Kerr Metric Forms and Parameters

Kerr spacetime is the two-parameter, asymptotically flat, stationary-axisymmetric vacuum family with mass $M$ and angular momentum $J$. Boyer–Lindquist coordinates display its commuting symmetries and separability, Kerr–Schild coordinates display its linear deformation of flat space, and ingoing or outgoing coordinates cross different oriented horizon branches. These forms describe one geometry only after the positive-mass convention, coordinate branch, and exceptional sets are supplied.

Required background. Frames, Coframes, and Anholonomy supplies the frame transformations used to verify the metric and its null directions.

Helpful background. The Einstein Equation supplies Ricci flatness; The Riemann Curvature Tensor supports invariant checks; Exact Solutions of Einstein’s Equations explains how to specify exact spacetime solutions; Causality and Global Structure supports extension language; Killing Fields and Isometry Groups supplies the stationary and axial generators.

Boyer–Lindquist form

Choose the positive-mass asymptotic end, so that $M>0$ and $m=GM>0$, and then define

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

Both $m$ and $a$ have dimensions of length in $c=1$ units. The dimensionless spin is $a_*=a/m=J/(GM^2)$. In coordinates $t,r,\theta,\phi$, with $0<\theta<\pi$ and $\phi\sim\phi+2\pi$, the line element is

$$\begin{aligned} ds^2={}&-\left(1-\frac{2mr}{\Sigma}\right)dt^2 -\frac{4mar\sin^2\theta}{\Sigma}\,dt\,d\phi +\frac{\Sigma}{\Delta}dr^2+\Sigma\,d\theta^2 \\ &+\left(r^2+a^2+\frac{2ma^2r\sin^2\theta}{\Sigma}\right) \sin^2\theta\,d\phi^2. \end{aligned}$$

The Killing fields are $\xi=\partial_t$ and $\eta=\partial_\phi$, with $\xi$ unit normalized at the asymptotically flat $r\to+\infty$ end. At large $r$, $g_{tt}=-1+2m/r+O(r^{-2})$ and $g_{t\phi}=-2J G\sin^2\theta/r+O(r^{-2})$, fixing the physical parameter interpretation. The symmetry-adapted transformation and metric are given in Boyer and Lindquist 1967, §II, pp. 270–271, Eqs. (2.11)–(2.15) and independently in Carter 1968, §1B, p. 1561, Eqs. (9)–(10).

The order of definitions matters. At $M=0$, $a=J/M$ is not a physical parameterization. One may nevertheless take the formal metric limit $m\to0$ while holding the coordinate parameter $a$ fixed; that separate limit is flat and has $J=Ma\to0$.

The chart is singular where $\Delta=0$ and the angular coordinates degenerate at the axis. For $m>0$, the metric extension is obstructed only where $\Sigma=0$, a distinction established by regular coordinates and invariants on later pages.

Two determinant checks are useful before any curvature calculation:

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

The first is finite at $\Delta=0$ away from the polar-coordinate axis, while the second shows that a Killing combination degenerates there. For $m>0$, the full metric determinant and curvature description fail at $\Sigma=0$. These algebraic facts already separate a symmetry-orbit boundary from the positive-mass singular locus.

Kerr–Schild form and its branch

Let $t_{\mathrm{KS}},x,y,z$ be Cartesian coordinates for

$$\eta=-dt_{\mathrm{KS}}^2+dx^2+dy^2+dz^2.$$

On the ingoing principal branch, Kerr admits

$$g=\eta+2H\,k\otimes k,$$

where

$$\begin{aligned} k={}&dt_{\mathrm{KS}} +\frac{rx+ay}{r^2+a^2}\,dx +\frac{ry-ax}{r^2+a^2}\,dy +\frac{z}{r}\,dz,\\ H={}&\frac{mr^3}{r^4+a^2z^2}=\frac{mr}{\Sigma}. \end{aligned}$$

The one-form $k$ is null with respect to both $\eta$ and $g$. The signed oblate coordinates obey

$$x+iy=(r+ia)e^{i\tilde\phi}\sin\theta, \qquad z=r\cos\theta,$$

and therefore

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

Equivalently, with $R^2=x^2+y^2+z^2$,

$$r^4-(R^2-a^2)r^2-a^2z^2=0.$$

The usual exterior sheet chooses

$$r^2=\frac12\left[ R^2-a^2+\sqrt{(R^2-a^2)^2+4a^2z^2} \right]$$

and $r\ge0$. Pulling $k$ back through the oblate map gives the compact expression

$$k=dt_{\mathrm{KS}}+dr-a\sin^2\theta\,d\tilde\phi.$$

These formulas reproduce Kerr 1963, p. 238, Eq. (5), Boyer and Lindquist 1967, §II, pp. 268–270, Eqs. (2.4)–(2.10), and Carter 1968, §1A, p. 1560, Eqs. (6)–(8). Debney, Kerr, and Schild derive the shared-null-vector ansatz and its rotating specialization in Debney, Kerr, and Schild 1969, §§5 and 7, pp. 1848–1854, Eqs. (5.1)–(5.4) and (7.1)–(7.17).

For $a\ne0$, the map is exceptional on the equatorial disk $z=0$, $x^2+y^2\le a^2$, where $r=0$. Its boundary $x^2+y^2=a^2$ is the locus $\Sigma=0$. Extending signed $r$ through the disk away from that boundary produces the familiar second oblate sheet; a single triple $(x,y,z)$ does not then label the global extended manifold uniquely. The rotation axis has the ordinary angular-coordinate degeneracy as well. This branch information is essential: “Cartesian” does not mean “one global chart.”

