Radial Teukolsky Equation and Physical Solutions

The separated radial Teukolsky equation becomes a physical problem only after its basis solutions, asymptotic phases, and flux normalization are declared. “Ingoing” and “upgoing” name global solutions by what is absent at a past boundary, not merely by the sign of a local exponential. This page fixes those meanings for $e^{-i\omega t+im\phi}$, derives the constant weighted Wronskian, and distinguishes retarded scattering data from quasinormal and bound-state conditions.

Required background. Frames, Coframes, and Anholonomy supplies the tetrad weights needed to interpret horizon asymptotics.

Helpful background. Black-Hole Geometry and Mechanics supplies Kerr horizon generators; PDEs and Green Functions for Gravitation supplies Green functions, Wronskians, and boundary-value problems; Formulations of General Relativity supplies characteristic fluxes and constraints.

Radial operator and characteristic frequencies

In the subextremal Kerr exterior define

$$\Delta=(r-r_+)(r-r_-), \qquad r_\pm=GM\pm\sqrt{(GM)^2-a^2},$$

and use the radial equation

$$\Delta^{-s}\frac{d}{dr} \left(\Delta^{s+1}\frac{dR}{dr}\right) +\left[ \frac{K^2-2is(r-GM)K}{\Delta} +4is\omega r-\lambda \right]R=T(r),$$

where $K=(r^2+a^2)\omega-am$ and $\lambda=A_{s\ell m}+a^2\omega^2-2am\omega$. The field convention and projected source normalization are those fixed in the derivation page; rescaling $R$ rescales every amplitude below.

Choose a tortoise coordinate by

$$\frac{dr_*}{dr}=\frac{r^2+a^2}{\Delta}.$$

Then $r_*\to-\infty$ at the future event horizon and $r_*\to+\infty$ at spatial infinity. The horizon generator is $\chi_H=\partial_t+\Omega_H\partial_\phi$, with $\Omega_H=a/(r_+^2+a^2)$, so the horizon-frame frequency is

$$k_H=\omega-m\Omega_H.$$

The factor $e^{-ik_Hr_*}$ is ingoing at the future horizon when combined with $e^{-i\omega t+im\phi}$ and a horizon-regular angular coordinate. Powers of $\Delta$ additionally depend on $s$ and on whether the Kinnersley or a regular tetrad component is used. Teukolsky gives these Frobenius behaviors and their physical interpretation at both boundaries Teukolsky 1973, § V, Eqs. (5.1)–(5.8), (5.10)–(5.11), printed pp. 642–643.

In and up bases

The radial powers can be derived rather than memorized. Put $d=r_+-r_-$, $D_H=r_+^2+a^2$ and $x=r-r_+$. Near the horizon, $\Delta=dx+O(x^2)$ and $r_*=(D_H/d)\log x+O(1)$. Inserting $R\sim x^p$ in the homogeneous equation gives, with $b_H=D_Hk_H/d$,

$$p(p+s)+b_H^2-isb_H=(p-ib_H)(p+s+ib_H)=0.$$

The roots give $e^{ik_Hr_*}$ and $\Delta^{-s}e^{-ik_Hr_*}$. At infinity, insert $R=e^{\eta i\omega r_*}r^p$, where $\eta=\pm1$. The first uncancelled coefficient, after division by $rR$, is $2i\omega[\eta(p+s+1)+s]$. Thus the incoming power is $p=-1$ and the outgoing power is $p=-2s-1$.

For real nonzero $\omega$ away from thresholds, fix an “in” solution by

$$R^{\mathrm{in}}\sim \begin{cases} B^{\mathrm{trans}}\Delta^{-s}e^{-ik_Hr_*},&r\to r_+,\\ B^{\mathrm{inc}}r^{-1}e^{-i\omega r_*} +B^{\mathrm{ref}}r^{-2s-1}e^{+i\omega r_*},&r\to\infty, \end{cases}$$

and an “up” solution by

$$R^{\mathrm{up}}\sim \begin{cases} C^{\mathrm{up}}e^{+ik_Hr_*} +C^{\mathrm{ref}}\Delta^{-s}e^{-ik_Hr_*},&r\to r_+,\\ C^{\mathrm{trans}}r^{-2s-1}e^{+i\omega r_*},&r\to\infty. \end{cases}$$

For $s=-2$, these reduce to the familiar $\Delta^2e^{-ik_Hr_*}$ transmitted factor and $r^3e^{+i\omega r_*}$ outgoing factor. Sasaki and Tagoshi give exactly this gravitational normalization and use it to construct the retarded sourced solution Sasaki–Tagoshi 2003, § 2.1, Eqs. (14)–(20), printed p. 9. Different rescaled variables have different powers; phases and fluxes, after reconstruction, are the invariant comparison.

