Smooth formal matter, EM tensors and observer quantities

Let $(M,g,N,o)$ be a smooth, Hausdorff, second-countable, boundaryless oriented and time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$. Here $N$ is the supplied smooth timelike field selecting the future cone, and $o$ is an independent orientation. Let $\mathrm{\nabla }$ be the unique torsion-free, $g$-compatible connection of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Use the curvature sign and contractions of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors; in particular $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab}$. The geometric and observer conventions are those of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and Oriented Lorentzian fields, observers and coordinate conventions. The tangent-bundle smooth structure inherits only the Axiom of Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$ from The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$).

Fix formal parameters $c>0$, ${\mu }_0>0$, $G_N>0$ and $\mathrm{\Lambda }\in \mathbb{R}$, with ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ and $\kappa =8\pi G_N\mathrm{/}{c}^4$. In length coordinates $x^0=ct$, all four coordinates have units metres and $g_{ab}$ is dimensionless. These are mathematical parameters with the displayed SI bookkeeping; they assert no measured value, exactness of ${\mu }_0$, or physical law.

A field strength $F$ is a real $C^2$ antisymmetric covariant two-tensor field. Its metric-raised components are $F^{ab}=g^{ac}{g}^{bd}{F}_{cd}$. For a future unit observer $n$ ($g(n,n)=-1$), the rest space is $n^{\perp }$ and the projection is $h_n(v)=v+g(n,v)n$. The observer electric and magnetic covectors are $E_{n,a}=c{F}_{ab}{n}^b$ and $B_{n,a}=-({\star }_{g}F{)}_{ab}{n}^b$, with the oriented Lorentzian Hodge convention of Observer electric/magnetic reconstruction and invariants. They annihilate $n$ and use the positive metric induced on the rest covector space. In a positively oriented orthonormal tetrad with time vector $n$, $F_{0i}=-E_i\mathrm{/}c$ and $F_{ij}={ϵ}_{ijk}{B}_k$.

For such an $F$, define

$T_{\mathrm{E}\mathrm{M}}^{ab}:=\frac{1}{{\mu }_0}(F^{ac}{F}^b{}_c-\frac{1}{4}{g}^{ab}{F}_{cd}{F}^{cd}),T_{ab}^{\mathrm{E}\mathrm{M}}:=g_{ac}{g}_{bd}{T}_{\mathrm{E}\mathrm{M}}^{cd}\mathrm{.}$

For any symmetric covariant tensor $S_{ab}$ and unit observer $n$, its observer energy density is $S_{ab}{n}^a{n}^b$. Its observer energy-current vector is $-S^a{}_b{n}^b$; the spatial energy flux measured by $n$ is the projection $h_n(-S^a{}_b{n}^b)$ into $n^{\perp }$. A nonzero causal vector is future-directed when it has negative inner product with $N$; future timelike means $g(v,v)<0$, and future null means the vector is nonzero and $g(v,v)=0$.

In four dimensions, define the weak energy condition (WEC) for $S$ by $S_{ab}{v}^a{v}^b\ge 0$ for every future timelike $v$; the null energy condition (NEC) by $S_{ab}{\mathrm{\ell }}^a{\mathrm{\ell }}^b\ge 0$ for every nonzero future null $\mathrm{\ell }$; and the dominant energy condition (DEC) by requiring $-S^a{}_b{v}^b$ to be future causal or zero for every future timelike $v$. Define the strong energy condition (SEC) by the WEC for $S_{ab}-\frac{1}{2}(g^{cd}{S}_{cd})g_{ab}$. These are definitions of tensor inequalities only; no general matter positivity premise is included.

A matter tensor $T_m^{ab}$ is a $C^1$ symmetric contravariant two-tensor field; a current $j^a$ is a $C^1$ vector field. The formal sourced Maxwell equations are $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$. The matter exchange equation is ${\mathrm{\nabla }}_a{T}_m^{ab}=F^b{}_c{j}^c$. The formal Einstein equation is

$G_{ab}+\mathrm{\Lambda }{g}_{ab}=\kappa (T_{m,ab}+T_{ab}^{\mathrm{E}\mathrm{M}})\mathrm{.}$

Here $T_{m,ab}$ is obtained by lowering both indices with $g$. Each displayed equation is a pointwise mathematical condition on the supplied fields with the regularities stated above. A formal Einstein–Maxwell–matter solution is a tuple satisfying whichever of these equations are explicitly assumed on the stated open domain; no existence or uniqueness of such a tuple is asserted.

Two matter forms used below are defined as follows. A charged-dust form has $T_m^{ab}=\rho U^a{U}^b$ and $j^a=\sigma U^a$, where $\rho ,\sigma \in C^1$, $\rho \ge 0$, and $U\in C^2$ is future timelike with $g(U,U)=-c^2$. On $\{\rho >0\}$ its charge-to-mass ratio is $\alpha =\sigma \mathrm{/}\rho$; it is not defined by division on $\{\rho =0\}$. A perfect-fluid form has

$T_m^{ab}=\frac{ϵ+p}{c^2}{U}^a{U}^b+p{g}^{ab},$

where $ϵ,p\in C^1$ and $U$ has the same timelike normalization. No equation of state, conductivity law, entropy law, or matter evolution theorem is part of this definition. A current may additionally be written $j^a=\sigma U^a+q^a$ with $q^a{U}_a=0$; an Ohmic relation $q^a=\chi E_U^a$, $\chi \ge 0$, is a separately imposed constitutive hypothesis, where $E_U^a=F^a{}_b{U}^b$.

