- •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
314 |
8 Conclusion of Volume 1 |
This is well established. For instance we have the representation:
γ0 |
|
0 |
1 |
|
γ1,2,3 |
|
|
0 |
σ1,2,3 |
|
|
− |
|
||||||
|
= 1 |
0 ; |
|
= |
σ1,2,3 |
0 |
|||
|
|
|
In D = 2ν one can construct the chirality matrix defined as follows:
ΓD+1 = αD Γ0Γ1Γ2 . . . ΓD−1; |
|αD |2 = 1 |
where αD is a phase-factor to be fixed in such a way that:
ΓD2+1 = 1
By direct evaluation one can verify that:
(A.2.9)
(A.2.10)
(A.2.11)
{Γa , ΓD+1} = 0 a = 0, 1, 2, . . . , D − 1 The normalization αD is easily derived. We have:
Γ0Γ1 . . . ΓD−1 = (−) 12 D(D−1)ΓD−1ΓD−2ΓD−1
so that imposing (A.2.11) results into the following equation for αD :
αD2 (−) 12 D(D−1)(−)(D−1) = 1
which has solution:
αD = 1 if ν = 2μ + 1 odd αD = i if ν = 2μ even
With the same token we can show that the chirality matrix is Hermitian:
ΓD†+1 = α (−) 12 D(D−1)(−)(D−1)Γ0Γ1Γ2 . . . ΓD−1 = ΓD+1
(A.2.12)
(A.2.13)
(A.2.14)
(A.2.15)
(A.2.16)
A.2.2 Odd Dimensions
When D = 2ν + 1 is an odd number, the Clifford algebra (A.2.1) can be represented by 2ν × 2ν matrices. It suffices to take the matrices Γa corresponding to the even case D = D − 1 and add to them the matrix ΓD = iΓD +1, which is anti-Hermitian and anti-commutes with all the other ones.
A.3 The Charge Conjugation Matrix
Since Γa and their transposed ΓaT satisfy the same Clifford algebras it follows that there must be a similarity transformation connecting these two representations of
A Spinors and Gamma Matrix Algebra
Table A.1 Structure of charge conjugation matrices in various space-time dimensions
Charge conjugation matrices
DC+ = C+ (real)
4C+T = −C+; C+2 = −1
5C+T = −C+; C+2 = −1
6C+T = −C+; C+2 = −1
7
8C+T = C+; C+2 = 1
9C+T = C+; C+2 = 1
10C+T = C+; C+2 = 1
11
315
C− = C− (real)
C−T = −C−; C−2 = −1
C−T = C−; C−2 = 1 C−T = C−; C−2 = 1 C−T = C−; C−2 = 1
C−T = −C−; C−2 = −1 C−T = −C−; C−2 = −1
the same algebra on the same carrier space. Such statement relies on Schur’s lemma and it is proved in the following way. We introduce the notation:
Γa1 |
...an |
|
Γ a1 Γa2 . . . Γan |
|
1 |
|
( )δP ΓaP (a1) . . . ΓaP (an) |
(A.3.1) |
|
≡ |
] = n |
|
|
||||||
|
|
[ |
! |
− |
|
||||
|
|
|
|
|
|
P |
|
where P denotes the sum over the n! permutations of the indices and δP the parity of permutation P , i.e. the number of elementary transpositions of which it is composed. The set of all matrices 1, Γa , Γa1a2 , . . . , Γa1...aD constitutes a finite group of 2[D/2]-dimensional matrices. Furthermore the groups generated in this way by Γa , −Γa or ΓaT are isomorphic. Hence by Schur’s lemma two irreducible representations of the same group, with the same dimension and defined over the same vector space, must be equivalent, that is there must be a similarity transformation that connects the two. The matrix realizing such a similarity is called the charge conjugation matrix. Instructed by this discussion we define the charge conjugation matrix by means of the following equations:
C−Γa C−−1 = −ΓaT
(A.3.2)
C+Γa C+−1 = ΓaT
By definition C± connects the representation generated by Γa to that generated by ±ΓaT . In even dimensions both C− and C+ exist, while in odd dimensions only one of the two is possible. Indeed in odd dimensions ΓD−1 is proportional to Γ0Γ1 . . . ΓD−2 so that the C− and C+ of D − 1 dimensions yield the same result on ΓD−1. This decides which C exists in a given odd dimension.