The names describe causal scattering bases. $R^{\mathrm{in}}$ has no wave emerging from the past horizon; $R^{\mathrm{up}}$ has no wave incident from past null infinity. Teukolsky and Press emphasize that opposite-spin representations can be mathematically better conditioned in different asymptotic regions Teukolsky–Press 1974, §§II–IV, pp. 444–453.

Wronskian and Green function

Call the zeroth-order bracket in the raw radial equation $U_s$. Multiplying the homogeneous equation by $\Delta^s$ puts its derivative part in Sturm–Liouville form:

$$\frac{d}{dr}\left(\Delta^{s+1}\frac{dR}{dr}\right) +\Delta^s U_sR=0.$$

For any two solutions $R_1,R_2$ of the same $(s,\ell,m,\omega)$ equation,

$$W_s[R_1,R_2] =\Delta^{s+1} \left(R_1\frac{dR_2}{dr}-R_2\frac{dR_1}{dr}\right)$$

is independent of $r$. Differentiate it, substitute both equations, and the potential terms cancel. This weighted Wronskian is the normalization invariant needed for variation of parameters; the unweighted expression is generally not constant.

Let $W=W_s[R^{\mathrm{in}},R^{\mathrm{up}}]\ne0$ and, for definiteness, take $T$ to have compact radial support strictly inside the exterior. Distributions with that support are allowed. The ingoing/outgoing radial solution is

$$\begin{aligned} R(r)=\frac1W\bigg[& R^{\mathrm{up}}(r)\int_{r_+}^{r}R^{\mathrm{in}}(r')\Delta(r')^sT(r')\,dr'\\ &+R^{\mathrm{in}}(r)\int_{r}^{\infty}R^{\mathrm{up}}(r')\Delta(r')^sT(r')\,dr' \bigg]. \end{aligned}$$

The factors of $\Delta^s$ follow from the exact source normalization; importing a formula for $\Delta^{-2}T$ without translating the equation changes the amplitude. Outside the source the solution is outgoing at infinity and ingoing at the horizon. Noncompact sources require separate convergence and boundary hypotheses. The retarded time-domain field is obtained by causal Fourier–Laplace inversion, continuing the resolvent from the upper half-plane (the $\omega+i0$ prescription where its boundary value exists); a bare real-frequency formula does not select that contour.

Flux normalization and Starobinsky identities

A raw Teukolsky amplitude is not itself energy flux. We now make the conversion explicit for a vacuum gravitational in-scattering mode with real $\omega>0$, $k_H\ne0$, $\ell\ge2$ and regular angular eigenfunctions. The rates are quadratic, cycle-averaged Killing-energy fluxes; an eternal monochromatic wave does not have finite total radiated energy. Keep the chapter’s angular normalization:

$$\delta\Psi_4=\rho_{NP}^4 R_{-2}\,{}_{-2}S(\theta) e^{-i\omega t+im\phi}, \qquad \int_0^\pi |{}_sS|^2\sin\theta\,d\theta=1.$$

There is no $1/\sqrt{2\pi}$ in this separation. Hence the squared norm of $S e^{im\phi}$ on the full sphere is $2\pi$. The complex curvature field encodes two real strains; do not take its real part as though it were one real scalar and insert an extra factor $1/2$ into the following rates.

