Maxwell stress tensor in an observer frame

Let $(M,g,N,o)$ be the smooth, boundaryless, oriented and time-oriented Lorentzian four-manifold of Oriented Lorentzian fields, observers and coordinate conventions. Let $U\subseteq M$ be open, let $n$ be a smooth observer congruence on $U$, and let $F$ be a smooth real two-form. Fix $c,{\mu }_0>0$ and ${ϵ}_0:=({\mu }_0{c}^2{)}^{-1}$. Assume the fixed-sector variational hypotheses and tensor formula supplied conditionally by Compact potential and metric variation of the Maxwell action, namely

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

For the smooth observer covectors $E,B$ of Observer electric/magnetic reconstruction and invariants, define the rest projector $h^a{}_b:={\delta }_b^a+n^a{n}_b$ and the observer energy flux $S_n^a:=-c{h}^a{}_b{T}_{\mathrm{E}\mathrm{M}}^{bc}{n}_c$. Then:

(a) Tensor symmetries. $T_{\mathrm{E}\mathrm{M},ab}$ is symmetric and $g^{ab}{T}_{\mathrm{E}\mathrm{M},ab}=0$.

(b) Orthonormal observer components. In any positively oriented orthonormal tetrad with $E_0=n$ and spatial orientation ${ϵ}_{123}=+1$, the components are

$T_{\mathrm{E}\mathrm{M}}^{00}=\frac{1}{2}({ϵ}_0{\sum }_i{E}_i^2+\frac{1}{{\mu }_0}{\sum }_i{B}_i^2),T_{\mathrm{E}\mathrm{M}}^{0i}=\frac{(\mathbf{E}\times \mathbf{B}{)}_i}{{\mu }_{0}c},$

$T_{\mathrm{E}\mathrm{M}}^{ij}=-[{ϵ}_0(E_i{E}_j-\frac{1}{2}{\delta }_{ij}{\sum }_k{E}_k^2)+\frac{1}{{\mu }_0}(B_i{B}_j-\frac{1}{2}{\delta }_{ij}{\sum }_k{B}_k^2)]\mathrm{.}$

Here $(\mathbf{E}\times \mathbf{B}{)}_i:={ϵ}_{ijk}{E}_j{B}_k$. In particular, $T_{\mathrm{E}\mathrm{M},0i}=-T_{\mathrm{E}\mathrm{M}}^{0i}$ in this tetrad.

(c) Observer energy and flux. The measured energy density $u_n:=T_{\mathrm{E}\mathrm{M},ab}{n}^a{n}^b$ equals the nonnegative value in part (b), and the spatial vector $S_n$ has components $S_n^i=(\mathbf{E}\times \mathbf{B}{)}_i\mathrm{/}{\mu }_0$ and $S_n^0=0$.

(d) Geometric normalization. For every smooth real two-form $H$, define

$Q_g(H{)}_{ab}:=H_{ac}{H}_b{}^c-\frac{1}{4}{g}_{ab}{H}_{cd}{H}^{cd}\mathrm{.}$

For every real constant $\lambda$, $Q_g(\lambda H)={\lambda }^2{Q}_g(H)$. Fix also a formal constant $G_N>0$, and use the positive square-root convention of Explicit rational and real power regimes to set

$\alpha :=\sqrt{\frac{4\pi G_N}{{\mu }_0{c}^4}},\mathcal{F}:=\alpha F,{\kappa }_E:=\frac{8\pi G_N}{c^4}\mathrm{.}$

Then $\alpha >0$ and

${\kappa }_E{T}_{\mathrm{E}\mathrm{M},ab}=2Q_g(\mathcal{F}{)}_{ab}\mathrm{.}$

For formal SI dimensional bookkeeping in length coordinates $x^0=ct$, take dimensionless $g_{ab}$, $[c]=\mathrm{m}{\mathrm{s}}^{-1}$, $[G_N]={\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$, $[{\mu }_0]=\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-2}{\mathrm{A}}^{-2}$, and $[F_{ab}]=\mathrm{T}=\mathrm{W}\mathrm{b}{\mathrm{m}}^{-2}$. Then $[T_{\mathrm{E}\mathrm{M},ab}]=\mathrm{J}{\mathrm{m}}^{-3}$, $[\alpha ]={\mathrm{m}}^{-1}{\mathrm{T}}^{-1}$, $[{\mathcal{F}}_{ab}]={\mathrm{m}}^{-1}$, and both sides of the last identity have units ${\mathrm{m}}^{-2}$. Components in angular or other scaled coordinates acquire their units through the coordinate Jacobian. A two-form $\mathcal{F}$ integrated over an oriented two-surface has length units under these assignments. These conventions introduce no measured constant or physical field-law premise.

These are conditional mathematical tensor identities. The batch-2 connection carriers are authored drafts. No independent proof acceptance or native readiness is claimed here.

The sign $T_{\mathrm{E}\mathrm{M}}^{0i}>0$ is for the conventions $F_{0i}=-E_i\mathrm{/}c$, $F_{ij}={ϵ}_{ijk}{B}_k$, and signature $(-,+,+,+)$. The covariant component $T_{\mathrm{E}\mathrm{M},0i}$ has the opposite sign because lowering the time index multiplies it by $-1$. The formula is not a claim about the source or matter stress tensor.