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sid.inpe.br/mtc-m21b/2015/11.26.11.39-TDI
ON THE LINEAR REGIME OF THE
CHARACTERISTIC FORMULATION OF GENERAL
RELATIVITY IN THE MINKOWSKI AND
SCHWARZSCHILD’S BACKGROUNDS
Carlos Eduardo Cedeño Montaña
Doctorate Thesis of the Post Grad-
uation Course in Astrophysics, ad-
vised by Dr. José Carlos Neves de
Araújo, approved in February 17,
2016.
URL of the original document:
<http://urlib.net/8JMKD3MGP3W34P/3KLMGUB>
INPE
São José dos Campos
2016
http://urlib.net/xx/yy
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sid.inpe.br/mtc-m21b/2015/11.26.11.39-TDI
ON THE LINEAR REGIME OF THE
CHARACTERISTIC FORMULATION OF GENERAL
RELATIVITY IN THE MINKOWSKI AND
SCHWARZSCHILD’S BACKGROUNDS
Carlos Eduardo Cedeño Montaña
Doctorate Thesis of the Post Grad-
uation Course in Astrophysics, ad-
vised by Dr. José Carlos Neves de
Araújo, approved in February 17,
2016.
URL of the original document:
<http://urlib.net/8JMKD3MGP3W34P/3KLMGUB>
INPE
São José dos Campos
2016
http://urlib.net/xx/yy
Cataloging in Publication Data
Montaña, Carlos Eduardo Cedeño.
M762l On the linear regime of the characteristic formulation of
general relativity in the minkowski and schwarzschild’s back-
grounds / Carlos Eduardo Cedeño Montaña . – São José dos Cam-
pos : INPE, 2016.
xx + 161 p. ; ( sid.inpe.br/mtc-m21b/2015/11.26.11.39-TDI)
Thesis (Doctorate in Astrophysics) – Instituto Nacional de
Pesquisas Espaciais, São José dos Campos, 2015.
Guiding : Dr. José Carlos Neves de Araújo.
1. General relativity. 2. Characteristic formalism. 3. Gravita-
tional waves. 4. Linear regime. I.Title.
CDU 530.12:52-1
Esta obra foi licenciada sob uma Licença Creative Commons Atribuição-NãoComercial 3.0 Não
Adaptada.
This work is licensed under a Creative Commons Attribution-NonCommercial 3.0 Unported Li-
cense.
ii
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http://creativecommons.org/licenses/by-nc/3.0/deed.pt_BR
http://creativecommons.org/licenses/by-nc/3.0/deed.pt_BR
http://creativecommons.org/licenses/by-nc/3.0/
This text is dedicated specially to my father Ricardo, in
Memoriam, my mother Elicenia, my brother Ricardo and my
wife, Sandra. Thanks for always being with me.
v
ACKNOWLEDGEMENTS
I feel grateful to my parents by their continuous support during all instants of my
life. I appreciate very much their guidance and patience. I would like to express my
gratitude for furnishing me a real model to follow. I don’t know how to express my
deep grateful to my brother, who with his criticisms and conscientious reading, help
me to improve my text. Also, I owe a special mention to Sandra, my wife, who helps
me every day with her support, happiness and for encouraging me to improve in all
aspects. Thanks also go to her for all her suggestions and critical readings of my
manuscript. I would like to express my deep and sincere gratitude to my advisor
Dr. José Carlos N. de Araujo. His continuous support, his patient guidance and
enthusiastic encouragement during my PhD study and related research have been
valuable. For give me hope in the most difficult circumstances. I think that this
project would not have been possible without his advices and hope. I would also
like to thank the Brazilian agencies CAPES, FAPESP (2013/11990-1) and CNPq
(308983/2013-0) for the financial support.
vii
ABSTRACT
We present here the linear regime of the Einstein’s field equations in the
characteristic formulation. Through a simple decomposition of the metric variables
in spin-weighted spherical harmonics, the field equations are expressed as a system
of coupled ordinary differential equations. The process for decoupling them leads
to a simple equation for J - one of the Bondi-Sachs metric variables - known in
the literature as the master equation. Then, this last equation is solved in terms of
Bessel’s functions of the first kind for the Minkowski’s background, and in terms of
the Heun’s function in the Schwarzschild’s case. In addition, when a matter source
is considered, the boundary conditions across the time-like world tubes bounding
the source are taken into account. These boundary conditions are computed for
all multipole modes. Some examples as the point particle binaries in circular and
eccentric orbits, in the Minkowski’s background are shown as particular cases of this
formalism.
Keywords: General Relativity. Characteristic Formalism. Gravitational Waves.
Linear Regime.
ix
NO REGIME LINEAR DA FORMULAÇÃO CARACTERÍSTICA DA
RELATIVIDADE GERAL NOS FUNDOS DE MINKOWSKI E DE
SCHWARZSCHILD
RESUMO
Nós apresentamos aqui o regime linear das equações de campo de Einstein na
formulação característica. Através de uma decomposição simples das variáveis
métricas em harmônicos esféricos com peso de spin, as equações de campo são
expressas como um sistema de equações diferenciais ordinárias acopladas. O processo
de desacoplá-las leva a uma equação para J - uma das variáveis da métrica de Bondi-
Sachs - conhecida na literatura como equação mestre. Então, esta última equação é
resolvida em termos de funções de Bessel do primeiro tipo para o fundo de Minkowski
e em termos de funções de Heun no caso de Schwarzschild. Além disso, quando uma
fonte é considerada, as condições de contorno através do tubo de mundo limitando
a fonte é levada em conta. Essas condições de contorno são calculadas para todos os
modos multipolares. Alguns exemplos como binárias em órbita circular e excêntrica
no fundo de Minkowski são mostrados como casos particulares deste formalismo.
Palavras-chave: Relatividade Geral. Formalismo Característico. Ondas
Gravitationais. Regime Linear.
xi
LIST OF FIGURES
Page
2.1 Source and observer’s position. . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Binary system as viewed from the top. The coordinates xai of the particles
are indicated, as well as the angle νu with respect to the x axis. . . . . . 30
3.1 Stereographic coordinates construction: the equatorial plane is projected
from the south pole to the surface of the unit sphere. The interior points
to the equator are projected to the north hemisphere, whereas the exterior
points are projected to the south. . . . . . . . . . . . . . . . . . . . . . . 35
3.2 Coordinate atlas in the sphere. Coordinate lines as result of the mapping
of the plane maps contructed from the equator of the sphere. . . . . . . . 36
3.3 Coordinate lines of north hemisphere into the south region. The
equatorial line is indicated as a circle in black. . . . . . . . . . . . . . . 37
4.1 Space-timeM foliated in 3D - hypersurfaces Σ. . . . . . . . . . . . . . . 79
4.2 Change of the normal

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