Opposite-spin amplitudes

Write $Z_{\rm in}=B^{\rm inc}$, $Z_{\rm out}=B^{\rm ref}$ and $Z_H=B^{\rm trans}$ for the $s=-2$ coefficients. For the companion physical $\delta\Psi_0=R_{+2}\,{}_{+2}S e^{-i\omega t+im\phi}$, let $Y_{\rm in}$, $Y_{\rm out}$ and $Y_H$ multiply the respective branches $r^{-1}e^{-i\omega r_*}$, $r^{-5}e^{i\omega r_*}$ and $\Delta^{-2}e^{-ik_Hr_*}$. Put $\mu=GM$ and

$$\lambda=\lambda_{-2},\qquad \lambda_{+2}=\lambda-4, \qquad \kappa_H=\frac{r_+-r_-}{2D_H}, \qquad \epsilon_H=\frac{\kappa_H}{2}.$$

Choose the relative angular phases used by Teukolsky and Press. The Starobinsky constant is then specified by

$$\begin{aligned} |\mathcal C|^2={}&[(\lambda+2)^2+4am\omega-4a^2\omega^2] [\lambda^2+36am\omega-36a^2\omega^2]\\ &+(2\lambda+3)(96a^2\omega^2-48am\omega) +144\omega^2(\mu^2-a^2),\\ \operatorname{Im}\mathcal C={}&12\mu\omega,\\ \operatorname{Re}\mathcal C={}&+\sqrt{|\mathcal C|^2-(12\mu\omega)^2}. \end{aligned}$$

The angular adjoint identity makes the radicand nonnegative for the regular real-frequency eigenfunction; $\operatorname{Im}\mathcal C\ne0$ makes $\mathcal C$ nonzero in this sector. Notice the $\mu^2$ in the last term: the original printed Eq. (3.23) omits the mass square. The dimensionally consistent expression is printed in Sasaki–Tagoshi 2003, § 5.6, Eq. (176), p. 41, PDF.

Define $\mathcal D_\mp=\partial_r\mp iK/\Delta$. For the physical-field radial functions, the complete fourth-order identities are

$$\mathcal D_-^4R_{-2}=\frac{\mathcal C}{4}R_{+2}, \qquad \Delta^2\mathcal D_+^4(\Delta^2R_{+2})=4\bar{\mathcal C}R_{-2}.$$

These identities relate the same physical perturbation, not independently normalized arbitrary solutions. Teukolsky and Press’s intermediate negative-spin radial function is multiplied by $\mathcal C$ when forming their physical field; that factor has already been absorbed into our $R_{-2}$ Teukolsky–Press 1974, § III, Eqs. (3.22), (3.28)–(3.30), pp. 447–448, PDF. Applying the identities to the endpoint expansions gives

$$\begin{aligned} \mathcal C Y_{\rm in}&=64\omega^4Z_{\rm in},\\ 4\omega^4Y_{\rm out}&=\bar{\mathcal C}Z_{\rm out},\\ \mathcal C Y_H&=64D_H^4ik_H(k_H^2+4\epsilon_H^2) (4\epsilon_H-ik_H)Z_H. \end{aligned}$$

For example, $\mathcal D_-$ acting on the incoming infinity phase has leading factor $-2i\omega$; its fourth power and the outer factor four give $64\omega^4$. Near the horizon the four factors acting on $x^{2-iD_Hk_H/d}$ are $2-j-2iD_Hk_H/d$ for $j=0,1,2,3$. Their product gives the last line. The outgoing relation follows most directly from the second fourth-order identity Teukolsky–Press 1974, Eqs. (4.10)–(4.11), (4.42), pp. 449, 451, PDF.

Physical rates and superradiance

At infinity use retarded time $u_{\rm ret}=t-r_*$. In the Kinnersley frame the outgoing curvature obeys $\Psi_4=\partial_{u_{\rm ret}}^2(h_+-ih_\times)/2$. Integrating $\langle|\partial_{u_{\rm ret}}(h_+-ih_\times)|^2\rangle r^2/(16\pi G)$ over the sphere therefore gives the reflected rate below. The incoming Kinnersley curvature has the corresponding coefficient two, rather than one half. At the horizon one instead uses the regular-tetrad shear and the area balance. The symmetric null frame in Metric Reconstruction and Radiative Observables instead gives an outgoing coefficient one. Its boost to the asymptotic Kinnersley frame multiplies the outgoing null leg by $\sqrt2$ and the ingoing leg by $1/\sqrt2$, halving $\Psi_4$; physical strain and power do not change. With the endpoint conversions already applied, the rates are

$$\begin{aligned} \mathcal F_{\rm inc} &=\frac{|Y_{\rm in}|^2}{32G\omega^2} =\frac{128\omega^6}{G|\mathcal C|^2}|B^{\rm inc}|^2,\\ \mathcal F_{\rm ref} &=\frac{|B^{\rm ref}|^2}{2G\omega^2},\\ \mathcal F_H &=\frac{\omega|Y_H|^2} {32Gk_H(k_H^2+4\epsilon_H^2)D_H^3}\\ &=\frac{128D_H^5\omega k_H(k_H^2+4\epsilon_H^2)(k_H^2+16\epsilon_H^2)} {G|\mathcal C|^2}|B^{\rm trans}|^2. \end{aligned}$$

