- •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
324 |
8 Conclusion of Volume 1 |
In this example the test particle, placed at distance of only 10 Schwarzschild radii from the center falls into the singularity in just two revolutions if its eccentricity is different from zero, no matter how it is small.
B.2 Metrigravpack
This is a MATHEMATICA package for the calculation of the Riemann and Ricci tensors of an arbitrary (pseudo) Riemannian metric in arbitrary space-time dimensions using the standard tensor calculus. It is an interactive package that is initialized and then waits fur further inputs by the user.
Metric Gravity
In this section we provide a package to calculate Einstein equations for any given metric in arbitrary dimensions and using the metric formalism
Routines: Metrigrav This routine is devised to calculate the Levi Civita connection, the Riemann curvature and the Einstein Tensor for general manifolds in the metric formalism. The inputs are
(1)the dimension n
(2)the set of coordinates a n vector = coordi
(3)the set of differentials, a n vector = diffe
(4)the metric given as a quadratic differential ds2=g[[i,j ]] dxi dxj .
TO START this programme you type mainmetric and then you follow instructions
Mainmetric
,
,
mainmetric:=
Print["OK I calculate your space, Give me the data"]; Print["Give me the dimension of your space"]; mdim = Input["dimension = ?"];
Print["Your space has dimension n = ", mdim];
Print["Now I stop and you give me two vectors of dimension ", mdim]; Print["vector coordi = vector of coordinates"];
Print["vector diffe = vector of differentials"]; Print["Next you give me the metric as ds2 = "];
Print["Then to resume calculation you print metricresume"];
-;
-
,
firstres:= Print "I resume the calculation"
, [ ];
Print["First I extract the metric coefficients from your data"]; gg = Table[0, {i, 1, mdim}, {j, 1, mdim}];
B Mathematica Packages |
325 |
Do[{Do[{gg[[i,j]] = 21 (Coefficient[ds2, diffe[[i]] diffe[[j]]]); |
|
gg[[j,i]] = 21 (Coefficient[ds2, diffe[[i]] diffe[[j]]]); |
|
}, {j, i + 1, mdim}]; |
2 |
gg[[i,i]] = Coefficient[ds2, diffe[[i]]]; }, {i, 1, mdim}]; |
|
Print["Then I calculate the inverse metric"]; |
|
ggm = Simplify[Inverse[gg]]; Print["Done !"];
Print["and I calculate also the metric determinant"]; detto = Simplify[Det[gg]];
Print "Done"
[ ]; -;
-
Metricresume
metricresume:={ firstres; metrigrav; }
Routine Metrigrav
,
,
metrigrav:= holviel = diffe;
Print["I perform the calculation of the Christoffel symbols"]; Gam = Table[0, {i, 1, mdim}, {j, 1, mdim}, {k, 1, mdim}]; Do[
Do[
mdim
[{ = [ 1
Do Gam[[a,b,c]] Simplify 2 (ggm[[a,m]]
m=1
(∂coordi[[b]] gg[[m,c]]
+∂coordi[[c]] gg[[m,b]]
−∂coordi[[m]] gg[[b,c]])), Trig → True]; }, {c, 1, mdim}], {b, 1, mdim}],
{a, 1, mdim}];
mdim
= [ { } { }];
Conne Table holviel[[b]] Gam[[a,b,c]], a, 1, mdim , c, 1, mdim
b=1
Print["—————–"]; Print["I finished"];
Print["the Levi Civita connection is given by:"]; Do[Do[
Print["Γ [", i, j, "] = ", Conne[[i,j]]], {j, 1, mdim}], {i, 1, mdim}]; Print["Task finished"];
Print["The result is encoded in a tensor Gam[a,b,c]"]; Print["—————–"];
Print[" Now I calculate the Riemann tensor"];
Rie = Table[0, {a, 1, mdim}, {b, 1, mdim}, {f, 1, mdim}, {g, 1, mdim}];
Print["I tell you my steps :"];
Do[{Print[" a = ", a]; Do[{Print[" b = ", b]; Do[Do[{