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.

Let $(M,g,N)$ be a smooth Hausdorff second-countable boundaryless four-manifold with smooth Lorentz metric of signature $(-,+,+,+)$ and smooth time orientation. All definitions below are mathematical. A solution of a specified covariant field system is a tuple of fields of the regularity stated by that system satisfying its equations on its stated domain; this terminology alone supplies neither existence nor a global electromagnetic potential.

(a) Global causal and Cauchy conventions. Use exactly Causal curves, chronological and causal reachability, and inextendibility: a future causal curve is locally absolutely continuous in coordinate charts, has nonzero future causal derivative almost everywhere, and has no constant subinterval. Inextendibility and causal reachability $J^{\pm }$ have that item's endpoint-extension meanings. A Cauchy set meets every inextendible causal curve at exactly one parameter value: its inverse image under each such curve is a singleton, as in Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons. A spacelike hypersurface is a smooth embedded codimension-one submanifold with positive-definite induced metric; this local condition does not assert the global Cauchy property. Strong causality has the small-neighborhood no-leaving-and-reentering meaning of Chronology, causality, strong causality, global hyperbolicity and time functions. Global hyperbolicity means strong causality and compactness of every $J^{+}(p)\cap J^{-}(q)$, exactly as there; no equivalence theorem is added here.

(b) Null and Killing horizons. Here a hypersurface in $M$ is a smooth embedded three-dimensional submanifold, using the slice-chart convention recorded in Spacelike hypersurfaces, acausal sets, Cauchy sets, domains of dependence and Cauchy horizons clause (a); that clause's positive-definiteness condition is not imposed unless “spacelike” is stated. A smooth embedded hypersurface $H\subset M$ is null when at every $p\in H$ the radical $\{v\in T_{p}H:g_p(v,X)=0\text{ for every }X\in T_{p}H\}$ is nonzero. Equivalently, the slice-chart definition supplies near each point a smooth local defining function $f$ with $H=\{f=0\}$ and $df\mathrm{\ne }0$ on $H$, and $g^{-1}(df,df)=0$ on $H$. A specified smooth vector field $\xi$ is Killing when ${\mathcal{L}}_{\xi }g=0$, equivalently ${\mathrm{\nabla }}_{(a}{\xi }_{b)}=0$, using Killing fields of a Lorentzian metric and Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor. A Killing horizon for $\xi$ is a null hypersurface whose nonzero-generator part $H_{\ast }:=\{p\in H:{\xi }_p\mathrm{\ne }0\}$ is nonempty and dense in $H$, and on that part the specified field is tangent, null and normal: ${\xi }_p\in T_{p}H$ and $g({\xi }_p,X)=0$ for every $X\in T_{p}H$. A generator parameter or asymptotic normalization, when required, must be specified separately.

On $H_{\ast }$ the signed surface gravity for this precise $\xi$ is the unique smooth scalar ${\kappa }_H$ such that

$d(g(\xi ,\xi ){)}_p=-2{\kappa }_H(p){\xi }_p^{\mathrm{♭}}(p\in H_{\ast })\mathrm{.}$

This is an equality of ambient covectors at points of $H$, not merely of their pullbacks to $TH$. No constancy or positivity is asserted. Under a nonzero constant rescaling ${\xi }^{\mathrm{\prime }}=a\xi$, ${\kappa }_H^{\mathrm{\prime }}=a{\kappa }_H$. A bifurcation surface for $\xi$ consists of a specified smooth spacelike embedded two-dimensional submanifold $B\subset M$ on which $\xi =0$, together with two distinct specified smooth embedded null hypersurfaces $H_{+}$ and $H_{-}$ in a common neighborhood of $B$, each a Killing horizon for $\xi$, such that near each $p\in B$ their intersection is $B$ and $T_p{H}_{+}\cap T_p{H}_{-}=T_{p}B$. Write $H_{\pm ,\ast }:=\{q\in H_{\pm }:{\xi }_q\mathrm{\ne }0\}$. At $p\in B$, a surface-gravity value is defined only if the two branch scalars on $H_{+,\ast }$ and $H_{-,\ast }$ each extend smoothly to a neighborhood of $p$ in their respective hypersurfaces and those extensions agree at $p$; the value is their common value. Such a value is unique whenever it exists, since each $H_{\pm ,\ast }$ is dense in its horizon branch. No bifurcation surface or such extension is asserted to exist, and the defining equation at $\xi =0$ alone determines no value.

