Near-Extremal Scaling Parameters and Throat Regions

Near-extremal perturbation theory contains several small quantities that are often denoted by the same symbol: distance from extremality, distance from the outer horizon, detuning from the synchronous frequency, inverse observation time, and sometimes ordinary low frequency. They describe different operations. This page derives a dictionary from Kerr, identifies the radial regions, and shows exactly what must be held fixed to obtain NHEK or near-NHEK. The central lesson is simple: extremality and a near-horizon zoom are independent limits. A nondegenerate near-extremal throat keeps their ratio finite; exact NHEK instead starts on the extremal family.

Helpful background. Black-Hole Geometry and Mechanics supplies the Kerr family; its Kerr Horizon and Ergoregion Anatomy derives the exact horizons and surface gravity, while Near-Horizon Extreme Kerr Geometry constructs the geometric throat; Formulations of General Relativity supplies regular horizon coordinates; PDEs and Green Functions for Gravitation supplies singular limits; Events, Intervals, and Minkowski Geometry fixes time and frequency orientation; and Black-Hole Perturbation Theory supplies the shared mode convention.

Near-extremality from Kerr charges

Let $M>0$ be the ADM mass. Set $c=1$, retain $G$ through the length $\mu=GM$, and write

$$a=\frac{J}{M}, \qquad \chi=\frac{a}{\mu}=\frac{J}{GM^2}.$$

Choose the azimuthal orientation so that $0\leq\chi\leq1$. The opposite orientation follows by $\widetilde\phi\to-\widetilde\phi$ and $m\to-m$. The Kerr horizons, horizon angular velocity, and surface gravity are

$$r_\pm=\mu\pm\sqrt{\mu^2-a^2}, \qquad \Omega_H=\frac{a}{r_+^2+a^2}, \qquad \kappa=\frac{r_+-r_-}{2(r_+^2+a^2)}.$$

The normalization matters in the last two quantities. Here the horizon generator is $k=\partial_{\widetilde t}+\Omega_H\partial_{\widetilde\phi}$, with $\partial_{\widetilde t}$ unit-normalized at asymptotically flat infinity. With that normalization, $\Omega_H$ and $\kappa$ are physical properties of the Kerr solution.

Two equivalent dimensionless measures of near-extremality are

$$\varepsilon=\sqrt{1-\frac{a^2}{\mu^2}}, \qquad \sigma=\frac{r_+-r_-}{r_+}.$$

Both are determined by the Kerr charges, and both vanish precisely at extremality, but they are not numerically identical. Using $r_+=\mu(1+\varepsilon)$ and $r_+^2+a^2=2\mu r_+$ gives the exact dictionary

$$\sigma=\frac{2\varepsilon}{1+\varepsilon}, \qquad 2\mu\kappa=\frac{\varepsilon}{1+\varepsilon}=\frac{\sigma}{2}, \qquad 2\mu\Omega_H =\sqrt{\frac{1-\varepsilon}{1+\varepsilon}} =\sqrt{1-\sigma}.$$

Thus $T_H=\kappa/(2\pi)$ and the long near-horizon time scale is $\kappa^{-1}$. The symbols $\varepsilon$ and $\sigma$ are reparameterizations of the same Kerr family. Because $\sigma=2\varepsilon+O(\varepsilon^2)$, they give the same leading power-law exponent, but not the same normalization or subleading coefficients. An exact coefficient or held-fixed combination must therefore name the parameter used. The zoom parameter $\lambda$ introduced below is different again: it is an arbitrary coordinate magnification, not another physical measure of spin.

The cap, throat, and exterior

Before changing coordinates, define the dimensionless Boyer–Lindquist radius

$$x=\frac{\widetilde r-r_+}{r_+}.$$

The factorization

$$\Delta=(\widetilde r-r_+)(\widetilde r-r_-) =r_+^2x(x+\sigma)$$

makes three radial regimes visible.

