We derive the equation governing the axial-perturbations in the space-time of a non-rotating uncharged primordial black hole (PBH), produced in early Universe, whose metric has been taken as the generalized McVittie metric. The generalized McVittie metric is a cosmological black hole metric, proposed by Faraoni and Jacques in 2007 (Phys. Rev. D 76:063510, 2007). This describes the space-time of a Schwarzschild black hole embedded in FLRW-Universe, while allowing its mass-change. Our derivation has basic similarities with the procedure of derivation of Chandrasekhar, for deriving the Regge-Wheeler equation for Schwarzschild metric (Chandrasekhar The Mathematical Theory of Black holes, Oxford University Press, New York, 1983); but it has some distinct differences with that due to the complexity and time-dependency of the generalized McVittie metric. We show that after applying some approximations which are very well valid in the early radiation-dominated Universe, the overall equation governing the axial perturbations can be separated into radial and angular parts, among which the radial part is the intended one, as the angular part is identical to the case of Schwarzschild metric as expected. We identify the potential from the Schrödinger-like format of the equation and draw some physical interpretation from it.

我们推导出了控制早期宇宙中产生的不旋转不带电的原始黑洞 （PBH） 时空轴向扰动的方程，其度量被用作广义 McVittie 度量。广义麦克维蒂度量是一种宇宙学黑洞度量，由 Faraoni 和 Jacques 于 2007 年提出 （Phys. Rev. D 76：063510， 2007）。这描述了嵌入 FLRW-Universe 中的史瓦西黑洞的时空，同时允许其质量变化。我们的推导与 Chandrasekhar 的推导过程有基本的相似之处，即推导出施瓦西度量的 Regge-Wheeler 方程（Chandrasekhar The Mathematical Theory of Black holes，牛津大学出版社，纽约，1983 年）;但由于广义 McVittie 度量的复杂性和时间依赖性，它与此有一些明显的差异。我们表明，在应用了一些在早期辐射主导的宇宙中非常有效的近似值之后，控制轴向扰动的总体方程可以分为径向部分和角度部分，其中径向部分是预期的部分，因为角度部分与预期的 Schwarzschild 度量的情况相同。我们从方程的类似薛定谔的格式中确定了潜力，并从中得出一些物理解释。