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# carlos_eduardo_cedeno_montana_2016

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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,