论文标题
昆特猜想的进步
Progress on the Kundt conjecture
论文作者
论文摘要
昆特的猜想指出，任意维度的洛伦兹（Lorentzian）歧管的特征是其标量多项式曲率不变式（SPI）允许非扭曲，非剪切和非膨胀（简称kundt，kundt）无效的null null Geodesics。该猜想已被证明针对尺寸3和4。时空不得以SPI为特征的必要条件是，Riemann Tensor的所有协变量均具有II型或更为特殊的null Arignment分类。在任意维度中，我们证明，当某个通用条件成立时，或者当无曲线的Ricci orweyl张量是真正的III型或N时，该特性确实需要存在昆特无效的一致性，从而确认了这些情况下昆特猜想的有效性。我们还通过删除规律性假设，并表明仅需要第三个协变量才能获得结果，我们还可以通过删除规律性假设来实现维度3和4的结果。我们证明中的一个关键工具是作用于相关升压订单相对于无效方向的张量的新的双线图。
The Kundt conjecture states that a Lorentzian manifold of arbitrary dimension which is not characterized by its scalar polynomial curvature invariants (SPIs) allows for a non-twisting, non-shearing and non-expanding (in short, Kundt) null congruence of geodesics. The conjecture has been proven for dimensions 3 and 4. A necessary condition for a spacetime not to be characterized by SPIs is that all covariant derivatives of the Riemann tensor are of aligned type II or more special in the null alignment classification. In arbitrary dimensions, we prove that this property indeed requires the presence of a Kundt null congruence when a certain genericity condition holds, or when the trac-free Ricci orWeyl tensor is of genuine type III or N, thus confirming the validity of the Kundt conjecture in these cases. We also strenghten the results for dimensions 3 and 4 by removing regularity assumptions and showing that only the third covariant derivative is needed to obtain the results. A key tool in our proofs is a new bilinear map acting on tensors of related boost orders relative to a null direction.
