Kerr Spacetime and Common Charts

Kerr spacetime needs different coordinates for different jobs. Boyer–Lindquist coordinates expose stationarity and separability, ingoing and outgoing Kerr coordinates cross selected horizon branches, and Cartesian Kerr–Schild coordinates are horizon regular and asymptotically Cartesian. These forms describe the same four-dimensional asymptotically flat vacuum geometry only after their parameter conventions, domains, and principal-null orientation are fixed.

Helpful background. Lorentzian Metrics and Causal Cones supplies causal classification; Exact Solutions of Einstein’s Equations supplies the chart-domain and invariant-check method.

Boyer–Lindquist chart and spin regimes

For physical mass $M\gt0$, with $c=1$ and $G$ explicit, define

$$m=GM, \qquad a=\frac{J}{M}, \qquad a_*=\frac{a}{m}=\frac{J}{GM^2},$$ $$\Sigma=r^2+a^2\cos^2\theta, \qquad \Delta=r^2-2mr+a^2, \qquad A=r^2+a^2.$$

Both $m$ and $a$ have dimensions of length. In Boyer–Lindquist coordinates,

$$\begin{aligned} ds^2={}&-\frac{\Delta}{\Sigma} \left(dt-a\sin^2\theta\,d\phi\right)^2 +\frac{\Sigma}{\Delta}\,dr^2+\Sigma\,d\theta^2 \\ &+\frac{\sin^2\theta}{\Sigma} \left(A\,d\phi-a\,dt\right)^2 . \end{aligned}$$

In the black-hole regimes $\lvert a\rvert\leq m$, the usual exterior chart has $t\in\mathbb R$, $r\gt r_+$, $0\lt\theta\lt\pi$, and $\phi\sim\phi+2\pi$. For $\lvert a\rvert\gt m$, $r_+$ is not real; the positive-$r$ Boyer–Lindquist block still contains the asymptotically flat end but no black-hole exterior or horizon. Separate angular charts cover the rotation axis. The metric and parameter interpretation are reviewed in Bičák 2000, §4.1, printed pp. 42–46, PDF and Frolov, Krtouš, and Kubizňák 2017, §3.1, printed pp. 41–42, PDF.

The radial roots are

$$r_\pm=m\pm\sqrt{m^2-a^2}.$$

They are distinct for $\lvert a\rvert\lt m$, coincide at $r=m$ when $\lvert a\rvert=m$, and are not real when $\lvert a\rvert\gt m$. Only the first two regimes contain a Kerr black hole. In the eternal asymptotically flat extension, $r_+$ is the outer event and Killing horizon; identifying an event horizon uses that global extension, not merely the equation $\Delta=0$. On the rotating subextremal branch $0\lt\lvert a\rvert\lt m$, $r_-$ is an inner Cauchy and Killing horizon. At $a=0$, the formal lower root $r_-=0$ is the Schwarzschild central curvature singularity, not a horizon.

Four superficially similar loci have different meanings:

Condition	Meaning
$\Delta=0$	A Boyer–Lindquist coordinate pole and a candidate Killing-horizon radius; regular charts cross it.
$g_{tt}=-(1-2mr/\Sigma)=0$	For $a\ne0$, a stationary-limit surface outside the outer horizon away from the axis; at $a=0$, it coincides with the outer horizon at every angle.
$\Sigma=0$	For $a\ne0$, the ring locus $r=0$, $\theta=\pi/2$, where curvature is singular; at $a=0$, the singularity is the central locus $r=0$.
$\sin\theta=0$	Degeneracy of the azimuthal coordinate on the regular rotation axis.

Indeed,

$$\det g=-\Sigma^2\sin^2\theta, \qquad \det(g_{AB})_{A,B=t,\phi}=-\Delta\sin^2\theta.$$

The full determinant has no zero at a horizon away from the angular axis and the ring, even though $g_{rr}=\Sigma/\Delta$ diverges there.

