Tentative coframe, Cartan torsion and curvature, and the curvature residual of the dust reduction

Work in the reduction conventions of Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns, with a smooth tentative frame $(e_0,e_1,e_2,e_3)$, $e_0=U$, dual coframe ${\theta }^i(e_j)={\delta }_j^i$ and smooth metric-compatible tentative connection coefficients ${\mathrm{\Gamma }}_i{}^k{}_j$. The connection acts on every tensor index by the tensor product rule and annihilates $\gamma$; torsion-freeness is not assumed. A differential $k$-form is a smooth alternating covariant $k$-tensor. The wedge is its alternating tensor product, normalized so that for a $p$-form $\alpha$ and a $q$-form $\beta$, $\alpha \wedge \beta$ is the sum over $(p,q)$ shuffles with their signs. The exterior derivative is specified, for smooth fields $X_0,\dots ,X_k$, by $d\alpha (X_0,\dots ,X_k)={\sum }_{a=0}^k(-1{)}^a{X}_a(\alpha (X_0,\dots ,{\widehat{X}}_a,\dots ,X_k))+{\sum }_{a<b}(-1{)}^{a+b}\alpha ([X_a,X_b],X_0,\dots ,{\widehat{X}}_a,\dots ,{\widehat{X}}_b,\dots ,X_k)\mathrm{.}$ Here a hat deletes a slot, and the undeleted slots in the second sum retain their order. These definitions fix the local frame-coordinate calculus.

(a) Connection forms. The connection $1$-forms are ${\omega }^i{}_j:={\mathrm{\Gamma }}_k{}^i{}_j{\theta }^k$, and the torsion $2$-forms and curvature $2$-forms are the Cartan expressions ${\mathrm{\Theta }}^i:=d{\theta }^i+{\omega }^i{}_j\wedge {\theta }^j,{\mathrm{\Omega }}^i{}_j:=d{\omega }^i{}_j+{\omega }^i{}_k\wedge {\omega }^k{}_j\mathrm{.}$ Components are denoted ${\mathrm{\Theta }}^i=\frac{1}{2}{\mathrm{\Theta }}^i{}_{jk}{\theta }^j\wedge {\theta }^k$ and similarly for $\mathrm{\Omega }$. The actual curvature uses the frame bracket in its component formula: $R^i{}_{jkl}={\mathrm{\partial }}_k{\mathrm{\Gamma }}_l{}^i{}_j-{\mathrm{\partial }}_l{\mathrm{\Gamma }}_k{}^i{}_j+{\mathrm{\Gamma }}_k{}^i{}_m{\mathrm{\Gamma }}_l{}^m{}_j-{\mathrm{\Gamma }}_l{}^i{}_m{\mathrm{\Gamma }}_k{}^m{}_j-{\mathrm{\Gamma }}_m{}^i{}_j[e_k,e_l{]}^m$, not the antisymmetrized connection coefficients alone, precisely so that no torsion conclusion is assumed.

(b) Curvature residual. With the prescribed curvature $R^{\mathrm{p}\mathrm{r}\mathrm{e}}$ of Schouten tensor, prescribed curvature, electric-magnetic Weyl variables and the reconstruction formula in the dust reduction the curvature residual $2$-forms are ${\mathrm{\Delta }}^i{}_j:={\mathrm{\Omega }}^i{}_j-\frac{1}{2}{R}^{\mathrm{p}\mathrm{r}\mathrm{e}i}{}_{jkl}{\theta }^k\wedge {\theta }^l\mathrm{.}$

(c) Bianchi and matter residuals. Using the reconstructed $C$ and prescribed $S$ of Schouten tensor, prescribed curvature, electric-magnetic Weyl variables and the reconstruction formula in the dust reduction, define $F_{jkl}:={\mathrm{\nabla }}_i(C^i{}_{jkl}-{\delta }^i{}_{[k}{S}_{l]j}),q_j:={\mathrm{\nabla }}_i(r{U}^i{U}_j),q:=U(r)+r\chi ,$ with brackets of weight $\frac{1}{2}$. The full flux covector satisfies $q_j=q{U}_j+r{\mathrm{\nabla }}_U{U}_j$, since ${\mathrm{\nabla }}_i{U}^i=\chi$ in the acceleration-free gauge. In particular $q=0$ and ${\mathrm{\nabla }}_{U}U=0$ imply $q_j=0$ even where $r=0$. This normalization uses the coefficient $r$ of the source tensor and does not divide by $r$ or $\kappa$.

(d) Zero quantities and reduced premises. Write ${\mathcal{T}}^i{}_{jk}:={\mathrm{\Theta }}^i(e_j,e_k)$, also denoted $T^i{}_{jk}$ when unambiguous; this torsion tensor is distinct from the stress-energy tensor. The zero quantities are $z:=({\mathcal{T}}^i{}_{jk},{\mathrm{\Delta }}^i{}_{jkl},F_{jkl},q_j)\mathrm{.}$ The reduced transport equations impose only ${\mathcal{T}}^i{}_{0a}=0$ and ${\mathrm{\Delta }}^i{}_{j0a}=0$, and density transport imposes $q=0$; the gauge gives ${\mathrm{\nabla }}_{U}U=0$. Weyl evolution imposes the symmetric spatial evolution projections of $F$. The reduced evolution does not itself impose every component of $z=0$. The subsidiary theorem propagates the remaining zero quantities from constrained initial data.

For vector-valued forms define $D_{\omega }\alpha =d\alpha +\omega \wedge \alpha$; for an endomorphism-valued $k$-form define $D_{\omega }\alpha =d\alpha +\omega \wedge \alpha -(-1{)}^k\alpha \wedge \omega$, with matrix multiplication of endomorphism indices. These conventions specify the Cartan identities used in Subsidiary system of the dust reduction: Cartan identities, closed homogeneous residual system and vanishing of the residuals.