任意维数下Wald定理的一个证明
A Proof for a Theorem of Wald in Arbitrary Dimensions
摘要：
静态的，轴对称的解决方案形成了一大类重要的黑洞在经典GR。在四维，其最一般的度量的存在依赖于这样一个事实，即二维子空间的切空间在每一个点上的向量正交的时间平移和旋转Killing场是可积的。这是由Wald通过应用Frobenius定理首次证明的。本文给出了Wald在任意维数下对这一定理的初等证明，给出了D-2正交交换Killing向量场的D维真空Einstein方程最一般解的度量估计.
英文摘要：
Static, axisymmetric solutions form a large class of important black holes in classical GR. In four dimensions, the existence of their most general metric ansatz relies on the fact that two-dimensional subspaces of the tangent space at each point spanned by vectors orthogonal to the time-translation and rotation Killing fields are integrable. This was first proved by Wald via an application of Frobenius theorem. In this note, we furnish an elementary proof for this theorem by Wald in arbitrary dimensions which yields the metric ansatz for the most general solution of the D-dimensional vacuum Einstein equations that admits D-2 orthogonal and commuting Killing vector fields.
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