Field Equations in Near-Horizon Extreme Kerr

Near-horizon extreme Kerr reorganizes a Kerr field into representations of $SL(2,\mathbb R)\times U(1)$. For a massless scalar, separation is elementary: the angular eigenvalue becomes an effective charged-$AdS_2$ mass, and the radial equation becomes a Whittaker equation. The enhanced symmetry fixes much of the local throat response, but it does not choose the boundary condition at the throat mouth or determine the amplitude seen at asymptotically flat infinity.

Helpful background. Black-Hole Geometry and Mechanics supplies extremal Kerr and its horizon; PDEs and Green Functions for Gravitation supplies separation and boundary-value problems; Formulations of General Relativity supplies scalar and Teukolsky equations; Events, Intervals, and Minkowski Geometry fixes Fourier orientation; Black-Hole Perturbation Theory supplies the Kerr mode conventions.

NHEK geometry and the separated field

Write $\mu=GM$ and begin with exact extremal Kerr, $a=\mu$ and $\kappa=0$. Before solving the field equation, scale the original Boyer–Lindquist coordinates as

$$\widetilde r=\mu(1+\lambda R),\qquad \widetilde t=\frac{2\mu}{\lambda}T,\qquad \widetilde\phi=\Phi+\frac{T}{\lambda},$$

and take $\lambda\to0$ with $(T,R,\theta,\Phi)$ fixed. With the site signature,

$$ds^2=2\mu^2\Gamma(\theta) \left[ -R^2dT^2+\frac{dR^2}{R^2}+d\theta^2 +\Lambda(\theta)^2(d\Phi+R\,dT)^2 \right],$$

where $\Gamma=(1+\cos^2\theta)/2$ and $\Lambda=2\sin\theta/(1+\cos^2\theta)$. This coordinate limit, including its corotating angular shift, is obtained in Bardeen and Horowitz 1999, § 2, pp. 3–7.

For the massless equation $\Box_g\Psi=0$, use the inherited mode phase

$$\Psi=e^{-i\widehat\omega T+im\Phi} S_{\ell m}(\theta)\mathcal R(R).$$

The throat frequency is $\widehat\omega=2\mu(\omega-m\Omega_H)/\lambda$ before the limit. Thus this field equation describes the near zone $\widetilde r-r_+=O(\lambda\mu)$ and the synchronous band $\omega-m\Omega_H=O(\lambda/\mu)$; it is not a fixed-radius, fixed-detuning approximation.

Angular and radial equations

The separation can be derived without guessing an effective potential. Put $\mathcal A=2\mu^2\Gamma$. The nonzero inverse metric components are

$$\begin{aligned} g^{TT}&=-\frac1{\mathcal A R^2},& g^{T\Phi}&=\frac1{\mathcal A R},& g^{\Phi\Phi}&=\frac{\Lambda^{-2}-1}{\mathcal A},\\ g^{RR}&=\frac{R^2}{\mathcal A},& g^{\theta\theta}&=\frac1{\mathcal A},& \sqrt{-g}&=\mathcal A^2\Lambda. \end{aligned}$$

Since $\Gamma\Lambda=\sin\theta$, the divergence formula $\Box_g\Psi=(-g)^{-1/2}\partial_a(\sqrt{-g}\,g^{ab}\partial_b\Psi)$ yields

$$\begin{aligned} \mathcal A\Box_g={}&\partial_R(R^2\partial_R) -R^{-2}\partial_T^2+2R^{-1}\partial_T\partial_\Phi\\ &+\frac1{\sin\theta}\partial_\theta(\sin\theta\,\partial_\theta) +(\Lambda^{-2}-1)\partial_\Phi^2. \end{aligned}$$