Deriving ingoing and outgoing Kerr coordinates

The horizon-penetrating transformations follow directly from the displayed metric. For the ingoing chart define

$$dv=dt+\frac{A}{\Delta}\,dr, \qquad d\widetilde\phi=d\phi+\frac{a}{\Delta}\,dr.$$

Using $A-a^2\sin^2\theta=\Sigma$ gives

$$\begin{aligned} dt-a\sin^2\theta\,d\phi &=dv-a\sin^2\theta\,d\widetilde\phi -\frac{\Sigma}{\Delta}\,dr,\\ A\,d\phi-a\,dt &=A\,d\widetilde\phi-a\,dv. \end{aligned}$$

Substitution cancels the $\Sigma\,dr^2/\Delta$ term exactly and yields

$$\begin{aligned} ds^2={}&-\frac{\Delta}{\Sigma} \left(dv-a\sin^2\theta\,d\widetilde\phi\right)^2 +2\left(dv-a\sin^2\theta\,d\widetilde\phi\right)dr \\ &+\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma} \left(A\,d\widetilde\phi-a\,dv\right)^2. \end{aligned}$$

Every coefficient is finite at a regular root of $\Delta$. This chart crosses the future outer horizon and is adapted to the ingoing principal-null family. Reversing both signs,

$$du=dt-\frac{A}{\Delta}\,dr, \qquad d\overline\phi=d\phi-\frac{a}{\Delta}\,dr,$$

instead gives

$$\begin{aligned} ds^2={}&-\frac{\Delta}{\Sigma} \left(du-a\sin^2\theta\,d\overline\phi\right)^2 -2\left(du-a\sin^2\theta\,d\overline\phi\right)dr \\ &+\Sigma\,d\theta^2 +\frac{\sin^2\theta}{\Sigma} \left(A\,d\overline\phi-a\,du\right)^2, \end{aligned}$$

which crosses the past outer horizon and follows the outgoing principal-null family. In either regular chart, $\det g=-\Sigma^2\sin^2\theta$; its nonzero value at $\Delta=0$ independently checks that the Boyer–Lindquist pole has been removed.

The differential definitions are exact locally because their coefficients depend only on $r$. On any radial interval not containing a root, set

$$r_*(r)=\int\frac{A}{\Delta}\,dr, \qquad \Phi_*(r)=\int\frac{a}{\Delta}\,dr,$$

so that

$$(v,\widetilde\phi)=(t+r_*,\phi+\Phi_*), \qquad (u,\overline\phi)=(t-r_*,\phi-\Phi_*).$$

For $0\lt\lvert a\rvert\lt m$, the primitives contain separate logarithms of $\lvert r-r_+\rvert$ and $\lvert r-r_-\rvert$, with an additive constant and a branch chosen independently on each radial interval. At extremality, $\Delta=(r-m)^2$; the pair of primitives contains pole terms, and $r_*$ also contains a logarithm. Thus the differential formulas remain the useful definition of the regular charts; one must not try to cross a horizon by assigning a finite value to a divergent Boyer–Lindquist primitive.

Cartesian Kerr–Schild chart

Starting from the ingoing chart, choose

$$T=v-r, \qquad x+iy=(r+ia)e^{i\widetilde\phi}\sin\theta, \qquad z=r\cos\theta.$$

Then $r$ is the oblate coordinate determined on the exterior branch by

$$\frac{x^2+y^2}{r^2+a^2}+\frac{z^2}{r^2}=1,$$

and the metric becomes

$$g_{\mu\nu}=\eta_{\mu\nu}+2Hk_\mu k_\nu, \qquad H=\frac{mr}{\Sigma}=\frac{mr^3}{r^4+a^2z^2},$$

where $\eta=-dT^2+dx^2+dy^2+dz^2$ and

$$k_\mu dx^\mu =dT+\frac{rx+ay}{r^2+a^2}\,dx +\frac{ry-ax}{r^2+a^2}\,dy +\frac{z}{r}\,dz.$$

Direct substitution gives $\eta^{\mu\nu}k_\mu k_\nu=0$. The matrix determinant lemma therefore gives $\det g=\det\eta=-1$, another check that no $\Delta=0$ coordinate pole remains. Cartesian Kerr–Schild coordinates also remove the polar-axis degeneracy and approach Cartesian inertial coordinates at infinity, making them useful for field-equation checks and numerical domains. The implicit oblate radius and selected principal-null orientation must still be recorded. The Kerr–Schild form and its flat and nonrotating limits are developed in Frolov, Krtouš, and Kubizňák 2017, §§3.6–3.7, printed pp. 59–64, PDF.

