Spinor, Newman-Penrose, and GHP Methods

Null-tetrad methods reorganize general relativity around two null directions and a complex transverse screen. Connection and curvature tensors become scalars with direct optical, algebraic, and radiative interpretations; Einstein and Bianchi identities become coupled first-order directional equations. The gain is greatest when the null directions are geometrically preferred, but only if normalization, curvature signs, and frame freedom remain explicit throughout the calculation.

Helpful background. The Einstein equation supplies the field equation projected in this chapter; Hypersurface normals and tangential projectors helps relate null directions to hypersurfaces; Frames, coframes, and anholonomy supplies tetrad connections; The Riemann curvature tensor fixes curvature projection; Timelike congruences and spatial projection supplies the expansion–shear–twist language whose null form appears here.

Enter this chapter

The chapter follows one sequence. First construct and normalize the null tetrad. Next project the connection to obtain spin coefficients and project curvature to obtain Weyl and Ricci scalars. Then use frame commutators, Ricci identities, Einstein’s equation, and the Bianchi identity to derive scalar field equations. Finally separate the full local Lorentz freedom from the residual spin-boost freedom and replace ordinary directional derivatives by GHP-covariant ones.

This is a local calculus. A smooth preferred pair of null directions need not admit one global complex transverse vector; the sphere already supplies the familiar obstruction. GHP quantities solve the patching problem by transforming with definite weights on overlaps. They do not make a singular choice of principal null directions smooth, and they do not replace spacetime diffeomorphism covariance.

The chapter uses spinor structure implicitly: the five complex Weyl scalars are the five dyad components of the totally symmetric Weyl spinor. The spinor viewpoint explains their binomial mixing under null rotations and their boost-spin weights. A full development of abstract two-spinor geometry, spin structures, and spinor bundles is beyond this chapter’s scope, as are spacetime-specific Teukolsky equations.

Fixed normalization and sign discipline

From a future-directed, positively oriented orthonormal tetrad $(e_0,e_1,e_2,e_3)$, define

$$\ell=\frac{e_0+e_1}{\sqrt2},\qquad n=\frac{e_0-e_1}{\sqrt2},\qquad m=\frac{e_2+i e_3}{\sqrt2},\qquad \bar m=\frac{e_2-i e_3}{\sqrt2}.$$

The full normalization used on every page is

$$\ell^2=n^2=m^2=\bar m^2=0, \qquad \ell\cdot n=-1, \qquad m\cdot\bar m=1,$$

with all real-transverse mixed products zero. Consequently,

$$g_{ab}=-\ell_a n_b-n_a\ell_b +m_a\bar m_b+\bar m_a m_b, \qquad \epsilon_{abcd}\ell^a n^b m^c\bar m^d=i.$$

The bar always means complex conjugation. The metric signature is $(-,+,+,+)$, and the Riemann convention is the site’s global convention. Scalar names alone do not determine a sign across Newman–Penrose sources. A paper using $\ell\cdot n=+1$ and $m\cdot\bar m=-1$ cannot be translated by visually matching $\rho$, $\Psi_2$, or $\Phi_{00}$; each quantity must be reconstructed from its declared contraction and index order. The chapter therefore defines spin coefficients and curvature scalars before using them. The opposite-sign source convention is visible in Gómez López and Quiroga 2017, §II.A–B, equations (3)–(28), pp. 2–3, PDF, while this chapter’s translation is checked against the explicit definitions in Andersson et al. 2022, §3.1, Definitions 3.1–3.2, equations (3.1)–(3.3), PDF.

