Teukolsky Master Equation

The Teukolsky master equation decouples massless integer-spin fields on a fixed Kerr background into one spin-weighted scalar equation. The differential operator alone is not a complete specification: a usable calculation must also fix the principal tetrad, master variable, Fourier phase, separation constant, projected source, and boundary branches. This record gives that complete package for the subextremal exterior.

Kerr and Kinnersley data

Use Boyer–Lindquist coordinates $(t,r,\theta,\phi)$, signature $(-,+,+,+)$, physical mass $M$, and $a=J/M$ with

$$\Sigma=r^2+a^2\cos^2\theta, \qquad \Delta=r^2-2GMr+a^2, \qquad \lvert a\rvert\lt GM,\quad r\gt r_+.$$

Order the Kinnersley tetrad as $(\ell^a,n^a,m^a,\bar m^a)$ and choose

$$\begin{aligned} \ell^\mu&=\left(\frac{r^2+a^2}{\Delta},1,0,\frac{a}{\Delta}\right),\\ n^\mu&=\frac{1}{2\Sigma}\left(r^2+a^2,-\Delta,0,a\right),\\ m^\mu&=\frac{1}{\sqrt2\,(r+ia\cos\theta)} \left(ia\sin\theta,0,1,\frac{i}{\sin\theta}\right). \end{aligned}$$

Thus $\ell\!\cdot n=-1$, $m\!\cdot\bar m=1$, and $g^{ab}=-2\ell^{(a}n^{b)}+2m^{(a}\bar m^{b)}$. Define

$$\rho_{NP}=-\frac{1}{r-ia\cos\theta}, \quad \Psi_0=-C_{abcd}\ell^am^b\ell^cm^d, \quad \Psi_4=-C_{abcd}n^a\bar m^bn^c\bar m^d.$$

For the integer-spin cases, the common master variable is

$s$	$\psi_s$
$+2$	$\delta\Psi_0$
$-2$	$\rho_{NP}^{-4}\delta\Psi_4$
$+1$	$\phi_0$
$-1$	$\rho_{NP}^{-2}\phi_2$
$0$	$\Phi$

Here $\phi_0,\phi_2$ are the extreme Newman–Penrose Maxwell scalars and $\Phi$ is a massless scalar. Teukolsky’s paper uses the opposite metric signature and geometrized units; the tetrad products, curvature signs, field rescalings, and the restored factors of $GM$ above are one translated package, not independent choices Teukolsky 1973, § IV, Eqs. (4.1)–(4.7) and Table 1, printed pp. 639–640, PDF.

Boyer–Lindquist master equation

Write the prepared source as $T_s$: it is a spin-weighted combination of tetrad projections and derivatives of the physical source, not a raw component of $T_{ab}$. For gravitational fields, $T_s$ is geometrized: after restoring the site’s explicit $G$, $T_{-2}=G\widehat T_{\rm phys}$, which equals the source $\widehat T$ of Sasaki and Tagoshi in their $G=1$ units. Scalar and electromagnetic sources instead retain their own declared coupling normalization. Their mostly-plus equation is ${}_s\mathcal O\psi_s=4\pi\Sigma\widehat T$, while the differential expression displayed below satisfies $\mathcal T_s=-{}_s\mathcal O$ Sasaki–Tagoshi 2003, § 2.1, Eqs. (3)–(6), printed pp. 7–8, PDF. The operator and source definitions therefore fix the paired normalization

$$\mathcal T_s[\psi_s]=-4\pi\Sigma T_s,$$

where

$$\begin{aligned} \mathcal T_s[\psi_s]={}& \left[\frac{(r^2+a^2)^2}{\Delta}-a^2\sin^2\theta\right]\partial_t^2\psi_s +\frac{4GMar}{\Delta}\,\partial_t\partial_\phi\psi_s\\ &+\left(\frac{a^2}{\Delta}-\frac{1}{\sin^2\theta}\right)\partial_\phi^2\psi_s -\Delta^{-s}\partial_r\!\left(\Delta^{s+1}\partial_r\psi_s\right)\\ &-\frac{1}{\sin\theta}\partial_\theta\!\left(\sin\theta\,\partial_\theta\psi_s\right) -2s\left[\frac{a(r-GM)}{\Delta}+\frac{i\cos\theta}{\sin^2\theta}\right]\partial_\phi\psi_s\\ &-2s\left[\frac{GM(r^2-a^2)}{\Delta}-r-ia\cos\theta\right]\partial_t\psi_s +(s^2\cot^2\theta-s)\psi_s . \end{aligned}$$

