Dirac and Maxwell Separability

Separability beyond spin zero requires more than inserting a product ansatz into coordinate components. Spinors and electromagnetic fields must be expressed in a frame aligned with Kerr’s repeated principal null directions, and their components carry spin- and boost-weight. In that adapted description, the principal Killing–Yano geometry supplies commuting operators and the type-D curvature removes otherwise obstructing couplings.

Required background. Killing Fields and Isometry Groups supplies the Fourier mode generators; Affine and Variational Geodesics supplies the principal-null and geometric-optics interpretation.

Helpful background. Einstein’s Equation distinguishes vacuum Kerr from electrovac Kerr–Newman; The Riemann Curvature Tensor supplies algebraic curvature; Exact Solutions of Einstein’s Equations supplies the type-D backgrounds; Frames, Coframes, and Anholonomy supplies null tetrads and spin connections.

The adapted principal frame

Choose a Newman–Penrose tetrad $(\ell^a,n^a,m^a,\bar m^a)$ with $\ell^a$ and $n^a$ along the repeated principal null directions. Normalize it by

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

with all other inner products zero. A residual spin-boost transformation changes tetrad components but not the underlying fields, so any separated variable must be accompanied by its tetrad normalization. In a Kinnersley-type tetrad on Kerr, algebraic speciality makes only the background Weyl scalar $\Psi_2$ nonzero and eliminates key spin coefficients. The directional derivatives then split into combinations depending on $r$ and $\theta$ after the common factor $e^{-i\omega t+im\phi}$ is removed.

This frame alignment is a mechanism, not a coordinate-invariant definition of radiation. Extreme tetrad components can vanish or rescale under allowed frame changes, while invariant conclusions require the full field or appropriately controlled tetrads.

Dirac symmetry and separation

For a spinor $\psi$ of mass $\mu_D$, use the Clifford convention of the geometry volume, $\{\gamma^a,\gamma^b\}=2g^{ab}\mathbf1$. In this convention the curved-space equation is

$$(\gamma^a\nabla_a+\mu_D)\psi=0.$$

The absent factor $i$ is important for signature $(-,+,+,+)$ with this Clifford sign. In flat spacetime, multiplication by $\gamma^a\partial_a-\mu_D$ gives $(\Box-\mu_D^2)\psi=0$, hence the physical dispersion relation $\omega^2=|\mathbf k|^2+\mu_D^2$. The often-used form $(i\widehat\gamma^a\nabla_a-\mu_D)\psi=0$ is equivalent if $\widehat\gamma^a=i\gamma^a$, so that $\{\widehat\gamma^a,\widehat\gamma^b\}=-2g^{ab}\mathbf1$; the matrix and operator conventions must change together. The displayed equation and the compatible separated first-order pairs are given in Frolov, Krtouš, and Kubizňák 2017, §3.5, Eqs. (3.77)–(3.85).

The Killing–Yano two-form $f_{ab}$ produces a first-order differential symmetry operator which, with its appropriate chirality factor, commutes with the Dirac operator. Its eigenvalue supplies the separation constant. In the principal spin frame, a four-component mode can be arranged from products of two radial functions and two angular functions; the Dirac equation becomes coupled first-order radial and angular pairs. Squaring and recombining those pairs gives second-order equations, but the first-order system retains the cleanest control of signs and mass couplings.

The essential statement is invariant: separability follows from a nontrivial commuting first-order operator generated by $f_{ab}$, not simply from Petrov type D. Frolov, Krtouš, and Kubizňák 2017, §§6.4–6.5, pp. 127–133, Open PDF develop this operator account. Page’s explicit Kerr–Newman treatment gives the separated massive equations in Page 1976, pp. 1509–1510.

Maxwell fields and Teukolsky variables

For a source-free test field on vacuum Kerr,

$$\nabla_aF^{ab}=0, \qquad \nabla_{[a}F_{bc]}=0.$$

The extreme Newman–Penrose scalars

$$\phi_0=F_{ab}\ell^am^b, \qquad \phi_2=F_{ab}\bar m^a n^b$$

are the extreme field components, but the two variables entering Teukolsky’s unified master equation are not normalized symmetrically. In the Kinnersley tetrad define the Newman–Penrose spin coefficient

$$\rho_{\mathrm{NP}}=-\frac{1}{r-ia\cos\theta}.$$

This $\rho_{\mathrm{NP}}$ is unrelated to the Weyl cylindrical coordinate used elsewhere in the volume. The spin-$+1$ and spin-$-1$ master variables are

$$\psi_{+1}=\phi_0, \qquad \psi_{-1}=\rho_{\mathrm{NP}}^{-2}\phi_2.$$

Raw $\phi_2$ has spin weight $-1$, but the position-dependent factor is essential for it to satisfy the standard spin-$-1$ master equation in this tetrad. With the mode factor $e^{-i\omega t+im\phi}$, each $\psi_s$ separates into a radial function and a spin-weighted spheroidal harmonic. The intermediate component $\phi_1$ and the full Maxwell tensor are reconstructed using the remaining first-order equations and compatibility conditions. The complete variable dictionary appears in Teukolsky 1973, Table 1, printed p. 641, ADS PDF, within the derivation in §§II–IV, printed pp. 638–644.

A complementary potential construction uses the principal tensor to build a polarization operator acting on a separated scalar potential. It clarifies the common hidden geometry and extends naturally to higher-dimensional Kerr–NUT–(A)dS metrics, but gauge equivalence and mode completeness must be checked separately.

Limits of the statement

The established separation applies to test Dirac and Maxwell fields on the specified type-D backgrounds. It does not by itself prove mode completeness, reconstruct every stationary exceptional mode, choose physical boundary conditions, or establish stability. On Kerr–Newman, electromagnetic and gravitational perturbations are coupled; the uncoupled test-Maxwell statement cannot be promoted to full Einstein–Maxwell perturbations without additional analysis.

An independent check is the Schwarzschild limit. Spin-weighted spheroidal harmonics reduce to spin-weighted spherical harmonics, while the radial equations acquire spherical parity relations. A generic deformation that preserves stationarity and axisymmetry but destroys the repeated-principal alignment or Killing–Yano equation retains $(\omega,m)$ modes yet generally loses the radial–angular split.

Exercise

Under a constant tetrad spin rotation $m^a\mapsto e^{i\chi}m^a$, determine the phases of $\phi_0$ and $\phi_2$.

Solution

The conjugate vector transforms as $\bar m^a\mapsto e^{-i\chi}\bar m^a$, while $\ell^a$ and $n^a$ are fixed. Hence $\phi_0\mapsto e^{i\chi}\phi_0$ and $\phi_2\mapsto e^{-i\chi}\phi_2$. Their spin weights are therefore $+1$ and $-1$. This also shows why a single tetrad component is not a frame-independent scalar.

The coefficient $\rho_{\mathrm{NP}}$ has spin weight zero, so the rescaled master variable $\rho_{\mathrm{NP}}^{-2}\phi_2$ retains spin weight $-1$.

References

• Frolov, Valeri P., Pavel Krtouš, and David Kubizňák. “Black Holes, Hidden Symmetries, and Complete Integrability.” Living Reviews in Relativity 20 (2017): 6. DOI; Open PDF.
• Page, Don N. “Dirac Equation Around a Charged, Rotating Black Hole.” Physical Review D 14 (1976): 1509–1510. DOI.
• Teukolsky, Saul A. “Perturbations of a Rotating Black Hole. I. Fundamental Equations for Gravitational, Electromagnetic, and Neutrino-Field Perturbations.” Astrophysical Journal 185 (1973): 635–647. DOI; ADS PDF.