Herein we explore the non-equatorial constant-r (“quasi-circular”) geodesics (both timelike and null) in the Painlevé–Gullstrand variant of the Lense–Thirring spacetime recently introduced by the current authors. Even though the spacetime is not spherically symmetric, shells of constant-r geodesics still exist. Whereas the radial motion is (by construction) utterly trivial, determining the allowed locations of these constant-r geodesics is decidedly non-trivial, and the stability analysis is equally tricky. Regarding the angular motion, these constant-r orbits will be seen to exhibit both precession and nutation — typically with incommensurate frequencies. Thus this constant-r geodesic motion, though integrable in the precise technical sense, is generically surface-filling, with the orbits completely covering a symmetric equatorial band which is a segment of a spherical surface, (a so-called “spherical zone”), and whose latitudinal extent is governed by delicate interplay between the orbital angular momentum and the Carter constant. The situation is qualitatively similar to that for the (exact) Kerr spacetime — but we now see that any physical model having the same slow-rotation weak-field limit as general relativity will still possess non-equatorial constant-r geodesics.

在这里，我们探索了当前作者最近引入的 Lense-Thirring 时空的 Painlevé-Gullstrand 变体中的非赤道常数-r（“准圆形”）测地线（类时和零）。即使时空不是球对称的，常数r测地线的壳仍然存在。尽管径向运动（通过构造）完全是微不足道的，但确定这些常数r测地线的允许位置绝对不是微不足道的，稳定性分析同样棘手。关于角运动，这些常数r轨道将被视为同时表现出进动和章动——通常具有不相称的频率。因此这个常数-r测地线运动，虽然在精确的技术意义上是可积分的，但通常是表面填充，轨道完全覆盖一个对称的赤道带，该赤道带是球面的一部分（所谓的“球区”），其纬度范围受轨道角动量和卡特常数之间微妙的相互作用控制。这种情况在性质上与（精确的）克尔时空相似——但我们现在看到，任何具有与广义相对论相同的慢转弱场极限的物理模型仍将具有非赤道常数r测地线。