CONICET | Buscador de Institutos y Recursos Humanos

Round null surfaces in Kerr space-timeMarcos A. Argañaraz and Osvaldo M. MoreschiFacultad de Matemática Astronomía, Física y Computación (FaMAF),Universidad Nacional de Córdoba,Instituto de Física Enrique Gaviola (IFEG), CONICET,Ciudad Universitaria,(5000) Córdoba, Argentina.While the Kerr metric has deservedly been one of the most studied exact solutions, there appears to be a peculiar lackof natural null coordinates to describe a dual-null foliation of the space-time, meaning two families of null hypersurfacesintersecting in a two-parameter family of transverse spatial surfaces, such that the horizons are two of the hypersurfaces. Wepresent a new denition for null coordinates, that we call u (out-going) and v (in-going), which are naturally adapted to thehorizons. Our denition involves a dierential equation which we solve numerically.In our construction there naturally appear a family of spheres that are parameterized by rs, which are the intersections ofthe null coordinates u and v. They can also be characterized in a coordinate independent way, by the intrinsic and extrinsicGHP curvature, given by KGaussian = Q¯GHP + QGHP and KExtrinsic = iQ¯GHP − QGHP , with Q = σσ0 − ρρ0 − Ψ2 givenin terms of the spin coecients of the GHP formalism. In the gure below, we show the smooth behavior of these curvaturesthrough their numerical computation on a surface characterized by rs, where (r, θ, φ) are in Boyer-Lindquist coordinates anda is the Kerr parameter.Our work improves several attempts that can be found in the literature. A remarkable one is developed in [Hayward(2004)],where the null hypersurfaces they construct do not include the null geodesics along the axis of symmetry. This is due to thefact that their construction does not give a smooth hypersurface at the poles. In order to compare with ours coordinates, from[Hayward(2004)], we consider the null function u∗ = t ∗ −r∗. Where the analog to our natural spheres are the intersectionof u∗ with the Boyer-Lindquist coordinate t; that can be parameterized by rsH. In the following graph one can be seen thatfor rsH there is a discontinuity in the derivatives at (θ = 0), while for rs it is clearly smooth.Our approach is more related to the work in [Pretorius and Israel(1998)], whose treatment only covers the northernhemisphere, but also their expressions fail to deal with the north pole, and are very dicult to compute, even numerically.Our new coordinates gives a new insight and are useful in the study of Kerr solution and the Kerr stability open problem.We plan to use them, in further works of Kerr perturbations.