Another important property of the charge conjugation matrix follows from iterating (A.3.2). Using Schur’s lemma one concludes that C± = αC±T so that iterating again we obtain α2 = 1. In other words C+ and C− are either symmetric or antisymmetric. We do not dwell on the derivation which can be obtained by explicit iterative construction of the gamma matrices in all dimensions and we simply collect below the results for the properties of C± in the various relevant dimensions (see Table A.1).
316 |
8 Conclusion of Volume 1 |
A.4 Majorana, Weyl and Majorana-Weyl Spinors
The Dirac conjugate of a spinor ψ is defined by the following operation:
ψ |
≡ ψ†Γ0 |
(A.4.1) |
and the charge conjugate of ψ is defined as:
ψc = C ψT |
(A.4.2) |
where C is the charge conjugation matrix. When we have such an option we can either choose C+ or C−. By definition a Majorana spinor λ satisfies the following condition:
λ = λc = CΓ0T λ |
(A.4.3) |
Equation (A.4.3) is not always self-consistent. By iterating it a second time we get the consistency condition:
C Γ0T C = Γ0 |
(A.4.4) |
There are two possible solutions to this constraint. Either C− is antisymmetric or C+ is symmetric. Hence, in view of the results displayed above, Majorana spinors exist only in
D = 4, 8, 9, 10, 11 |
(A.4.5) |
In D = 4, 10, 11 they are defined using the C− charge conjugation matrix while in D = 8, 9 they are defined using C+.
Weyl spinors, on the contrary, exist in every even dimension; by definition they are the eigenstates of the ΓD+1 matrix, corresponding to the +1 or −1 eigenvalue. Conventionally the former eigenstates are named left-handed, while the latter are named right-handed spinors:
ΓD+1ψ: L |
; = ±ψ: L ; |
(A.4.6) |
R |
R |
|
In some special dimensions we can define Majorana-Weyl spinors which are both eigenstates of ΓD+1 and satisfy the Majorana condition (A.4.3). In order for this to be possible we must have:
C Γ0T ΓD+1ψ = ΓD+1ψ |
(A.4.7) |
which implies: |
|
C Γ0T ΓD+1Γ0T C −1 = ΓD+1 |
(A.4.8) |
A Spinors and Gamma Matrix Algebra |
317 |
With some manipulations the above condition becomes:
C ΓD+1C −1 = −ΓDT+1 |
(A.4.9) |
which can be checked case by case, using the definition of ΓD+1 as product of all the other gamma matrices. In the range 4 ≤ D ≤ 11 the only dimension where (A.4.9) is satisfied is D = 10 which is the critical dimensions for superstrings. This is not a pure coincidence.
Summarizing we have:
Spinors in 4 ≤ D ≤ 11
D |
Dirac |
Majorana |
Weyl |
Majora-Weyl |
|
|
|
|
|
4 |
Yes |
Yes |
Yes |
No |
5 |
Yes |
No |
No |
No |
6 |
Yes |
No |
Yes |
No |
7 |
Yes |
No |
No |
No |
8 |
Yes |
Yes |
Yes |
No |
9 |
Yes |
Yes |
No |
No |
10 |
Yes |
Yes |
Yes |
Yes |
11 |
Yes |
Yes |
No |
No |
|
|
|
|
|
A.5 A Particularly Useful Basis for D = 4 γ -Matrices
In this section we construct a D = 4 gamma matrix basis which is convenient for various purposes. Let us first specify the basis and then discuss its convenient properties.
In terms of the standard matrices (A.2.8) we realize the so(1, 3) Clifford algebra:
{γa , γb} = 2ηab; ηab = diag(+, −, −, −) |
(A.5.1) |
by setting:
γ0 = σ1 σ3; |
γ1 = iσ2 σ3 |
|
γ2 = i1 σ2; |
γ3 = iσ3 σ3 |
(A.5.2) |
γ5 = 1 σ1; |
C = iσ2 1 |
|
where γ5 is the chirality matrix and C the generators of the Lorentz algebra and nice 4 × 4 matrices. Explicitly we
is the charge conjugation matrix. In this basis so(1, 3), namely γab are particularly simple get: