Schouten tensor, prescribed curvature, electric-magnetic Weyl variables and the reconstruction formula in the dust reduction

Work in the reduction conventions of Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns: $\gamma$ is the signature $(+---)$ metric, $U=u$ the prescribed unit future timelike vector (not necessarily the hypersurface normal), $r=\rho c^2$, the frame $(e_0,e_1,e_2,e_3)$ is $\gamma$-orthonormal with $e_0=U$, $\gamma (e_i,e_j)={\eta }_{ij}$ on spatial pairs, and indices are moved with $\gamma$.

(a) Schouten tensor. The reduction Schouten tensor of $\gamma$ is $S_{ij}:={\mathrm{Ric}}_{ij}-\frac{1}{6}{S}_{\gamma }{\gamma }_{ij},$ where $\mathrm{Ric}$ and $S_{\gamma }={\gamma }^{ab}{\mathrm{Ric}}_{ab}$ are the Ricci tensor and scalar curvature of $\gamma$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). If the dust field equation $G(\gamma )-\mathrm{\Lambda }\gamma =\kappa r{U}^{\mathrm{♭}}\otimes U^{\mathrm{♭}}$ holds, then $S_{ij}=\kappa r(U_i{U}_j-\frac{1}{3}{\gamma }_{ij})-\frac{1}{3}\mathrm{\Lambda }{\gamma }_{ij}\mathrm{.}$ Define the Weyl tensor of the metric-selected connection by $C_{ijkl}:=R_{ijkl}-K(S,\gamma {)}_{ijkl},K(S,\gamma {)}_{ijkl}:=\frac{1}{2}(S_{ik}{\gamma }_{jl}-S_{il}{\gamma }_{jk}+S_{jl}{\gamma }_{ik}-S_{jk}{\gamma }_{il})\mathrm{.}$ Here $R$ is the lowered curvature tensor. The curvature symmetries and Ricci symmetry of The algebraic symmetries of the curvature tensor and the symmetry of the Ricci and Einstein tensors transfer from $g$ to $\gamma =-g$ because their lowered curvatures differ by a constant minus sign. Symmetry of $S$ and $\gamma$ gives the same two skew-symmetries, pair symmetry and cyclic identity for $K(S,\gamma )$ by expanding its four terms. Moreover ${\mathrm{tr}}_{\gamma }S=S_{\gamma }\mathrm{/}3$ and ${\gamma }^{ik}K(S,\gamma {)}_{ijkl}=S_{jl}+\frac{1}{2}({\mathrm{tr}}_{\gamma }S){\gamma }_{jl}={\mathrm{Ric}}_{jl},{\gamma }^{ik}{C}_{ijkl}=0.$ The other metric traces vanish by the curvature symmetries. Thus $C$ is a smooth curvature-symmetric, trace-free covariant four-tensor, and the Schouten decomposition $R=C+K(S,\gamma )$ holds by definition. A tensor with these algebraic symmetries and zero metric traces is called Weyl-type, whether or not it is the curvature part of a metric. The constant-curvature specialization is checked in the Remarks.

(b) Prescribed curvature. The prescribed curvature tensor of a configuration $(C,E,B,r)$ is the tensor field with frame components $R^{\mathrm{p}\mathrm{r}\mathrm{e}}{}_{ijkl}:=C_{ijkl}+\frac{1}{2}(S_{ik}{\gamma }_{jl}-S_{il}{\gamma }_{jk}+S_{jl}{\gamma }_{ik}-S_{jk}{\gamma }_{il}),S_{ij}\text{ as in (a)},$ where $C=C[E,B]$ is the Weyl tensor reconstructed from the electric and magnetic variables in (d). A tentative reduced configuration is a smooth assignment $(e_a{}^{\mu },{\mathrm{\Gamma }}_a{}^i{}_j,E_{ij},B_{ij},r)$ in the gauge of Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns. Equality of its actual curvature with $R^{\mathrm{p}\mathrm{r}\mathrm{e}}$ is the curvature constraint, rather than an assumption on the independent unknowns. Reduced evolution enforces its time components; the residual theorem establishes the remaining components from constrained initial data. This distinction keeps the tentative connection available before torsion-freeness and the full curvature constraint are proved.

