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We derive the equation governing the axial-perturbations in the space-time ofa non-rotating uncharged primordial black hole (PBH), produced in earlyUniverse, whose metric is taken as the generalized McVittie metric. Thegeneralized McVittie metric is a cosmological black hole metric, proposed by V.Faraoni and A. Jacques in 2007 [Phys. Rev. D 76, 063510 (2007)]. This describesthe space-time of a Schwarzschild black hole embedded in FLRW-Universe, whileallowing its mass-change. Our derivation has basic similarities with theprocedure of derivation of S. Chandrasekhar, for deriving the Regge-Wheelerequation for Schwarzschild metric [S. Chandrasekhar, The Mathematical Theory ofBlack holes ; Oxford University Press (1983)] ; but it has some distinctdifferences with that due to the complexity and time-dependency of thegeneralized McVittie metric. We show that after applying some approximationswhich are very well valid in the early radiation-dominated Universe, theoverall equation governing the axial perturbations can be separated into radialand angular parts, among which the radial part is the intended one, as theangular part is identical to the case of Schwarzschild metric as expected. Weidentify the potential from the Schr\"odinger-like format of the equation anddraw some physical interpretation from it.

Evolution of axial perturbations in space-time of a non-rotating uncharged primordial black hole