Vacuum sets $T_s=0$. A nonzero source must be accompanied by its support and tetrad projection; a time-domain problem additionally needs initial or characteristic data and a retarded, advanced, or other causal prescription. Teukolsky’s original $+---$ equation pairs the coordinate coefficients with his own Table 1 source symbol Teukolsky 1973, § IV, Eq. (4.7) and Table 1, printed pp. 640–641, PDF. That symbol must not be identified with the mostly-plus prepared source by name alone: the metric, tetrad contractions, curvature scalars, master field, and source are translated as one package.

Separation and eigenvalue dictionary

Fix the Fourier and azimuthal phase by

$$\psi_s= \int_{\mathbb R}\frac{d\omega}{2\pi} \sum_{\ell,m} R_{s\ell m\omega}(r)\, S_{s\ell m}(\theta;c)\, e^{-i\omega t+im\phi}, \qquad c=a\omega .$$

For real $c$, take $\int_0^\pi\lvert S\rvert^2\sin\theta\,d\theta=1$; the unnormalized factor $e^{im\phi}$ then has azimuthal norm $2\pi$. Define the direct angular constant $\Lambda$, the Teukolsky constant $A$, and the radial constant $\lambda$ by

$$A_{s\ell m}=\Lambda_{s\ell m}-s, \qquad \lambda_{s\ell m}=\Lambda_{s\ell m}-s+c^2-2cm.$$

The angular equation is

$$\frac{1}{\sin\theta}\frac{d}{d\theta} \left(\sin\theta\frac{dS}{d\theta}\right) +\left[ c^2\cos^2\theta-2cs\cos\theta -\frac{(m+s\cos\theta)^2}{\sin^2\theta} +\Lambda_{s\ell m}(c) \right]S=0,$$

with regularity at both poles and $\ell\ge\max(\lvert m\rvert,\lvert s\rvert)$. With $K=(r^2+a^2)\omega-am$, the matching radial equation is

$$\Delta^{-s}\frac{d}{dr} \left(\Delta^{s+1}\frac{dR}{dr}\right) +\left[ \frac{K^2-2is(r-GM)K}{\Delta} +4is\omega r-\lambda_{s\ell m} \right]R =T_{s\ell m\omega}(r).$$

For real frequency, define the radial source coefficient by the positive expansion

$$4\pi\Sigma T_s =\int_{\mathbb R}\frac{d\omega}{2\pi} \sum_{\ell,m} T_{s\ell m\omega}(r)S_{s\ell m}(\theta;c)e^{-i\omega t+im\phi}.$$

Equivalently, because the polar harmonic has unit norm and $e^{im\phi}$ has norm squared $2\pi$,

$$T_{s\ell m\omega}(r) =\frac{1}{2\pi}\int_{\mathbb R}dt\int d\Omega\, e^{i\omega t-im\phi}S_{s\ell m}^*(\theta;c)\, 4\pi\Sigma T_s(t,r,\theta,\phi).$$

Substitution then gives $\mathcal T_s[RSe^{-i\omega t+im\phi}] =-e^{-i\omega t+im\phi}(S\mathcal L_r[R]+R\mathcal A_\theta[S])$, so the angular equation $\mathcal A_\theta[S]=0$ leaves the displayed radial equation with the positive source $T_{s\ell m\omega}$. These separated equations and their shared constant follow from Teukolsky 1973, § IV, Eqs. (4.8)–(4.10), printed p. 641, PDF; the gravitational specialization and explicit source projection are in Sasaki–Tagoshi 2003, § 2.1, Eqs. (8)–(13), printed pp. 8–9, PDF.

