Kerr Invariants and the Ring Singularity

For positive-mass Kerr, horizons are zeros of a coordinate-dependent radial polynomial, but the curvature singularity is visible invariantly. On the rotating branch $a\ne0$, a principal-frame complex Weyl invariant has a denominator that vanishes only when $r=0$ and $\cos\theta=0$, the ring. At $a=0$, the same condition collapses to the Schwarzschild center $r=0$. Scalar invariants also encode mass and spin locally, although discrete ambiguities and exceptional surfaces require several invariants or Cartan data for a complete identification.

Required background. Frames, Coframes, and Anholonomy supplies the principal tetrad used to derive the invariant formulas.

Helpful background. The Einstein Equation supplies Ricci flatness; The Riemann Curvature Tensor supplies scalar contractions; Exact Solutions of Einstein’s Equations supplies the identification hierarchy; Causality and Global Structure distinguishes curvature blowup from global inextendibility; Killing Fields and Isometry Groups supplies the horizon candidates.

A complex curvature diagnostic

For vacuum Kerr, with $\epsilon_{tr\theta\phi}\gt0$, define the left dual on the first index pair by

$$({*C})_{abcd}=\frac12\epsilon_{ab}{}^{ef}C_{efcd},$$

and write

$$K=C_{abcd}C^{abcd}, \qquad P=C_{abcd}{*C}^{abcd}.$$

The two conjugate complex invariants are

$$\begin{aligned} \mathcal W_-&=K-iP =48\Psi_2^2 =\frac{48m^2}{(r-ia\cos\theta)^6},\\ \mathcal W_+&=K+iP =\overline{\mathcal W_-} =\frac{48m^2}{(r+ia\cos\theta)^6}. \end{aligned}$$

Here the type-D page fixes $\Psi_2=+m/(r-ia\cos\theta)^3$. Abdelqader and Lake define the two quadratic invariants and print the conjugate combination $\mathcal W_+$ in Abdelqader and Lake 2015, arXiv v4, § II, printed p. 2, Eqs. (1)–(2) and footnote 1; Appendix A, printed p. 6, Eq. (A.2), Open PDF. Translating their convention to the site’s orientation and left dual gives the pair above. For the four-dimensional Weyl tensor the standard left and right duals agree, so moving the star between index pairs alone does not interchange them. Reversing orientation or choosing the opposite sign for the dual does; the modulus and singular locus are unchanged.

The ordinary Kretschmann scalar is their common real part,

$$C_{abcd}C^{abcd} =\frac{48m^2 (r^6-15a^2r^4\cos^2\theta+15a^4r^2\cos^4\theta-a^6\cos^6\theta)} {(r^2+a^2\cos^2\theta)^6}.$$

This is the $Q=0$ specialization of Henry 2000, arXiv v1 PDF p. 6, unnumbered Kerr–Newman Kretschmann formula, Open PDF. The parity-odd scalar fixed by the same convention is

$$P=-\frac{96m^2ar\cos\theta \left(3r^4-10a^2r^2\cos^2\theta+3a^4\cos^4\theta\right)} {(r^2+a^2\cos^2\theta)^6}.$$

The parity-odd numerator can vanish on regular surfaces, so a zero of $P$ does not mean flatness. Possible curvature blowup is confined to $\Sigma=r^2+a^2\cos^2\theta=0$; the modulus below resolves cancellations between the two real invariants.

Ring singularity and regular horizons

For real $(r,\theta)$ with $a\ne0$, $\Sigma=0$ requires

$$r=0, \qquad \theta=\frac\pi2.$$

In Kerr–Schild Cartesian coordinates this locus is $x^2+y^2=a^2$, $z=0$. The modulus

$$|\mathcal W_-|=|\mathcal W_+| =\sqrt{K^2+P^2} =\frac{48m^2}{\Sigma^3}$$

diverges there for $m\ne0$. This proves scalar-curvature blowup along every approach to the ring, even though $K$ or $P$ separately can stay finite along specially tuned curves. Lake demonstrates this complementary-invariant point in Lake 2003, arXiv v3 PDF p. 1, Eqs. (1)–(2) and the concluding paragraph, Open PDF. Kerr’s original asymptotically Cartesian form is analytic except at the same ring, $x^2+y^2=a^2$, $z=0$, in Kerr 1963, p. 238, Eq. (5).

