Understanding initial data for black hole collisions

Numerical relativity, applied to collisions of black holes, starts with initial data for black holes already in each other's strong field. The initial hypersurface data typically used for computation is based on mathematical simplifying prescriptions, such as conformal flatness of the 3-geometry and longitudinality of the extrinsic curvature. In the case of head on collisions of equal mass holes, there is evidence that such prescriptions work reasonably well, but it is not clear why, or whether this success is more generally valid. Here we study these questions by considering the ``particle limit'' for head on collisions of nonspinning holes. Einstein's equations are linearized in the mass of the small hole, and described by a single gauge invariant spacetime function psi, for each multipole. The resulting equations have been solved by numerical evolution for collisions starting from various initial separations, and the evolution is studied on a sequence of hypersurfaces. In particular, we extract hypersurface data, that is psi and its time derivative, on surfaces of constant background Schwarzschild time. These evolved data can then be compared with ``prescribed'' data, evolved data can be replaced by prescribed data on any hypersurface, and evolved further forward in time, a gauge invariant measure of deviation from conformal flatness can be evaluated, etc. The main findings of this study are: (i) For holes of unequal mass the use of prescribed data on late hypersurfaces is not successful. (ii) The failure is likely due to the inability of the prescribed data to represent the near field of the smaller hole. (iii) The discrepancy in the extrinsic curvature is more important than in the 3-geometry. (iv) The use of the more general conformally flat longitudinal data does not notably improve this picture.Comment: 20 pages, REVTEX, 26 PS figures include

Global Prescribed Mean Curvature foliations in cosmological spacetimes II

This paper is devoted to the investigation of global properties of prescribed mean curvature (PMC) foliations in cosmological space-times with local U(1)xU(1) symmetry and matter described by the Vlasov equation. It turns out that these space-times admit a global foliation by PMC surfaces as well, but the techniques to achieve this goal are more complex than in the cases considered in Paper I [Henkel (2002)

Parabolic methods for the construction of spacelike slices of prescribed mean curvature in cosmological spacetimes

Spacelike hypersurfaces of prescribed mean curvature in cosmological space times are constructed as asymptotic limits of a geometric evolution equation. In particular, an alternative, constructive proof is given for the existence of maximal and constant mean curvature slices

Spacelike convex surfaces with prescribed curvature in (2+1)-Minkowski space

We prove existence and uniqueness of solutions to the Minkowski problem in any domain of dependence $D$ in $(2+1)$-dimensional Minkowski space, provided $D$ is contained in the future cone over a point. Namely, it is possible to find a smooth convex Cauchy surface with prescribed curvature function on the image of the Gauss map. This is related to solutions of the Monge-Amp\`ere equation $\det D^2 u(z)=(1/\psi(z))(1-|z|^2)^{-2}$ on the unit disc, with the boundary condition $u|_{\partial\mathbb{D}}=\varphi$, for $\psi$ a smooth positive function and $\varphi$ a bounded lower semicontinuous function. We then prove that a domain of dependence $D$ contains a convex Cauchy surface with principal curvatures bounded from below by a positive constant if and only if the corresponding function $\varphi$ is in the Zygmund class. Moreover in this case the surface of constant curvature $K$ contained in $D$ has bounded principal curvatures, for every $K<0$. In this way we get a full classification of isometric immersions of the hyperbolic plane in Minkowski space with bounded shape operator in terms of Zygmund functions of $\partial \mathbb{D}$. Finally, we prove that every domain of dependence as in the hypothesis of the Minkowski problem is foliated by the surfaces of constant curvature $K$, as $K$ varies in $(-\infty,0)$.Comment: 45 pages, 17 figures. Final version, improved presentation and details of some proof