Region	Scaling	What it resolves
Horizon cap	$x=O(\sigma)$	The separation of the two horizons and finite-$\kappa$ dynamics
Intermediate throat	$\sigma\ll x\ll1$	A long scale-free interval and the generic near/far overlap
Exterior scale	$x=O(1)$	Departure from the near zone and the route to asymptotic infinity

The logarithmic throat length follows rather than being an analogy. On a constant-$\widetilde t$ Boyer–Lindquist slice at fixed $(\theta,\widetilde \phi)$, let $\delta=r_+-r_-$ and $y=\widetilde r-r_+$. Integrate from $y=c_{\rm in}\delta$ to $y=c_{\rm out}\mu$, where $c_{\rm in}>0$ and $0<c_{\rm out}\ll1$ are fixed as $\sigma\to0$. Since $g_{\widetilde r\widetilde r}=\Sigma/\Delta$,

$$L_\theta =\int_{c_{\rm in}\delta}^{c_{\rm out}\mu} \sqrt{\frac{(r_++y)^2+a^2\cos^2\theta} {y(y+\delta)}}\,dy.$$

For $y\ll\mu$ and $\sigma\ll1$, the radial kernel has the uniform form

$$\sqrt{g_{\widetilde r\widetilde r}} =\frac{\mu\sqrt{1+\cos^2\theta}}{\sqrt{y(y+\delta)}} \left[1+O\!\left(\frac{y}{\mu}\right)+O(\sigma)\right].$$

Unlike a $1/y$ expansion, this expression remains valid at the cap edge $y=O(\delta)$. Its leading primitive is exact:

$$\int\frac{dy}{\sqrt{y(y+\delta)}} =2\operatorname{arsinh}\sqrt{\frac{y}{\delta}}.$$

Since $\delta=\sigma r_+=\sigma\mu[1+O(\sigma)]$, evaluation at the two endpoints gives

$$L_\theta =\mu\sqrt{1+\cos^2\theta}\log\frac{1}{\sigma}+O(\mu).$$

Changing the fixed endpoints changes the $O(\mu)$ term, not the coefficient of the logarithm. This is a useful geometric diagnostic, not a scalar notion of distance: it refers to a particular spacelike slice. Every $\sigma>0$ throat is finite on that slice, while the exactly extremal horizon is infinitely far down it. The metric ingredients and the infinite-distance extremal limit follow from Bardeen and Horowitz 1999, § 2, pp. 3–4.

The exact-extremal zoom: NHEK

At exact extremality, introduce Boyer–Lindquist coordinates with tildes and scaled, corotating coordinates by

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

then take $\lambda\to0$ at fixed $(T,R,\theta,\Phi)$. The angular shift equals $\Omega_H\widetilde t$ because extremal Kerr has $\Omega_H=1/(2\mu)$. In the site signature $(-,+,+,+)$, the result is

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

where

$$\Gamma(\theta)=\frac{1+\cos^2\theta}{2}, \qquad \Lambda(\theta)=\frac{2\sin\theta}{1+\cos^2\theta}.$$

This is the Poincaré patch of NHEK. The exterior throat has $R>0$, $R=0$ is its Poincaré horizon, $\Phi$ has period $2\pi$, and $R\to\infty$ is the timelike throat mouth. The apparent divergence of $g_{RR}$ at $R=0$ is a coordinate-horizon effect, not a curvature singularity. NHEK is a nonsingular vacuum solution with $SL(2,\mathbb R)\times U(1)$ isometry, but it is not the asymptotically flat Kerr exterior. A physical Kerr perturbation must still be matched through the mouth. The coordinate limit and enhanced symmetry are derived in Bardeen and Horowitz 1999, § 2, eqs. (2.5)–(2.14), pp. 4–6.

