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a thin shell, a circular point particle binary system of unequal masses,
and a generalisation to this last model including eccentricity. These gravitational
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radiating systems were treated and solved from the formalism developed from
the perturbations for the metrics mentioned above. In order to present that,
perturbations to a generic space-time at first and higher order are shown in chapter
2. Gravitational wave equations for these orders are obtained as well as their
respective eikonal equations. Additionally, chapter 2 introduces the Green functions
and the multipolar expansion. In chapter 3, the eth formalism is explained in detail,
separately from the characteristic formulation. It is an efficient method to regularise
angular derivative operators. The spin-weighted spherical harmonics are introduced
from the usual harmonics through successive applications of the eth derivatives.
In chapter 4 the initial value problem, the ADM formulation and the outgoing
characteristic formalism with and without eth expressions are shown. In chapter 5
the linear regime in the outgoing characteristic formulation is obtained, the field
equations are simplified and solved analytically. In order to do that a differential
equation (the master equation) for J , a Bondi-Sachs variable, is found. This equation
is solved for the Minkowski (Schwarzschild) background in terms of Hypergeometric
(Heun) functions. Finally in chapter 6, two examples are presented, the point particle
binaries without and with eccentricity. At the end, the conclusions and some final
considerations are discussed.
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2 LINEAR REGIME OF THE EINSTEIN’S FIELD EQUATIONS AND
GRAVITATIONAL WAVES
This chapter explores the linear regime of Einstein’s field equations and the
gravitational waves. In general, the linearisation of the Einstein’s field equations
is performed assuming a flat background (BUONANNO, 2007; CATTANI, 2010a;
CATTANI, 2010b; CATTANI, 2010c). However, this approximation turns inapplicable
to the cases in which strong fields are involved. Eisenhart in 1926 and Komar in
1957 made perturbations to the metric tensor at the first order showing how the
gravitational waves are propagated away from the sources (REGGE;WHEELER, 1957).
Regge and Wheeler (1957) considered small perturbations in a spherical symmetric
space-time to explore the stability of the Schwarzschild’s solution, obtaining a
radial wave equation in presence of an effective gravitational potential, namely the
Reege-Wheeler equation, which appears for odd-parity perturbations. On the other
hand, Zerilli (1970) made even-parity perturbations obtaining a different radial wave
equation, namely the Zerilli equation obeying a different effective potential. After
that, by using the vector and tensor harmonics, Moncrief extended the Zerilli’s
works to the Reissner-Nordström exterior space-time and to stellar models by
using a perfect fluid stress-energy tensor (MONCRIEF, 1974c; MONCRIEF, 1974d;
MONCRIEF, 1974b; MONCRIEF, 1974a). Brill and Hartle (1964) explored the stability
of the Geons, which are objects composed of electromagnetic fields held together
by gravitational attraction in the linear regime of the field equations, off the flat
space-time, but considering spherical symmetry and asymptotically flat space-times.
Isaacson found a generalisation to the gravitational wave equation when an arbitrary
background is considered. He proved that the gravitational waves for high and low
frequencies are found by performing perturbations to distinct orders in the metric
tensor (ISAACSON, 1968a; ISAACSON, 1968b).
Here some of the aspects of the linearisation approximation to first and higher
orders are examined. By using the Wentzel-Kramers-Brillouin approximation
(WKB) the eikonal equation is found, relating the tensor of amplitudes to the
metric perturbations with its propagation vector. After that, in the Minkowski’s
background, the gravitational waves are expressed in terms of the Green’s functions.
In addition, a multipolar expansion is made as usual. Finally, following (PETERS;
MATHEWS, 1963), the quadrupole radiation formula is used to find the energy lost
by emission of gravitational waves by a binary system of unequal masses.
The convention used here with respect to the indices is: xµ represent coordinates,
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µ = 1 for temporal components, µ = i, j, k, · · · for spatial coordinates. The adopted
signature is +2.
Finally, it is worth mentioning that the linearisation process of the Einstein’s
field equation presented in this section and the general results shown here are
important because we linearise the characteristic equations in the same way, by just
perturbing the Bondi-Sachs metric. The equations obtained for these perturbations
(See Chapters 5 and 6) are equivalent to those obtained in this section. For this
reason it is not surprise that in the characteristic formulation we obtain radiative
solutions and that they are characterised by the Bondi’s News function.
