Principal Null Directions and Type D

For $m\ne0$, Kerr is algebraically special at every regular point: its Weyl tensor has two repeated principal null directions. In a tetrad aligned with those directions, four of the five Newman–Penrose Weyl scalars vanish and only the Coulomb scalar $\Psi_2$ remains. This type-D pattern is invariant even though the individual scalar values depend on conventions and tetrad normalization. At $m=0$, the Kerr form is flat and the Weyl type degenerates to O.

Required background. Frames, Coframes, and Anholonomy supplies null frames and frame transformations.

Helpful background. The Einstein Equation supplies the vacuum hypothesis; The Riemann Curvature Tensor supplies the Weyl tensor; Exact Solutions of Einstein’s Equations supplies invariant classification; Causality and Global Structure supports domain qualifications; Killing Fields and Isometry Groups supplies the stationary symmetries.

Principal null directions

At one point, a null vector $k^a$ is a principal null direction of the Weyl tensor when

$$k_{[e}C_{a]bc[d}k_{f]}k^bk^c=0.$$

Equivalently, choose a normalized null tetrad $(\ell,n,m,\bar m)$ with

$$\ell\!\cdot n=-1, \qquad m\!\cdot\bar m=1,$$

all other products zero, and the tetrad orientation agreeing with $\epsilon_{0123}\gt0$. Curvature uses the site’s Riemann tensor. When a dual is needed, use the left Weyl dual

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

Define all five Newman–Penrose Weyl scalars by

$$\begin{aligned} \Psi_0&=-C_{abcd}\ell^am^b\ell^cm^d, &\Psi_1&=-C_{abcd}\ell^am^b\ell^cn^d,\\ \Psi_2&=-C_{abcd}\ell^am^b\bar m^cn^d, &\Psi_3&=-C_{abcd}\ell^an^b\bar m^cn^d,\\ \Psi_4&=-C_{abcd}\bar m^an^b\bar m^cn^d.&& \end{aligned}$$

Under a null rotation about $n$ with complex parameter $b$, candidate principal directions are roots of

$$\Psi_0+4b\Psi_1+6b^2\Psi_2+4b^3\Psi_3+b^4\Psi_4=0.$$

Four distinct roots give Petrov type I. Coincident roots yield types II, D, III, or N; vanishing Weyl curvature is type O. This multiplicity statement is tetrad invariant.

A principal tetrad for Kerr

For Boyer–Lindquist Kerr, a Kinnersley-type contravariant tetrad is

$$\ell^\mu=\left(\frac{r^2+a^2}{\Delta},1,0,\frac{a}{\Delta}\right),$$ $$n^\mu=\frac{1}{2\Sigma} \left(r^2+a^2,-\Delta,0,a\right),$$ $$m^\mu=\frac{1}{\sqrt2\,(r+ia\cos\theta)} \left(ia\sin\theta,0,1,i\csc\theta\right).$$

As displayed, this coordinate tetrad is defined only where $\Sigma\ne0$, $\Delta\ne0$, and $\sin\theta\ne0$. The $1/\Delta$ factors are a Boyer–Lindquist frame singularity; a boost and a horizon-regular coordinate change produce a smooth principal frame at a selected horizon. The $\csc\theta$ factor is the usual polar-frame singularity on the axis and is removed by using an angular frame patch. Direct contraction with the curvature and Newman–Penrose definitions above gives

$$\Psi_0=\Psi_1=\Psi_3=\Psi_4=0, \qquad \Psi_2=+\frac{m}{(r-ia\cos\theta)^3}.$$

The sign deserves an explicit source translation. Teukolsky uses the opposite metric signature, $\ell\cdot n=+1$ and $m\cdot\bar m=-1$, and writes $\rho=-1/(r-ia\cos\theta)$ and $\Psi_2^{\mathrm T}=m\rho^3$ in Teukolsky 1973, § IV, printed p. 640, Eqs. (4.2)–(4.6). Under the overall metric-sign change to the site’s normalization, the same contravariant tetrad has the dot products stated above and the all-lowered Weyl tensor changes sign, so $\Psi_2=-\Psi_2^{\mathrm T}=+m/(r-ia\cos\theta)^3$. Independently, Godazgar et al. use the site signature but define $\Psi_2^{\mathrm G}=C_{\ell m\bar m n}$ rather than the page’s $-C_{\ell m\bar m n}$; their Kerr limit is $\Psi_2^{\mathrm G}=-m/(r-ia\cos\theta)^3$. See Godazgar et al. 2021, § 2, printed p. 7, Eq. (2.16), and Appendix E, printed p. 41, Eqs. (E.7)–(E.9), Open PDF.

