The weakest known criterion for local rotational symmetry (LRS) in spacetimes of Petrov type D is due to Goode and Wainwright (Gen Rel Grav 18:315, 1986). Here it is shown, using methods related to the Cartan-Karlhede procedure, to be equivalent to local spatial rotation invariance of the Riemann tensor and its first derivatives. Conformally flat spacetimes are similarly studied and it is shown that for almost all cases the same criterion ensures LRS. Only for conformally flat accelerated perfect fluids are three curvature derivatives required to ensure LRS, showing that Ellis’s original condition for that case is necessary as well as sufficient.

彼得罗夫 D 型时空中的局部旋转对称性 (LRS) 的最弱已知标准是古德和温赖特 (Gen Rel Grav 18:315, 1986) 提出的。这里显示，使用与 Cartan-Karlhede 过程相关的方法，等效于黎曼张量及其一阶导数的局部空间旋转不变性。共形平坦时空也进行了类似的研究，结果表明，对于几乎所有情况，相同的标准确保了 LRS。只有对于共形平坦加速完美流体，才需要三个曲率导数来确保 LRS，这表明 Ellis 在这种情况下的原始条件是必要的也是充分的。