- •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
1.8 Criticism of Special Relativity: Opening the Road to General Relativity |
31 |
of Minkowski space-time. Applying Noether’s theorem as stated in (1.7.3) we can construct the corresponding conserved Noether current:
|
|
γ μψ |
(1.7.7) |
j μ = eψ |
|||
This is the electric current which, by coupling to the electromagnetic gauge potential Aμ, gives rise to Electrodynamics.
As a second example let us consider the case of a scalar field ϕ(x), whose standard action was written in (1.6.9). In Chap. 5 we shall reconsider this action from the point of view of its gravitational coupling and we shall rewrite it in the vielbein formalism. Here it is just considered as the starting point of a dynamical Poincaré invariant field theory in Minkowski space. Its invariance under space-time translations is evident and this leads to the conservation of an associated current, the stressenergy tensor. Let us compute this current following Noether theorem. Naming Pρ the generators of space-time translations, as we already did above, the infinitesimal transformations are as follows:
1 |
+ εμPμ |
xμ = xμ + δxμ; |
δxμ = ερ ρμ; |
ρμ = δρμ |
(1.7.8) |
1 + εμPμ ϕ = ψ + 0; |
δψ = ερ Θρ ; |
Θρ = 0 |
|
||
The fact that Θρ = 0 signalizes that translations are just the opposite case with respect to that considered before. Translations are purely space-time symmetries and the momentum operator Pμ has no non-trivial action in the space of fields. Applying formula (1.7.3) to the present case we obtain the conserved Noether current of translations:
T μ |
= |
1 |
∂μϕ∂ρ ϕ |
− |
1 |
δμ∂σ ϕ∂σ ϕ |
+ |
W (ϕ) |
(1.7.9) |
||
|
|
|
|||||||||
4 |
2 |
||||||||||
ρ |
|
ρ |
|
|
|||||||
which, as we are going to see in Chap. 5 coincides with the definition of the stress energy tensor as variation of the matter action with respect to the metric. Indeed it suffices to lower the first index of the calculated current with the Minkowski metric ημσ and we obtain the symmetric tensor:
|
≡ |
ρ = |
4 |
|
− |
2 |
|
|
+ |
|
|
Tρσ |
|
ησ μT μ |
1 |
∂ρ ϕ∂σ ϕ |
|
1 |
ηρσ ∂ |
ϕ∂ aϕ |
|
W (ϕ) |
(1.7.10) |
|
|
|
|
|
which can be confronted with the result (5.6.41) obtained in Sect. 5.6.4.
1.8Criticism of Special Relativity: Opening the Road to General Relativity
Let us now consider Special Relativity in retrospective.
Through the implications of Maxwell Electromagnetic Theory and by means of a complicated historical path, Special Relativity arrives at the unification of time and
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1 Special Relativity: Setting the Stage |
space into Minkowski space-time and replaces the Galileo group with the Lorentz group as the correct group of transformations that relate one inertial reference frame to the other. Special Relativity encodes a spectacular conceptual advance yet it does not solve, rather it shares with Classical Newtonian Physics the logical weakness of being funded on circular reasoning. Indeed both in Newtonian Physics and in Special Relativity we adhere to the following way of arguing:
•We have fundamental laws of Nature that apply only in special reference frames, the inertial ones.
•How are the inertial frames defined?
•As those where the fundamental laws of Nature that we have constructed apply.
Furthermore while Maxwell Theory is automatically Lorentz covariant, gravitation, as described by Newton’s law of universal attraction is by no means Lorentz covariant and needs to be revised in order to be reconciled with relativity.
Where to start in order to overcome these two problems?
For those who know a little bit of differential geometry, and the reader of the next chapter will be such a person, something which immediately appears very specific and probably too restrictive is the character of Minkowski space-time. It is, in the language of next chapter, a manifold, actually a Riemannian manifold, but it is also an affine variety, namely it is a vector space. Physically this means that although we have given up absolute space-distance, we have not yet given up absolute space-time separation of events. Given any two events xμ, yμ we can still define their absolute separation as:
2(x, y) = (x − y, x − y) |
(1.8.1) |
where (, ) denotes the Minkowskian scalar product.
