Abstract

We construct dynamical many-black-hole spacetimes with well-controlled asymptotic behavior as solutions of the Einstein vacuum equation with positive cosmological constant. We accomplish this by gluing Schwarzschild–de Sitter or Kerr–de Sitter black hole metrics into neighborhoods of points on the future conformal boundary of de Sitter space, under certain balance conditions on the black hole parameters. We give a self-contained treatment of solving the Einstein equation directly for the metric, given the scattering data we encounter at the future conformal boundary. The main step in the construction is the solution of a linear divergence equation for trace-free symmetric 2-tensors; this is closely related to Friedrich’s analysis of scattering problems for the Einstein equation on asymptotically simple spacetimes.

摘要

我们构建了具有良好控制的渐近行为的动力学多黑洞时空，作为具有正宇宙常数的爱因斯坦真空方程的解。我们通过在黑洞参数的某些平衡条件下将 Schwarzschild-de Sitter 或 Kerr-de Sitter 黑洞度量粘合到 de Sitter 空间的未来共形边界上的点的邻域中来实现这一点。考虑到我们在未来共形边界处遇到的散射数据，我们给出了直接为度量求解爱因斯坦方程的独立处理。构造的主要步骤是求解无迹对称 2-张量的线性发散方程；这与弗里德里希在渐近简单时空上对爱因斯坦方程的散射问题的分析密切相关。