Choosing a chart

Chart	Best suited to	Domain and limitation
Boyer–Lindquist $(t,r,\theta,\phi)$	Stationarity, asymptotics, orbit equations, and separation of variables	The standard exterior has $r\gt r_+$; the components do not cross $\Delta=0$, and angular charts degenerate on the axis.
Ingoing Kerr $(v,r,\theta,\widetilde\phi)$	Future-horizon crossing and infalling fields or worldlines	Regular at the future outer horizon away from $\Sigma=0$; it is not the chart adapted to the past branch.
Outgoing Kerr $(u,r,\theta,\overline\phi)$	Past-horizon crossing and outgoing principal-null propagation	Regular at the past outer horizon away from $\Sigma=0$; it is not the chart adapted to the future branch.
Cartesian Kerr–Schild $(T,x,y,z)$	Horizon-penetrating, asymptotically Cartesian, and numerical work	The exterior branch of implicit $r$ must be chosen; the curvature singularity is a ring for $a\ne0$ and the central locus $r=0$ for $a=0$, while the displayed sign selects one principal-null orientation.

The complete derivation and parameter dictionary are in Kerr Metric Forms and Parameters. For the global qualification of horizons, stationary limits, and negative-$r$ blocks, see Kerr Domains, Horizons, and Extensions.

Independent checks and limiting cases

• Solving $\Delta=0$ gives the quoted $r_\pm$ and separates subextremal, extremal, and superextremal regimes by the discriminant $m^2-a^2$.
• The substitutions above cancel all inverse powers of $\Delta$ before any horizon limit is taken, while the regular-chart determinant remains $-\Sigma^2\sin^2\theta$.
• As $a\to0$, Boyer–Lindquist form becomes Schwarzschild, $d\widetilde\phi=d\overline\phi=d\phi$, and the advanced and retarded maps become the Eddington–Finkelstein transformations.
• As $\lvert a\rvert\to m$, the two roots merge without making the ingoing or outgoing metric coefficients singular; only the integrated nonextremal logarithmic formula must be replaced.
• At fixed $a$, taking $M\to0^+$ gives $m\to0$ and $J=aM\to0$. At the endpoint, $a$ is no longer the ratio $J/M$ but remains as the limiting metric parameter labeling the oblate coordinates. Then $H\to0$ and the geometry is Minkowski away from the oblate-coordinate branch disk; that disk is a coordinate branch set, not a curvature singularity. This flat limit is displayed explicitly in Frolov, Krtouš, and Kubizňák 2017, §3.6, printed pp. 59–60, PDF.

Common pitfalls

A stationary-limit surface is not the horizon. The former solves $g_{tt}=0$; the latter lies at a root of $\Delta$ and is generated by a co-rotating Killing field. For nonzero spin they coincide only on the rotation axis; at $a=0$ they coincide at every angle.

A finite determinant does not make one component chart regular. The Boyer–Lindquist determinant is finite at $\Delta=0$, but $g_{rr}$ still has a pole. The explicit horizon-penetrating transformation supplies the needed overlap.

Horizon regularity does not remove a curvature singularity. For positive-mass Kerr, ingoing, outgoing, and Kerr–Schild charts cross a selected horizon branch, but none removes the curvature singularity at $\Sigma=0$: a ring for $a\ne0$ and the central locus $r=0$ for $a=0$. The fixed-$a$ zero-mass endpoint described above is instead flat away from its coordinate branch disk.

References

• Bičák, Jiří. “The Role of Exact Solutions of Einstein’s Equations in the Developments of General Relativity and Astrophysics: Selected Themes.” In Einstein’s Field Equations and Their Physical Implications, edited by Bernd G. Schmidt, 1–126. Lecture Notes in Physics 540. Springer, 2000. arXiv:gr-qc/0004016.
• Frolov, Valeri P., Pavel Krtouš, and David Kubizňák. “Black Holes, Hidden Symmetries, and Complete Integrability.” Living Reviews in Relativity 20 (2017): 6. DOI: 10.1007/s41114-017-0009-9. Open PDF.