Einstein’s intuition was that in order to remove circular reasoning and formulate laws of Nature that apply in any reference frame, one had to give up the notion of absolute space-time distances. What we are allowed to do is just to measure the length of any curve drawn in space-time that should not be required to be an affine manifold rather just a manifold. The rule to calculate such distances is encoded in the metric tensor of Riemannian geometry and Einstein discovered that such a geometrical object is nothing else but the gravitational field. To arrive at these conclusions Einstein studied differential geometry that had independently, slowly developed for about eighty years and was coming to maturity just at the dawn of the new century. The same has to do the student and to help him in this task the next two chapters have been written.
References
1.Maxwell, J.C.: A dynamical theory of the electromagnetic field. Philos. Trans. R. Soc. Lond. 155, 459–512 (1865)
2.Maxwell, J.C.: A Treatise on Electricity and Magnetism. Clarendon Press, Oxford (1873)
References |
33 |
3.Lorentz, H.A.: Electromagnetic phenomena in a system moving with any velocity smaller than that of light. Proc. Acad. Sci. Amst. 6, 809–831 (1904)
4.Einstein, A.: Zur Elektrodynamik bewegter Körper. Ann. Phys. 17, 891 (1905)
5.Lorentz, H.A.: Simplified theory of electrical and optical phenomena in moving systems. Proc. Acad. Sci. Amst. 1, 427–442 (1899)
6.Poincaré, H.: La thorie de Lorentz et le principe de réaction. Arch. Néerl. Sci. Exactes Nat. V, 253–278 (1900)
7.Cartan, É.: Les groupes projectifs qui ne laissent invariante aucune multiplicit plane. Bull. Soc. Math. Fr. 41, 53–96 (1913)
8.Pauli, W.: Zur Quantenmechanik des magnetischen Elektrons. Z. Phys. 43 (1927)
9.Dirac, P.M.: The quantum theory of the electron. Proc. R. Soc. Lond. A 117, 610–624 (1928)
10.Noether, E.: Invariante variationsprobleme. Nachr. König. Ges. Wiss. Gött., Math.-Phys. Kl., 235–257 (1918)
Chapter 2
Basic Concepts About Manifolds and Fibre
Bundles
Mathematics, the Queen of Sciences. . .
Carl Friedrich Gauss
2.1 Introduction
General Relativity is founded on the concept of differentiable manifolds. The mathematical model of space-time that we adopt is given by a pair (M , g) where M is a differentiable manifold of dimension D = 4 and g is a metric, that is a rule to calculate the length of curves connecting points of M . In physical terms the points of M take the name of events while every physical process is a continuous succession of events. In particular the motion of a point-like particle is represented by a world-line, namely a curve in M while the motion of an extended object of dimension p is given by a d = p + 1 dimensional world-volume obtained as a continuous succession of p-dimensional hypersurfaces Σp M .
Therefore, the discussion of such physical concepts is necessarily based on a collection of geometrical concepts that constitute the backbone of differential geometry. The latter is at the basis not only of General Relativity but of all Gauge Theories by means of which XX century Physics obtained a consistent and experimentally verified description of all Fundamental Interactions.
The central notions are those which fix the geometric environment:
•Differentiable Manifolds
•Fibre-Bundles
and those which endow such environment with structures accounting for the measure of lengths and for the rules of parallel transport, namely:
•Metrics
•Connections
Once the geometric environments are properly mathematically defined, the metrics and connections one can introduce over them turn out to be the structures which encode the Fundamental Forces of Nature.
The present chapter introduces Differentiable Manifolds and Fibre-Bundles while the next one is devoted to a thorough discussion of Metrics and Connections.
P.G. Frè, Gravity, a Geometrical Course, DOI 10.1007/978-94-007-5361-7_2, |
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