Now use $\Lambda^{-2}-1=\csc^2\theta-2+\sin^2\theta/4$ and the separated phase. Grouping the angular terms into a constant $K_{\ell m}$ gives

$$\frac{1}{\sin\theta}\frac{d}{d\theta} \left(\sin\theta\frac{dS_{\ell m}}{d\theta}\right) +\left[ K_{\ell m}-\frac{m^2}{\sin^2\theta} -\frac{m^2}{4}\sin^2\theta \right]S_{\ell m}=0$$

and

$$\frac{d}{dR}\left(R^2\frac{d\mathcal R}{dR}\right) +\left[ \frac{(\widehat\omega+mR)^2}{R^2} +m^2-K_{\ell m} \right]\mathcal R=0.$$

These equations, in the source’s equivalent NHEK normalization, appear in Amsel et al. 2009, § 3.1, Eqs. (3.1)–(3.2), p. 16 (PDF). The dictionary to the angular Teukolsky eigenvalue is

$$K_{\ell m}=A_{0\ell m}(m/2)+\frac{m^2}{4}.$$

Here $A_{0\ell m}$ is an eigenvalue, not the metric factor $\mathcal A$. For $m=0$, the angular equation is the spherical-harmonic equation and $K_{\ell0}=\ell(\ell+1)$. For $m\ne0$, the $m^2\sin^2\theta/4$ term deforms the spectrum; importing a spherical $\ell(\ell+1)$ would give the wrong radial weight.

The combination $\widehat\omega+mR$ shows the charged-$AdS_2$ structure. Azimuthal momentum $m$ acts as an effective charge coupled to the fibration $d\Phi+R\,dT$. This is a representation-theoretic reorganization of the four-dimensional field, not an additional physical electromagnetic charge.

Two elementary checks protect the separation from normalization errors. First, for $m=0$ the fibration drops out of both ordinary differential equations: the angular eigenvalue is $\ell(\ell+1)$ and the radial equation has the standard neutral-$AdS_2$ form. Second, the simultaneous dilation $R\mapsto cR$, $T\mapsto T/c$ preserves the metric and sends a separated frequency to $c\widehat\omega$; the radial equation depends only on $\widehat\omega/R$ and is covariant under exactly that transformation. A term depending on $\widehat\omega$ alone, without the compensating radial scaling, would signal that the original Kerr frequency was inserted without converting to the corotating throat time.

The differential Casimir

The symmetry statement is an operator identity, not just a resemblance between radial powers. Acting on scalars, choose the Killing fields

$$\begin{aligned} H_+&=\partial_T,\qquad H_0=T\partial_T-R\partial_R,\\ H_-&=(T^2+R^{-2})\partial_T-2TR\partial_R-2R^{-1}\partial_\Phi. \end{aligned}$$

The last term in $H_-$ preserves the angular fibre; omitting it preserves only the two-dimensional base. Direct commutation gives $[H_0,H_+]=-H_+$, $[H_0,H_-]=H_-$ and $[H_+,H_-]=2H_0$. Therefore

$$\begin{aligned} \mathcal C &=H_0^2-\frac12(H_+H_-+H_-H_+)\\ &=\partial_R(R^2\partial_R)-R^{-2}\partial_T^2 +2R^{-1}\partial_T\partial_\Phi. \end{aligned}$$

On a separated solution, $\mathcal C\Psi=(K_{\ell m}-2m^2)\Psi$. This fixes the weight relation $h(h-1)=K_{\ell m}-2m^2$. The source uses $\eta_{-1}=H_-/2$, with correspondingly different bracket normalization; see Amsel et al. 2009, Eqs. (2.7)–(2.8), p. 6 (PDF).