2.1 First Order Perturbations
In this section, we will explore in some detail the linear regime of the Einstein’s
field equations when an arbitrary background is considered. Despite the derivation
of the wave equation at first order does not differ from that in which a Minkowski’s
background is taken into account, additional terms related to the background
Riemann tensor and the correct interpretation of the D’Alembertian is shown.
We follow the same convention and procedures exposed by Isaacson (1968a) and
subsequently used in (MISNER et al., 1973)
As a starting point, perturbations to an arbitrary background (0)gµν at first order are
chosen, i.e.,
gµν =
(0)
gµν + �
(1)
gµν , (2.1)
where � is a parameter that measures the perturbation, satisfying �� 1. It is worth
stressing that it guaranties that the second term is smaller than the first, because
the characteristic length of such perturbations, λ, must be very small compared to
the characteristic length of the radius of curvature of the background, L. This limit
is known as the high frequency approximation (ISAACSON, 1968a).
Considering that the inverse metric gµν is given as a background term plus a first
order perturbation with respect to the background, i.e.,
gµν =
(0)
gµν + �
(1)
gµν , (2.2)
and that gµνgην =
(0)
gµν
(0)
gην = δ ηµ , then
gµσg
σν = (0)gµσ
(0)
gσν + �
(
(1)
gσν
(0)
gµσ +
(1)
gµσ
(0)
gσν
)
+O(�2). (2.3)
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Therefore, the perturbation of the inverse metric is given by
(1)
gην = −(1)gµσ
(0)
gµη
(0)
gσν . (2.4)
As a result, the Christoffel’s symbols of the first kind, reads
Γµνγ =
1
2 (gµν,γ + gγµ,ν − gνγ,µ) , (2.5)
where the comma indicates partial derivative. These symbols can be separated as a
term referred to the background plus a perturbation, namely,
Γµνγ =
(0)
Γµνγ + �
(1)
Γµνγ, (2.6)
where
(i)
Γµνγ =
1
2
(
(i)
gµν,γ +
(i)
gγµ,ν −
(i)
gνγ,µ
)
, i = 0, 1. (2.7)
Thus, the Christoffel’s symbols of the second kind,
Γµνγ = gµσΓσνγ, (2.8)
can also be separated (ISAACSON, 1968a) as,
Γµνγ =
(0)
Γµνγ + �
(1)
Γµνγ +O(�2), (2.9)
where,
(0)
Γµνγ =
(0)
gµσ
(0)
Γσνγ and
(1)
Γµνγ =
(0)
gµσ
(1)
Γσνγ +
(1)
gµσ
(0)
Γσνγ. (2.10)
Consequently the Riemann’s tensor is written as a term associated to the background
plus a term corresponding to a perturbation, i.e.,
Rµ νγδ =
(0)
Rµ νγδ + �
(1)
Rµ νγδ, (2.11)
where the background Riemann tensor is given by
(0)
Rµ νγδ = 2
(0)
Γµν[δ,γ] + 2
(0)
Γσν[δ
(0)
Γµγ]σ, (2.12)
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and the term associated with the perturbation reads
(1)
Rµ νγδ = 2
(1)
Γµν[δ,γ] + 2
(1)
Γµσ[γ
(0)
Γσδ]ν + 2
(1)
Γσν[δ
(0)
Γµγ]σ. (2.13)
As usual, the square brackets indicate anti-symmetrisation, i.e.,
A[α1···αn] =
1
n!�
β1···βn
α1···αn Aβ1···βn , (2.14)
where � β1···βnα1···αn is the generalised Levi-Civita permutation symbol (MISNER et al.,
1973).
From
(1)
Γµνδ:γ, where the colon indicates covariant derivative associated with the
background metric (0)g µν , one obtains
(1)
Γµν[δ,γ] =
(1)
Γµν[δ:γ] −
(0)
Γµσ[γ
(1)
Γσδ]ν +
(0)
Γσν[γ
(1)
Γµδ]σ +
(0)
Γσ[γδ]
(1)

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