The astrophysical importance of the Kerr spacetime cannot be overstated. Of the currently known exact solutions to the Einstein field equations, the Kerr spacetime stands out in terms of its direct applicability to describing astronomical black hole candidates. In counterpoint, purely mathematically, there is an old classical result of differential geometry, due to Darboux, that all 3-manifolds can have their metrics recast into diagonal form. In the case of the Kerr spacetime the Boyer–Lindquist coordinates provide an explicit example of a diagonal spatial 3-metric. Unfortunately, as we demonstrate herein, Darboux diagonalization of the spatial 3-slices of the Kerr spacetime is incompatible with simultaneously putting the Kerr metric into unit-lapse form while retaining manifest axial symmetry. This no-go theorem is somewhat reminiscent of the no-go theorem to the effect that the spatial 3-slices of the Kerr spacetime cannot be chosen to be conformally flat.

中文翻译：

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克尔时空中空间 3 度量的 Darboux 对角化

克尔时空的天体物理学重要性怎么强调都不为过。在目前已知的爱因斯坦场方程的精确解中，克尔时空在描述天文黑洞候选者的直接适用性方面脱颖而出。相反，纯粹在数学上，由于 Darboux，微分几何有一个古老的经典结果，即所有 3 流形都可以将其度量重新转换为对角线形式。在 Kerr 时空的情况下，Boyer-Lindquist 坐标提供了一个对角线空间 3 度量的明确示例。不幸的是，正如我们在此证明的那样，克尔时空的空间 3 切片的达布对角化与同时将克尔度量置于单位递减形式同时保持明显的轴对称性是不兼容的。