Einstein–Maxwell primitive quantities and SI conventions

Adopt SI units and the following model-local quantity roles. Here the selected Einstein–Maxwell framework combines the stated Lorentzian geometry with a classical electromagnetic two-form and current; this definition itself supplies no gravitational equation, matter action, or empirical claim. A primitive is an input or independent field variable of this formulation; a derived quantity is fixed by a stated construction from its inputs. Neither label is universal across physical theories, and unit choice alone does not determine it.

The spacetime data are a smooth, Hausdorff, second-countable, boundaryless four-manifold $M$, a smooth Lorentzian metric $g$ of signature $(-,+,+,+)$, a smooth time-orientation field, and an independent orientation, exactly as in Oriented Lorentzian fields, observers and coordinate conventions. The metric, its time orientation, and the manifold orientation are separate data. An observer at $p\in M$ is a future unit timelike vector $n\in T_{p}M$ with $g(n,n)=-1$. A smooth observer congruence is a smooth field of such vectors. A local tetrad is a smooth basis $(e_0,e_1,e_2,e_3)$ with metric matrix $\mathrm{diag}(-1,1,1,1)$; an observer congruence, a coordinate chart, and a tetrad are different objects. Tetrads and observer fields need only exist locally; this definition assumes no global inertial chart, global tetrad, or hypersurface-orthogonal congruence.

The electromagnetic field-strength variable $F$ is a smooth real two-form, hence an alternating covariant tensor of type $(0,2)$, on an open subset of $M$, as in Oriented Lorentzian fields, observers and coordinate conventions. It is primitive in this formulation. For an observer congruence $n$, its electric and magnetic observer covectors are derived by

$E_a:=c{F}_{ab}{n}^b,B_a:=-({\star }_{g}F{)}_{ab}{n}^b\mathrm{.}$

Here ${\star }_g$ is the unique Hodge map characterized pointwise by $\alpha \wedge {\star }_g\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{v}\mathrm{o}\mathrm{l}}_g$, using the metric and independent orientation, as in Lorentzian exterior calculus and Hodge star. Both covectors annihilate $n$ and lie in the rest covector space $\{\alpha \in T_p^{\ast }M:\alpha (n)=0\}$, whose induced metric is positive definite. In a positively oriented future orthonormal tetrad with $e_0=n$ and ${ϵ}_{123}=+1$, their spatial components obey the fixed dictionary $F_{0i}=-E_i\mathrm{/}c$ and $F_{ij}={ϵ}_{ijk}{B}_k$ from Observer electric/magnetic reconstruction and invariants. Thus $E$ and $B$ are observer-dependent derived fields, not globally defined observer-independent three-vectors on a curved spacetime. Their tetrad components have units $\mathrm{V}{\mathrm{m}}^{-1}$ and tesla (T), respectively; the components of $F$ in a length-coordinate chart have unit T. No global potential is assumed; when a local potential $A_a$ is used, its components have unit $\mathrm{T}\mathrm{m}$ and a scalar potential $\phi$ has unit V.

For any chosen observer $n$, a smooth four-current is a vector field $j$ whose local decomposition is $j=c\rho n+J$, where $\rho :=-g(n,j)\mathrm{/}c$ is the observer charge density and $J:=j-c\rho n$ is orthogonal to $n$. In the associated positively oriented orthonormal tetrad its components are $j^0=c\rho$ and $j^i=J^i$. The units are $[\rho ]=\mathrm{C}{\mathrm{m}}^{-3}$, $[J^i]=[j^a]=\mathrm{A}{\mathrm{m}}^{-2}$, and $[q]=\mathrm{C}$ for a charge parameter supplied by a matter model. The current is a prescribed input or is supplied by an explicitly stated matter model; it is not inferred from the Maxwell equation alone.

