- •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
306
Table 7.1 Data of the binary pulsar system PSR1913+16
7 Gravitational Waves and the Binary Pulsars
Right ascension |
19 h 13 m 12.4 s |
Declination |
+16◦01 08 |
Distance |
21,000 light years |
Mass of detected pulsar |
1.441 × MSun |
Mass of companion |
1.387 × MSun |
Rotational period of detected pulsar |
59.02 ms |
Diameter of each neutron star |
20 km |
Orbital period |
7.752 h |
Eccentricity |
0.62 |
Semilatus rectum |
1.95 × 106 km |
7.4.3 The Fate of the Binary System
Defining |
|
|
|
|
α ≡ u(G, c) · g(μ1, μ2) · f(e) |
(7.4.46) |
|||
whose numerical value is: |
|
|
|
|
α = −0.625128 × 10−4 |
5 |
(7.4.47) |
||
s 3 |
||||
The revolution period obeys the following differential equation |
|
|||
dT |
5 |
|
|
|
|
|
+ αT − 3 = 0 |
|
(7.4.48) |
|
dt |
|
which is immediately integrated. Considering as initial the present instant of time and fixing the boundary condition at T (0) = T0 = 7.751 h we obtain:
8tα 3/8
T (t) = T08/3 + (7.4.49)
3
hence the period constantly decreases while the orbit radius shrinks and eventually T will reduce to zero when the two stars come so close as to coalesce. From (7.4.49) we can get a rough estimate of the time needed to reach coalescence. Such an estimate is determined by solving T (tf ) = 0 for tf . We obtain:
tf = |
3T08/3 |
= 4.2983 × 1015 s = 1.382 × 108 years |
(7.4.50) |
8α |
In other words the two neutron stars will fall one on top of the other in about 140 millions of years. Clearly the quadrupole approximation will loose its validity when approaching coalescence. At short distances the non-linear nature of Einstein equations will play an essential role and the only known methods to calculate gravitational radiation in such situations are numerical.
7.4 Quadruple Radiation from the Binary Pulsar System |
307 |
Fig. 7.16 The double pulsar system PSR J0737-3039A/B, in the representation of an artist
7.4.4 The Double Pulsar
December 12th 2003, on journals and on the Internet appeared the official announcement of a new exciting discovery. An international team of radio-astronomers, including a strong and driving group of Italians,7 found the the Double Pulsar system officially named PSR J0737-3039A/B (see Fig. 7.16). This system is quite similar to the system PSR1913+16 but it has some additional features that make it an extraordinary precise laboratory to test General Relativity and Neutron Star Physics under extreme strong field conditions. The relevant data are displayed in Table 7.2. The first feature, in contrast with the case of PSR1913+16 is that both members of PSR J0737-3039A/B are pulsars and therefore they are both directly detectable. The second notable feature of the system is its extreme narrowness which emphasizes all General Relativity effects. The orbit has low eccentricity but the semilatus rectum is less than a million of Kilometers which results in a revolution period of just 2.4 hours. The periastron advance is accordingly very high and its measured value perfectly fits the predictions of General Relativity. Similarly the shortness of the revolution period allowed a rapid measure of its shrinking with very high statistics and the indirect evidence of the emission of gravitational waves was tested once again in excellent agreement with General Relativity. Using the measured parameters we
7The Italian Team participating to the discovery is constituted by members of INAF, the Istituto Nazionale di Astrofisica, belonging to the Cagliari Pulsar Group and to the Universities of Cagliari and Bologna, including Marta Burgay, Andrea Possenti and Nichi d’Amico. The main international partners of the collaboration were the Jodrell Bank pulsar group in Manchester, the ATNF pulsar group in Sydney (Australia) and the Swinburne pulsar group also in Australia (Melbourne). Finally also the European Pulsar Timing Array collaboration was involved. The radio-telescope used for the discovery is the Parkes radio telescope in Australia (see Fig. 7.17). Further observations were carried on at the Northern Cross radio-telescope near Bologna in Italy and in other European radio observatories.