arXiv 0704.04352007-04gr-qchep-th

高维D型爱因斯坦时空

Type D Einstein spacetimes in higher dimensions

Vojtech Pravda, Alena Pravdova, Marcello Ortaggio

备注：: 24 pages. v2: new appendix on geometric optics of higher dimensional Kerr-NUT-AdS black holes, minor changes in the text. To appear in Class.Quant.Grav
发表信息：: Class.Quant.Grav.24:4407-4428,2007
DOI：: 10.1088/0264-9381/24/17/009

摘要：

我们证明了高维静态时空都是Weyl型G，I_i，D或O.这也适用于静态时空，如果满足额外的条件，如大多数已知的黑洞/环的解决方案。(The当Killing生成器为空时，结论会发生变化，例如在Killing水平线处。接下来，我们证明了同样的Weyl类型表征了具有一维洛伦兹（类时）因子的翘曲乘积时空，而具有二维洛伦兹因子的翘曲时空则限于D或O类型。通过探索比安基恒等式，我们分析了上述类别中最简单的非平凡情形--D型真空时空，它可能具有一个宇宙学常数，但放弃了时空是静态的、静止的或弯曲的假设。它表明，对于“一般的”D型真空时空，相应的主零方向在任何维度上都是测地的（这也适用于II型时空）。然而，对于n>=5，可能存在允许非测地多重主零方向的D型时空的特殊情况，我们在任何n>=7中给出这样的例子。进一步的研究被限制在5维，其中D型Weyl张量由3 × 3矩阵Φ ij描述。在"twistfree“（A_{ij} =0）主零测地线的情形下，我们证明了在”generic“情形下Φ_{ij}是对称的，并且Φ_{ij}的本征向量与展开矩阵S_{ij}的本征向量重合，从而为我们提供了时空的三个优选类空方向.当放松无扭条件而假设Φ ij是对称的时，也得到了类似的结果。作为D型真空时空的具体例子，对任意维的n=5的Myers-Perry黑洞和Kerr-NUT-AdS度规进行了简要的研究.

英文摘要：

We show that all static spacetimes in higher dimensions are of Weyl types G, I_i, D or O. This applies also to stationary spacetimes if additional conditions are fulfilled, as for most known black hole/ring solutions. (The conclusions change when the Killing generator becomes null, such as at Killing horizons.) Next we demonstrate that the same Weyl types characterize warped product spacetimes with a one-dimensional Lorentzian (timelike) factor, whereas warped spacetimes with a two-dimensional Lorentzian factor are restricted to the types D or O. By exploring the Bianchi identities, we then analyze the simplest non-trivial case from the above classes - type D vacuum spacetimes, possibly with a cosmological constant, dropping, however, the assumptions that the spacetime is static, stationary or warped. It is shown that for ``generic'' type D vacuum spacetimes the corresponding principal null directions are geodetic in any dimension (this applies also to type II spacetimes). For n>=5, however, there may exist particular cases of type D spacetimes which admit non-geodetic multiple principal null directions and we present such examples in any n>=7. Further studies are restricted to five dimensions, where the type D Weyl tensor is described by a 3x3 matrix Φ_{ij}. In the case with ``twistfree'' (A_{ij}=0) principal null geodesics we show that in a ``generic'' case Φ_{ij} is symmetric and eigenvectors of Φ_{ij} coincide with those of the expansion matrix S_{ij}, providing us with three preferred spacelike directions of the spacetime. Similar results are also obtained when relaxing the twistfree condition and assuming instead that Φ_{ij} is symmetric. The n=5 Myers-Perry black hole and Kerr-NUT-AdS metrics in arbitrary dimension are briefly studied as specific examples of type D vacuum spacetime.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文