arXiv 2103.064552021-03gr-qchep-thmath-phmath.MP

Simon张量的替代方法

An alternative to the Simon tensor

Masato Nozawa

备注：: v2: 17 pages, accepted version in CQG
发表信息：: Class. Quantum Grav. 38 155001 (2021)
DOI：: 10.1088/1361-6382/ac0a87

摘要：

西蒙张量在真空时空的固定类中给出了克尔-NUT家族的局部特征。我们发现，一个对称的和无痕张量的商空间中的固定Killing轨迹提供了一个有用的替代西蒙张量。我们的张量不同于Simon张量的空间对偶，并且更明显地说明了三维商空间的几何性质。给出了广义Simon张量为零的度量的重建过程。我们给出了一个四维描述这个张量的库仑部分的虚自对偶Weyl张量，这对应于推广的三个指标张量火星定义。这使我们能够为Kerr-NUT族建立一个新的和简单的判据：Ernst势的梯度成为虚自对偶Weyl张量的库仑部分的非零本征向量。我们还讨论了${\rm SU}(1,2)$协变扩张的阻塞张量到Einstein-Maxwell系统的Kerr-Newman-NUT家庭的一个内在的特征。

英文摘要：

The Simon tensor gives rise to a local characterization of the Kerr-NUT family in the stationary class of vacuum spacetimes. We find that a symmetric and traceless tensor in the quotient space of the stationary Killing trajectory offers a useful alternative to the Simon tensor. Our tensor is distinct from the spatial dual of the Simon tensor and illustrates the geometric property of the three dimensional quotient space more manifest. The reconstruction procedure of the metric for which the generalized Simon tensor vanishes is spelled out in detail. We give a four dimensional description of this tensor in terms of the Coulomb part of the imaginary selfdual Weyl tensor, which corresponds to the generalization of the three-index tensor defined by Mars. This allows us to establish a new and simple criterion for the Kerr-NUT family: the gradient of the Ernst potential becomes the non-null eigenvector of the Coulomb part of the imaginary selfdual Weyl tensor. We also discuss the ${\rm SU}(1,2)$ covariant extension of the obstruction tensor into the Einstein-Maxwell system as an intrinsic characterization of the Kerr-Newman-NUT family.

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