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```vectors to Σ. The difference δna only provides
information about the change in the direction of the vectors, because
they are normalised. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
4.3 Representation of two successive hypersurfaces and the displacement
vector βµ (red). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
4.4 Null coordinates construction. Tangent and normal vectors to the null
hypersurfaces emanated from a time-like world tube Γ. . . . . . . . . . . 88
4.5 Space-time M foliated in 2D - null hypersurfaces Σ. (a) Null cones
emanating from a central time-like geodesic. (b) Null cones as emanating
from a central time-like world tube. . . . . . . . . . . . . . . . . . . . . . 89
4.6 Space-timeM foliated in 2D - null hypersurfaces Σ. Section showing the
space-like for t constant and characteristic hypersurfaces corresponding
to the retarded time u constant. . . . . . . . . . . . . . . . . . . . . . . . 90
5.1 Sketch of the world tube generated by the thin shell. Here we note the
two regions (r < r0 and r > r0) in which the space-time is divided. . . . . 114
5.2 Metric variables as a function of the compactified coordinate s for a thin
shell of r = r0, centred at the origin. (a) β0 := β0(s), (b) J0 := J0(s), (c)
U0 := U0(s), (d) w0 := w0(s) . . . . . . . . . . . . . . . . . . . . . . . . . 120
6.1 Binary system with the world tubes of each orbit extended along the
direction of the retarded time, separating the space-time into three regions.122
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6.2 Real part of some components of the metric functions ( l = m = 2 )
versus the compactified coordinate s (see the text) for a binary system
with M1 = 1/2, M2 = 1. The angular velocity is computed by means of
Kepler’s third law. Here r1 and r2 are referred to the center of mass of
the system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
6.3 Snapshots of the metric variables as seen from the equatorial plane
(θ = π/2), as a function of s and φ for u = π/2. HereM1 = 16,M2 = 4/3,
r1 = 1/13, r2 = 12/13, R0 = 1/2 and ν = 2
√
13/3. (a) β(s, φ), (b) J(s, φ),
(c) U(s, φ) and (d) W (s, φ) = w(s, φ)(1− s2)/(s2R20). . . . . . . . . . . . 128
6.4 (a) Eccentric binary system with the world tubes of their orbits extended
along the central time-like geodesic. (b) Top view of the point particle
binary system, where the angular position φ̃ is indicated. . . . . . . . . . 132
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LIST OF ABBREVIATIONS
IVP – Intial Value Problem
CIVP – Characteristic Intial Value Problem
WKB – Wentzel-Kramers-Brillouin
BSSN – Baumgarte-Shapiro-Shibata-Nakamura
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LIST OF SYMBOLS
Greek indices α, β, γ, · · · – run from 1 to 4
Capital Latin indices A,B,C, · · · – run from 3 to 4
Upper indices (i), i = 0, 1, 2, · · · – Indicates the perturbation order
Gµν – Einstein’s tensor
Tµν – Stress-energy tensor
Rµν – Ricci’s tensor
Rµνγδ – Riemann’s tensor of type (0, 4)
Rµνδγ – Riemann’s tensor of type (1, 3)
,α or ∂α – Partial derivative
;α or ∇α – Covariant derivative
:α or
(0)
∇α – Covariant derivative referred to the background
Γαβγ – Christoffel’s symbols of the first kind
Γαβγ – Christoffel’s symbols of the second kind
|A or 4A – Covariant derivative referred to qAB
‖A or ∇A – Covariant derivative referred to hAB
qA – Dyads related to the vectors in TpS
L – Legendrian operator
ð – Eth operator
ð – Eth bar operator
sYlm – Spin-weighted spherical harmonic
sZlm – Spin-weighted spherical harmonic
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CONTENTS
Page
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 LINEAR REGIME OF THE EINSTEIN’S FIELD EQUATIONS
AND GRAVITATIONAL WAVES . . . . . . . . . . . . . . . . . . 5
2.1 First Order Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Higher Order Perturbations . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Green’s Functions for the Flat Background and Perturbations of First
