标题： 广义相对论中具有Hopf结构的精确真空解 显示英文标题

标题： Exact vacuum solution with Hopf structure in general relativity

摘要： 展示了一个真空爱因斯坦方程的精确解，其结构基于Hopf纤维化。 该解采用一个测地线零向量场，该场定义了一个扭曲的共形，并以Kerr-Schild形式出现在度规中。 该解属于Petrov类型D，并涉及两个参数。 值得注意的是，所得时空是正则的，没有曲率奇点。 Kretschmann标量和Chern-Pontryagin标量均不为零，并在整个时空中保持有限。 此外，Newman-Penrose Weyl标量$\Psi_2$同时具有非零实部和虚部，反映了引力场拓扑非平凡的性质。 该时空还包含两个Killing向量场和一个Killing-Yano张量，该张量诱导了一个相关的Killing张量，揭示了其隐藏对称性。 推导过程简单且自洽，提供了一种透明且几何引导的方法来寻找广义相对论中的新精确解。 显示英文摘要

摘要： An exact solution to the vacuum Einstein equations is presented, whose structure is based on the Hopf fibration. The solution employs a geodesic null vector field that defines a twisting congruence and appears in the metric in Kerr-Schild form. This solution is of Petrov type D and involves two parameters. Remarkably, the resulting spacetime is regular, with no curvature singularities. Both the Kretschmann scalar and the Chern-Pontryagin scalar are nonzero and remain finite throughout the spacetime. In addition, the Newman-Penrose Weyl scalar $\Psi_2$ possesses both nonzero real and imaginary parts, reflecting the topologically nontrivial nature of the gravitational field. The spacetime also admits two Killing vector fields and a Killing-Yano tensor, which induces an associated Killing tensor, revealing its hidden symmetry. The derivation is simple and self-contained, offering a transparent and geometrically guided approach to finding new exact solutions in general relativity.