Derivation of the Teukolsky Equation

The Teukolsky equation is remarkable because a rotating black hole admits a single decoupled equation for each extreme-spin curvature component even though individual metric components remain coupled. Decoupling is not a generic feature of curved backgrounds: it uses vacuum Petrov type D, a principal null tetrad, and the background Newman–Penrose identities in a precise way. This page exposes that mechanism, gives the translated master operator, and separates it using the fixed phase $e^{-i\omega t+im\phi}$.

Required background. Frames, Coframes, and Anholonomy supplies null tetrads, connection coefficients, and frame gauge freedom.

Helpful background. Black-Hole Geometry and Mechanics supplies the Kerr exterior; PDEs and Green Functions for Gravitation supplies operator domains and characteristic data; Formulations of General Relativity supplies linearized equations, gauge, and constraints.

This derivation continues from Kinnersley Tetrads and Newman–Penrose Curvature. Its tetrad normalization, spin coefficients, type-D curvature identities, and $\rho_{NP}$ convention are assumed below.

Why the extreme components decouple

Let $(\ell,n,m,\bar m)$ be the principal Kinnersley tetrad with $\ell\!\cdot n=-1$ and $m\!\cdot\bar m=1$. Here $\ell^a$ is a tetrad vector; the same glyph later labels the angular quantum number, whose meaning is fixed by its subscript or summation context. On the Kerr background,

$$\kappa=\sigma=\lambda_{\mathrm{NP}}=\nu=0, \qquad \Psi_0=\Psi_1=\Psi_3=\Psi_4=0,$$

while $\Psi_2\ne0$. We will eliminate three auxiliary perturbations from two Bianchi identities and one Ricci identity. This does not set those perturbations to zero. The calculation uses the vacuum equations, type D and principal alignment; generic matter backgrounds or generic tetrads do not inherit its cancellation.

The three starting equations

Write $\phi_j=\delta\Psi_j$, $\kappa_1=\delta\kappa$ and $\sigma_1=\delta\sigma$ for variations. In the equations below, however, $\delta=m^a\nabla_a$ is the Newman–Penrose directional derivative; $D=\ell^a\nabla_a$ and $\Delta_{\mathrm{NP}}=n^a\nabla_a$. All spin coefficients and $\Psi_2$ on their right-hand sides are background quantities. For compactness set $\rho=\rho_{NP}$. In the connection and Weyl conventions of the preceding page, the three linearized identities are

$$\begin{aligned} (\bar\delta+\pi-4\alpha)\phi_0 -(D-4\rho-2\epsilon)\phi_1-3\Psi_2\kappa_1&=0,\\ (\Delta_{\mathrm{NP}}-4\gamma+\mu_{\mathrm{NP}})\phi_0 -(\delta-4\tau-2\beta)\phi_1-3\Psi_2\sigma_1&=0,\\ (D-\rho-\bar\rho-3\epsilon+\bar\epsilon)\sigma_1 -(\delta-\tau+\bar\pi-\bar\alpha-3\beta)\kappa_1 +\phi_0&=0. \end{aligned}$$

These are the vacuum identities of Teukolsky 1973, § II, Eqs. (2.6)–(2.8), pp. 637–638, PDF, translated to the site’s Weyl contractions. In particular, $\Psi_2=GM/(r-ia\cos\theta)^3$ is the negative of Teukolsky’s background Weyl scalar. That translation changes the Ricci identity’s curvature sign as well as the signs multiplying $\kappa_1$ and $\sigma_1$.