Order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.4 Multipolar Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5 Gravitational Radiation from Point Particle Binary System . . . . . . . . 29
3 THE Eth FORMALISM AND THE SPIN-WEIGHTED
SPHERICAL HARMONICS . . . . . . . . . . . . . . . . . . . . . 33
3.1 Non-conformal Mappings in the Sphere . . . . . . . . . . . . . . . . . . . 34
3.2 Stereographic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.3 Decomposition of the Metric of the Unit Sphere . . . . . . . . . . . . . . 38
3.4 Transformation Rules for Vectors and One-forms . . . . . . . . . . . . . . 40
3.5 Transformation Rules for the Dyads and Spin-weight . . . . . . . . . . . 41
3.6 Spin-weighted Scalars and Spin-weight . . . . . . . . . . . . . . . . . . . 43
3.7 Raising and Lowering Operators . . . . . . . . . . . . . . . . . . . . . . . 45
3.8 Transforming the Coordinate Basis . . . . . . . . . . . . . . . . . . . . . 47
3.9 Legendrian Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.10 The ð and ð in Spherical Coordinates . . . . . . . . . . . . . . . . . . . . 60
3.11 Integrals for the Angular Manifold . . . . . . . . . . . . . . . . . . . . . 61
3.12 Spin-weighted Spherical Harmonics sYlm . . . . . . . . . . . . . . . . . . 69
3.13 Spin-weighted Spherical Harmonics sZlm . . . . . . . . . . . . . . . . . . 74
4 THE INITIAL VALUE PROBLEM AND THE NON-LINEAR
REGIME OF THE EINSTEIN’S FIELD EQUATIONS . . . . . 77
4.1 The Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2 Arnowitt-Desser-Misner Formulations (ADM) . . . . . . . . . . . . . . . 78
4.2.1 (ADM) formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.2.2 The Baumgarte-Shibata-Shapiro-Nakamura (BSSN) Equations . . . . . 85
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4.3 Outgoing Characteristic Formulation . . . . . . . . . . . . . . . . . . . . 87
4.3.1 The Bondi-Sachs Metric . . . . . . . . . . . . . . . . . . . . . . . . . . 87
4.3.2 Characteristic Initial Value Problem . . . . . . . . . . . . . . . . . . . 89
4.3.3 The Einstein’s Field Equations . . . . . . . . . . . . . . . . . . . . . . 90
4.4 The Einstein’s Field Equations in the Quasi-Spherical Approxi-mation . 91
4.5 The Einstein’s Field equations Using the Eth Formalism . . . . . . . . . 94
5 LINEAR REGIME IN THE CHARACTERISTIC FORMULA-
TION AND THE MASTER EQUATION SOLUTIONS . . . . . 97
5.1 Einstein’s Field Equations in the linear . . . . . . . . . . . . . . . . . . 98
5.2 Harmonic Decomposition and Boundary Problem . . . . . . . . . . . . . 99
5.3 The Master Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5.4 Families of Solutions to the Master Equation . . . . . . . . . . . . . . . . 103
5.4.1 The Minkowski’s Background . . . . . . . . . . . . . . . . . . . . . . . 104
5.4.2 The Schwarzschild’s Background . . . . . . . . . . . . . . . . . . . . . 107
5.5 Families of Solutions for l = 2 . . . . . . . . . . . . . . . . . . . . . . . . 109
5.6 Thin Shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
6 APPLICATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
6.1 Point Particle Binary System with Different Masses . . . . . . . . . . . . 121
6.1.1 Gravitational Radiation from the Binary System . . . . . . . . . . . . 128
6.2 Eccentric Point Particle Binary System . . . . . . . . . . . . . . . . . . . 131
6.2.1 Gravitational Radiation Emitted by the Binary . . . . . . . . . . . . . 134
7 CONCLUSIONS, FINAL REMARKS AND PERSPECTIVES . 137
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
Appendix A - Explicit Form for the ð and ð Operators in
Stereographic Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . 155
Appendix B - Angular Operators ∂θθ, ∂θφ and ∂φφ in terms of ð and ð 159
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1 INTRODUCTION
The high complexity of the Einstein’s field equations, given their non-linearity, makes
impossible to find analytical solutions valid for all gravitational systems. However,