The SI constants and coordinate units are fixed as follows. The SI defining constant $c>0$ (the speed of light in vacuum) is exactly $299792458\mathrm{m}{\mathrm{s}}^{-1}$. Independently of the flat physical postulate, this Einstein–Maxwell formulation chooses that SI constant as its primitive conversion parameter in $E_a=c{F}_{ab}{n}^b$, $j=c\rho n+J$, and $x^0=ct$. This explicitly adopted model-local normalization is not deduced from the SI unit definition or from a flat-spacetime assumption. The ideal-light and clock interpretation supplied by Adopted Minkowski physical model of the electromagnetic framework applies only when this model is restricted to flat Minkowski spacetime; it supplies no ideal-clock or light-ray law on a curved $M$. The elementary charge $e$ is exactly $1.602176634\times {10}^{-19}\mathrm{C}$. These unit definitions do not imply that charge is quantized. The magnetic constant ${\mu }_0>0$ has unit $\mathrm{N}{\mathrm{A}}^{-2}$ and is experimentally determined in the revised SI, so no exact numerical value for ${\mu }_0$ is adopted. Define the electric constant (vacuum permittivity) ${\epsilon }_0:=({\mu }_0{c}^2{)}^{-1}$ in $\mathrm{F}{\mathrm{m}}^{-1}$; hence ${\epsilon }_0{\mu }_0{c}^2=1$, and ${\epsilon }_0$ has the same relative standard uncertainty as ${\mu }_0$ because $c$ is exact. The exact component equation ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ is dimensionally consistent: both sides have unit $\mathrm{T}{\mathrm{m}}^{-1}$ in a length-coordinate frame.

In a length-coordinate chart all four coordinate values are assigned metres, with $x^0=ct$ when a time coordinate $t$ in seconds is selected. This is local coordinate bookkeeping, not a global inertial-coordinate claim. In such a chart $g_{ab}$ is dimensionless, the Levi-Civita coefficients ${\mathrm{\Gamma }}^a{}_{bc}$ have unit ${\mathrm{m}}^{-1}$, and curvature components have unit ${\mathrm{m}}^{-2}$, as in Oriented Lorentzian fields, observers and coordinate conventions. Components in other coordinate units transform by their Jacobians.

For the gravitational sector, take Newton's gravitational constant $G_N>0$ with unit ${\mathrm{m}}^3\mathrm{k}{\mathrm{g}}^{-1}{\mathrm{s}}^{-2}$ and the cosmological constant parameter $\mathrm{\Lambda }\in \mathbb{R}$ with unit ${\mathrm{m}}^{-2}$ as primitive parameters of the selected model; no numerical value for either is used here. The coupling normalization ${\kappa }_E:=8\pi G_N\mathrm{/}{c}^4$ is derived from these parameters and has unit $\mathrm{m}{\mathrm{J}}^{-1}$. This definition records the convention for a later Einstein equation; it asserts no Einstein equation. A separate positive normalization ${\kappa }_0$ with unit weber (Wb) is a chosen flux conversion reserved for a later explicitly specified $U(1)$ formulation, where $U(1):=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ under multiplication. It is not ${\kappa }_E$, an electric charge, ${\mu }_0$, or Planck's constant, and it implies no charge quantization. Test-particle rest mass $m>0$ in kg and charge $q$ in C are externally supplied matter parameters, not quantities derived by classical electromagnetism.

These are SI and model-definition conventions only. They neither assert empirical measurements beyond the stated SI defining constants and measured status of ${\mu }_0$, nor identify similarly named quantities in another framework.

Within the primitive/derived convention stated above, $c$ and ${\mu }_0$ are primitive inputs of this formulation; a numerical ${\mu }_0$ remains a qualified empirical input. The exact SI constant $e$ is included only to specify the coulomb and is not a parameter in the selected field equations. The optional ${\kappa }_0$ is an independent chosen primitive normalization convention. These role labels are local to this model and do not identify physical charge with integer multiples of $e$.

Adopt the following physical field law for this Einstein–Maxwell model. Let $M$ be a smooth, oriented, time-oriented Lorentzian four-manifold with metric $g$ of signature $(-,+,+,+)$ as in Einstein–Maxwell primitive quantities and SI conventions, let $U\subseteq M$ be open, let $F\in {\mathrm{\Omega }}^2(U)$ be a smooth real electromagnetic two-form, and let $j$ be a smooth four-current vector field on $U$. Using the Levi-Civita connection ${\mathrm{\nabla }}^g$ of $g$, the SI constants ${\mu }_0>0$ and $c>0$, and the field/current conventions of Einstein–Maxwell primitive quantities and SI conventions, postulate

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

This is the selected smooth vacuum constitutive law. The first equation is the homogeneous field equation; the second is the sourced field equation with the sign fixed by $F_{0i}=-E_i\mathrm{/}c$ and $j=(c\rho ,J)$ in a positively oriented inertial tetrad. No global potential is assumed: although $F=dA$ implies $dF=0$ when a smooth potential exists, the postulate applies to the two-form itself and does not assert that such an $A$ exists globally.

