Property of ν

4.6 Property of $ν$

Update

In this section, we discuss the property of the solution of Equation (136

) which we recapitulate:

The numerical evaluation of this equation was not done very much before. Leaver [64

] briefly mentioned a numerical implementation of a code to obtain $ν$ by solving Equation (174

). In the Schwarzschild case, Tagoshi and Nakamura [99 ] solved Equation (174

) numerically to obtain $ν$ . They evaluated the homogeneous solution numerically based on Leaver’s method [64 ] by using the value of $ν$ obtained numerically. Later, Fujita and Tagoshi [40 ] tried a numerical implementation of the MST method. They found that, as $ω$ becomes large, it becomes impossible to obtain a solution of (174

) if $ν$ is restricted to a real number. In a subsequent paper [41 ], they found that when the real solution ceases to exist, a complex solution appears. They also found that when $ν$ is complex, the real part of $ν$ is always either an integer or half-integer. As an example, we show the value of $ν$ as a function of $M ω$ in Table 1.

The fact that we only have an integer or half-integer as the real part of $ν$ is strongly suggested from the property of $g (ν) n$ [37 ]. Let us summarize the argument. We first convert $ν$ to $y$ as $ν = p∕2 + iy$ , where $p$ is an arbitrary integer and $y$ is an arbitrary complex number. We note that for an arbitrary integer $n$ ,

where $∗$ denotes complex conjugation. From these relations, we find that if $ν = p∕2 + iy$ is a solution of $gn(ν ) = 0$ , we have

where we used the relation, $gn+1(ν) = gn(ν + 1)$ . We see that in this case $′ ∗ ν = p∕2 − p − 2n − 1 + iy$ is also an solution of $gn = 0$ . As already discussed at the end of Section 4.2, when $ν$ is a solution of $gn = 0$ , $ν + k$ and $− ν − 1 + k$ with an arbitrary integer $k$ are also solutions. We assume that there are no other solutions. Although we do not have a formal proof of it, numerical investigations suggest that this is so. Under this assumption, since both $ν = p∕2 + iy$ and $′ ∗ ν = p∕2 − p − 2n − 1 + iy$ are solutions of $gn = 0$ , we have two possibilities about the property of $y$ :

In the former case, $y$ is real. In this case, $ν$ is complex with real part $p∕2$ (integer or half-integer). In the later case, $y$ is pure imaginary. In this case, $ν$ is real.

It becomes possible to determine $ν$ in the wide range of $ω$ by allowing $Im (ν) ⁄= 0$ . The MST formalism is now very useful in the fully numerical evaluation of homogeneous solutions of the Teukolsky equation. As a first step, Fujita, Hikida and Tagoshi [38 ] considered generic bound geodesic orbits around a Kerr black hole and evaluate the energy and angular momentum flux to infinity as well as the rate of change of the Carter constant in a wide range of orbital parameters.

The critical value of $ω$ when $ν$ becomes complex is not very small. The complex $ν$ does not appear in the analytic evaluation of $ν$ in the low frequency expansion in powers of $𝜖 = 2M ω$ . Thus, at the first glance, it seems impossible to express the complex $ν$ in the power series expansion of $𝜖$ . However, Hikida et al. [52 ] pointed out that it is possible to evaluate $2 sin (πν)$ very accurately in terms of the power series expansion of $𝜖$ , even if $ω$ is larger than a critical value and $ν$ is complex. Such an analytical expression of $ν$ is very useful in the numerical root finding of Equation (174 ) as well as in the analytical calculation of the homogeneous solutions.

Table 1: The value of $ν$ for various value of $M ω$ in the case $s = − 2$ , $l = m = 2$ and $q = 0$ .

$M ω$	Re $(ν)$	Im $(ν)$
0.1	1.9793154547208	0.0000000000000
0.2	1.9129832302687	0.0000000000000
0.3	1.7792805424199	0.0000000000000
0.4	1.5000000000000	0.1862468531447
0.5	1.5000000000000	0.3618806153941
0.6	1.7878302655744	0.0000000000000
0.7	2.0000000000000	0.8003377636925
0.8	2.0000000000000	1.1099466644118
0.9	2.0000000000000	1.3699138540831
1.0	2.0000000000000	1.6085538776570
2.0	2.0000000000000	3.6867890278893
3.0	2.0000000000000	5.5939000509184