标题： 关于超紧致天体弯曲时空中的光环数量 显示英文标题

标题： On the number of light rings in curved spacetimes of ultra-compact objects

摘要： 在一篇非常有趣的文章中，Cunha、Berti 和 Herdeiro 最近声称，具有光环的超紧致天体——即爱因斯坦场方程的自引力无视界解——必须至少拥有{\it 两个}（且通常为偶数个）光环，其中内侧的一个为{\it 稳定}。在目前这篇紧凑的文章中，我们明确证明了虽然这一引人注目的定理通常是正确的，但在存在退化光环的情况下存在一个重要例外，在球对称静态情况下，这些退化光环由简单的无量纲关系$8\pi r^2_{\gamma}(\rho+p_{\text{T}})=1$特征化（这里$r_{\gamma}$是光环的半径，$\{\rho,p_{\text{T}}\}$分别是物质场的能量密度和横向压力）。属于这一独特族的超紧致天体可以具有{\it 奇数}个数量级的光环。 作为一个具体的例子，我们证明了具有无量纲紧凑性的球对称恒星，即 $M/R=1/3$，仅拥有 {\it 一个}个光环，并且有趣的是，该光环被证明是 {\it 不稳定}。 显示英文摘要

摘要： In a very interesting paper, Cunha, Berti, and Herdeiro have recently claimed that ultra-compact objects, self-gravitating horizonless solutions of the Einstein field equations which have a light ring, must possess at least {\it two} (and, in general, an even number of) light rings, of which the inner one is {\it stable}. In the present compact paper we explicitly prove that, while this intriguing theorem is generally true, there is an important exception in the presence of degenerate light rings which, in the spherically symmetric static case, are characterized by the simple dimensionless relation $8\pi r^2_{\gamma}(\rho+p_{\text{T}})=1$ [here $r_{\gamma}$ is the radius of the light ring and $\{\rho,p_{\text{T}}\}$ are respectively the energy density and tangential pressure of the matter fields]. Ultra-compact objects which belong to this unique family can have an {\it odd} number of light rings. As a concrete example, we show that spherically symmetric constant density stars with dimensionless compactness $M/R=1/3$ possess only {\it one} light ring which, interestingly, is shown to be {\it unstable}.