The correlated near-extremal zoom

Now let the black hole approach extremality while zooming. Define

$$q=\frac{\sigma}{\lambda}, \qquad \widetilde r=r_++\lambda\mu R, \qquad \widetilde t=\frac{2\mu}{\lambda}T, \qquad \widetilde\phi=\Phi+\Omega_H\widetilde t,$$

and take $\lambda\to0$ while $q$ and $(T,R,\theta,\Phi)$ stay fixed. Direct substitution into the horizon function gives

$$\frac{\Delta}{\lambda^2\mu^2} =R\left[R+q\frac{r_+}{\mu}\right] \longrightarrow R(R+q).$$

Thus fixed finite $q$ retains both roots in the scaled chart. Direct expansion of the full Kerr metric gives

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

The outer and inner horizons are at $R=0$ and $R=-q$. Because the azimuth is corotating with the actual outer horizon, $\partial_T$ is the horizon generator and has scaled surface gravity and temperature

$$\kappa_T=\frac{q}{2}, \qquad T_T=\frac{q}{4\pi}.$$

The conventional coordinates in the cited source use

$$4\pi T_R=q, \qquad \phi_{\mathrm B} =\widetilde\phi-\frac{\widetilde t}{2\mu} \longrightarrow\Phi-\frac{q}{2}T.$$

Accordingly, $d\phi_{\mathrm B}+(R+q/2)dT=d\Phi+R\,dT$. This explains why its displayed near-NHEK metric and the actual-horizon-corotating form above differ by a constant term in the angular fiber.

For $q=0$, this metric reduces to the NHEK Poincaré patch. For fixed $q>0$ it is the finite-temperature near-NHEK patch. Its maximal extension is diffeomorphic to NHEK, but the thermal time, horizon patch, and the boundary where an exterior solution is attached are different data; the singular coordinate map at that boundary cannot be ignored in a scattering problem. This correlated limit and the conventional near-NHEK coordinates are derived in Bredberg, Hartman, Song, and Strominger 2010, § 3, eqs. (3.1)–(3.9), pp. 6–7.

The chapter overview instead uses variables tied directly to the physical gap scale,

$$X=\frac{x}{\sigma}, \qquad U=\kappa\widetilde t, \qquad W=\frac{\omega-m\Omega_H}{\kappa}.$$

For fixed $q>0$, the two notations are locked together in the correlated limit:

$$X\longrightarrow\frac{R}{q}, \qquad U=\frac{q}{2}T, \qquad \widehat\omega=\frac{q}{2}W.$$

Choosing the convenient zoom $\lambda=\sigma$ sets $q=1$, so $X\to R$, $U=T/2$, and $W=2\widehat\omega$. No physics depends on that choice. Taking $\lambda\to0$ at fixed nonzero $\sigma$ instead sends $q\to\infty$ and does not produce a finite metric in this throat chart.

Synchronous detuning is not ordinary low frequency

The shared mode phase is $e^{-i\omega\widetilde t+im\widetilde\phi}$. In the corotating scaled coordinates it becomes

$$e^{-i\widehat\omega T+im\Phi}, \qquad \widehat\omega=\frac{2\mu}{\lambda}(\omega-m\Omega_H).$$

Consequently, a finite-frequency throat excitation is a synchronous double scaling:

$$\omega-m\Omega_H=O(\lambda/\mu), \qquad \widehat\omega=\frac{2\mu}{\lambda}(\omega-m\Omega_H)=O(1).$$

At fixed $0<q<\infty$, this is equivalently fixed $W$. Fixed exterior frequency away from $m\Omega_H$ oscillates infinitely rapidly in $T$ and does not define a finite throat mode. Conversely, fixed $\widehat\omega$ occupies a shrinking frequency band for an asymptotic observer. Bardeen and Horowitz exhibit this frequency scaling at exact extremality in Bardeen and Horowitz 1999, § 3, eq. (3.3), p. 7.

This is not the usual low-frequency limit $\mu\omega\to0$. For $m\neq0$, fixed $W$ near extremality instead gives

$$\mu\omega\longrightarrow\frac{m}{2}.$$

Only for $m=0$ can synchronous and low-frequency scalings overlap, and even then their relative rate must be stated. The symbols answer different questions:

Quantity made small or fixed	Meaning
$\sigma\to0$	Approach extremal Kerr
$x\to0$	Approach the outer horizon in a chosen Kerr spacetime
$X=x/\sigma$ fixed	Follow the finite-width horizon cap
$W=(\omega-m\Omega_H)/\kappa$ fixed	Follow the synchronous frequency band
$\mu\omega\to0$	Take the ordinary long-wavelength limit
$U=\kappa\widetilde t$ fixed	Resolve the long near-extremal time scale

The matching region also has a transparent expression. With $x\sim\lambda R$ and finite $q$, the overlap $\sigma\ll x\ll1$ becomes

$$q\ll R\ll\frac{1}{\lambda}.$$

The lower inequality leaves the horizon cap; the upper one remains inside the near zone. This is the interval used on Matched Asymptotic Expansions Across the Throat.