The postulate adopts one curved-spacetime continuation as a physical model assumption. The flat Minkowski and special-relativity suppliers Adopted Minkowski physical model of the electromagnetic framework, Adopted inertial covariance of the model equations, The microscopic vacuum Maxwell equations: smooth and weak branches, and Tensor Maxwell equations equal the four laboratory Maxwell equations fix the inherited SI, inertial, and flat-limit conventions; they do not impose a globally flat metric on $M$ or deduce this curved field law. General-relativity sources describe covariant Maxwell formulations in their own unit and sign conventions; those treatments motivate the chosen formulation but neither mathematical structure nor source authority proves that this is nature's law. No empirical result is adopted here.

The model has no magnetic-charge term and no material excitation pair such as $D,H$. It applies only to smooth $F$ and $j$ on the stated domain; point and sheet sources, distribution-valued fields, and products of distributions require separate definitions and are outside this postulate.

Let $I\subseteq \mathbb{R}$ be a nonempty open interval, $\mathrm{\Omega }\subseteq {\mathbb{R}}^3$ a nonempty open region, and $U=I\times \mathrm{\Omega }$ a flat inertial Minkowski model with metric $\eta =\mathrm{diag}(-1,1,1,1)$, orientation $d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$, and coordinates $x^0=ct$ and $x^i$ in metres. Let $E,B:U\to {\mathbb{R}}^3$ be jointly $C^1$, with every additional derivative displayed below existing and continuous, and let $\rho ,J$ be continuous. Use $F_{0i}=-E_i\mathrm{/}c$, $F_{ij}={ϵ}_{ijk}{B}_k$, $F^{0i}=E_i\mathrm{/}c$, $F^{i0}=-E_i\mathrm{/}c$, $F^{ij}={ϵ}_{ijk}{B}_k$, $j^0=c\rho$, $j^i=J^i$, and ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ as fixed by Einstein–Maxwell primitive quantities and SI conventions.

On this flat specialization, consider the first-order tensor systems

$(\mathrm{T}\mathrm{M}1){\mathrm{\partial }}_{\mu }{F}^{\mu \nu }=-{\mu }_0{j}^{\nu }(\nu =0,1,2,3),(\mathrm{T}\mathrm{M}2){\mathrm{\partial }}_{\lambda }{F}_{\mu \nu }+{\mathrm{\partial }}_{\mu }{F}_{\nu \lambda }+{\mathrm{\partial }}_{\nu }{F}_{\lambda \mu }=0(\lambda ,\mu ,\nu =0,1,2,3)\mathrm{.}$

For smooth $F,j$ these are the flat coordinate expressions of Smooth curved vacuum Maxwell constitutive adoption; at the stated $C^1$/continuous regularity they are well-defined component systems, without extending that postulate's physical scope. They hold if and only if the four adopted smooth-branch SI vacuum Maxwell equations hold on $U$:

$\mathrm{div}E=\rho \mathrm{/}{\epsilon }_0,\mathrm{div}B=0,\mathrm{curl}E=-{\mathrm{\partial }}_{t}B,\mathrm{curl}B={\mu }_{0}J+c^{-2}{\mathrm{\partial }}_{t}E\mathrm{.}$

Here ${\mathrm{\partial }}_t$ is at fixed spatial position and $\mathrm{div}$ and $\mathrm{curl}$ use the positively oriented Euclidean spatial basis. In addition, for the tensor contraction $f^{\nu }:=F^{\nu \lambda }{j}_{\lambda }$, its time and spatial components satisfy

$f^0=(J\cdot E)\mathrm{/}c,f^i=\rho E_i+(J\times B{)}_i\mathrm{.}$

If, in addition, $(\rho ,J)$ is the source of an explicitly specified matter model on $U$ to which the continuum coupling of The Lorentz coupling: external force on a test charge and the continuum force and power densities applies, then $c{f}^0$ and $(f^i{)}_{i=1}^3$ reproduce that model's adopted electromagnetic power and force densities. Every application of this physical interpretation must specify that matter model and verify the supplier's source and coupling hypotheses. For a prescribed current alone the displayed formulas are algebraic identities and assert no matter dynamics. The tensor systems have exactly eight independent scalar equations: four sourced components and four independent homogeneous components. All these are conditional component identities for the selected smooth model; no empirical premise is used. The result checks only the Maxwell subsystem on a flat background and the local force dictionary, not the coupled Einstein equation.