(c) Electric and magnetic Weyl variables. For a smooth Weyl-type tensor $C$ of type $(0,4)$ with the curvature symmetries write $C^{\ast }{}_{ijkl}:=\frac{1}{2}{C}_{ij}{}^{pq}{ϵ}_{klpq}$ for its right dual (Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns clause (c) for ${ϵ}_{ijkl}$). The electric and magnetic parts with respect to $U$ are the spatial fields $E_{ij}:=C_{0i0j},B_{ij}:=C^{\ast }{}_{0i0j},$ with frame indices; they are symmetric and spatial in all indices and trace-free with respect to $h$ whenever $C$ has the Weyl symmetries.

(d) Reconstruction formula. Define the dependent tensor $C_{ijkl}:=2(l_{j[k}{E}_{l]i}-l_{i[k}{E}_{l]j})-2(U_{[k}{B}_{l]p}{ϵ}^p{}_{ij}+U_{[i}{B}_{j]p}{ϵ}^p{}_{kl}),$ with brackets of weight $\frac{1}{2}$ and $l_{ij}=h_{ij}-U_i{U}_j$; this expression is the reconstruction of a Weyl tensor from a pair $(E,B)$ of symmetric spatial trace-free fields. The verification that the reconstructed tensor has the curvature antisymmetries, pair symmetry, cyclic identity and zero traces — and that its electric and magnetic parts recover $(E,B)$ — is the content of Reconstruction of a Weyl tensor from its electric and magnetic parts: symmetries, trace-freeness and recovery.

The definition fixes the objects and conventions only; the evolution and subsidiary equations, the initial-data determination and the local existence theorem are proved in the items named above and in The reduced dust evolution equations: Weyl system, trace preservation, transport equations and symmetric hyperbolicity and Local Einstein-dust evolution: smooth constrained data with prescribed velocity admit a smooth local development, unique up to a data-fixing isometry.

• Normalization check. For a constant-curvature metric with $R_{ijkl}=K({\gamma }_{ik}{\gamma }_{jl}-{\gamma }_{il}{\gamma }_{jk})$ one has ${\mathrm{Ric}}_{ij}=3K{\gamma }_{ij}$, $S_{\gamma }=12K$ and $S_{ij}=3K{\gamma }_{ij}-2K{\gamma }_{ij}=K{\gamma }_{ij}$, so the decomposition of (a) reads $C+\frac{1}{2}(K{\gamma }_{ik}{\gamma }_{jl}-K{\gamma }_{il}{\gamma }_{jk}+K{\gamma }_{jl}{\gamma }_{ik}-K{\gamma }_{jk}{\gamma }_{il})=C+K({\gamma }_{ik}{\gamma }_{jl}-{\gamma }_{il}{\gamma }_{jk})=R$, as required for a metric of constant curvature. The design's original display used the same Schouten normalization with a differently normalized bracket expression; the normalization above is the one under which the Kulkarni--Nomizu identity and this check both hold, and the discrepancy is recorded in the batch notes for owner review.
• Vacuum sign check of (a). For $r=0$, tracing $G(\gamma )-\mathrm{\Lambda }\gamma =0$ with ${\gamma }^{-1}$ in dimension four gives $-S_{\gamma }-4\mathrm{\Lambda }=0$, hence $S_{\gamma }=-4\mathrm{\Lambda }$ and $\mathrm{Ric}=-\mathrm{\Lambda }\gamma$. Therefore $S_{ij}=-\mathrm{\Lambda }{\gamma }_{ij}+\frac{2}{3}\mathrm{\Lambda }{\gamma }_{ij}=-\frac{1}{3}\mathrm{\Lambda }{\gamma }_{ij}$, exactly as in (a). In the constant-curvature vacuum case the preceding normalization check has $K=-\mathrm{\Lambda }\mathrm{/}3$. Returning to $g=-\gamma$ gives $\mathrm{Ric}(g)=\mathrm{\Lambda }g$ and $S_g=4\mathrm{\Lambda }$, the usual de Sitter signs when $\mathrm{\Lambda }>0$; the reduction signs reflect the metric flip.