(c) Sections, area and ergoregions. A horizon section is a specified smooth spacelike embedded two-dimensional submanifold $S$ of the horizon, with positive induced metric $h$. Its area density is ${\mu }_h=\sqrt{\det (h_{ij})}\mathrm{\mid }d{y}^{1}d{y}^2\mathrm{\mid }$ of Riemannian volume density. Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)) for intrinsic density integration, as in Riemannian volume of a compactly supported smooth density. Its possibly infinite area is the nonnegative extended value

${\mathrm{Area}}_g(S):=\sup \{{\int }_S\chi {\mu }_h:\chi \in C_c^{\mathrm{\infty }}(S),0\le \chi \le 1\}\in [0,+\mathrm{\infty }],$

with the extended order of The extended real line $\bar{\mathbb{R}}=\mathbb{R}\cup \{-\mathrm{\infty },+\mathrm{\infty }\}$, its order, and the arithmetic that is left undefined. On compact $S$ this is ${\int }_S{\mu }_h$; orientation is unnecessary. The density glues by The riemannian volume density is coordinate independent. A stationary field here is a specified Killing field timelike on a specified nonempty open region; no normalization is inferred from this adjective. Its ergoregion in the chosen model is $\{g(\xi ,\xi )>0\}$, its stationary-limit set is $\{g(\xi ,\xi )=0\}$, and its ergosurface is the relative boundary of its ergoregion. Every ergosurface point lies in the stationary-limit set; the converse requires a separate sign-change argument. Nullness, Killing-horizon status and escape-boundary status are separate claims; a coordinate root alone proves none of them.

(d) Selected-model escape. Fix a specified open spacetime model $U\subseteq M$ and specified smooth radius observable $r:U\to \mathbb{R}$. An admissible escape path from $p$ is a continuous map $\gamma :[0,T)\to U$, $0<T\le +\mathrm{\infty }$, with $\gamma (0)=p$, such that for every compact interval $[a,b]\subset [0,T)$ there is a finite subdivision $a=t_0<\cdots <t_m=b$ and, for each $j$, a spacetime chart $(V_j,x_j)$ with $\gamma ([t_{j-1},t_j])\subset V_j$ for which $x_j\circ \gamma$ is $C^1$ on $[t_{j-1},t_j]$; its derivative is required to be nonzero and future causal in the chart interior and in the one-sided sense at subdivision endpoints. Define

${\mathcal{E}}_{U,r}:=\{p\in U:\mathrm{\exists }\gamma \text{ admissible from }p\mathrm{\forall }R>0\mathrm{\exists }t\in [0,T):r(\gamma (t))>R\}\mathrm{.}$

The selected escape boundary is ${\mathrm{\partial }}_U{\mathcal{E}}_{U,r}$ in the relative topology of $U$. This explicitly selected piecewise-$C^1$ path class is contained in the global causal-curve class of (a); it does not replace that class in Cauchy definitions. The definition includes no completion of $U$ or extension of $M$.

(e) Approached metric extensions and curvature obstructions. A $C^2$ nondegenerate metric extension of an open model $(U,g)$ consists of a smooth Hausdorff second-countable four-manifold $\tilde{M}$, a $C^2$ Lorentz metric $\tilde{g}$ of the same signature, and a smooth open embedding $j:U\to \tilde{M}$ with $j^{\ast }\tilde{g}=g$. An approached added point is specified either by $p\in \tilde{M}\setminus j(U)$ and a sequence $x_n\in U$ with $j(x_n)\to p$, or by such a $p$ and a continuous curve $\sigma :[0,1)\to U$ with $j(\sigma (t))\to p$ as $t\uparrow 1$. An extension obstruction at specified approach data means no such extension and added point realize that same sequence or curve limit. A polynomial complete curvature contraction is a scalar formed from a finite sum of tensor products of copies of the Riemann curvature tensor and the inverse metric, with every tensor index contracted. If its absolute values are unbounded along the specified sequence (as $n\to \mathrm{\infty }$), or along the specified curve (as $t\uparrow 1$), then that approach obstructs a $C^2$ nondegenerate metric extension. This criterion concerns curvature without its covariant derivatives. A smooth metric extension is the same definition with smooth $\tilde{g}$. These definitions assert no field equation and no additional global extension property.