Whittaker solutions and conformal weight

Define

$$\nu^2=\frac14+K_{\ell m}-2m^2, \qquad z=-\frac{2i\widehat\omega}{R}.$$

The radial equation becomes

$$\frac{d^2\mathcal R}{dz^2} +\left[-\frac14+\frac{im}{z}+\frac{1/4-\nu^2}{z^2}\right]\mathcal R=0,$$

with independent solutions $M_{im,\nu}(z)$ and $W_{im,\nu}(z)$ away from exceptional parameter values. At the throat mouth $R\to\infty$,

$$\mathcal R \sim A R^{-1/2+\nu}+B R^{-1/2-\nu}.$$

Equivalently, the two $SL(2,\mathbb R)$ weights are

$$h_\pm=\frac12\pm\nu, \qquad h_\pm(h_\pm-1)=K_{\ell m}-2m^2.$$

Real $\nu$ gives two real power laws. Imaginary $\nu$ gives oscillatory logarithmic behavior and belongs to the principal-series regime. Neither case alone proves an instability. A time-evolution statement also needs a domain, a conserved flux, and an inverse-frequency prescription. Exceptional values $2\nu\in\mathbb Z$ require a limiting construction and can introduce logarithms, so formulas obtained by dividing by gamma functions must not be continued through them blindly.

Boundary conditions and matching data

The Poincaré horizon is $R=0$. Initially take $\operatorname{Im}\widehat\omega\gt0$, $\widehat\omega\ne0$ and $2\nu\notin\mathbb Z$. Choose $\nu$ with nonnegative real part and positive imaginary part when $\nu^2\lt0$; set $h=1/2+\nu$. Use real $\log R$ and continue $\operatorname{Log}(-2i\widehat\omega)$ from the right half-plane. Real-frequency retarded data are its boundary values, not a fresh choice of branch below the real axis.

Future-regular coordinates $v=T-1/R$ and $\varphi=\Phi-\log R$ give

$$ds^2=\mathcal A\left[-R^2dv^2+2\,dv\,dR+d\theta^2 +\Lambda^2(d\varphi+R\,dv)^2\right].$$

Since $W_{im,\nu}(z)\sim e^{-z/2}z^{im}$, its singular phase in the original coordinates cancels the coordinate change. A radial solution with unit future-regular field amplitude is

$$\mathcal R_H(R)=(-2i\widehat\omega)^{-im} W_{im,\nu}(-2i\widehat\omega/R).$$

Indeed the complete field tends to $e^{-i\widehat\omega v+im\varphi}S_{\ell m}$ at the future horizon. The exact Whittaker connection identity gives both mouth coefficients:

$$\begin{aligned} A_H&=(-2i\widehat\omega)^{1-h-im} \frac{\Gamma(2h-1)}{\Gamma(h-im)},\\ B_H&=(-2i\widehat\omega)^{h-im} \frac{\Gamma(1-2h)}{\Gamma(1-h-im)},\\ \mathcal R_H&=A_H R^{h-1}[1+O(R^{-1})] +B_H R^{-h}[1+O(R^{-1})]. \end{aligned}$$

The gamma functions here are Euler’s function, not the angular warp $\Gamma(\theta)$. These are coefficients of two complete Frobenius series: descendants of the larger branch can exceed the smaller branch, so one must not extract $B_H$ by subtracting only a single leading power. The Whittaker connection and horizon conditions are given in Amsel et al. 2009, Eqs. (3.5)–(3.8) and (3.11)–(3.12), pp. 17–19 (PDF) and NIST DLMF, § 13.14, Eq. (13.14.33). Dividing the coefficients gives

$$\frac{B_H}{A_H}=(-2i\widehat\omega)^{2h-1} \frac{\Gamma(1-2h)\Gamma(h-im)} {\Gamma(2h-1)\Gamma(1-h-im)}.$$

The other endpoint, $R\to\infty$, is a timelike throat boundary, not asymptotically flat null infinity. A standalone NHEK problem must prescribe a flux condition relating $A$ and $B$. An embedded Kerr problem instead obtains that relation by matching to the far-zone solution.

