Electrovacuum tensor Cauchy system

Let $(M,g,N,o)$ be a smooth, Hausdorff, second-countable, oriented and time-oriented four-manifold with smooth Lorentz metric of signature $(-,+,+,+)$, as in Oriented Lorentzian fields, observers and coordinate conventions. Use its metric-selected connection and curvature convention. Fix $c>0$, $G_N>0$, ${\mu }_0>0$, and $\mathrm{\Lambda }\in \mathbb{R}$, and put

$\kappa :=\frac{8\pi G_N}{c^4}>0.$

Within this formulation, $c$, $G_N$, ${\mu }_0$, and $\mathrm{\Lambda }$ are primitive model parameters: respectively, the time-to-length conversion parameter, the gravitational coupling parameter, the electromagnetic normalization parameter identified with vacuum permeability in the SI convention, and the coefficient of the cosmological term. Their values are not derived by this definition, and no numerical value or empirical exactness claim is made here. The coupling $\kappa =8\pi G_N\mathrm{/}{c}^4$ is derived from $G_N$ and $c$. For the SI interpretation, $c$ has units $\mathrm{m}{\mathrm{s}}^{-1}$, $G_N$ has units ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, ${\mu }_0$ has units $\mathrm{N}{\mathrm{A}}^{-2}$, $\kappa$ has units ${\mathrm{s}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{m}}^{-1}$, and $\mathrm{\Lambda }$ has units ${\mathrm{m}}^{-2}$. In a length-coordinate chart $x^0=ct$, metric components are dimensionless and curvature components have units ${\mathrm{m}}^{-2}$.

For a smooth real two-form $F\in {\mathrm{\Omega }}^2(M)$, define

$Q_{ab}(g,F):=F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd},T_{ab}^{\mathrm{E}\mathrm{M}}:={\mu }_0^{-1}{Q}_{ab}(g,F)\mathrm{.}$

Indices are raised with $g^{-1}$; ${\star }_g$ is the Lorentzian Hodge star of Lorentzian exterior calculus and Hodge star. The electromagnetic stress tensor $T^{\mathrm{E}\mathrm{M}}$ is a symmetric covariant two-tensor; in an orthonormal frame its components have energy-density units $\mathrm{J}{\mathrm{m}}^{-3}$. If $n$ is a future unit normal and $({\mathfrak{e}}_1,{\mathfrak{e}}_2,{\mathfrak{e}}_3)$ is a positively oriented orthonormal spatial triad, define $e_i:=F({\mathfrak{e}}_i,n)$ and $b_i:=\frac{1}{2}{ϵ}_i{}^{jk}F({\mathfrak{e}}_j,{\mathfrak{e}}_k)$. By Observer electric/magnetic reconstruction and invariants, these are $E_i\mathrm{/}c$ and $B_i$ in the SI observer convention; both have units tesla.

For sign clarity only, a sourced extension would write ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{J}^b$ for a smooth current vector $J$; the source-free system below is the case $J=0$.

The source-free electrovacuum tensor Cauchy system on an open set $U\subseteq M$ is the system for a smooth Lorentz metric $g$ and smooth real two-form $F$ on $U$ given by

$G_{ab}[g]+\mathrm{\Lambda }{g}_{ab}=\kappa T_{ab}^{\mathrm{E}\mathrm{M}}[g,F],dF=0,d({\star }_{g}F)=0.$

Here $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab}$ is the Einstein tensor. A spacelike initial hypersurface is a smooth spacelike embedded three-manifold $i:\mathrm{\Sigma }\hookrightarrow U$ with induced Riemannian metric $h=i^{\ast }g$ and future unit normal $n$, using Spacelike initial data, the constraint equations, developments and maximal developments. Its second fundamental form is $K(X,Y)=g({\mathrm{\nabla }}_{X}n,Y)$ for tangent vectors $X,Y$. The notation $K=\frac{1}{2}{\mathcal{L}}_{n}h$ requires an induced-metric family: locally near each $p\in \mathrm{\Sigma }$, define $\mathrm{\Phi }(t,q)={\gamma }_{q,n_q}(t)$ by the ambient geodesic with initial position $q$ and initial tangent $n_q$. Smooth dependence of geodesics and the inverse function theorem give, after shrinking around $(0,p)$, a local diffeomorphism $\mathrm{\Phi }:(-ϵ,ϵ)\times V\to U_p$; its slices remain spacelike after a further shrinking. Put $h_t:={\mathrm{\Phi }}_t^{\ast }g$ on $V$. In local slice coordinates $y^i$, differentiating $(h_t{)}_{ij}=g_{ab}(\mathrm{\Phi }(t,y)){\mathrm{\partial }}_i{\mathrm{\Phi }}^a{\mathrm{\partial }}_j{\mathrm{\Phi }}^b$ at $t=0$ gives ${{\mathrm{\partial }}_t(h_t{)}_{ij}\mid }_0=n^c{\mathrm{\partial }}_c{g}_{ab}{\mathrm{\partial }}_i{\mathrm{\Phi }}^a{\mathrm{\partial }}_j{\mathrm{\Phi }}^b+g_{cb}({\mathrm{\partial }}_i{n}^c){\mathrm{\partial }}_j{\mathrm{\Phi }}^b+g_{ac}{\mathrm{\partial }}_i{\mathrm{\Phi }}^a({\mathrm{\partial }}_j{n}^c)=g({\mathrm{\nabla }}_{{\mathrm{\partial }}_i}n,{\mathrm{\partial }}_j\mathrm{\Phi })+g({\mathrm{\nabla }}_{{\mathrm{\partial }}_j}n,{\mathrm{\partial }}_i\mathrm{\Phi })=2K_{ij},$ using metric compatibility, torsion-freeness and symmetry of $K$. Thus ${\mathcal{L}}_{n}h$ denotes only this initial derivative of the local induced-metric family; no global normal foliation or uniform time width is assumed. A hypersurface is a Cauchy hypersurface in $U$ exactly when every inextendible causal curve $\gamma :I\to U$ meets it at exactly one parameter value, that is, ${\gamma }^{-1}[i(\mathrm{\Sigma })]$ has exactly one element, using the causal-curve convention of Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons. The local initial-data constructions require a spacelike initial hypersurface; they do not assume this global intersection property. The initial data are the induced $(h,K,e,b)$, where $e,b$ are the spatial fields just defined. No global electromagnetic potential is part of this definition.

In the SI convention the electromagnetic two-form has orthonormal-frame components in tesla, $T_{ab}^{\mathrm{E}\mathrm{M}}$ has units $\mathrm{J}{\mathrm{m}}^{-3}$, and every term of the Einstein equation has units ${\mathrm{m}}^{-2}$. The equations define a mathematical model; they do not assert that the model describes nature or that arbitrary Cauchy data have a development.