Round null surfaces in Kerr space-time

MARCOS A. ARGAÑARAZ; OSVALDO M. MORESCHI

Conferencia; GR22, 22th International Conference on General Relativity and Gravitation/13th Edoardo Amaldi Conference on Gravitational Waves; 2019

While the Kerr metric has deservedly been one of the moststudied exact solutions, there appears to be a peculiarlack of natural null coordinates to describe a dual-null foliation of the space-time, meaning two families of null hypersurfaces intersecting in a two-parameter family of transverse spatialsurfaces, such that the horizons are two of the hypersurfaces. We present a new definition for null coordinates, that we call $\mathbf{u}$ (out-going) and $\mathbf{v}$ (in-going), which are naturally adapted to thehorizons. Our definition involves a differential equation which we solve numerically.In our construction there naturally appear a family of spheres that are parameterizedby $r_s$, which are the intersectionsof the null coordinates $\mathbf{u}$ and $\mathbf{v}$. They can also be characterized in a coordinate independent way, by the intrinsic and extrinsic GHP curvature, given by $K_{Gaussian}=\bar{Q}_{GHP}+Q_{GHP}$ and $K_{Extrinsic}=i\left( \bar{Q}_{GHP}-Q_{GHP}\right) $, with $Q=\sigma\sigma'-\rho\rho'-\Psi_2$ given in terms of the spin coefficients of the GHP formalism. In the figure below, we show the smooth behavior of these curvatures through theirnumerical computation on a surface characterized by $r_{s}$, where ($r,\theta, \phi$) are in Boyer-Lindquist coordinates and $a$ is the Kerr parameter.Our work improves several attempts that can be found in the literature. A remarkable one isdeveloped in \cite{Hayward:2004ih}, where the null hypersurfaces they construct do not include the null geodesics along theaxis of symmetry.This is due to the fact that their construction does not give a smooth hypersurface at the poles.In order to compare with ours coordinates, from \cite{Hayward:2004ih}, we consider the null function $u* = t* - r*$. Where the analog to our natural spheres are the intersection of $u*$ with the Boyer-Lindquist coordinate $t$; that can be parameterized by $r_{sH}$. In the following graph one can be seen that for $r_{sH}$ there is a discontinuity inthe derivatives at ($\theta=0$), while for $r_{s}$ it is clearly smooth.Our approach is more related to the work in \cite{Pretorius:1998sf}, whose treatment only covers the northern hemisphere,but also their expressions fail to deal with the north pole,and are very difficult 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.