One matching step can be done directly from the scaled coordinates. Since $x=(\widetilde r-r_+)/r_+=\lambda R$ at exact extremality, denote the future-regular Whittaker solution by $\mathcal R_{\mathrm{in}}=\mathcal R_H$, with $A_{\mathrm{in}}=A_H$ and $B_{\mathrm{in}}=B_H$, and write its mouth expansion as

$$\mathcal R_{\mathrm{in}} \sim A_{\mathrm{in}}R^{-1/2+\nu} +B_{\mathrm{in}}R^{-1/2-\nu}.$$

The same field in the exterior overlap $\lambda\ll x\ll1$ is therefore

$$\mathcal R_{\mathrm{in}} \sim A_{\mathrm{in}}\lambda^{1/2-\nu}x^{-1/2+\nu} +B_{\mathrm{in}}\lambda^{1/2+\nu}x^{-1/2-\nu}.$$

If the far solution has coefficients $C$ and $D$ multiplying those two $x$-powers, equality in the overlap gives $C=A_{\mathrm{in}}\lambda^{1/2-\nu}$ and $D=B_{\mathrm{in}}\lambda^{1/2+\nu}$. Thus $D/C=(B_H/A_H)\lambda^{2\nu}$, also for complex nonexceptional $\nu$: positive $\lambda$ fixes its real logarithm. For imaginary $\nu$ this relative power has unit modulus and an oscillating phase, not small magnitude. This concrete match also shows why an arbitrary isolated-NHEK choice of $B/A$ is not exterior data. At $2\nu\in\mathbb Z$, the logarithmic limiting basis must replace this simple power-law division.

This distinction changes the allowed dynamics. Amsel and collaborators analyze an isolated NHEK domain: generalized Dirichlet conditions for their scalar model, and Teukolsky–Dirichlet conditions for gravitational perturbations, together with the full linearization-stability charge constraints imposed at all times. Their analysis excludes nontrivial scalar dynamics and bulk gravitational dynamics within those classes; more general gravitational boundary conditions are not settled by that calculation. See Amsel et al. 2009, §§ 4.5–5, pp. 31–32 (PDF). This conditional conclusion does not erase near-horizon excitations of an asymptotically flat Kerr solution, whose throat mouth carries matching data from the exterior.

The order of operations is therefore decisive. “Scale, impose an isolated NHEK boundary condition, then solve” and “solve the asymptotically flat Kerr problem, then take a near-zone limit” need not select the same ratio $B/A$. Likewise, taking $\widehat\omega$ fixed means $\omega\to m\Omega_H$ as the throat is formed; taking the exterior frequency fixed does not.

The preceding scaling definitions are collected in Near-Extremal Scaling Parameters and Throat Regions. The spectral consequence is developed next in Zero-Damped and Damped Mode Families.

Exercises

Insert $\mathcal R\sim R^p$ into the radial equation at $R\to\infty$ and derive the two powers and the quadratic Casimir eigenvalue.

Solution

At large $R$, $(\widehat\omega+mR)^2/R^2\to m^2$. The leading equation is

$$\frac{d}{dR}(R^2\mathcal R') +(2m^2-K_{\ell m})\mathcal R=0.$$

For $\mathcal R=R^p$ this gives $p(p+1)+2m^2-K_{\ell m}=0$, hence

$$p=-\frac12\pm \sqrt{\frac14+K_{\ell m}-2m^2} =-\frac12\pm\nu.$$

Writing the decaying exponent as $p=-h$ gives $h(h-1)=K_{\ell m}-2m^2$. Thus $h=1/2\pm\nu$, exactly the $SL(2,\mathbb R)$ Casimir relation. If $\nu$ is imaginary, both powers have the same real decay exponent but oscillate in $\log R$; that observation alone does not determine the boundary flux.

References

• Amsel, A. J., Horowitz, G. T., Marolf, D., and Roberts, M. M. (2009). “No dynamics in the extremal Kerr throat.” Journal of High Energy Physics 2009(09), 044. DOI.
• Bardeen, J. M., and Horowitz, G. T. (1999). “The Extreme Kerr throat geometry: A vacuum analog of $AdS_2\times S^2$.” Physical Review D 60, 104030. DOI.
• National Institute of Standards and Technology. n.d. Digital Library of Mathematical Functions. § 13.14, Whittaker functions, Eq. (13.14.33). Online reference.