Which limit gives which problem?

Question	Limit path	Variables kept finite	Result
Exterior extremal geometry	$\sigma\to0$	Ordinary Kerr coordinates	Extremal Kerr with its asymptotic region
Exact throat	$\lambda\to0$ with $\sigma=0$, or $\lambda,\sigma\to0$ with $q=\sigma/\lambda\to0$	$T,R,\Phi,\widehat\omega$	NHEK Poincaré patch
Near-extremal throat	$\lambda,\sigma\to0$ with $0<q<\infty$	$T,R,\Phi,\widehat\omega$	Near-NHEK thermal patch
Long transient	$\kappa\to0$ and $\widetilde t\to\infty$	$U=\kappa\widetilde t$	A finite scaling profile
Ordinary low frequency	$\mu\omega\to0$	Geometry and boundary data specified separately	A long-wavelength problem, generally not the throat problem

The exterior and throat extremal limits are both regular, but they retain different parts of the original family. A point at fixed $R$ has $\widetilde r-r_+=O(\lambda\mu)$ and collapses onto the horizon in exterior coordinates. Hence a field smooth for a throat observer can have large transverse derivatives for an exterior observer, and conversely. This far-observer/near-observer distinction is made explicit in Gralla, Zimmerman, and Zimmerman 2016, Appendix B, eqs. (B1), (B4), and (B6), pp. 6–7.

Late time is nonuniform for the same reason. For the toy contribution $F_\kappa(\widetilde t)=e^{-\kappa\widetilde t}$,

$$\lim_{\kappa\to0}\lim_{\widetilde t\to\infty}F_\kappa=0, \qquad \lim_{\widetilde t\to\infty}\lim_{\kappa\to0}F_\kappa=1,$$

whereas fixed $U=\kappa\widetilde t$ retains $e^{-U}$. Likewise, the synchronous limit need not commute with analytic continuation of the retarded frequency-domain response. The associated coalescing resonances are developed on Zero-Damped and Damped Mode Families. Noncommutation diagnoses a singular perturbation problem; it does not by itself establish an exponentially growing mode. The finite-$\kappa$ response and its extremal scaling are analyzed in Gralla, Zimmerman, and Zimmerman 2016, §§ II–III, pp. 2–5.

Common pitfalls

Setting $a=\mu$ too early. This is appropriate for the exact NHEK limit, but it erases the horizon gap that supports the finite-$q$ cap and its thermal time. Keep $\sigma/\lambda$ fixed when the near-extremal throat is the object.

Calling every small frequency “low frequency.” Synchronous detuning makes $\omega-m\Omega_H$ small. For $m\neq0$, the asymptotic frequency itself stays near $m/(2\mu)$ rather than zero.

Treating NHEK as the whole Kerr exterior. The throat limit removes asymptotic infinity. Boundary conditions imposed at the NHEK mouth and data inherited by matching to an asymptotically flat exterior are different problems.

The next step is to derive the separated equations in Field Equations in Near-Horizon Extreme Kerr.

Exercises

Starting from $r_\pm=\mu(1\pm\varepsilon)$, derive $\sigma$, $\kappa$, and $\Omega_H$ in terms of $\varepsilon$. Expand them through the first nonconstant correction near $\varepsilon=0$.

Solution

The horizon gap is $2\mu\varepsilon$, so

$$\sigma=\frac{2\mu\varepsilon}{\mu(1+\varepsilon)} =\frac{2\varepsilon}{1+\varepsilon}.$$

This relation shows that the two dimensionless measures vanish together but are not equal: their leading values differ by a factor of two, and their higher-order coefficients depend on which parameter is used.

