# Near-horizon metric

The near-horizon metric (NHM) refers to the near-horizon limit of the global metric of a black hole. NHMs play an important role in studying the geometry and topology of black holes, but are only well defined for extremal black holes.[1][2][3] NHMs are expressed in Gaussian null coordinates, and one important property is that the dependence on the coordinate $r$ is fixed in the near-horizon limit.

## NHM of extremal Reissner–Nordström black holes

The metric of extremal Reissner–Nordström black hole is

$ds^2\,=\,-\Big(1-\frac{M}{r}\Big)^2\,dt^2+\Big(1-\frac{M}{r}\Big)^{-2}dr^2+r^2\,\big(d\theta^2+\sin^2\theta\,d\phi^2 \big)\,.$

Taking the near-horizon limit

$t\mapsto \frac{\tilde{t}}{\epsilon}\,,\quad r\mapsto M+\epsilon\,\tilde{r}\,,\quad \epsilon\to 0\,,$

and then omitting the tildes, one obtains the near-horizon metric

$ds^2=-\frac{r^2}{M^2}\,dt^2+\frac{M^2}{r^2}\,dr^2+M^2\,\big(d\theta^2+\sin^2\theta\,d\phi^2 \big)$

## NHM of extremal Kerr black holes

The metric of extremal Kerr black hole ($M=a=J/M$) in Boyer–Lindquist coordinates can be written in the following two enlightening forms,[4][5]

$ds^2\,=\,-\frac{\rho_K^2\Delta_K}{\Sigma^2}\,dt^2+\frac{\rho_K^2}{\Delta_K}\,dr^2+\rho_K^2d\theta^2+\frac{\Sigma^2\sin^2\theta}{\rho_K^2}\big( d\phi-\omega_K\, dt \big)^2\,,$
$ds^2\,=\,-\frac{\Delta_K}{\rho_K^2}\,\big(dt-M\sin^2\theta d\phi \big)^2+\frac{\rho_K^2}{\Delta_K}\,dr^2+\rho_K^2 d\theta^2+\frac{\sin^2\theta}{\rho_K^2}\Big( Mdt-(r^2+M^2)d\phi \Big)^2\,,$

where

$\rho_K^2:=r^2+M^2\cos^2\theta\,,\;\; \Delta_K:=\big(r-M\big)^2\,,\;\; \Sigma^2:=\big(r+M^2\big)^2-M^2\Delta_K\sin^2\theta\,,\;\; \omega_K:=\frac{2M^2 r}{\Sigma^2}\,.$

Taking the near-horizon limit[6][7]

$t\mapsto \frac{\tilde{t}}{\epsilon}\,,\quad r\mapsto M+\epsilon\,\tilde{r}\,,\quad \phi\mapsto \tilde{\phi}+\frac{1}{2M\epsilon}\tilde{t}\,,\quad \epsilon\to 0\,,$

and omitting the tildes, one obtains the near-horizon metric (this is also called extremal Kerr throat[6] )

$ds^2\simeq \frac{1+\cos^2\theta}{2}\,\Big(-\frac{r^2}{2M^2}\,dt^2+\frac{2M^2}{r^2}\,dr^2+2M^2d\theta^2 \Big)+\frac{4M^2\sin^2\theta}{1+\cos^2\theta}\,\Big(d\phi +\frac{rdt}{2M^2}\Big)^2\,.$

## NHM of extremal Kerr–Newman black holes

Extremal Kerr–Newman black holes ($r_+^2=M^2+Q^2$) are described by the metric[4][5]

$ds^2=-\Big(1-\frac{2Mr-Q^2}{\rho_{KN}} \!\Big)dt^2-\frac{2a\sin^2\!\theta\,(2Mr-Q^2)}{\rho_{KN}}dt d\phi +\rho_{KN}\Big(\frac{dr^2}{\Delta_{KN}} + d\theta^2\Big)+\frac{ \Sigma^2 }{\rho_{KN}}d\phi^2,$

where

$\Delta_{KN}\,:=\, r^2-2Mr+a^2+Q^2\,,\;\; \rho_{KN}\,:=\,r^2+a^2\cos^2\!\theta\,,\;\;\Sigma^2\,:=\,(r^2+a^2)^2-\Delta_{KN} a^2\sin^2\theta\,.$

Taking the near-horizon transformation

$t\mapsto \frac{\tilde{t}}{\epsilon}\,,\quad r\mapsto M+\epsilon\,\tilde{r}\,,\quad \phi\mapsto \tilde{\phi}+\frac{a}{r^2_0\epsilon}\tilde{t}\,,\quad \epsilon\to 0\,,\quad \Big(r^2_0\,:=\,M^2+a^2\Big)$

and omitting the tildes, one obtains the NHM[7]

$ds^2\simeq \Big(1-\frac{a^2}{r_0^2}\sin^2\!\theta \Big)\left(-\frac{r^2}{r^2_0}dt^2+\frac{r^2_0}{r^2}dr^2+r^2_0d\theta^2 \right)+r^2_0\sin^2\!\theta\,\Big(1-\frac{a^2}{r_0^2} \sin^2\!\theta\Big)^{-1}\left( d\phi+\frac{2arM}{r^4_0}dt \right)^{-1}\,.$

## NHMs of generic black holes

In addition to the NHMs of extremal Kerr–Newman family metrics discussed above, all stationary NHMs could be written in the form[1][2][3][8]

$ds^2=(\hat{h}_{AB}G^A G^B-F)r^2 dv^2+2dvdr- \hat{h}_{AB}G^B r dv dy^A -\hat{h}_{AB}G^Ar dv dy^B+\hat{h}_{AB} dy^A dy^B$
$=-F\,r^2 dv^2+2dvdr+\hat{h}_{AB}\big(dy^A-G^A\,r dv \big)\big(dy^B-G^B\,r dv \big)\,,$

where the metric functions $\{F,G^A\}$ are independent of the coordinate r, $\hat{h}_{AB}$ denotes the intrinsic metric of the horizon, and $y^A$ are isothermal coordinates on the horizon.

Remark: In Gaussian null coordinates, the black hole horizon corresponds to $r=0$.