Infinity reflection is positive outward; horizon transmission is positive inward. The horizon expression includes the boost from the Kinnersley to the regular Hawking–Hartle tetrad and the shear response; it is not a raw Weyl amplitude squared Teukolsky–Press 1974, Eqs. (4.12)–(4.16), (4.38)–(4.44), pp. 449–451, PDF. If instead one uses the unit-sphere harmonic $\mathbb Y_s=S_s e^{im\phi}/\sqrt{2\pi}$, the radial coefficients become $\mathcal B=\sqrt{2\pi}B$. The familiar expression $|\mathcal B^{\rm ref}|^2/(4\pi G\omega^2)$ is then the same reflected rate, not a competing normalization.

The horizon rate has the sign of $k_H$ for $\omega>0$. Moreover,

$$\mathcal F_{J,H}=\frac m\omega\mathcal F_H, \qquad \mathcal F_\chi=\mathcal F_H-\Omega_H\mathcal F_{J,H} =\frac{k_H}{\omega}\mathcal F_H\ge0.$$

Thus $0\lt\omega\lt m\Omega_H$ permits negative $\partial_t$ Killing-energy flux into the hole while the generator flux stays nonnegative. This is not a negative local energy measured by every horizon-crossing observer.

The conserved relation proving balance

The unconjugated same-spin Wronskian used for the Green function is not itself an energy current. To obtain the physical balance, set

$$X_s=\Delta^{s/2}\sqrt{r^2+a^2}\,R_s, \qquad g_s=\frac{s(r-\mu)}{r^2+a^2} +\frac{r\Delta}{(r^2+a^2)^2}.$$

Direct substitution gives $\partial_{r_*}^2X_s+V_sX_s=0$, where

$$V_s=\frac{K^2-2is(r-\mu)K+\Delta(4is\omega r-\lambda_s)} {(r^2+a^2)^2}-g_s^2-\partial_{r_*}g_s.$$

For real frequency and $\lambda_{+2}=\lambda_{-2}-4$, these potentials satisfy $V_{+2}=\bar V_{-2}$. Consequently

$$\mathscr W=X_{+2}\partial_{r_*}\bar X_{-2} -\bar X_{-2}\partial_{r_*}X_{+2}$$

is constant: differentiation cancels the two identical potentials. Its endpoint values, in the physical amplitude convention above, are

$$\begin{aligned} \mathscr W_\infty &=2i\omega(Y_{\rm in}\bar Z_{\rm in} -Y_{\rm out}\bar Z_{\rm out}),\\ \mathscr W_H&=2D_H(4\epsilon_H+ik_H)Y_H\bar Z_H. \end{aligned}$$

Substitute the three Starobinsky amplitude relations and multiply the equality by $\omega/(iG\bar{\mathcal C})$. Every coefficient is then a physical flux coefficient:

$$\mathcal F_{\rm inc}-\mathcal F_{\rm ref}=\mathcal F_H.$$

This is the radial proof of gravitational scattering balance Teukolsky–Press 1974, § V, Eqs. (5.2)–(5.8), p. 452, PDF. For $B^{\rm inc}\ne0$, define energy reflection and transmission fractions:

$$\begin{aligned} \mathcal R_E&=\frac{|\mathcal C|^2}{256\omega^8} \left|\frac{B^{\rm ref}}{B^{\rm inc}}\right|^2,\\ \mathcal T_E&=\frac{D_H^5 k_H(k_H^2+4\epsilon_H^2)(k_H^2+16\epsilon_H^2)} {\omega^5}\left|\frac{B^{\rm trans}}{B^{\rm inc}}\right|^2,\\ \mathcal R_E+\mathcal T_E&=1. \end{aligned}$$

Superradiance means $\mathcal T_E\lt0$ and $\mathcal R_E\gt1$ when transmission is nonzero. The unequal dimensions of the raw coefficients provide a quick warning against identifying amplitude ratios with energy fractions: since $R_{-2}$ has dimension $L^2$, the dimensions of $B^{\rm inc},B^{\rm ref},B^{\rm trans}$ are respectively $L^3,L^{-1},L^{-2}$.