(f) Observers and spatial fields. Observers, smooth observer congruences and their integral curves, adapted charts and tetrads retain the distinct meanings of Oriented Lorentzian fields, observers and coordinate conventions. Relative to a future unit observer $n$, an electric or magnetic spatial covector is identified with a spatial vector only by raising with the positive rest metric $h=g+n^{\mathrm{♭}}\otimes n^{\mathrm{♭}}$ on $n^{\perp }$. Such spatial vectors $e,b$ are not tetrad basis vectors. No singular-source product or action is defined by this vocabulary.

For the geometric data convention, orient ${\mathbb{T}}_{2\pi }^3$ by the ordered dimensionless coordinates $({\theta }^1,{\theta }^2,{\theta }^3)$; the corresponding length-coordinate frame $({\mathrm{\partial }}_{X^1},{\mathrm{\partial }}_{X^2},{\mathrm{\partial }}_{X^3})$ has the same orientation.

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ as required by The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s and The Fourier basis and Parseval's identity on the finite torus; no use of the full Axiom of Choice is made here. Let ${\mathbb{T}}_{2\pi }^3$, its normalized measure, the componentwise real Sobolev spaces $H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$, and their weak-derivative and Fourier-coefficient conventions be exactly those of The 2 pi-periodic three-torus, its normalized measure, test functions and the Sobolev spaces H^s. In particular, $H^s$ is defined for integer $s\ge 0$, the weak-derivative norm is equivalent to the Fourier-weight norm by Periodic Fourier differentiation identity and equivalence of the derivative and Fourier-weight Sobolev norms, and the trigonometric characters are a complete orthonormal system by The Fourier basis and Parseval's identity on the finite torus. A field is a function with the domain and codomain declaration of A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain; Sobolev fields are its $L^2$-equivalence classes, not pointwise representatives.

For finite $m\ge 1$, open $O\subseteq {\mathbb{R}}^m$, compact $K\subseteq O$, $R\ge 0$, and integer $s\ge 5$, define the bounded admissible initial-state class

${\mathcal{S}}_s(K,R):=\{u_0\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m):u_0\text{ has its continuous representative valued in }K,\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{H^s}\le R\}\mathrm{.}$

The representative is unique because two continuous functions equal almost everywhere on the torus agree everywhere. For a smooth periodic first-order reduction of the electrovacuum equations, its specified finite list of real state components is identified with one such ${\mathbb{R}}^m$-valued function; the state-component convention is part of that reduction. In the geometric data convention below, the component arrays are specified separately. This is a definition of admissible function-space classes only; it asserts no existence result for a coupled Einstein-Maxwell reduction.

For the geometric data convention, write the dimensionless torus coordinate as ${\theta }^i\in \mathbb{R}\mathrm{/}(2\pi \mathbb{Z})$ and introduce a fixed length scale $L_0>0$ in metres. The corresponding length coordinates are $X^i=L_0{\theta }^i$ on ${\mathbb{T}}_{2\pi L_0}^3:={\mathbb{R}}^3\mathrm{/}(2\pi L_0\mathbb{Z}{)}^3$; the Sobolev variable remains $\theta$, and $X$ is not a second name for a coordinate on ${\mathbb{T}}_{2\pi }^3$. The arrays $h_{ij}(\theta )$ and $K_{ij}(\theta )$ are the covariant metric and second-fundamental-form components in the length-coordinate frame ${\mathrm{\partial }}_{X^i}$, with $h_{ij}=h_{ji}$ and $K_{ij}=K_{ji}$. Let ${\mathsf{e}}_A(h)$, $A=1,2,3$, be the positively oriented $h$-orthonormal frame obtained by Gram-Schmidt from the ordered global frame $({\mathrm{\partial }}_{X^1},{\mathrm{\partial }}_{X^2},{\mathrm{\partial }}_{X^3})$. For spatial vectors $e,b$, the electromagnetic component arrays in this convention are $e_A:=h(e,{\mathsf{e}}_A)$ and $b_A:=h(b,{\mathsf{e}}_A)$; these are the orthonormal observer components used in the Einstein-Maxwell first-order reduction. The frame coefficients are smooth functions of the positive matrix $(h_{ij})$. If $h\in H^{s+1}$ has range in a compact subset of the positive-definite matrices, the composition supplier gives these coefficients in $H^{s+1}$; since $s\ge 5$, the Sobolev-algebra supplier gives their products with coordinate vector components in $H^s$. The inverse frame matrix has the same property, so the coordinate and orthonormal component arrays describe the same $H^s$ vector fields.

