of Astronomy at the University of Massachusetts where he was Director of the Five College Radio Astronomy Observatory.
Nine year younger than his colleague, Russell A. Hulse was born in New York in 1950. He studied at the University of Massachusetts from which he received his Ph.D. in 1975. Presently he is staff member of the Princeton Plasma Physics Laboratory and also visiting professor at the University of Texas, Dallas.
At a very early stage of his scientific career, Hulse engaged with Taylor on a large scale survey for pulsars at the Arecibo facility and the result of their researches was the uncovering of the PSR1913+16 system. This latter is made up of a pulsar and a black companion star. The rotating neutron star emits impulses that are extremely regular and stable in the radio wave region. This allows for a careful reconstruction of the orbital motion of the two stars around their center of mass. Hulse and Taylor