Other physical domains

Boundary conditions define different problems on the same ODE:

• A retarded sourced solution is future-horizon ingoing and future-null-infinity outgoing, with the frequency contour fixed by causal continuation.
• A quasinormal mode satisfies the same homogeneous boundary directions but only at discrete complex $\omega$ where $W(\omega)=0$ on the stated analytic sheet.
• A scattering solution at real $\omega$ retains an incident component at one past boundary and measures reflected and transmitted flux.
• A massive bound or quasibound state would require a different field equation and an exponentially decaying infinity condition; the massless Teukolsky equation written here does not acquire that condition by relabeling its solutions.

Thresholds $\omega=0$ and $k_H=0$, extremality $r_+=r_-$, and complex frequency change the Frobenius structure and must be treated separately. The formulas above assume subextremality and ordinary nonzero characteristic frequencies.

Exercises

For $s=0$ and real $\omega>0$, suppose $R^{\mathrm{in}}\sim(e^{-i\omega r_*}+\mathcal R e^{i\omega r_*})/r$ at infinity and $R^{\mathrm{in}}\sim\mathcal T e^{-ik_Hr_*}$ at the horizon. Derive the constant-current relation and decide when reflected flux exceeds incident flux. Repeat the normalization using $u=\sqrt{r^2+a^2}R$.

Solution

For $s=0$ the potential is real at real $\omega$, so the conserved current is

$$J=\frac{\Delta}{2i} \left(R^*\frac{dR}{dr}-R\frac{dR^*}{dr}\right).$$

Since $\Delta\,d/dr=(r^2+a^2)d/dr_*$, the infinity $1/r$ falloff cancels the leading $r^2$ factor there. The incident/reflected cross terms cancel in the current, giving

$$J_\infty=\omega(|\mathcal R|^2-1),$$

where outward radial orientation is positive. At the horizon,

$$J_H=-D_Hk_H|\mathcal T|^2.$$

Constancy gives

$$1-|\mathcal R|^2 =\frac{D_Hk_H}{\omega}|\mathcal T|^2.$$

The horizon area factor is not a common factor that can be cancelled at both ends. For $u=\sqrt{r^2+a^2}R$, direct differentiation instead gives $J=\operatorname{Im}(\bar u\partial_{r_*}u)$ exactly. Its horizon amplitude is $\mathcal T_u=\sqrt{D_H}\mathcal T$, so the same result is $1-|\mathcal R|^2=(k_H/\omega)|\mathcal T_u|^2$. If $k_H\lt0$ and transmission is nonzero, then $|\mathcal R|^2\gt1$. For $s=\pm2$ the complex same-spin potential precludes this scalar-current argument; the opposite-spin conversion above is essential.

Set $a=0$ in the gravitational flux conversion. Show that the Starobinsky constant has the expected spherical polynomial and explain why the raw reflection amplitude still is not an energy fraction.

Solution

Now $\lambda=(\ell-1)(\ell+2)$ and $\lambda+2=\ell(\ell+1)$, so

$$|\mathcal C|^2=[(\ell-1)\ell(\ell+1)(\ell+2)]^2 +144(GM)^2\omega^2.$$

It is nonzero for $\ell\ge2$. The ratio $|B^{\rm ref}/B^{\rm inc}|^2$ has dimension $L^{-8}$; the factor $|\mathcal C|^2/(256\omega^8)$ is still needed to form $\mathcal R_E$. Since $\Omega_H=0$, $k_H=\omega>0$ and $\mathcal T_E\ge0$, giving $\mathcal R_E\le1$ as expected for Schwarzschild scattering.

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.
• Teukolsky, Saul A., and William H. Press. “Perturbations of a Rotating Black Hole. III. Interaction of the Hole with Gravitational and Electromagnetic Radiation.” Astrophysical Journal 193 (1974): 443–461. ADS record and PDF.