Weighted differentiation and cancellation

Introduce six first-order operators; products mean composition, with the rightmost operator acting first:

$$\begin{aligned} A&=D-3\epsilon+\bar\epsilon-4\rho-\bar\rho,\\ B&=\delta+\bar\pi-\bar\alpha-3\beta-4\tau,\\ C&=\Delta_{\mathrm{NP}}-4\gamma+\mu_{\mathrm{NP}}, &E&=\bar\delta+\pi-4\alpha,\\ P&=D-4\rho-2\epsilon, &Q&=\delta-4\tau-2\beta. \end{aligned}$$

The background equations $D\Psi_2=3\rho\Psi_2$ and $\delta\Psi_2=3\tau\Psi_2$, together with the product rule, turn the third identity into

$$A(\Psi_2\sigma_1)-B(\Psi_2\kappa_1)+\Psi_2\phi_0=0.$$

The other ingredient is the weighted type-D commutator. For constant $p,q$, its complete operator form is

$$\begin{aligned} &[D-(p+1)\epsilon+\bar\epsilon+q\rho-\bar\rho] (\delta-p\beta+q\tau)\\ &\quad-[\delta-(p+1)\beta-\bar\alpha+\bar\pi+q\tau] (D-p\epsilon+q\rho)=0. \end{aligned}$$

Taking $p=2,q=-4$ gives $AQ-BP=0$. Apply $A$ to the second Bianchi identity and subtract $B$ times the first. The result is

$$\begin{aligned} 0={}&(AC-BE)\phi_0-(AQ-BP)\phi_1\\ &-3\{A(\Psi_2\sigma_1)-B(\Psi_2\kappa_1)\}. \end{aligned}$$

The commutator removes $\phi_1$; the weighted Ricci identity replaces the braces by $-\Psi_2\phi_0$. We have obtained a closed equation,

$$(AC-BE+3\Psi_2)\phi_0=0.$$

This is the mechanism behind decoupling: the background curvature supplies exactly the weights that let the unwanted derivatives cancel Teukolsky 1973, § II, Eqs. (2.9)–(2.12), p. 638, PDF. Substituting the explicit Kinnersley vectors and connection coefficients gives the full coordinate operator below, including its zeroth-order term:

$$\mathcal T_{+2}=2\Sigma(AC-BE+3\Psi_2).$$

This equality includes derivatives of the operator coefficients, not just the principal part. The opposite extreme follows by the tetrad-prime interchange in the Newman–Penrose equations. Expressing it with the same coordinate operator also requires the $\rho_{NP}^{-4}$ rescaling below; it is not a bare change $s\mapsto-s$.

A short check isolates the curvature sign. At $a=0$, act on the constant test field $1$; this is an operator test, not a physical spin-two solution:

$$\begin{aligned} AC[1]&=-\frac2{r^2}-\frac{3GM}{r^3}, &BE[1]&=-\frac{1+2\cot^2\theta}{r^2},\\ 2r^2(AC-BE+3\Psi_2)[1]&=4\cot^2\theta-2. \end{aligned}$$

The result is precisely the coordinate operator’s $s=2$ zeroth-order term. Using $-3\Psi_2$ instead would leave an erroneous $-12GM/r$ contribution.

Field variable and source convention

In the site’s signature, define the master field by spin weight

$$\psi_s= \begin{cases} \delta\Psi_0, & s=+2,\\ \rho_{NP}^{-4}\delta\Psi_4, & s=-2, \end{cases}$$

where $\rho_{NP}=-(r-ia\cos\theta)^{-1}$. The rescaling in the negative-spin variable makes the two cases share one coordinate operator. Electromagnetic and scalar fields fit the same operator at $s=\pm1$ and $s=0$, with different field and source definitions.

