论文标题
赤道及时的圆形圆形轨道的拓扑，固定黑洞周围
Topology of equatorial timelike circular orbits around stationary black holes
论文作者
论文摘要
在通用黑洞的背景下，已成功地使用了一种拓扑方法来研究光环和空圆形轨道的性质。但是，对于赤道时型圆形轨道，与光环外壳完全不同，其半径密切取决于测试粒子的能量和角动量。这一事实似乎限制了拓扑处理的扩展到及时的圆形轨道。在本文中，我们确认角动量不会影响构造矢量的渐近行为，其零点表示赤道时代的圆形轨道。结果，可以构建一个表征赤道时类似圆形轨道的良好行为拓扑。我们的研究表明，临时圆形轨道的总拓扑数消失了通用黑洞，这取决于粒子的能量。值得注意的是，它表明，如果存在及时的圆形轨道，它们总是成对以固定角动量成对。同时，稳定且不稳定的时光圆形轨道具有正绕组或负数。特别有趣的是，边缘稳定的圆轨对应于构造载体零点的分叉点。此外，我们还检查了粒子能充当控制参数时的情况。结果表明，当粒子能的值是一个时，将会存在拓扑相变。低于此值，定时圆形轨道总是成对以固定能量。否则，我们将拥有一个不稳定的时间符号圆形轨道。我们进一步将治疗方法应用于Kerr黑洞。通用黑洞背景中给出的所有结果均已重现。这些强烈表明我们的拓扑方法可以推广到赤道时型圆形轨道。
A topological approach has been successfully used to study the properties of the light ring and the null circular orbit, in a generic black hole background. However, for the equatorial timelike circular orbit, quite different from the light ring case, its radius is closely dependent of the energy and angular momentum of a test particle. This fact seems to restrict the extension of the topological treatment to the timelike circular orbit. In this paper, we confirm that the angular momentum does not affect the asymptotic behavior of the constructed vector with its zero points denoting the equatorial timelike circular orbits. As a result, a well-behaved topology to characterize the equatorial timelike circular orbits can be constructed. Our study shows that the total topological number of the timelike circular orbits vanishes for a generic black hole, which is dependent of the energy of the particle. Significantly, it reveals that if there the timelike circular orbits exist, they always come in pairs for fixed angular momentum. Meanwhile, the stable and unstable timelike circular orbits have positive or negative winding number. Of particular interest is that the marginally stable circular orbit corresponds to the bifurcation point of the zero point of the constructed vector. Moreover, we also examine the case when the particle energy acts as the control parameter. It is shown that there will be topological phase transition when the value of the particle energy is one. Below this value, the timelike circular orbits always come in pairs for fixed energy. Otherwise, we will have one more unstable timelike circular orbit. We further apply the treatment to the Kerr black hole. All the results given in a generic black hole background are exactly reproduced. These strongly indicate that our topological approach can be generalized to the equatorial timelike circular orbits.
