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to the exact solutions, which are valid for some restricted geometries
and situations, the perturbative methods and the numerical relativity are two of
the most promising ways to solve the Einstein’s field equations in presence of strong
gravitational fields in a wide variety of matter configurations.
The holy grail of numerical relativity is to obtain the gravitational radiation patterns
produced by black hole - black hole (BH-BH), neutron star - neutron star (NS-NS)
or neutron star - black hole (NS-BH) binary systems, because of their relevance in
astrophysics. Actually, there are highly accurate and strongly convergent numerical
codes, capable of performing simulations of binaries taking into account the mass
and momentum transfer (FABER et al., 2006; LEHNER; PRETORIUS, 2014; KYUTOKU
et al., 2015), the hydrodynamic evolution (TANIGUCHI et al., 2005; BAUMGARTE et al.,
2013; MONTERO et al., 2014), the magneto-hydrodynamic evolution (FONT, 2008),
the electromagnetic and gravitational signatures produced by binaries (PALENZUELA
et al., 2013b; PALENZUELA et al., 2013a; KYUTOKU et al., 2015); and recently, the spin-
spin and the spin-orbit interactions in binary systems have been also studied (DAIN
et al., 2008; IORIO, 2012; ZLOCHOWER; LOUSTO, 2015).
All these advances were possible thanks to the Lichnerowicz, Choquet-Bruhat and
Geroch works (LICHNEROWICZ, 1944; FOURÈS-BRUHAT, 1952; CHOQUET-BRUHAT;
GEROCH, 1969), which opened the possibility to evolve a space-time from a
set of initial data; putting the principles of the Initial Value Problem (IVP)
(GOURGOULHON, 2007; ALCUBIERRE, 2008; BAUMGARTE; SHAPIRO, 2010) and
checking that this is a local and a global well-posed problem, that are necessary
conditions to guarantee stable numerical evolutions.
A different point of view to carry out the evolution of a given space-time was
proposed by Bondi et. al. in the 1960s decade (BONDI et al., 1962; SACHS, 1962). They
studied the problem of evolving a given metric, from the specification of it and its
first derivatives, by using the radiation coordinates, assuming that the initial data is
given on a null initial hypersurface and on a prescribed time-like world tube. This is
known as the Characteristic Initial Value Problem (CIVP) (STEWART; FRIEDRICH,
1982) and was effectively proved as a well-posed problem when the field equations
are written in terms only of first-order derivatives (FRITTELLI, 2005).
1
In the literature, there are essentially three possible ways to evolve space-times and
sources from a specific initial data, see e. g. (COOK, 2000; LEHNER, 2001; MARTÍ;
MÜLLER, 2003; GUNDLACH; MARTÍN-GARCÍA, 2007; WINICOUR, 2012; CARDOSO
et al., 2015) for detailed descriptions and status of the formalisms available in
numerical relativity. The first one is the Regge calculus, in which the space-
time is decomposed in a network of 4-dimensional flat simplices.1 The Riemann
tensor and consequently the field equations are expressed in a discrete version
of such atomic structures. It extends the calculus to the most general spaces
than differentiable manifolds (REGGE, 1961). The second are the Arnowitt-Deser-
Misner (ADM) based formulations in which the space-time is foliated into space-
like hypersurfaces which are locally orthogonal to the tangent vectors of a central
time-like geodesic (ARNOWITT et al., 1959; ARNOWITT et al., 1960a; ARNOWITT et al.,
1960b; YORK JR., 1971; YORK JR., 1979). The third are the characteristic formalisms,
which are based on the Bondi et. al. works in which the space-time is foliated into null
cones emanated from a central time-like geodesic or a world tube, and hypersurfaces
that are related to the unit sphere through diffeomorphisms (BONDI et al., 1962;
SACHS, 1962; WINICOUR, 1983; WINICOUR, 1984; WINICOUR, 2012).
Most of the recent work have been constructed using the ADM formalisms,2 whereas
the null cone formalisms are less known. One of the biggest problems in these last
formulations is their mathematical complexity. However, these formalisms result
particularly useful for constructing waveform extraction schemes, because they are
based on radiation coordinates. Impressive advances in the characteristic formulation
have been carried out recently, in particular in the development of matching
algorithms, which evolving from the Cauchy-Characteristic-Extraction (CCE) to
the Cauchy-Characteristic-Matching (CCM) (BISHOP et al., 1996; BISHOP et al., 2005;
REISSWIG et al., 2007; BABIUC et al., 2009; BABIUC et al., 2011; REISSWIG et al., 2011).
A cumbersome aspect of the null-cone formulation is the formation of caustics in
the non-linear regime, because at these points the coordinates are meaningless.
The caustics are formed when the congruences of light beams bend, focusing and
forming points where the coordinate system is not well defined. This problem is not
present in the CCM algorithms because the characteristic evolution is performed
1Simplices (Simplexes) are the generalisation of triangles for bi-dimensional and tetrahedron for
three-dimensional spaces to four or more dimensional spaces. In the Regge calculus these simplices
are supposed flat and the curvature is given just at the vertices of the structure, just like when a
sphere is covered using flat triangles.
2These formalisms are known also as 3+1 because of the form in which the field equations are
decomposed.
2
for the vacuum, where the light beams not bend outside of the time-like world
tube (WINICOUR, 2012). Therefore, the characteristic evolutions have been usually
performed only for the vacuum, considering the sources as bounded by such time-
like hypersurface. Inside of the time-like world tube, the matter is evolved from the
conservation laws. However, there are some works in which the gravitational collapse
of scalar fields, massive or not, are performed using only characteristic schemes,
but obeying restrictive geometries and taking into account the no-development
of caustics (GÓMEZ et al., 2007; BARRETO, 2014a; BARRETO, 2014b). At this
point it is worth mentioning that the finite difference schemes are not the unique
methods to solve efficiently the Einstein’s field equations. There are significative
advances in the spectral methods applied to the characteristic formulation using the
Galerkin method, see e.g. (RODRIGUES, 2008; LINHARES; OLIVEIRA, 2007; OLIVEIRA;
RODRIGUES, 2008; OLIVEIRA; RODRIGUES, 2011)
One way to calibrate these complex and accurate codes is to make tests of validity in
much simpler systems and geometries than those used in such kind of simulations. In
order to do so, toy models for these codes can be obtained with the linear version of
the field equations. Depending on the background, the linearised equations can lead
to several regimes of validity. One example of this is that the linear regime of the field
equations on a Minkowski or on a Schwarzschild’s background leads to waveforms
and behaviours of the gravitational fields completely different. There is a great
quantity of possibilities to perform approximations to the field equations. Among
them, there are different orders of the Post-Newtonian approximations, the post-
Minkowskian approximations, the approximations using spectral decompositions,
and so on.
Despite lack of real physical meaning near to the sources, the linear approximations
of the characteristic formulation of general relativity exhibit an interesting point of
view even from the theoretical perspective. It is possible to construct exact solutions
to the Einstein’s field equations for these space-times in a easy way. It allows us
to reproduce at first approximation some interesting features of simple radiative
systems. In the weak field limit, it is possible to write the field equations as a system
of coupled ordinary differential equations, that can be easily solved analytically.
Here we present exact solutions for space-times resulting from small perturbations
to the Minkowski and Schwarzschild’s space-times. Also, we construct three simple
toy models,

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