Tetrad, Newman-Penrose, and Spinor Symbols

This card fixes the basis order, normalization, orientation, and curvature convention for orthonormal and complex null tetrads. It defines the site’s Newman–Penrose derivatives, spin coefficients, Weyl scalars, and minimal two-spinor dictionary; derivations and field equations remain in the linked Library chapters. The metric signature throughout is $(-,+,+,+)$.

From an orthonormal tetrad to a null tetrad

In four spacetime dimensions, an orthonormal tetrad $e_{\hat a}{}^\mu$ and its dual coframe $\theta^{\hat a}{}_{\mu}$ obey

$$g(e_{\hat a},e_{\hat b})=\eta_{\hat a\hat b}, \qquad \theta^{\hat a}{}_{\mu}e_{\hat b}{}^\mu =\delta^{\hat a}{}_{\hat b}, \qquad \eta_{\hat a\hat b}=\operatorname{diag}(-1,1,1,1).$$

Coordinate and frame components are related by

$$V^{\hat a}=\theta^{\hat a}{}_{\mu}V^\mu, \qquad V^\mu=e_{\hat a}{}^\mu V^{\hat a},$$

and the two inverse matrices reconstruct the metric in covariant and contravariant form:

$$g_{\mu\nu} =\eta_{\hat a\hat b}\theta^{\hat a}{}_{\mu}\theta^{\hat b}{}_{\nu}, \qquad g^{\mu\nu} =\eta^{\hat a\hat b}e_{\hat a}{}^\mu e_{\hat b}{}^\nu.$$

These are exact frame identities, not a locally flat approximation. Carroll’s tetrad construction and component conversion are given in Carroll 1997, § 3, pp. 88–92, eqs. (3.114)–(3.129). The complete geometric treatment is Orthonormal Tetrads and Measured Components.

Choose $e_{\hat0}$ future directed and the real tetrad positively oriented. The site’s ordered complex null tetrad is

$$\ell^a=\frac{e_{\hat0}^a+e_{\hat1}^a}{\sqrt2}, \qquad n^a=\frac{e_{\hat0}^a-e_{\hat1}^a}{\sqrt2}, \qquad m^a=\frac{e_{\hat2}^a+i e_{\hat3}^a}{\sqrt2}, \qquad \bar m^a=\frac{e_{\hat2}^a-i e_{\hat3}^a}{\sqrt2}.$$

The round trip is

$$e_{\hat0}^a=\frac{\ell^a+n^a}{\sqrt2}, \qquad e_{\hat1}^a=\frac{\ell^a-n^a}{\sqrt2}, \qquad e_{\hat2}^a=\frac{m^a+\bar m^a}{\sqrt2}, \qquad e_{\hat3}^a=\frac{m^a-\bar m^a}{i\sqrt2}.$$

The bar means complex conjugation. The only nonzero inner products are

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

and the metric, identity map, and orientation reconstruct as

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

Together these reconstructions test normalization, conjugation, and orientation. Exact frame-change laws are developed in Null-Tetrad Gauge Transformations. The construction, normalization, and patching questions are developed in Null Tetrads and Normalization. For the residual boost-spin subgroup,

$$\ell\mapsto A\ell, \qquad n\mapsto A^{-1}n, \qquad m\mapsto e^{i\chi}m, \qquad \bar m\mapsto e^{-i\chi}\bar m, \qquad A\gt0.$$

Directional derivatives and the twelve connection scalars

The four Newman–Penrose directional derivatives are names for derivatives along the ordered tetrad:

$$D=\ell^a\nabla_a, \qquad \Delta=n^a\nabla_a, \qquad \delta=m^a\nabla_a, \qquad \bar\delta=\bar m^a\nabla_a.$$

On a scalar these reduce to directional partial derivatives. On a tensor or spinor, the connection and the order of projection still matter. In particular, $\Delta$ here is an operator; it is not the Kerr horizon polynomial or a finite-difference increment.

The site uses a prime-symmetric GHP family for the twelve Levi-Civita connection components. Eight transform homogeneously under a boost and spin:

$$\begin{aligned} \kappa&=-\ell^a m^b\nabla_a\ell_b, &\kappa'&=-\bar m^a n^b\nabla_b n_a,\\ \rho&=-m^a\bar m^b\nabla_b\ell_a, &\rho'&=-m^a\bar m^b\nabla_a n_b,\\ \sigma&=-m^a m^b\nabla_a\ell_b, &\sigma'&=-\bar m^a\bar m^b\nabla_b n_a,\\ \tau&=-m^a n^b\nabla_b\ell_a, &\tau'&=-\ell^a\bar m^b\nabla_a n_b. \end{aligned}$$