Let $T_s$ denote the Teukolsky source after tetrad projection and differential preparation, not a raw stress–energy component. For $s=-2$, it is the geometrized source $T_{-2}=G\widehat T_{\rm phys}$ built from the physical stress tensor; this is simply $\widehat T$ in the $G=1$ convention of Sasaki and Tagoshi. Scalar and electromagnetic sources retain their separately declared coupling normalization. Their operator obeys ${}_s\mathcal O=-\mathcal T_s$ for the coordinate expression used here, so their equation ${}_s\mathcal O\psi_s=4\pi\Sigma\widehat T$ becomes

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

The coordinate operator has the opposite sign to Sasaki–Tagoshi 2003, § 2.1, Eqs. (4)–(5), p. 7, PDF. Their source shorthand must also include the rescaling required by the master field. The complete ordered preparation and a signed circular-orbit check are given on Teukolsky Sources and Point-Particle Coupling. Teukolsky’s original master equation instead uses $+---$, geometrized units and its own Table 1 source symbol Teukolsky 1973, § IV, Eq. (4.7) and Table 1, pp. 640–641, PDF. Compare amplitudes only after translating the signature, master field, prepared source, Fourier measure and harmonic normalization together. The vacuum elimination above establishes the left-hand operator, not the matter-source coupling by itself.

Coordinate master operator

With $\Sigma=r^2+a^2\cos^2\theta$ and $\Delta=r^2-2GMr+a^2$, the source-normalized equation is

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

At $s=0$, every term linear or quadratic in $s$ vanishes, and multiplying the scalar wave equation $\Box\Phi=0$ by $-\Sigma$ gives this homogeneous operator. This scalar reduction is the cleanest sign check because it does not depend on Weyl-scalar conventions. The principal part is the scalar Kerr wave operator for every $s$; lower-order complex terms encode spin-curvature coupling.

Separation into radial and angular equations

For a stationary, axisymmetric background use

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

Define

$$K=(r^2+a^2)\omega-am, \qquad \lambda_{\mathrm{sep}} =A_{s\ell m}(a\omega)+a^2\omega^2-2am\omega,$$

where the angular eigenvalue convention is fixed by the next displayed equation. Define the radial source by the positive expansion

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

For one mode, direct substitution into the coordinate operator gives

$$\mathcal T_s\!\left[RSe^{-i\omega t+im\phi}\right] =-e^{-i\omega t+im\phi} \left(S\mathcal L_r[R]+R\mathcal A_\theta[S]\right).$$

Thus $\mathcal A_\theta[S]=0$ and the four-dimensional minus sign produce the positive radial source below:

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

and

$$\frac{1}{\sin\theta}\frac{d}{d\theta} \left(\sin\theta\frac{dS}{d\theta}\right) +\left[ a^2\omega^2\cos^2\theta-2a\omega s\cos\theta -\frac{(m+s\cos\theta)^2}{\sin^2\theta} +s+A_{s\ell m} \right]S=0.$$

Regularity at both poles quantizes $A_{s\ell m}$ for integer $m$, $\ell\ge\max(|m|,|s|)$. The radial domain and its horizon/infinity conditions are separate information; separation alone does not choose a retarded, advanced, scattering, or quasinormal solution. Sasaki and Tagoshi give the corresponding $s=-2$ operator, source projection, positive radial equation, and Green-function normalization Sasaki–Tagoshi 2003, § 2.1, Eqs. (4)–(20), pp. 7–9, PDF. The compact Teukolsky Master Equation records the same convention package for lookup.

Adjoint structure and limits

The $s$ and $-s$ equations are linked by weighted adjoint relations rather than naive complex conjugation. Let $\mathcal E$ be the linearized Einstein operator, $\mathcal C_s$ curvature extraction and $\mathcal P_s$ source preparation. Their decoupling identity and its formal adjoint are

$$\begin{aligned} \mathcal T_s\mathcal C_s&=\mathcal P_s\mathcal E,\\ \mathcal E^\dagger\mathcal P_s^\dagger &=\mathcal C_s^\dagger\mathcal T_s^\dagger. \end{aligned}$$

For the Einstein Hessian’s self-adjoint pairing, an adjoint solution $\mathcal T_s^\dagger\Phi_H=0$ therefore generates $h^{\mathrm{rec}}=2\operatorname{Re}(\mathcal P_s^\dagger\Phi_H)$ with $\mathcal E[h^{\mathrm{rec}}]=0$. This is the operator argument for Hertz reconstruction, introduced by Wald 1978, pp. 203–206. The adjoints depend on a specified volume measure and field pairing, with boundary terms removed for compactly supported tests or treated explicitly on the selected domain. The algebra does not establish a global inverse, completeness or a regular gauge.

