arXiv 2001.037772020-01-11gr-qc

类时射线的微扰偏转角

Perturbative deflection angles of timelike rays

Yujie Duan, Weiyu Hu, Ke Huang, Junji Jia

备注：: 41 pages, 5 figures
DOI：: 10.1088/1361-6382/ab852c

摘要：

在引力场中，质量为非零的光场和类时粒子的测地线都发生了偏转。本文应用文献\cite{Jia:2020dap}中发展的微扰方法计算了四种时空中弱场极限下零射线和类时射线的偏转角。我们得到了Barbian时空到$m/b$的第十一阶的偏转角，其中$m$是ADM质量，$b$是碰撞参量，以及Hayward，Janis-Newman-Winicour和Einstein-Born-Infeld时空到第九，第七和第十一阶的偏转角.分析了碰撞参量$b$、速度$v$和时空参数对四个时空中偏转角的影响。结果表明，一般情况下，微扰偏转角依赖于且仅依赖于度量函数的渐近性态，且与阶相关。此外，它表明，虽然这些偏转角计算在大的$b/m$限制，其最小有效$b$可以小到几个$m$的，只要该命令是足够高的。在这些碰撞参量下，还发现偏转角本身很大。随着速度的减小，在所研究的所有时空中的偏转角都增大。对于给定的$b$，如果时空参数允许临界速度$v_c$，那么当$v$减小到$v_c$时，微扰偏转角将偏离其真实值。我们还发现，如果时空参数的变化只能在小半径而不能在大半径处定性地改变时空，那么这些时空参数将不会引起偏转角的定性变化，尽管其值仍然受到定量的影响。讨论了该工作的应用和可能的扩展。

英文摘要：

Geodesics of both lightrays and timelike particles with nonzero mass are deflected in a gravitational field. In this work we apply the perturbative method developed in Ref. \cite{Jia:2020dap} to compute the deflection angle of both null and timelike rays in the weak field limit for four spacetimes. We obtained the deflection angles for the Bardeen spacetime to the eleventh order of $m/b$ where $m$ is the ADM mass and $b$ is the impact parameter, and for the Hayward, Janis-Newman-Winicour and Einstein-Born-Infeld spacetimes to the ninth, seventh and eleventh order respectively. The effect of the impact parameter $b$, velocity $v$ and spacetime parameters on the deflection angle are analyzed in each of the four spacetimes. It is found that in general, the perturbative deflection angle depends on and only on the asymptotic behavior of the metric functions, and in an order-correlated way. Moreover, it is shown that although these deflection angles are calculated in the large $b/m$ limit, their minimal valid $b$ can be as small as a few $m$'s as long as the order is high enough. At these impact parameters, the deflection angle itself is also found large. As velocity decreases, the deflection angle in all spacetime studied increases. For a given $b$, if the spacetime parameters allows a critical velocity $v_c$, then the perturbative deflection angle will deviate from its true value as $v$ decreases to $v_c$. It is also found that if the variation of spacetime parameters can only change the spacetime qualitatively at small but not large radius, then these spacetime parameter will not cause a qualitative change of the deflection angle, although its value is still quantitatively affected. The application and possible extension of the work are discussed.

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