The Kerr identity $r_+^2+a^2=2\mu r_+$ then gives

$$\kappa=\frac{r_+-r_-}{4\mu r_+} =\frac{\sigma}{4\mu}.$$

Finally,

$$2\mu\Omega_H =\frac{a}{r_+} =\sqrt{\frac{1-\varepsilon}{1+\varepsilon}}.$$

The positive square root follows from the chosen spin orientation. It also checks that the horizon slows from its extremal value as the horizon gap opens.

Thus

$$\sigma=2\varepsilon+O(\varepsilon^2), \qquad \kappa=\frac{\varepsilon}{2\mu}+O(\varepsilon^2/\mu), \qquad \Omega_H=\frac{1-\varepsilon+O(\varepsilon^2)}{2\mu}.$$

Use $x=(\widetilde r-r_+)/r_+$ to recover the leading logarithmic throat length at fixed latitude. Which part of the answer depends on the chosen endpoints?

Solution

Because $\Delta=r_+^2x(x+\sigma)$ and $d\widetilde r=r_+dx$, the leading near-extremal kernel is uniform through the cap edge:

$$dL_\theta =\mu\sqrt{1+\cos^2\theta} \frac{dx}{\sqrt{x(x+\sigma)}} \left[1+O(x)+O(\sigma)\right].$$

The leading integral can be evaluated without using the intermediate-region expansion at its lower endpoint:

$$\int\frac{dx}{\sqrt{x(x+\sigma)}} =2\operatorname{arsinh}\sqrt{\frac{x}{\sigma}}.$$

Integrating from $x=c_{\rm in}\sigma$ to fixed small $x=c_{\rm out}$ therefore yields

$$L_\theta =\mu\sqrt{1+\cos^2\theta}\log\frac{1}{\sigma}+O(\mu).$$

The constants $c_{\rm in}$ and $c_{\rm out}$ shift only the finite $O(\mu)$ term. The latitude fixes the coefficient because the chosen Boyer–Lindquist slice is not spherically symmetric.

For the correlated limit, derive $X\to R/q$, $U=qT/2$, and $\widehat\omega=qW/2$. Then explain why fixed $W$ is not ordinary low frequency when $m\neq0$.

Solution

The radial definitions give

$$X=\frac{x}{\sigma} =\frac{\lambda\mu R}{r_+\sigma} \longrightarrow\frac{R}{q}.$$

Using $\kappa=\sigma/(4\mu)$ and $\widetilde t=2\mu T/\lambda$ gives

$$U=\kappa\widetilde t=\frac{q}{2}T.$$

Similarly,

$$\widehat\omega =\frac{2\mu}{\lambda}(\omega-m\Omega_H) =\frac{q}{2}\frac{\omega-m\Omega_H}{\kappa} =\frac{q}{2}W.$$

At fixed $W$, the detuning vanishes with $\kappa$, while $\Omega_H\to1/(2\mu)$. Therefore $\mu\omega\to m/2$, which is nonzero for $m\neq0$. Ordinary low frequency would instead require $\mu\omega\to0$.

Reverse the two limits for $F_\kappa(\widetilde t)=e^{-\kappa\widetilde t}$, and find the scaling limit that preserves a nontrivial profile.

Solution

At every fixed $\kappa>0$, taking $\widetilde t\to\infty$ first gives zero, so

$$\lim_{\kappa\to0}\lim_{\widetilde t\to\infty}F_\kappa=0.$$

At every fixed $\widetilde t$, taking $\kappa\to0$ first gives one, so

$$\lim_{\widetilde t\to\infty}\lim_{\kappa\to0}F_\kappa=1.$$

Holding $U=\kappa\widetilde t$ fixed instead preserves the nontrivial scaling function $e^{-U}$. The disagreement shows nonuniformity; it is not an instability at any fixed positive $\kappa$.

References

• 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. Open PDF.
• Bredberg, I., Hartman, T., Song, W., and Strominger, A. (2010). “Black Hole Superradiance From Kerr/CFT.” Journal of High Energy Physics 2010(04), 019. DOI. Open PDF.
• Gralla, S. E., Zimmerman, A., and Zimmerman, P. (2016). “Transient Instability of Rapidly Rotating Black Holes.” Physical Review D 94, 084017. DOI. Open PDF.