List of Footnotes

1 In the literature the polar perturbations are also called even-parity because they are characterized by their behavior under parity operations as discussed earlier, and in the same way the axial perturbations are called odd-parity. We will stick to the polar/axial terminology since there is a confusion with the definition of the parity operation, the reason is that to most people, the words “even” and “odd” imply that a mode transforms under π as (− 1)2n or (− 1)2n+1 respectively (for n some integer). However only the polar modes with even ℓ have even parity and only axial modes with even ℓ have odd parity. If ℓ is odd, then polar modes have odd parity and axial modes have even parity. Another terminology is to call the polar perturbations spheroidal and the axial ones toroidal. This definition is coming from the study of stellar pulsations in Newtonian theory and represents the type of fluid motions that each type of perturbation induces. Since we are dealing both with stars and black holes we will stick to the polar/axial terminology.
2 The definition of S (− s2,𝜃) ℓ|m| on the complex plane is made unique by fixing certain conventions about the branch cuts. The exceptional points are the beginnings of branch cuts. R(l,m,s) as a solution of (40View Equation) is defined also at the exceptional points; just the series does not exist there.
3 For neutron stars the frequencies depend not only on the mass and rotation rate, but also on the radius and the equation of state.
4 Chandrasekhar and Ferrari [59Jump To The Next Citation Point] have also reduced the time independent perturbation equations (using a different gauge) into a 5th order system which involves only the spacetime perturbations with the fluid perturbations being calculated via algebraic relations from the spacetime perturbations. It was later proven by Ipser and Price [112, 169] that this system of ODEs can be reduced to the standard equations in the Regge–Wheeler gauge. That this is possible is apparent from equations (51View Equation, 52View Equation, 53View Equation, 54View Equation) and it is discussed in [6]