Weyl, Ricci, and Scalar Curvature Components

The curvature tensor becomes a finite set of complex scalars after projection onto a normalized null tetrad. Five Weyl scalars carry the ten real components of conformal curvature; a Hermitian array of Ricci scalars plus one real trace scalar carries the ten real components of the Ricci tensor. This repackaging exposes principal null directions and radiative components, but it is meaningful only after the tetrad, curvature sign, and contraction order are fixed explicitly.

Required background. Frames, coframes, and anholonomy supplies tetrad projection; The Riemann curvature tensor fixes the Riemann and Weyl tensors; The Einstein equation relates Ricci components to matter.

Helpful background. Hypersurface normals and tangential projectors helps interpret adapted null directions; Timelike congruences and spatial projection supplies the congruence setting in which these components are often used.

Five Weyl scalars

Use $(\ell,n,m,\bar m)$ with $\ell\cdot n=-1$ and $m\cdot\bar m=1$. For compactness write $C(x,y,z,w)=C_{abcd}x^ay^bz^cw^d$. This page defines

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

These equations, rather than the names $\Psi_i$, are the convention. With the site’s covariant Weyl tensor and $(-,+,+,+)$ metric, they agree numerically with Andersson et al. 2022, §3.1, Definition 3.2, equations (3.3a)–(3.3b), PDF after translating that source’s opposite metric signature. They are also equivalent, using Weyl symmetries and tracelessness, to the form in Gómez López and Quiroga 2017, §II.B, equation (28), p. 3, PDF, again only after its null normalization is translated.

Under a boost-spin transformation, the five weights $(b,s)$ are

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

in increasing order from $\Psi_0$ to $\Psi_4$. Thus $\Psi_2$ is invariant under this residual subgroup, while $\Psi_0$ and $\Psi_4$ scale oppositely. This does not make the individual scalars invariant under null rotations, which change the selected pair of null directions.

Ricci projections and the trace

Define the independent Ricci scalars by

$$\begin{aligned} \Phi_{00}&=\tfrac12 R_{ab}\ell^a\ell^b, &\Phi_{01}&=\tfrac12 R_{ab}\ell^a m^b, &\Phi_{02}&=\tfrac12 R_{ab}m^a m^b,\\ \Phi_{11}&=\tfrac14 R_{ab}(\ell^a n^b+m^a\bar m^b), &\Phi_{12}&=\tfrac12 R_{ab}n^a m^b, &\Phi_{22}&=\tfrac12 R_{ab}n^a n^b, \end{aligned}$$

together with $\Phi_{10}=\bar\Phi_{01}$, $\Phi_{20}=\bar\Phi_{02}$, and $\Phi_{21}=\bar\Phi_{12}$. The trace is recorded as

$$\Lambda_{\mathrm{NP}}=\frac{R}{24}.$$

The subscript matters: $\Lambda_{\mathrm{NP}}$ is not the cosmological constant in the Einstein equation. The two remaining real tetrad contractions reconstruct as

$$R_{ab}\ell^a n^b=2\Phi_{11}-6\Lambda_{\mathrm{NP}}, \qquad R_{ab}m^a\bar m^b=2\Phi_{11}+6\Lambda_{\mathrm{NP}}.$$

These formulas follow immediately from $R=-2R_{\ell n}+2R_{m\bar m}$ and independently check the factors of two. The three real diagonal entries, three complex off-diagonal entries, and the trace contain ten real numbers, exactly the dimension of a symmetric rank-two tensor in four dimensions.

When Einstein’s equation is imposed, these Ricci scalars are algebraic projections of the stress tensor. In vacuum with cosmological constant $\Lambda_{\mathrm{c}}$, one has $R_{ab}=\Lambda_{\mathrm{c}}g_{ab}$, so all $\Phi_{ij}$ vanish and $\Lambda_{\mathrm{NP}}=\Lambda_{\mathrm{c}}/6$. Calling such a spacetime “Ricci flat” would be incorrect unless $\Lambda_{\mathrm{c}}=0$.

Algebraic and limiting checks

For constant sectional curvature $K$,

$$R_{abcd}=K(g_{ac}g_{bd}-g_{ad}g_{bc}),$$

so $C_{abcd}=0$, every $\Psi_i$ vanishes, every $\Phi_{ij}$ vanishes, and $\Lambda_{\mathrm{NP}}=K/2$. This simultaneously checks traceless Weyl projection and Ricci reconstruction.

For a Ricci-flat Petrov type-D spacetime, a principal tetrad can be chosen so that

$$\Psi_0=\Psi_1=\Psi_3=\Psi_4=0, \qquad \Psi_2\ne0.$$

In the usual principal Schwarzschild tetrad compatible with the present definitions, $\Psi_2=+GM/r^3$. The sign follows from the site’s Riemann convention together with the displayed leading minus in $\Psi_2=-C(\ell,m,\bar m,n)$; the Kerr principal-null calculation gives the complete source translation. The statement that only $\Psi_2$ survives is tied to principal alignment; a null rotation generally repopulates the other four scalars without changing the Weyl tensor. Likewise, complex conjugation does not identify $\Psi_0$ with $\Psi_4$ in an arbitrary tetrad. Reality of $C_{abcd}$ is already encoded in the reconstruction from the five complex numbers.

A useful component count is therefore an independent diagnostic: five complex Weyl scalars give ten real degrees of freedom, while the Ricci set gives nine trace-free degrees plus one trace. If a proposed convention needs extra unconstrained components, or if it imposes a conjugacy relation between $\Psi_0$ and $\Psi_4$, it has discarded valid curvature data.

Exercises

Project the constant-curvature tensor above onto the null tetrad and verify the stated values of the Weyl and Ricci scalars.

Solution

In four dimensions the contraction gives $R_{ab}=3K g_{ab}$ and $R=12K$. Subtracting the Ricci traces from $R_{abcd}$ therefore gives $C_{abcd}=0$, so all five $\Psi_i$ vanish. Every off-diagonal or repeated-null Ricci projection vanishes because the corresponding metric product vanishes. Moreover, $R_{\ell n}=-3K$ and $R_{m\bar m}=3K$, so $\Phi_{11}=(-3K+3K)/4=0$. Finally $\Lambda_{\mathrm{NP}}=12K/24=K/2$. The reconstruction equations return $R_{\ell n}=-3K$ and $R_{m\bar m}=3K$, completing the check.

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.