Thus the data component spaces mean $(h_{ij})\in H^{s+1}({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^9)$ with values in symmetric positive matrices, $(K_{ij})\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^9)$ with values in symmetric matrices, and $(e_A),(b_A)\in H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^3)$. A smooth periodic first-order state also includes the specified component functions of that reduction and the first spatial metric derivatives; in the above length-coordinate convention $v_{kij}:={\mathrm{\partial }}_{X^k}{h}_{ij}=L_0^{-1}{\mathrm{\partial }}_{{\theta }^k}{h}_{ij}$ and $K_{ij}$ carry one inverse-length factor. For a fixed field scale $F_0>0$ in tesla, the dimensionless physical component arrays are $h_{ij}$, $L_0{K}_{ij}$, $L_0{v}_{kij}$, $e_A\mathrm{/}{F}_0$, and $b_A\mathrm{/}{F}_0$. Each fixed rescaling is invertible and preserves the finite-component Sobolev topology. These are component and unit conventions, not additional field equations.

Assume ${\mathrm{A}\mathrm{C}}_{\omega }$. Let $s\ge 5$ be an integer, let $O\subseteq {\mathbb{R}}^m$ be open, and let $A^{\mu },B$ ($\mu =0,1,2,3$) be a periodic symmetric-hyperbolic coefficient family on $I\times {\mathbb{T}}_{2\pi }^3\times O$ in the exact sense of Periodic first-order symmetric hyperbolic systems: coefficients, admissible values, data classes and solutions: the coefficient maps are smooth, each $A^{\mu }$ is a real symmetric matrix, and $A^0$ is positive definite. Let $K\subseteq \mathrm{int}{K}^{\mathrm{\prime }}\subseteq O$ be compact and let $R\ge 0$. Then there are $T>0$ and $C<\mathrm{\infty }$, uniform over all $u_0\in {\mathcal{S}}_s(K,R)$ of Periodic high-Sobolev state spaces, such that the system

$A^0(t,x,u){\mathrm{\partial }}_{t}u+{\sum }_{i=1}^3{A}^i(t,x,u){\mathrm{\partial }}_{i}u+B(t,x,u)=0$

has a unique Sobolev solution $u:(-T,T)\to H^s({\mathbb{T}}_{2\pi }^3;{\mathbb{R}}^m)$ with $u(0)=u_0$, values in $K^{\mathrm{\prime }}$, $u\in L^{\mathrm{\infty }}{H}^s\cap C^0{H}^{s-1}$, and ${\mathrm{\partial }}_{t}u\in L^{\mathrm{\infty }}{H}^{s-1}$. The corresponding norm and time-derivative bounds are those in Local existence of Sobolev and classical solutions of a periodic symmetric hyperbolic system, uniform on bounded H^s data. Solutions are unique in the stated bounded classical class and depend continuously on initial data in $H^{s-1}$ by Uniqueness of classical solutions and continuous dependence in H^{s-1} for periodic symmetric hyperbolic systems. If $u_0$ is smooth, the same solution is smooth on this same interval by Smooth data give smooth solutions of a periodic symmetric hyperbolic system on the existence interval.

This theorem supplies only the analytic conclusion for a system whose coefficient hypotheses have already been verified. It does not assert that a particular gauge-fixed Einstein-Maxwell reduction has these matrices; that is a separate mathematical obligation.

