Null-Tetrad Gauge Transformations

A null tetrad is not unique. Boosts and spins preserve a chosen pair of null directions, whereas two classes of null rotations change one direction while holding the other fixed. These transformations leave the metric and curvature tensors unchanged but alter their tetrad components. Tracking that distinction separates invariant geometry from frame-dependent simplifications such as setting four Weyl scalars to zero in a principal type-D tetrad.

Required background. Frames, coframes, and anholonomy supplies local Lorentz transformations; The Riemann curvature tensor supplies the tensor whose components transform; The Einstein equation distinguishes frame choices from spacetime field equations.

Helpful background. Hypersurface normals and tangential projectors helps identify adapted null frames; Timelike congruences and spatial projection supplies the congruence interpretation of changing null directions.

Boosts and spins

Keep the normalization $\ell\cdot n=-1$ and $m\cdot\bar m=1$. The transformation

$$\ell'=A\ell,\qquad n'=A^{-1}n,\qquad m'=e^{i\chi}m, \qquad A>0,$$

preserves each real null direction. Properly weighted quantities transform homogeneously. Selected examples are

$$\begin{aligned} (\kappa)_{\mathrm{new}}&=A^2e^{i\chi}\kappa,& \sigma_{\mathrm{new}}&=Ae^{2i\chi}\sigma,& \rho_{\mathrm{new}}&=A\rho,& \tau_{\mathrm{new}}&=e^{i\chi}\tau,\\ (\Psi_0)_{\mathrm{new}}&=A^2e^{2i\chi}\Psi_0,& (\Psi_2)_{\mathrm{new}}&=\Psi_2,& (\Psi_4)_{\mathrm{new}}&=A^{-2}e^{-2i\chi}\Psi_4. \end{aligned}$$

For local $A(x)$ and $\chi(x)$, the four boost-spin connection coefficients transform inhomogeneously. For example, direct substitution into this chapter’s definition gives

$$\epsilon_{\mathrm{new}} =A\left(\epsilon+\frac12D\ln A+\frac{i}{2}D\chi\right).$$

This derivative term is why ordinary directional derivatives of weighted scalars are not themselves weighted. The boost-spin transformations compose pointwise by multiplying $A$ and adding $\chi$.

The two null rotations

A null rotation about $\ell$ with complex parameter $a$ is

$$\begin{aligned} \ell'&=\ell,& m'&=m+\bar a\ell,& \bar m'&=\bar m+a\ell,& n'&=n+a m+\bar a\bar m+|a|^2\ell. \end{aligned}$$

With the Weyl-scalar contractions fixed earlier, binomial expansion gives

$$\begin{aligned} \Psi_0'&=\Psi_0,\\ \Psi_1'&=\Psi_1+a\Psi_0,\\ \Psi_2'&=\Psi_2+2a\Psi_1+a^2\Psi_0,\\ \Psi_3'&=\Psi_3+3a\Psi_2+3a^2\Psi_1+a^3\Psi_0,\\ \Psi_4'&=\Psi_4+4a\Psi_3+6a^2\Psi_2+4a^3\Psi_1+a^4\Psi_0. \end{aligned}$$

A rotation about $n$ with parameter $b$ is

$$\begin{aligned} n'&=n,&m'&=m+b n,& \bar m'&=\bar m+\bar b n,& \ell'&=\ell+\bar b m+b\bar m+|b|^2n. \end{aligned}$$

It produces the reversed polynomial sequence:

$$\begin{aligned} \Psi_4'&=\Psi_4,\\ \Psi_3'&=\Psi_3+b\Psi_4,\\ \Psi_2'&=\Psi_2+2b\Psi_3+b^2\Psi_4,\\ \Psi_1'&=\Psi_1+3b\Psi_2+3b^2\Psi_3+b^3\Psi_4,\\ \Psi_0'&=\Psi_0+4b\Psi_1+6b^2\Psi_2+4b^3\Psi_3+b^4\Psi_4. \end{aligned}$$

The formulas follow from multilinearity and are therefore an independent check on parameter conjugations. The transformations in Gómez López and Quiroga 2017, §II.A, equations (11)–(18), pp. 2–3, PDF use a different parameter naming and the opposite metric dyad; matching the displayed tetrad transformation must precede comparison of scalar polynomials. Rotations about the same fixed direction add their complex parameters, but rotations about opposite directions do not commute. Together with boost-spin transformations, they generate the connected local Lorentz freedom of an oriented, time-oriented null tetrad.

Frame gauge is not spacetime gauge

A tetrad transformation changes the basis at a fixed point while keeping $g_{ab}$, $C_{abcd}$, and the spacetime point fixed. A diffeomorphism instead maps points and pulls back tensor fields. Coordinate changes are passive descriptions of that spacetime freedom. These operations may be combined in a calculation, but they answer different questions and have different gauge conditions. Andersson and collaborators explicitly separate diffeomorphism gauge from frame gauge in Andersson et al. 2022, §1, Definitions 1.1–1.2, and §3.1, PDF.

Individual Weyl scalars are tetrad components, not spacetime scalar invariants. Two useful Lorentz-invariant combinations in the present normalization are

$$I=\Psi_0\Psi_4-4\Psi_1\Psi_3+3\Psi_2^2,$$

and

$$J=\det\!\begin{pmatrix} \Psi_4&\Psi_3&\Psi_2\\ \Psi_3&\Psi_2&\Psi_1\\ \Psi_2&\Psi_1&\Psi_0 \end{pmatrix}.$$

Their values do not depend on the null tetrad, although their overall normalization is convention dependent. Vanishing patterns, by contrast, are meaningful only after an alignment statement. At points where principal null directions merge or a chosen frame becomes singular, an alignment gauge can cease to be smooth even while $I$ and $J$ remain regular.

Exercises

Begin with type-D curvature in a principal tetrad, so that only $\Psi_2=\psi$ is nonzero. Apply a null rotation about $\ell$ and verify that $I$ and $J$ are unchanged.

Solution

The polynomial laws give $\Psi_0'=\Psi_1'=0$, $\Psi_2'=\psi$, $\Psi_3'=3a\psi$, and $\Psi_4'=6a^2\psi$. Therefore

$$I'=3\psi^2,$$

which equals its principal-frame value. Substitution into the determinant gives $J'=-\psi^3$, also its original value. The rotated tetrad has nonzero $\Psi_3'$ and $\Psi_4'$, demonstrating that those components are frame dependent even though the curvature tensor and its invariants have not changed. The same calculation with the rotation about $n$ populates $\Psi_0'$ and $\Psi_1'$ instead.

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.
• 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.