The harmonic gauge vector, the reduced Einstein equation and the Einstein residual

Let $(M,g,N)$ be a time-oriented smooth Lorentzian four-manifold (Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model) with metric-selected connection $\mathrm{\nabla }$ (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) and let $U\subseteq M$ be an open set carrying a fixed smooth background metric $\bar{g}$ with Levi-Civita connection $\bar{\mathrm{\nabla }}$ (Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields), whose Christoffel symbols in any chart are ${\bar{\mathrm{\Gamma }}}^a{}_{bc}$ (Christoffel symbols of an affine connection); the same symbol ${\mathrm{\Gamma }}^a{}_{bc}$ denotes those of $g$. Let $\kappa >0$ and $\mathrm{\Lambda }\in \mathbb{R}$ be constants and let $T$ be a smooth symmetric covariant $2$-tensor field on $U$ (Smooth covariant 2-tensor fields on a smooth manifold), the matter stress-energy tensor, with $T=g^{ab}{T}_{ab}$ its metric trace (The inverse metric, musical isomorphisms and index raising).

(a) Gauge vector. The harmonic-gauge vector field of the pair $(g,\bar{g})$ is the smooth vector field $H^a:=g^{bc}({\mathrm{\Gamma }}^a{}_{bc}-{\bar{\mathrm{\Gamma }}}^a{}_{bc})\mathrm{.}$ The difference of the connection coefficients is a $(1,2)$-tensor, so $H$ is a well-defined smooth vector field independent of the chart used to express the symbols (Tensor fields of type (r,s) on a Lorentzian four-manifold and their components).

(b) Reduced Einstein equation. The reduced Einstein equation for the pair $(g,T)$ with respect to $\bar{g}$ is the second-order quasilinear system ${\mathrm{Ric}}_{ab}-{\mathrm{\nabla }}_{(a}{H}_{b)}=\kappa (T_{ab}-\frac{1}{2}T{g}_{ab})+\mathrm{\Lambda }{g}_{ab},$ where ${\mathrm{Ric}}_{ab}$ is the Ricci tensor of $g$ (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors), $H_b=g_{bc}{H}^c$ and ${\mathrm{\nabla }}_{(a}{H}_{b)}=\frac{1}{2}({\mathrm{\nabla }}_a{H}_b+{\mathrm{\nabla }}_b{H}_a)$.

(c) Reduced solutions. A reduced solution on $U$ is a pair $(g,T)$ of a smooth Lorentzian metric and a smooth symmetric $2$-tensor on $U$ satisfying (b) for the fixed background $\bar{g}$.

(d) Einstein residual. The Einstein residual of a pair $(g,T)$ is the smooth symmetric $2$-tensor $E_{ab}:=G_{ab}+\mathrm{\Lambda }{g}_{ab}-\kappa T_{ab},G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab},$ with $S=g^{ab}{\mathrm{Ric}}_{ab}$ the scalar curvature (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors). The pair solves the Einstein equation exactly when $E=0$.

(e) Constraint equations. If $g$ is defined on a neighborhood of a spacelike hypersurface $\mathrm{\Sigma }$ with future unit normal field $\mathfrak{n}$ and tangential frame $(e_i)$ (Spacelike initial data, the constraint equations, developments and maximal developments), the constraint residuals are the functions $E(\mathfrak{n},\mathfrak{n})$ and $E(\mathfrak{n},e_i)$ on $\mathrm{\Sigma }$; that these are the normal and normal-tangential projections of the field equation, and that they agree with the Hamiltonian and momentum constraint equations of the initial data, is recorded in The normal and normal-tangential projections of the field equation are the constraint equations and Spacelike initial data, the constraint equations, developments and maximal developments.

The definition introduces the objects only. The principal part of (b), its first-order symmetric hyperbolic reduction, the identity expressing $E$ through $H$, the wave equation satisfied by $H$ and the propagation of the constraint residuals are proved in Principal part of the reduced Einstein operator and its first-order symmetric hyperbolic reduction, Local existence for the reduced Einstein equation from Cauchy data on a spacelike slice, The Einstein residual through the gauge vector, and the subsidiary wave equation satisfied by the gauge vector and Propagation of the harmonic gauge and of the Einstein constraints: a constrained reduced solution is an Einstein solution.