AI 中文总结
该研究推导了静态球对称时空中极端相对论性及亚光速常速运动者的最速降线轨迹方程，求解了两类典型时空的轨迹并分析了其特征，拓展了最速降线问题的适用场景。
AI 中文摘要
本研究探讨静态球对称（SSS）时空中，具有恒定局域速度的运动者所面临的最速降线问题。针对极端相对论性运动者，推导了一般SSS度规下的最速降线轨迹（BT）方程，并以积分形式给出其形式解。随后将该结果应用于两类典型时空：有限边界的奇异等温球（SIS）和相对论性普卢默质量分布。对于SIS时空，边界内的BT可解析求解，且始终呈弯曲形态；当物态方程指数w增大时，BT的转向半径r₀及总运动时间均随之增加。对于边界外的BT，中心天体质量越大，r₀和总运动时间越大。对于相对论性普卢默模型，当初末点的半径相当或小于质量分布核心区尺寸时，BT为过原点的直线；当端点远在核心区外时，BT发生弯曲，且核心越致密，转向半径越大。我们进一步将研究扩展到亚光速恒定速度v的运动者，通过推广的费马原理证明，此类运动者的BT在光学度规下与极端相对论性运动者的测地线相同，总运动时间按1/v的比例缩放。
英文摘要
This work investigates the brachistochrone problem for a traveler with constant local velocity within static and spherically symmetric (SSS) spacetimes. The brachistochrone trajectory (BT) equations for ultra-relativistic travelers are derived for general SSS metrics, and the solution is formally obtained in an integral form. We then apply the result to two representative spacetimes corresponding to the singular isothermal sphere (SIS) with a finite boundary and the relativistic Plummer mass profile, respectively. For the SIS spacetime, the BT inside the boundary is solved analytically and found always to bend. As the equation of state index $w$ increases, the turning radius $r_0$ of the BT, and consequently the total time, also increase. For the BT outside the boundary, it is found that the heavier the central object, the larger the $r_0$ and the total travel time. For the relativistic Plummer model, the BT will be a straight line passing through the origin when the initial and final points' radii are comparable or smaller than the size of the core region of the mass distribution. When the end points lie well outside the core region, the BT bends, exhibiting a larger turning radius for a more concentrated core. We then extend the consideration to travelers with subluminal constant velocity $v$ and show through the generalized Fermat's principle that the BT will be the same geodesic in the optical metric as ultra-relativistic travelers, with the total travel time scaled by a factor of $1/v$.
Comments9 pages, 4 figures