Two limits guard the derivation. First, $s=0$ reproduces the scalar Kerr wave equation. Second, $a\to0$ gives spin-weighted spherical harmonics and, after the appropriate field transformations, the Schwarzschild radial equations. An additional exact check is an anti-linear symmetry at fixed $s$. If $(R,S)$ solve the homogeneous separated equations at $(m,\omega,A)$, then

$$\begin{aligned} \widetilde m&=-m,&\widetilde\omega&=-\bar\omega,\\ \widetilde A&=\bar A, &\widetilde\lambda_{\mathrm{sep}}&=\bar\lambda_{\mathrm{sep}},\\ \widetilde R(r)&=\overline{R(r)}, &\widetilde S(\theta)&=\overline{S(\pi-\theta)} \end{aligned}$$

solve the transformed equations. Direct substitution verifies both operators. The conjugation in $-\bar\omega$ matters for complex frequency: its imaginary part is preserved. This is an operator identity, not a boundary-condition or spectral theorem; opposite-spin relations require additional weights and the Teukolsky–Starobinsky identities.

Common pitfalls

Treating decoupling as reconstruction. Solving for one extreme Weyl scalar determines radiative curvature in the stated sector. It does not automatically supply a globally regular metric, its gauge, or the stationary mass and angular-momentum perturbations.

Comparing separation constants by symbol. Authors call different shifted quantities $A$, $E$, or $\lambda$. Here $\lambda_{\mathrm{sep}}$ is distinct from the Newman–Penrose spin coefficient $\lambda_{\mathrm{NP}}$. Compare the complete angular and radial equations, not the letter.

Letting the source float. A projected radial source includes tetrad contractions, differential operators, Fourier factors, and harmonic normalization. A missing $2\pi$, $\rho_{NP}$ power, or orientation changes amplitudes even if the homogeneous equation is correct.

Exercises

Set $s=0$ and $a=0$ in the separated equations. Show that the angular equation gives the spherical eigenvalue and reduce the radial equation to the Schwarzschild scalar potential using $u=rR$ and $dr_*/dr=f^{-1}$, where $f=1-2GM/r$.

Solution

At $s=a=0$, the angular equation becomes

$$\frac1{\sin\theta}\frac{d}{d\theta} \left(\sin\theta\frac{dS}{d\theta}\right) +\left(A_{0\ell m}-\frac{m^2}{\sin^2\theta}\right)S=0.$$

Regularity at $\theta=0,\pi$ therefore gives $A_{0\ell m}=\ell(\ell+1)$. The radial equation becomes

$$\frac{d}{dr}\left(\Delta\frac{dR}{dr}\right) +\left(\frac{r^4\omega^2}{\Delta}-\ell(\ell+1)\right)R=0, \qquad \Delta=r^2f.$$

Put $R=u/r$ and use $d/dr_*=f\,d/dr$. Straight differentiation gives

$$\frac{d^2u}{dr_*^2} +\left[\omega^2-f\left(\frac{\ell(\ell+1)}{r^2}+\frac{2GM}{r^3}\right)\right]u=0.$$

This is the standard Schwarzschild scalar equation. It simultaneously verifies the sign of the radial kinetic term, the $GM$ restoration, the eigenvalue convention, and the Fourier sign. A translated master equation that fails this reduction has not been made consistent merely by relabeling its Weyl scalar.

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.
• Wald, Robert M. “Construction of Solutions of Gravitational, Electromagnetic, or Other Perturbation Equations from Solutions of Decoupled Equations.” Physical Review Letters 41 (1978): 203–206. DOI.