Dust reduction variables: signature flip, frame, connection, kinematics, comoving-parallel gauge and unknowns

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold with signature $(-+++)$ and metric-selected connection $\mathrm{\nabla }$. Fix mathematical parameters $c>0$, $\kappa >0$ and $\mathrm{\Lambda }\in \mathbb{R}$. On an open set $U_0\subseteq M$, prescribe a smooth function $\rho \ge 0$ and a smooth future-directed unit timelike field $u$, with $g(u,u)=-1$. Set $\epsilon :=\rho c^2$ and $T^{ab}:=\epsilon u^a{u}^b$. On the open positive subset $\{\rho >0\}$, the tuple $(\epsilon ,u)$, rather than $(\rho ,u)$, is exactly the dust model of The dust matter model: rest density, four-velocity, stress-energy and dust system: that supplier's density is the coefficient of $u\otimes u$. Here $\rho$ denotes its mass-normalized parameter and $\epsilon$ its energy-normalized coefficient. On zero-density regions the smooth unit field is prescribed separately; the geodesic equation is imposed there as well and is not deduced from conservation of the zero tensor. This is the smooth geodesic-dust extension, without a free boundary or a division by density. Make the following reduction conventions, all local to an open set $\mathcal{U}\subseteq U_0$:

(a) Signature flip. Put $\gamma :=-g$, a smooth metric of signature $(+---)$ on $\mathcal{U}$, with inverse ${\gamma }^{-1}=-g^{-1}$; all indices in the reduction are raised and lowered with $\gamma$ (The inverse metric, musical isomorphisms and index raising). Put $U:=u,r:=\rho c^2\mathrm{.}$ Then $\gamma (U,U)=1$. Whenever the mathematical model equation $G(g)+\mathrm{\Lambda }g=\kappa T$ holds, it reads $G(\gamma )-\mathrm{\Lambda }\gamma =\kappa r{U}_{\gamma }^{\mathrm{♭}}\otimes U_{\gamma }^{\mathrm{♭}}$ with the same constant $\kappa$ and the Einstein tensor of $\gamma$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). Indeed the constant metric flip leaves the connection and mixed curvature unchanged, so Ricci is unchanged, the scalar changes sign and $G(\gamma )=G(g)$; both lower factors in $T$ change sign together. The original metric is recovered by $g=-\gamma$. Physical identification of these mathematical parameters is made by the consuming physical model, not assumed by this definition.

(b) Orthonormal frame. Choose a smooth orthonormal frame $(e_0,e_1,e_2,e_3)$ on $\mathcal{U}$ with $e_0=U$ and $\gamma (e_i,e_j)={\eta }_{ij}:=\mathrm{diag}(1,-1,-1,-1{)}_{ij}$ (Local frame fields, coordinate frames and tetrad fields, Observer congruences, their observer curves and adapted charts). Its connection coefficients are defined by ${\mathrm{\nabla }}_i{e}_j={\mathrm{\Gamma }}_i{}^k{}_j{e}_k,{\mathrm{\Gamma }}_{ijk}:={\gamma }_{kl}{\mathrm{\Gamma }}_i{}^l{}_j=-{\mathrm{\Gamma }}_{ikj},$ and the frame components of $U$ are $U^i={\delta }_0^i$. The spatial projection is the endomorphism $h^i{}_j:={\delta }_j^i-U^i{U}_j$, whose image is $U^{\perp }=\mathrm{span}(e_1,e_2,e_3)$ and whose kernel is $\mathbb{R}U$. Its lowered bilinear form and the auxiliary covariant tensor are $h_{ij}:={\gamma }_{ij}-U_i{U}_j,l_{ij}:=h_{ij}-U_i{U}_j\mathrm{.}$ On spatial pairs $h$ and $l$ both restrict to $\gamma$; they are covariant two-tensors, while $h^i{}_j$ is the projection map.

(c) Kinematics. With $\mathrm{\nabla }$ the metric connection of $\gamma$ (Metric compatibility, torsion freedom and the metric-selected connection class, Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields) put ${\chi }_{ij}:=h_i{}^k{\mathrm{\nabla }}_k{U}_j,\chi :=h^{ij}{\chi }_{ij},{\chi }^{ij}:=h^{ik}{h}^{jl}{\chi }_{kl},$ and let $D$ denote fully projected covariant differentiation: for a spatial tensor field $T$, $(DT{)}_{i\dots }:=(h_i{}^j\cdots ){\mathrm{\nabla }}_j(T_{\dots })$, so $D_i{h}_{jk}=0$ and $D_i{ϵ}_{jkl}=0$. Let ${ϵ}_{ijkl}$ be the $\gamma$-volume tensor with ${ϵ}_{0123}=1$ and set ${ϵ}_{ijk}:={ϵ}_{0ijk}$ (so ${ϵ}_{123}=1$).

(d) Comoving-parallel gauge. Assume the coordinates $(x^0=s,x^1,x^2,x^3)$ on $\mathcal{U}$ are comoving ($e_0={\mathrm{\partial }}_s$ and $x^{\alpha }$ constant along the flow lines of $e_0$, $s$ a parameter on them) and parallel (${\mathrm{\Gamma }}_0{}^i{}_j=0$, i.e. the spatial frame is parallelly transported along $U$). This gauge is admissible for tilted configurations: the initial surface $s=0$ is the given spacelike surface and need not be orthogonal to $U$.

(e) Unknowns. The unknowns of the reduction are the coefficients $u_{\mathrm{r}\mathrm{e}\mathrm{d}}:=(e_a{}^{\mu },{\mathrm{\Gamma }}_a{}^i{}_j,E_{ij},B_{ij},r),a=1,2,3,$ where $E$ and $B$ are independent smooth symmetric spatial covariant $2$-tensor fields, with $E_{0i}=B_{0i}=0$ and ${\sum }_{a=1}^3{E}_{aa}={\sum }_{a=1}^3{B}_{aa}=0$ in this orthonormal frame. Their slots are thus defined without assuming curvature reconstruction, evolution or residual vanishing. Their interpretation as electric and magnetic Weyl components is supplied after these reduction conventions are fixed.

During the independent reduced evolution, the frame and the coefficients ${\mathrm{\Gamma }}_a{}^i{}_j$ are tentative smooth unknowns. The coefficients define a metric-compatible connection by ${\widehat{\mathrm{\nabla }}}_{e_a}{e}_j={\mathrm{\Gamma }}_a{}^i{}_j{e}_i$ and the tensor product rule; impose the same coefficient antisymmetry and parallel gauge, but do not assume its torsion vanishes or that it is the metric-selected connection. In reduction calculations $\mathrm{\nabla }$ denotes this tentative connection. The residual theorem establishes that it equals the metric-selected connection for constrained solutions. The frame is required to remain a basis, so it defines the tentative metric by its constant orthonormal components.