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arXiv 1903.12486gr-qc
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Schwarzschild时空中定域静坐标下的逃逸、束缚和俘获测地线

Escape, bound and capture geodesics in local static coordinates in Schwarzschild spacetime

Yaoguang Wang, Xionghui Liu, Nan Yang, Jiawei Liu, Junji Jia

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26 pages, 10 figures
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摘要：

根据类时粒子的起始半径$r$、速度$v$和在定域静态观测者参考系中与径向向外方向的夹角$α$，分析了Schwarzschild时空中类时粒子的经典测地线.利用有效势的三种情况，求解了$(r,~v,~α)$参数空间中的逃逸轨道、束缚轨道和俘获轨道区域。结果表明，当半径小于$4M$或速度大于$c/\sqrt{2}$时，一般不存在束缚轨道.当粒子半径大于$4M$（或速度小于$c/\sqrt{2}$）时，当速度（或半径）从零（或$2M$）开始增大时，粒子总是被捕获，直到达到一个临界值$v_{\mathrm{crit1}}$（或$r_{\mathrm{crit1}}$），粒子的束缚轨道开始出现在双推锥结构之间的$α=π/2$周围.当速度（或半径）增加到另一个临界值$v_{\mathrm{crit2}}$（或$r_{\mathrm{crit2}}$）时，向外锥体中的束缚方向和逃逸方向变为逃逸方向，只留下向内锥体将捕获方向和束缚方向分开。当速度（或半径）增加到其渐近值时，这个圆锥的角度将增加到其渐近值。简要讨论了这些结果在黑洞阴影、黑洞吸积和航天器导航中的应用。

英文摘要：

The classical geodesics of timelike particles in Schwarzschild spacetime is analyzed according to the particle starting radius $r$, velocity $v$ and angle $α$ against the radial outward direction in the reference system of an local static observer. The region of escape, bound and capture orbits in the parameter space of $(r,~v,~α)$ are solved using the three cases of the effective potential. It is found that generally for radius smaller than $4M$ or velocity larger than $c/\sqrt{2}$ there will be no bound orbits. While for fixed radius larger than $4M$ (or velocity smaller than $c/\sqrt{2}$), as velocity (or radius) increase from zero (or $2M$), the particle is always captured until a critical value $v_{\mathrm{crit1}}$ (or $r_{\mathrm{crit1}}$) when the bound orbit start to appear around $α=π/2$ between a double-napped cone structure. As the velocity (or radius) increases to another critical value $v_{\mathrm{crit2}}$ (or $r_{\mathrm{crit2}}$) then the bound directions and escape directions in the outward cone become escape directions, leaving only the inward cone separating the capture and bound directions. The angle of this cone will increase to its asymptotic value as velocity (or radius) increases to its asymptotic value. The implication of these results in shadow of black holes formed by massive particles, in black hole accretion and in spacecraft navigation is briefly discussed.

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