Horizon and infinity branches

Let

$$r_\pm=GM\pm\sqrt{(GM)^2-a^2}, \qquad \frac{dr_*}{dr}=\frac{r^2+a^2}{\Delta}, \qquad \Omega_H=\frac{a}{r_+^2+a^2}, \quad k_H=\omega-m\Omega_H .$$

For the Kinnersley-normalized radial function and ordinary nonzero characteristic frequencies, the four local homogeneous branches are

$$\begin{array}{c|c} r\to r_+ & R_H^{\rm in}\sim\Delta^{-s}e^{-ik_Hr_*}, \qquad R_H^{\rm out}\sim e^{+ik_Hr_*}\\[2pt] r\to\infty & R_\infty^{\rm in}\sim r^{-1}e^{-i\omega r_*}, \qquad R_\infty^{\rm out}\sim r^{-2s-1}e^{+i\omega r_*}. \end{array}$$

A retarded sourced solution selects $R_H^{\rm in}$ and $R_\infty^{\rm out}$ together with the retarded frequency contour. A quasinormal mode imposes the same directions on a homogeneous solution at a discrete complex frequency and on a declared analytic sheet; scattering instead retains an incident branch. Teukolsky gives the Frobenius powers, phases, and physical branch selection in Teukolsky 1973, § V, Eqs. (5.1)–(5.8), (5.10)–(5.11), printed pp. 642–643, PDF, while the $s=-2$ in/up solutions and retarded Green function are given in Sasaki–Tagoshi 2003, § 2.1, Eqs. (14)–(20), printed p. 9, PDF.

The Boyer–Lindquist Kinnersley tetrad is itself singular at $\Delta=0$. Horizon regularity must therefore be checked after passing to ingoing Kerr coordinates and a regular spin-boosted tetrad; the displayed powers of $\Delta$ are frame weights, not by themselves curvature singularities.

Formula-level checks and scope

• At $s=0$, the homogeneous operator equals $-\Sigma\Box\Phi$ in the site’s signature.
• At $c=0$, $\Lambda_{s\ell m}=\ell(\ell+1)-s^2$ and $A_{s\ell m}=\ell(\ell+1)-s(s+1)$; for $s=-2$, $\lambda=(\ell-1)(\ell+2)$.
• At $a\to0$, $\Delta\to r(r-2GM)$ and $K\to r^2\omega$, giving the Schwarzschild spin-weighted equation; obtaining a Regge–Wheeler variable still requires a field transformation.
• Acting with $\chi_H=\partial_t+\Omega_H\partial_\phi$ on the fixed mode phase gives $-ik_H$, guarding the sign in both horizon exponentials.
• For $s=-2$, the selected retarded powers become $\Delta^2e^{-ik_Hr_*}$ at the horizon and $r^3e^{+i\omega r_*}$ at infinity.
• For two homogeneous radial solutions, $W_s=\Delta^{s+1}(R_1R_2'-R_2R_1')$ is constant. Failure of this check usually signals a mismatched radial rescaling or separation constant.

This card assumes supported massless integer-spin fields ($s=0,\pm1,\pm2$) on a fixed, subextremal vacuum Kerr geometry. Extremality, $\omega=0$, $k_H=0$, massive fields, Kerr–Newman, and complex-frequency angular normalization need separate limiting or adjoint prescriptions. Extreme Weyl scalars also omit gauge and Kerr-parameter completion data, so the equation alone does not reconstruct a complete metric perturbation.

For the corresponding full treatments, see Kinnersley Tetrads and Newman–Penrose Curvature, Derivation of the Teukolsky Equation, Radial Teukolsky Equation and Physical Solutions, Angular Teukolsky Equation and Separation Constants, and Teukolsky Sources and Point-Particle Coupling.

References

• Sasaki, Misao, and Hideyuki Tagoshi. “Analytic Black Hole Perturbation Approach to Gravitational Radiation.” Living Reviews in Relativity 6 (2003): article 6. DOI. Open PDF.
• Teukolsky, Saul A. “Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations.” Astrophysical Journal 185 (1973): 635–647. DOI. ADS PDF.