史瓦西时空中的单摆
A Simple Pendulum in Schwarzschild Spacetime
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中文总结 AI 辅助
该研究推导了史瓦西时空中单摆的小振动周期，复现牛顿极限结果，对比早期研究指出弱场外两问题不等价。
中文摘要 AI 辅助
我们确定了史瓦西时空中单摆的小振动周期。瞬态衰减后，偏移的摆绳与诱导空间度量的测地线重合，据此可确定摆锤径向提升量对角度振幅的二次阶关系，进而得到位于摆锤平衡位置$r=r_2$、支点在$r_1$处的观测者测得的周期：$T_{(2)}=4\pi\frac{r_2^2}{c r_s}\sqrt{N_2(N_1-N_2)}$，其中$N_i=N(r_i)$，$N(r)=\sqrt{1-r_s/r}$为史瓦西时滞函数。该结果由决定钟速和引力红移的同一函数表达，在牛顿极限下可复现经典结果$T=2\pi\sqrt{L_0/g}$。我们用时有效摆长解释了近视界处的周期，并将本文处理方式与早期研究采用的坐标长度约束对比，指出在远离经典弱场极限时，两个问题不等价。
英文摘要
We determine the small-oscillation period of a simple pendulum in Schwarzschild spacetime. After transients decay, the displaced rope coincides with a geodesic of the induced spatial metric. This determines the radial lift of the bob to quadratic order in the angular amplitude and leads to the period measured by an observer at the bob's equilibrium position, $r=r_2$, with the fulcrum at $r_1$, \[ T_{(2)}=4π\frac{r_2^2}{c r_s}\sqrt{N_2(N_1-N_2)}, \] where $N_i=N(r_i)$ and $N(r)=\sqrt{1-r_s/r}$ is the Schwarzschild lapse. The result is therefore expressed in terms of the same function that determines clock rates and gravitational redshifts. In the Newtonian limit, we reproduce the classical result $T=2π\sqrt{L_0/g}$. We interpret the period close to the horizon in terms of an effective pendulum length. We compare our treatment with the coordinate-length constraint adopted in an earlier study and argue that the two problems are inequivalent away from the classical, weak-field limit.