The other four are boost-spin connection coefficients and transform inhomogeneously when the frame change varies with position:

$$\begin{aligned} \beta={}&\frac12m^a\bar m^b\nabla_a m_b +\frac12\ell^a m^b\nabla_b n_a,\\ \beta'={}&-\frac12\bar m^a\bar m^b\nabla_b m_a -\frac12\ell^a\bar m^b\nabla_b n_a,\\ \epsilon={}&\frac12\ell^a\bar m^b\nabla_a m_b +\frac12\ell^a\ell^b\nabla_b n_a,\\ \epsilon'={}&-\frac12\bar m^a n^b\nabla_b m_a -\frac12\ell^a n^b\nabla_b n_a. \end{aligned}$$

These contractions are the definitions. They agree with the modern prime-symmetric formulas in Andersson et al. 2022, § 3.1, Definition 3.1, eqs. (3.1a–d), (3.2a–b) after translating that paper’s $(+,-,-,-)$ package. Translate the complete convention package contraction by contraction; do not attach one guessed sign to every familiar name.

Older NP notation usually replaces the six primed labels as follows:

Site label	Common older label	Why the contraction still wins
$\kappa'$	$-\nu$	A source may also reverse the null normalization or the defining sign
$\sigma'$	$-\lambda$	Complex-conjugation placement varies with the priming convention
$\rho'$	$-\mu$	The same letter $\mu$ is also widely used for an index or mass scale
$\tau'$	$-\pi$	This $\pi$ is a connection scalar, not the number $3.14159\ldots$
$\beta'$	$-\alpha$	Both are frame-connection components, not invariant scalars
$\epsilon'$	$-\gamma$	The label $\gamma$ can also denote a metric or Lorentz factor

These signed aliases use the same mostly-plus NP/GHP package as this card and follow Spiers, Pound, and Moxon 2023, Appendix B, p. 17, eqs. (B4)–(B6). They are not a license to translate another source by its labels alone: match that source’s contractions, tetrad order, signature, curvature sign, and prime operation to the definitions above. For constant $A$ and $\chi$, representative weights are $\rho\mapsto A\rho$, $\sigma\mapsto Ae^{2i\chi}\sigma$, and $\kappa\mapsto A^2e^{i\chi}\kappa$. Position-dependent transformations add derivatives of $A$ and $\chi$ to $\epsilon$, $\epsilon'$, $\beta$, and $\beta'$ because those four are connection coefficients. The full optical interpretation, including $\theta_{(\ell)}=-2\operatorname{Re}\rho$, is developed in Spin Coefficients and Null-Congruence Geometry.

Weyl scalars and their frame weights

Write $C(x,y,z,w)=C_{abcd}x^ay^bz^cw^d$. With the site’s Riemann convention, the five Weyl scalars are

$$\begin{aligned} \Psi_0&=-C(\ell,m,\ell,m), &\Psi_1&=-C(\ell,m,\ell,n),\\ \Psi_2&=-C(\ell,m,\bar m,n), &\Psi_3&=-C(\ell,n,\bar m,n),\\ \Psi_4&=-C(\bar m,n,\bar m,n).&& \end{aligned}$$

The primed-family curvature definitions are cross-checked against Andersson et al. 2022, § 3.1, Definition 3.2, eqs. (3.3a–b), with the same complete signature translation.

Their boost and spin weights $(b,s)$, in order from $\Psi_0$ to $\Psi_4$, are

$$(2,2),\quad(1,1),\quad(0,0),\quad(-1,-1),\quad(-2,-2).$$

Thus $\Psi_2$ is spin-boost invariant but not invariant under a general null rotation; $\Psi_0$ and $\Psi_4$ need not be complex conjugates. Their geometric development is in Weyl, Ricci, and Scalar Curvature Components.

Minimal two-spinor bridge

Unprimed capital indices $A,B,\ldots$ label a complex two-dimensional spin space; primed indices $A',B',\ldots$ label its complex-conjugate space. Fix the antisymmetric spin metric by

$$\epsilon_{AB}=-\epsilon_{BA}, \qquad \epsilon_{01}=-1, \qquad \epsilon^{01}=+1,$$

with raising and lowering

$$\xi_A=\epsilon_{AB}\xi^B, \qquad \xi^A=\epsilon^{AB}\xi_B.$$

For an ordered dyad $(o^A,\iota^A)$, the site’s normalization is

$$o_A\iota^A=1, \qquad o^A\iota_A=-1.$$

Let $\varsigma^a{}_{AA'}$ denote the soldering form, fixed here by

$$g_{ab}\varsigma^a{}_{AA'}\varsigma^b{}_{BB'} =-\epsilon_{AB}\bar\epsilon_{A'B'}.$$

Then the dyad generates the normalized null tetrad:

$$\ell^{AA'}=o^A\bar o^{A'}, \qquad n^{AA'}=\iota^A\bar\iota^{A'}, \qquad m^{AA'}=o^A\bar\iota^{A'}, \qquad \bar m^{AA'}=\iota^A\bar o^{A'}.$$