Therefore both $\ell$ and $n$ are repeated principal null directions and Kerr with $m\ne0$ is type D on its regular domain. Kerr’s original paper identifies the algebraically special structure in Kerr 1963, pp. 237–238. Its asymptotically Cartesian form also makes the $m=0$ degeneration manifestly flat in Kerr 1963, p. 238, Eq. (5); then $\Psi_2=0$, the Weyl tensor is type O, and the parameter $a$ only labels oblate spheroidal coordinates.

Congruence geometry and electrovac alignment

Extending $\ell$ and $n$ as vector fields gives two null congruences. In Kerr they are geodesic and shear free, although they expand and twist. The vacuum Goldberg–Sachs theorem relates this behavior to repeated Weyl principal directions, subject to its smoothness and non-conformal-flat hypotheses. Algebraic speciality is thus tied to a differential congruence structure, not merely to four vanishing components in one lucky frame.

For Kerr–Newman, the Maxwell and Weyl principal directions are aligned: in a principal Newman–Penrose frame only the Coulomb Maxwell scalar and the candidate $\Psi_2$ are nonzero. In the site’s conventions,

$$\Psi_2= \frac{m(r+ia\cos\theta)-q^2} {(r-ia\cos\theta)^3(r+ia\cos\theta)}, \qquad q^2=G(Q_e^2+Q_m^2).$$

Thus Kerr–Newman is generically type D, but the statement is pointwise. At a regular zero of this scalar the Weyl tensor is type O, not D; for $m\gt0$ and $q\ne0$, this occurs where $r=q^2/m$ and $a\cos\theta=0$. The Maxwell-sourced Ricci tensor generally remains nonzero there, so type O does not mean a flat spacetime. The electrovac scalar and alignment follow, after the scalar-definition translation just described, from Godazgar et al. 2021, Appendix E, printed p. 41, Eqs. (E.7)–(E.9), Open PDF.

The spacetime is electrovac rather than vacuum, so one must use the aligned electrovac extension rather than quote the vacuum Goldberg–Sachs theorem without checking Ricci terms. Frolov, Krtouš, and Kubizňák discuss the principal null directions and canonical type-D structure in Frolov, Krtouš, and Kubizňák 2017, §§ 3.2–3.3, pp. 43–49.

The value of $\Psi_2$ changes under sign and tetrad conventions, but its pattern and the root multiplicities do not. A tetrad boost $\ell\to A\ell$, $n\to A^{-1}n$ leaves $\Psi_2$ unchanged; a spin $m\to e^{i\vartheta}m$ also leaves it unchanged. These checks help distinguish true type D from a coordinate or normalization artifact.

Algebraic speciality is pointwise. It neither proves that a principal tetrad extends smoothly through every chart boundary nor supplies a global Killing field, horizon, or separability theorem. Kerr has those additional structures, but each requires its own differential or global check. Conversely, a singular principal tetrad at $\Delta=0$ does not change the Petrov type when a regular boosted frame exists.

Exercise

Use the principal-frame Weyl scalars to read the multiplicity of the principal-null-direction polynomial.

Solution

With only $\Psi_2\ne0$, the polynomial in a rotation about $n$ is $6\Psi_2b^2$. Thus $b=0$ is a double root representing $\ell$. The polynomial has degree two rather than four because the remaining two roots coincide at the point at infinity of the complex projective parameter; they represent the repeated direction $n$. Hence the multiplicity pattern is $(2,2)$, Petrov type D.

References

• 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.
• Godazgar, Hadi, Mahdi Godazgar, Ricardo Monteiro, David Peinador Veiga, and C. N. Pope. “Asymptotic Weyl Double Copy.” arXiv:2109.07866v2 [hep-th] (2021). 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.
• Teukolsky, Saul A. “Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations.” The Astrophysical Journal 185 (1973): 635–647. DOI: 10.1086/152444.