The massless fixed-$a$ limit is an immediate discriminant. When $m=0$, $H=0$ wherever the oblate chart is regular, and $g=\eta$ there; the resulting flat metric extends smoothly across the chart’s disk and ring loci. Those loci then record only the failure of the oblate parameterization or of an imposed two-sheeted construction: they are not a black hole or a ring curvature singularity. For $m>0$ and $a\ne0$, by contrast, $\Sigma=0$ is a genuine curvature singularity.

Ingoing and outgoing charts

Starting from Boyer–Lindquist coordinates, define the ingoing differentials

$$dv=dt+\frac{r^2+a^2}{\Delta}dr, \qquad d\phi_+=d\phi+\frac{a}{\Delta}dr.$$

Substitution cancels every $1/\Delta$ pole and gives

$$\begin{aligned} ds^2={}&-\left(1-\frac{2mr}{\Sigma}\right)dv^2 +2\,dv\,dr+\Sigma\,d\theta^2 -2a\sin^2\theta\,dr\,d\phi_+\\ &-\frac{4mar\sin^2\theta}{\Sigma}\,dv\,d\phi_+ +\left(r^2+a^2+\frac{2ma^2r\sin^2\theta}{\Sigma}\right) \sin^2\theta\,d\phi_+^2. \end{aligned}$$

This ingoing chart is regular on the future outer-horizon branch and, in the rotating subextremal extension, on the corresponding ingoing branch of the inner horizon. The outgoing differentials

$$du=dt-\frac{r^2+a^2}{\Delta}dr, \qquad d\phi_-=d\phi-\frac{a}{\Delta}dr$$

instead regularize the time-reversed branches. Multiplying one Kerr–Schild null form by $-1$ does not exchange these charts, because $k\otimes k$ is unchanged; ingoing and outgoing forms are adapted to the two distinct repeated principal congruences. Neither principal chart alone contains the bifurcation two-surfaces of the simple horizons. Boyer and Lindquist construct the two principal charts and their overlaps in Boyer and Lindquist 1967, §II–III, pp. 271–275, Eqs. (2.16)–(3.11); Carter gives the paired transformations in Carter 1968, §1B–C, pp. 1561–1564, Eqs. (9) and (14)–(23).

The integrated logarithms require a separate branch choice on each radial interval. Frolov, Krtouš, and Kubizňák compare Boyer–Lindquist, canonical, and principal-null forms in Frolov, Krtouš, and Kubizňák 2017, arXiv:1705.05482v3, §3.2, pp. 43–50, Open PDF.

For $m>0$, the exact metric exists for every real $a$, but the asymptotically flat black-hole regime is $|a|\le m$; field equations alone do not enforce that inequality. The Schwarzschild endpoint $a=0$, the rotating cases $a\ne0$, and the massless fixed-$a$ limit must be distinguished before interpreting roots, disks, or rings.

Changing $a\to-a$ with $\phi\to-\phi$ is an isometry that reverses the chosen axial orientation. Thus the sign of $J$ has meaning only relative to that orientation, while $|a|$ controls the radial root regime. This check also explains why scalar invariants can be even in $a$ unless an oriented pseudoscalar is included.

Exercise

Take $a\to0$ at fixed $(t,r,\theta,\phi)$ and physical mass $M>0$. Then compare it with $m\to0$ at fixed coordinate parameter $a$. Which features survive in each limit?

Solution

For $a\to0$ at fixed $m>0$, $\Sigma\to r^2$, $\Delta\to r(r-2m)$, and the Boyer–Lindquist metric becomes Schwarzschild in static coordinates. The implicit oblate relation becomes $r^2=x^2+y^2+z^2$, while Kerr–Schild form becomes the Schwarzschild Kerr–Schild chart. Ingoing differentials reduce to $dv=dt+f^{-1}dr$ and $d\phi_+=d\phi$, the ingoing Eddington–Finkelstein transformation.

For $m\to0$ at fixed $a$, the Boyer–Lindquist components are Minkowski space in oblate coordinates and $H\to0$ makes the Cartesian Kerr–Schild metric exactly $\eta$. Because $a=J/M$ is not defined at $M=0$, this is a formal coordinate-parameter limit with $J=Ma\to0$, not a rotating massless black hole.

References

• Boyer, Robert H., and Richard W. Lindquist. “Maximal Analytic Extension of the Kerr Metric.” Journal of Mathematical Physics 8, no. 2 (1967): 265–281. DOI: 10.1063/1.1705193.
• Carter, Brandon. “Global Structure of the Kerr Family of Gravitational Fields.” Physical Review 174, no. 5 (1968): 1559–1571. DOI: 10.1103/PhysRev.174.1559.
• Debney, George C., Roy P. Kerr, and Alfred Schild. “Solutions of the Einstein and Einstein–Maxwell Equations.” Journal of Mathematical Physics 10, no. 10 (1969): 1842–1854. DOI: 10.1063/1.1664769.
• 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; arXiv:1705.05482v3.
• Kerr, Roy P. “Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics.” Physical Review Letters 11, no. 5 (1963): 237–238. DOI: 10.1103/PhysRevLett.11.237.