On an open set $V$, a local potential for $F$ is a supplied smooth one-form $A$ with $dA=F$. Such a potential need not exist globally. A local gauge change is $A\mapsto A+d\lambda$ for a supplied smooth scalar $\lambda$; this leaves $F$ unchanged. No global bundle, potential, or point-source construction is asserted.

In the stated length-coordinate SI interpretation, $[F_{ab}]=\mathrm{T}$, $[A_a]=\mathrm{T}\mathrm{m}$, $[j^a]=\mathrm{A}{\mathrm{m}}^{-2}$, $[T_m]=[T_{\mathrm{E}\mathrm{M}}]=\mathrm{J}{\mathrm{m}}^{-3}$, $[\rho ]=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$, $[\sigma ]=\mathrm{C}{\mathrm{m}}^{-3}$, $[\alpha ]=\mathrm{C}\mathrm{k}{\mathrm{g}}^{-1}$, $[ϵ]=[p]=\mathrm{J}{\mathrm{m}}^{-3}$, and $[\chi ]=\mathrm{S}{\mathrm{m}}^{-1}$. The equalities ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ and $\kappa =8\pi G_N\mathrm{/}{c}^4$ make the displayed tensor equations dimensionally consistent. No distributional product or stress tensor for a singular point source is defined.

Square brackets denote SI units for components in the length-coordinate convention $x^0=ct$ supplied by Oriented Lorentzian fields, observers and coordinate conventions. The formal parameters have assignments

$[c]=\mathrm{m}{\mathrm{s}}^{-1},[{\mu }_0]=\mathrm{N}{\mathrm{A}}^{-2}=\mathrm{k}\mathrm{g}\mathrm{m}{\mathrm{s}}^{-2}{\mathrm{A}}^{-2},[G_N]={\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2},[\mathrm{\Lambda }]={\mathrm{m}}^{-2}\mathrm{.}$

Consequently,

$[{\epsilon }_0]=[({\mu }_0{c}^2{)}^{-1}]={\mathrm{A}}^2{\mathrm{s}}^4\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{m}}^{-3}=\mathrm{F}{\mathrm{m}}^{-1},[\kappa ]=[8\pi G_N\mathrm{/}{c}^4]={\mathrm{s}}^2\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{m}}^{-1}\mathrm{.}$

These assignments check the displayed equations directly: $[F_{ab}]=\mathrm{T}=\mathrm{k}\mathrm{g}{\mathrm{s}}^{-2}{\mathrm{A}}^{-1}$, so both ${\mathrm{\nabla }}_a{F}^{ab}$ and ${\mu }_0{j}^b$ have units $\mathrm{T}{\mathrm{m}}^{-1}$; $[F{]}^2\mathrm{/}[{\mu }_0]=\mathrm{k}\mathrm{g}{\mathrm{m}}^{-1}{\mathrm{s}}^{-2}=\mathrm{J}{\mathrm{m}}^{-3}$, as required for $T_{\mathrm{E}\mathrm{M}}$; and the length-coordinate curvature assignments of Oriented Lorentzian fields, observers and coordinate conventions, together with the Einstein-tensor definition of The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors, give $[G_{ab}]={\mathrm{m}}^{-2}$, matching both $[\mathrm{\Lambda }{g}_{ab}]$ and $[\kappa T_{m,ab}]=[\kappa T_{ab}^{\mathrm{E}\mathrm{M}}]={\mathrm{m}}^{-2}$. These are formal dimensional assignments, not measured values or claims that ${\mu }_0$ has an exact numerical value.

• This is a definition of formal mathematical systems and observer contractions. It does not adopt Einstein, Maxwell, dust, or fluid equations as physical postulates and does not infer that the equations have a solution.
• In the length-coordinate SI convention, the projection $h_n(-S^a{}_b{n}^b)$ is the spatial component of the observer energy-current. For a stress-energy tensor whose components have units $\mathrm{J}{\mathrm{m}}^{-3}$, this component also has units $\mathrm{J}{\mathrm{m}}^{-3}$; the physical energy-flux vector (power per area) is $c$ times it and has units $\mathrm{W}{\mathrm{m}}^{-2}$. Thus the spatial component is the physical energy flux divided by $c$.
• The perfect-fluid inequalities below are not imposed by the definition; only the named energy-condition results establish the stated parameter ranges. In particular, there is no general matter positivity axiom.
• The Carroll and Tong sources use $c=1$; their tensor forms are cited only as comparisons. The present SI normalization and all later contractions are stated and checked locally.