Choosing the phase for the spin-weighted spheroidal functions

Abstract

The spin-weighted spheroidal functions are the eigenfunctions of the angular Teukolsky equation. They are a generalization of the widely used spin-weighted spherical functions, and are extremely important in the area of black-hole perturbation theory. Like other special functions, they have an inherent phase ambiguity and need to be phase fixed to be uniquely defined. Clearly, such a phase choice does not have a direct physical impact. But, a poorly constructed phase choice could hinder the extraction, from mode information, of physical information about a system. To date, possible phase choices for the spin-weighted spheroidal functions have received little attention. Here, we clearly define and extensively explore two useful phase fixing schemes, and we propose that the spherical-limit phase-fixing scheme be adopted as the default phase-fixing scheme for the spin-weighted spheroidal functions.

Figures (43)

• Figure 1: The magnitude of the first 11 eigenvalues for $s=0$ and $m=1$ along a sequence of values of $c=|c|e^{-i\frac{\pi}{3}}$. The bottom sequence corresponds to $L=0$($\ell=1$), while the top sequence corresponds to $L=10$($\ell=11$). Notice that the sequences that would be labeled $L=5$ and $L=6$ in the spherical limit($c=0$) belong to the family of solutions which have leading order asymptotic behavior $-c^2$ and could be labeled as $\hat{L}=0$ and $\hat{L}=1$. The sequences that would be labeled $L=0-4$ and $L=7$, in the spherical limit have entered the asymptotic regime and belong to the family of solutions which have leading order asymptotic behavior $ic(2\bar{L}+1)$ and could be labeled $\bar{L}=0-5$. The upper 3 sequences have not entered the asymptotic regime in this figure and cannot yet be classified.
• Figure 2: Phase difference between $\mathcal{P}_{\text{CZ-SL}}$ and $\mathcal{P}_{\text{Math}}$ for ${}_{{}_{0}}S^{}_{21}(x;|c|e^{-i\frac{\pi}{3}})$. The discontinuities in the phase occur as the expansion coefficient with the largest magnitude changes as $|c|$ increases. For $|c|\lesssim10.6$, ${}_{{}_{0}}\mathcal{A}^{}_{221}(c)$ has the largest magnitude. For $10.6\lesssim|c|\lesssim26.2$, ${}_{{}_{0}}\mathcal{A}^{}_{421}(c)$ has the largest magnitude, and for $26.2\lesssim|c|<30$, ${}_{{}_{0}}\mathcal{A}^{}_{621}(c)$ has the largest magnitude.
• Figure 3: The first 10 eigenvalues for $s=-2$ and $m=2$ along a sequence of values of $c$ obtained from the QNM mode sequence $\{2,2,0\}$. The plot displays $|{}_{{}_{-2}}A^{}_{\ell2}(c(a))-2|$ so that the eigenvalues appear in their sorted order. The values are displayed as functions of the dimensionless angular momentum $a/M$ which is related to the oblateness parameter by $c(a)=a\omega^+_{220}(a)$, where $\omega^+_{220}(a)$ is the complex mode frequency along the QNM sequence. The eigenvalue sequence corresponding to the actual $\{2,2,0\}$ mode sequences is displayed as the thick black line.
• Figure 4: The function ${}_{{}_{-2}}S^{}_{22}(x;0.662-0.0547i)$ where the upper plot has been phase fixed using $\mathcal{P}_{\text{SL-C}}$ and the bottom using $\mathcal{P}_{\text{CZ-SL}}$. Note that there is very little difference between the two phase fixing schemes in this case.
• Figure 5: The phase difference between the functions ${}_{{}_{-2}}S^{}_{22}(x;a\omega^+_{220}(a))$ which have been phase fixed using different schemes. In the top plot, the functions have been phase fixed using $\mathcal{P}_{\text{SL-C}}$ and $\mathcal{P}_{\text{CZ-SL}}$. In the bottom plot, the functions have been phase fixed using $\mathcal{P}_{\text{SL-C}}$ and $\mathcal{P}_{\text{SL-Ind}}$.