For rotating subextremal Kerr, $0\lt\lvert a\rvert\lt m$, both roots $r_\pm=m\pm\sqrt{m^2-a^2}$ are positive and $\Sigma$ remains positive at either one for every $\theta$. All polynomial curvature invariants are therefore finite there, and ingoing or outgoing coordinates remove the metric-component pole. At the nonrotating endpoint $a=0$, the outer root $r_+=2m$ is regular, but $r_-=0$ is the Schwarzschild curvature singularity, not an inner horizon; indeed $K=48m^2/r^6$ diverges there. At extremality $\lvert a\rvert=m$, the repeated root $r=m$ also has $\Sigma=m^2(1+\cos^2\theta)\gt0$. The ring and horizon diagnoses are independent: $\Sigma=0$ controls curvature blowup, while $\Delta=0$ controls the degeneracy of the symmetry-adapted radial chart only when no simultaneous cancellation or singular endpoint changes the interpretation.

Abdelqader and Lake construct a broader set of Kerr invariants and show how a dimensionless combination detects the two rotating horizons in Abdelqader and Lake 2015, § III.A, printed pp. 2–3, Eqs. (8)–(12). Their detector carries an explicit factor of $a^2\sin^2\theta$ and degenerates in the Schwarzschild limit, where a different invariant is required. Such invariant horizon detectors are more sophisticated than testing a curvature scalar for divergence; a regular horizon need not be a level set singled out by every scalar.

Recovering parameters and checking limits

The phase and magnitude of $\mathcal W_-$ constrain $m$ and the complex combination $r-ia\cos\theta$, but one scalar alone does not identify which real coordinates produced it. Gradient invariants add scale and directional information. In a generic Kerr region, a finite collection can recover the dimensionless local combinations and discriminate Kerr from neighboring families; on symmetry planes, axes, and degenerate surfaces, special handling is required.

Three limits check the formula. As $a\to0$, the imaginary part vanishes and $\mathcal W_\pm\to48m^2/r^6$, the Schwarzschild Kretschmann scalar. As $m\to0$ at fixed Kerr–Schild Cartesian point away from the oblate coordinate branch set, curvature vanishes and the metric is flat. At large positive $r$,

$$\mathcal W_- =48m^2r^{-6} \left[1+\frac{6ia\cos\theta}{r}+O(r^{-2})\right],$$

so the leading term fixes mass magnitude and the $+6ia\cos\theta/r$ correction carries the spin orientation in the stated left-dual convention. The conjugate $\mathcal W_+$ has the opposite parity-odd sign.

Scalar divergence proves failure of a smooth curvature-bounded extension through the ring, but stronger statements about geodesic completeness or low-regularity inextendibility require their own hypotheses. An invariant calculation must keep those different singularity notions distinct.

Exercise

Why does $C_{abcd}C^{abcd}=0$ at one Kerr point fail to prove local flatness, while $|\mathcal W_-|\to\infty$ proves curvature blowup?

Solution

The real contraction is only one component of the complex invariant and can vanish by cancellation while the pseudoscalar $C_{abcd}{*C}^{abcd}$ remains nonzero. Indeed Kerr is type D with $\Psi_2\ne0$ at generic zeros of the real part. By contrast, $|\mathcal W_-|=\sqrt{K^2+P^2}$ is the nonnegative magnitude of both contractions; its divergence forces at least one scalar polynomial curvature invariant to be unbounded.

References

• Abdelqader, Majd, and Kayll Lake. “Invariant Characterization of the Kerr Spacetime: Locating the Horizon and Measuring the Mass and Spin of Rotating Black Holes Using Curvature Invariants.” Physical Review D 91, no. 8 (2015): 084017. arXiv:1412.8757v4 [gr-qc]. DOI: 10.1103/PhysRevD.91.084017. Open PDF.
• Henry, Richard C. “Kretschmann Scalar for a Kerr-Newman Black Hole.” The Astrophysical Journal 535, no. 1 (2000): 350–353. arXiv:astro-ph/9912320v1. DOI: 10.1086/308819. Open PDF.
• 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.
• Lake, Kayll. “Invariants of the Kerr Vacuum.” arXiv:gr-qc/0302087v3 (2003). Open PDF.