Route through the six methods

1. Null Tetrads and Normalization constructs the complex frame, reconstructs the metric and orientation, and identifies boosts, spins, and null rotations.
2. Spin Coefficients and Null-Congruence Geometry defines all twelve connection components. It distinguishes the full screen-area expansion $\theta_{(\ell)}=-2\operatorname{Re}\rho$ from the averaged Newman–Penrose expansion $-\operatorname{Re}\rho$.
3. Weyl, Ricci, and Scalar Curvature Components packages the twenty independent real curvature components into five complex Weyl scalars, nine trace-free Ricci components, and one trace scalar.
4. Newman–Penrose Field and Bianchi Equations derives representative commutator, optical, Einstein, and type-D Bianchi equations and tests them in flat and Schwarzschild geometries.
5. Null-Tetrad Gauge Transformations gives both null-rotation polynomials and separates tetrad gauge from diffeomorphism gauge.
6. Geroch–Held–Penrose Calculus assigns $\{p,q\}$ types, constructs four weighted derivatives, and states precisely the residual covariance they preserve.

The order matters pedagogically. Curvature components cannot be compared before the tetrad contractions are fixed; field equations cannot be checked before connection and curvature conventions agree; GHP operators cannot be constructed before the inhomogeneous connection coefficients and homogeneous weights are known.

Checks that survive a long calculation

Normalization should be checked by reconstructing $g_{ab}$, not merely by verifying one inner product. Curvature components should pass dimension counting: five complex Weyl scalars carry ten real components, while the Ricci set carries ten. Every GHP equation should have one common $\{p,q\}$ type. Flat spacetime should make all curvature scalars vanish but need not make spin coefficients or frame commutators vanish. Constant curvature should make the Weyl scalars and trace-free Ricci scalars vanish while leaving $R/24$ nonzero. In a principal type-D vacuum tetrad only $\Psi_2$ survives, but a null rotation repopulates other Weyl scalars without changing the curvature invariants.

These checks detect different errors. Weight matching detects a missing frame factor but not a common curvature-sign error. A flat limit detects spurious curvature but not an incorrect factor of two in the expansion. The area identity $\theta_{(\ell)}=\ell(\ln\sqrt q)=-2\operatorname{Re}\rho$ supplies that independent factor-of-two check.

Review the chapter

In Minkowski spacetime, use the spherical null tetrad

$$\ell=\frac{\partial_t+\partial_r}{\sqrt2},\qquad n=\frac{\partial_t-\partial_r}{\sqrt2},\qquad m=\frac1{\sqrt2r}(\partial_\theta+i\csc\theta\,\partial_\phi).$$

Verify its normalization, find $\rho$, compare the averaged and area expansions, identify the Weyl scalars, and check the GHP types in the optical equation.

Solution

The spherical metric gives $\ell\cdot n=-1$, $m\cdot\bar m=1$, and all remaining pairings zero. Direct connection projection yields $\rho=-1/(\sqrt2r)$, with $\kappa=\sigma=0$. Hence the averaged Newman–Penrose expansion is $-\operatorname{Re}\rho=1/(\sqrt2r)$, while the full area expansion is

$$\theta_{(\ell)}=-2\operatorname{Re}\rho=\frac{\sqrt2}{r} =\ell(\ln r^2).$$

Flatness gives $\Psi_0=\cdots=\Psi_4=0$ and $\Phi_{00}=0$. Since $\rho$ has type $\{1,1\}$, $\mathsf P\rho$ and $\rho^2$ have type $\{2,2\}$. The vanished shear product $\sigma\bar\sigma$ and $\Phi_{00}$ also have type $\{2,2\}$, so the equation $\mathsf P\rho=\rho^2$ is weight consistent. In this affine, spin-parallel tetrad $\mathsf P=D$, and both sides equal $1/(2r^2)$.

References

• Andersson, Lars, Thomas Bäckdahl, Pieter Blue, and Siyuan Ma. “Nonlinear Radiation Gauge for Near Kerr Spacetimes.” Communications in Mathematical Physics 396 (2022): 45–90. DOI.
• Geroch, Robert, A. Held, and Roger Penrose. “A Space-Time Calculus Based on Pairs of Null Directions.” Journal of Mathematical Physics 14, no. 7 (1973): 874–881. DOI.
• Gómez López, L. A., and G. D. Quiroga. “Asymptotic Structure of Spacetime and the Newman–Penrose Formalism: A Brief Review.” arXiv:1711.11381v1 (2017). arXiv.