The standard Lense–Thirring metric is a century-old slow-rotation large-distance approximation to the gravitational field outside a rotating massive body, depending only on the total mass and angular momentum of the source. Although it is not an exact solution to the vacuum Einstein equations, asymptotically the Lense–Thirring metric approaches the Kerr metric at large distances. Herein we shall discuss a specific variant of the standard Lense–Thirring metric, carefully chosen for simplicity, clarity, and various forms of improved mathematical and physical behaviour, (to be more carefully defined in the body of the article). We shall see that this Lense–Thirring variant can be viewed as arising from the linearization of a suitably chosen tetrad representing the Kerr spacetime. In particular, we shall construct an explicit unit-lapse Painlevé–Gullstrand variant of the Lense–Thirring spacetime, one that has flat spatial slices, a very simple and physically intuitive tetrad, and extremely simple curvature tensors. We shall verify that this variant of the Lense–Thirring spacetime is Petrov type I, (so it is not algebraically special), but nevertheless possesses some very straightforward timelike geodesics, (the “rain” geodesics). We shall also discuss on-axis and equatorial geodesics, ISCOs (innermost stable circular orbits) and circular photon orbits. Finally, we wrap up by discussing some astrophysically relevant estimates, and analyze what happens if we extrapolate down to small values of r; verifying that for sufficiently slow rotation we explicitly recover slowly rotating Schwarzschild geometry. This Lense–Thirring variant can be viewed, in its own right, as a “black hole mimic”, of direct interest to the observational astronomy community.

中文翻译：

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Lense-Thirring时空的Painlevé-Gullstrand形式

标准的Lense-Thirring度量标准是对旋转的大质量物体外部的引力场有百年历史的慢速旋转大距离近似值，仅取决于源的总质量和角动量。尽管它不是真空爱因斯坦方程的精确解，但渐近Lense-Thirring度量在很远的距离上接近Kerr度量。在这里，我们将讨论一个特定的变体标准Lense-Thirring度量标准，经过仔细选择，以简化，清晰和改进数学和物理行为的各种形式（要在文章正文中进行更仔细的定义）。我们将看到，这伦泽-Thirring变体可从代表克尔时空适当选择的四分体的线性化所引起观看。特别是，我们将构造一个清晰的Lense-Thirring时空单位间隔Painlevé-Gullstrand变体，该变体具有平坦的空间切片，非常简单且在物理上直观的四倍体和极其简单的曲率张量。我们将验证Lense-Thirring时空的这种变体是彼得罗夫类型I，（因此它不是代数上的特殊性），但仍具有一些非常简单的时间性测地线（“雨”测地线）。我们还将讨论轴上和赤道测地线，ISCO（最内层的稳定圆形轨道）和圆形光子轨道。最后，我们通过讨论一些与天体有关的估计来总结，并分析如果我们推断出较小的r值会发生什么。验证对于足够慢的旋转，我们明确恢复了缓慢旋转的Schwarzschild几何形状。可以将这种Lense-Thirring变体本身视为观测天文学界直接感兴趣的“黑洞模拟物”。