Applying the soldering form gives the corresponding spacetime vectors:

$$\ell^a=o^A\bar o^{A'}\varsigma^a{}_{AA'}, \qquad n^a=\iota^A\bar\iota^{A'}\varsigma^a{}_{AA'}, \qquad m^a=o^A\bar\iota^{A'}\varsigma^a{}_{AA'}, \qquad \bar m^a=\iota^A\bar o^{A'}\varsigma^a{}_{AA'}.$$

The minus sign in the soldering identity makes this bridge reproduce $\ell\cdot n=-1$ and $m\cdot\bar m=1$. The executable xAct derivation gives the spin metric and dyad in García-Parrado 2009, § 2, pp. 3–5, inputs (18)–(28) and the dyad–null-tetrad bridge in § 2, p. 10, inputs (78)–(90). It uses the opposite signature package, so the site identity is translated and then checked by reconstruction.

Define the totally symmetric Weyl spinor $\Psi_{ABCD}$ so that its dyad components equal the tensor contractions above:

$$\begin{aligned} \Psi_0&=\Psi_{ABCD}o^Ao^Bo^Co^D, &\Psi_1&=\Psi_{ABCD}o^Ao^Bo^C\iota^D,\\ \Psi_2&=\Psi_{ABCD}o^Ao^B\iota^C\iota^D, &\Psi_3&=\Psi_{ABCD}o^A\iota^B\iota^C\iota^D,\\ \Psi_4&=\Psi_{ABCD}\iota^A\iota^B\iota^C\iota^D.&& \end{aligned}$$

This is the entire two-spinor scope of the card. A local dyad does not by itself establish the existence of a global spin structure, and these symbols must not be conflated with the four-component Clifford/Dirac conventions used elsewhere. For that geometry, continue to Frames, Tetrads, Spinors, and Clifford Geometry.

Checks and symbol collisions

Three checks catch different classes of error:

1. Reconstruct $g_{ab}$ and $\epsilon_{abcd}\ell^an^bm^c\bar m^d$ from the tetrad. This detects a wrong normalization, conjugation, or orientation.
2. A constant Cartesian null tetrad on Minkowski spacetime has all twelve spin coefficients and all five Weyl scalars equal to zero. In a spherical null tetrad on the same flat spacetime, every $\Psi_i$ vanishes although $\rho=-1/(\sqrt2r)$ is nonzero. A nonzero connection component is not curvature.
3. In a principal Schwarzschild tetrad compatible with these definitions, $\Psi_2=+GM/r^3$ and the other four Weyl scalars vanish. The Kerr principal-null calculation gives the complete source-sign translation. A null rotation can repopulate them without changing the Weyl tensor.

Five complex $\Psi_i$ carry ten real components, exactly the component count of the four-dimensional Weyl tensor. This count is independent of the flat and type-D checks and exposes an invalid conjugacy condition.

Symbol	Meaning on this card	Frequent collision
$\ell^a$	First real null vector	The letter $l$ or the numeral $1$
$m^a$	Complex transverse null vector	A mass parameter or azimuthal mode number $m$
$\rho$	NP connection scalar	Mass density or a radial coordinate
$\sigma$	NP shear scalar	Soldering form, conductivity, or cross section
$\kappa$	NP nongeodesicity scalar	Surface gravity or gravitational coupling
$\epsilon$	NP connection scalar	Spin metric $\epsilon_{AB}$ or spacetime volume tensor $\epsilon_{abcd}$
$\Delta$	Derivative $n^a\nabla_a$	Kerr polynomial, Laplacian, or finite difference
$\delta$	Derivative $m^a\nabla_a$	A variation or Dirac delta distribution
$\Lambda_{\mathrm{NP}}$	Curvature trace $R/24$ on the curvature card	Cosmological constant $\Lambda$
$\Psi_i$	Tetrad components of Weyl curvature	A wavefunction or gauge-invariant master field

A page using any collision must redeclare the local meaning before the first consequential formula. For full NP identities and GHP derivatives, use Spinor, Newman–Penrose, and GHP Methods. For the Kerr-adapted Kinnersley specialization, use Kinnersley Tetrads and Newman–Penrose Curvature.

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; arXiv:2108.03148v2.
• Carroll, Sean M. Lecture Notes on General Relativity. arXiv:gr-qc/9712019 (1997). arXiv.
• García-Parrado Gómez-Lobo, Alfonso. The N.P. and G.H.P. Formalisms: An Application of Spinors and xCoba. Executable xAct notebook documentation (2009). Official PDF.
• Spiers, Andrew, Adam Pound, and Jordan Moxon. “Second-Order Teukolsky Formalism in Kerr Spacetime: Formulation and Nonlinear Source.” Physical Review D 108 (2023): 064002. DOI; arXiv:2305.19332.