Riemann curvature invariants are important in general relativity because they encode the geometrical properties of spacetime in a manifestly coordinate-invariant way. Fourteen such invariants are required to characterize four-dimensional spacetime in general, and Zakhary and McIntosh showed that as many as seventeen can be required in certain degenerate cases. We calculate explicit expressions for all seventeen of these Zakhary–McIntosh curvature invariants for the Kerr–Newman metric that describes spacetime around black holes of the most general kind (those with mass, charge, and spin), and confirm that they are related by eight algebraic conditions (dubbed syzygies by Zakhary and McIntosh), which serve as a useful check on our results. Plots of these invariants show richer structure than is suggested by traditional (coordinate-dependent) textbook depictions, and may repay further investigation.

中文翻译：

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带电和旋转黑洞的曲率不变量

黎曼曲率不变性在广义相对论中很重要，因为它们以明显的坐标不变的方式编码时空的几何特性。通常，需要十四个这样的不变量来表征四维时空，Zakhary和McIntosh表明，在某些退化的情况下，可能需要多达十七个。我们针对所有Kerr-Newman度量的Zakhary-McIntosh曲率不变量中的全部17个计算显式表达式，这些度量描述了最一般的黑洞（具有质量，电荷和自旋的黑洞）周围的时空，并确认它们与8代数条件（Zakhary和McIntosh称其为“ sysygies”），可以有效地检验我们的结果。