Let $F$ be a smooth real two-form on a smooth oriented Lorentzian four-manifold of signature $(-,+,+,+)$, with the connection, Hodge star, and tensor $Q_{ab}=F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd}$ of Electrovacuum tensor Cauchy system. For a smooth vector field $J$, suppose

$dF=0,{\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{J}^b\mathrm{.}$

Then $Q_{ab}$ is symmetric and trace-free, and the SI-normalized stress tensor $T_{ab}^{\mathrm{E}\mathrm{M}}={\mu }_0^{-1}{Q}_{ab}$ satisfies

${\mathrm{\nabla }}^a{T}_{ab}^{\mathrm{E}\mathrm{M}}=-F_{bc}{J}^c\mathrm{.}$

In the source-free case $J=0$, ${\mathrm{\nabla }}^a{T}_{ab}^{\mathrm{E}\mathrm{M}}=0$. If $n$ is a future unit normal and $({\mathfrak{e}}_1,{\mathfrak{e}}_2,{\mathfrak{e}}_3)$ a positively oriented orthonormal spatial triad, put $e_i:=F({\mathfrak{e}}_i,n)$, $b_i:=\frac{1}{2}{ϵ}_i{}^{jk}F({\mathfrak{e}}_j,{\mathfrak{e}}_k)$, and $(e\times b{)}_i:={ϵ}_{ijk}{e}_j{b}_k$. Then

$T^{\mathrm{E}\mathrm{M}}(n,n)=\frac{\mathrm{\mid }e{\mathrm{\mid }}^2+\mathrm{\mid }b{\mathrm{\mid }}^2}{2{\mu }_0},T^{\mathrm{E}\mathrm{M}}(n,{\mathfrak{e}}_i)=-\frac{(e\times b{)}_i}{{\mu }_0}\mathrm{.}$

These are pointwise tensor identities. In the SI interpretation $e=E\mathrm{/}c$ and $b=B$ have units tesla; no measurement or physical applicability claim is included.

Let $(M,g,N,o)$ be the Lorentzian geometry and use the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions. Let $A_a$ be a smooth one-form, $A^a:=g^{ab}{A}_b$, and $F:=dA$, so $F_{ab}={\mathrm{\nabla }}_a{A}_b-{\mathrm{\nabla }}_b{A}_a$ by torsion-freeness. Define ${\mathrm{\square }}_g{A}_b:={\mathrm{\nabla }}^a{\mathrm{\nabla }}_a{A}_b$ and $\mathrm{div}A:={\mathrm{\nabla }}_a{A}^a$. Then

${\mathrm{\nabla }}^a{F}_{ab}={\mathrm{\square }}_g{A}_b-{\mathrm{\nabla }}_b(\mathrm{div}A)-{\mathrm{Ric}}_{bc}{A}^c\mathrm{.}$

Consequently, if ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{J}^b$, the Lorenz condition $\mathrm{div}A=0$ makes the potential equation

${\mathrm{\square }}_g{A}_b-{\mathrm{Ric}}_{bc}{A}^c=-{\mu }_0{J}_b\mathrm{.}$

The curvature term is part of the identity; this is not an uncoupled collection of scalar metric wave equations. The statement is local in the sense that it assumes a potential $A$ is already defined on the open set under consideration; it does not assert a global potential for every closed $F$.

Let $U$ be a coordinate neighborhood whose image in ${\mathbb{R}}^4$ is star-shaped about the origin, and let $F$ be a smooth real two-form on $U$ with $dF=0$. In those coordinates define

$A_a(x):={\int }_0^{1}t{x}^b{F}_{ba}(tx)dt\mathrm{.}$

Then $A$ is a smooth one-form on $U$ and $dA=F$ there. This is a local potential statement only.

For the gauge assertion, let $g$ be a smooth Lorentz metric on a neighborhood of a smooth spacelike hypersurface patch $S$, and let $A$ be a smooth one-form already defined on that neighborhood. Near each point of $S$ there is a smaller neighborhood and a smooth scalar $\chi$ such that

$A^{\mathrm{\prime }}:=A+d\chi ,d{A}^{\mathrm{\prime }}=dA,{\mathrm{\nabla }}^a{A}_a^{\mathrm{\prime }}=0.$

Assume Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ for the symmetric-hyperbolic supplier Local well-posedness and finite-speed independence of the extension for symmetric hyperbolic systems on a coordinate neighbourhood. The gauge proof constructs local Gaussian-normal coordinates with $x^0=0$ on the slice, $g_{00}=-1$, $g_{0i}=0$, and $(g^{ij})$ positive definite. No global potential or global gauge choice is asserted. When the Lorenz condition is combined with the Maxwell equation for a potential, Lorenz potential wave contains Ricci coupling retains the Ricci term.