Independent equation-based Einstein–Maxwell coupling

Let $M$ be the smooth, boundaryless, oriented and time-oriented Lorentzian four-manifold of Einstein–Maxwell primitive quantities and SI conventions, with metric $g$ of signature $(-,+,+,+)$ and the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions. Let $U\subseteq M$ be open. Take a smooth real two-form $F$ on $U$. For each application, fix one explicitly named smooth classical matter-model instance $\mathcal{M}=({\mathcal{C}}_m,{\mathcal{E}}_m,{\mathsf{J}}_m,{\mathsf{T}}_m)\mathrm{.}$ For each admitted pair $(g,F)$, ${\mathcal{C}}_m[g,F]$ is a specified set of smooth matter-field configurations on $U$, including their field bundles and regularity; ${\mathcal{E}}_m[g,F;\mathrm{\Psi }]$ is the explicit system of equations and constraints that defines the admissible configurations; and ${\mathsf{J}}_m[g,F;\mathrm{\Psi }]$ and ${\mathsf{T}}_m[g,F;\mathrm{\Psi }]$ are specified maps from those configurations to, respectively, smooth vector fields and smooth symmetric covariant two-tensors on $U$. For an admissible configuration $\mathrm{\Psi }\in {\mathcal{C}}_m[g,F]$, set $j:={\mathsf{J}}_m[g,F;\mathrm{\Psi }]$ and $T^m:={\mathsf{T}}_m[g,F;\mathrm{\Psi }]$. A consuming item must name the selected model and state the exact field variables, equations, constraints, regularity, and additional domain or initial-data hypotheses it uses. Those matter equations and constraints are premises of that model instance; they are not supplied or proved by this postulate. In particular, arbitrary independent smooth $j$ and $T^m$ are not a matter-model instance merely because their types match.

There is also a named zero-matter instance ${\mathcal{M}}_0$: its configuration set is the singleton $\{\ast \}$, it has no matter-field variables or additional constraints, and ${\mathsf{J}}_0(\ast )=0$, ${\mathsf{T}}_0(\ast )=0$. Whenever a use sets $j=0$ and $T^m=0$ without additional matter fields, it is this source-free instance.

For an admissible $\mathrm{\Psi }$, adopt $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ under the adopted convention of Smooth curved vacuum Maxwell constitutive adoption. This equation-based postulate then adds the gravitational field law below, independently of any matter action.

Define the Einstein tensor by $G_{ab}:={\mathrm{Ric}}_{ab}-\frac{1}{2}R{g}_{ab}$, using the curvature contractions of Oriented Lorentzian fields, observers and coordinate conventions, and define the smooth-field electromagnetic stress tensor by

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

For constant cosmological parameter $\mathrm{\Lambda }\in \mathbb{R}$ and ${\kappa }_E:=8\pi G_N\mathrm{/}{c}^4$, adopt the additional equation-based gravitational field law

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

Here $c>0$, $G_N>0$, and ${\mu }_0>0$ have the SI roles and units fixed in Einstein–Maxwell primitive quantities and SI conventions; in length coordinates, $G_{ab}$ and $\mathrm{\Lambda }{g}_{ab}$ have units ${\mathrm{m}}^{-2}$, $T^{\mathrm{E}\mathrm{M}}$ and $T^m$ have units $\mathrm{J}{\mathrm{m}}^{-3}$, and ${\kappa }_E$ has units $\mathrm{m}{\mathrm{J}}^{-1}$. The source-free and sourced Maxwell equations remain smooth-field postulates: point or sheet sources, distribution-valued products, material excitation laws, and a global potential are outside this scope.

This is an equation-based adoption. It does not assume or imply the existence of a matter action and is not derived from an action principle. The sources give formulations in their own conventions; they motivate comparison but do not establish a physical law or a solution-existence theorem.

For the action-based Einstein–Maxwell branch, choose one actual smooth matter configuration space $\mathcal{C}$ on the spacetime of Einstein–Maxwell primitive quantities and SI conventions, a smooth right principal $U(1)$ bundle $\pi :P\to M$, and a real differentiable functional $S_m[g,\omega ,\mathrm{\Psi }]$ on its declared domain, where $\mathrm{\Psi }\in \mathcal{C}$ and $\omega$ is a smooth principal connection on that fixed bundle. Use a supplied cover and fixed transition data with normalized real local connection representatives $a_i$; on phase-lift neighborhoods their convention is $a_j=a_i+d{f}_{ij}$ when $g_{ij}=e^{i{f}_{ij}}$. With the fixed normalization ${\kappa }_0>0$ of Einstein–Maxwell primitive quantities and SI conventions, the local expression for $\omega$ in a circle coordinate $z\in U(1)$ is $-i{z}^{-1}dz+a_i$. Define the physical local potentials $A_i:={\kappa }_0{a}_i$. They obey $A_j=A_i+{\kappa }_{0}d{f}_{ij}$, and their local exterior derivatives patch to the global physical curvature $F$, with $F{\mathrm{\mid }}_{U_i}=d{A}_i={\kappa }_{0}d{a}_i$ and $dF=0$. A global real potential is not assumed. The field class, transition data, gauge action, matter equations, and compact variation class must be specified for the chosen matter model.

More precisely, use the matter-field and differentiability definitions in Gauge/diffeomorphism matter identity and field exchange: supply a smooth finite-dimensional fiber bundle $E\to M$, take $\mathcal{C}={\mathrm{\Gamma }}^{\mathrm{\infty }}(E)$, and specify a real functional domain $\mathcal{D}$ of metric, fixed-sector connection and section triples. Compact variations are jointly smooth curves in $\mathcal{D}$ fixed off a common compact set. Require the freezing of any two components to remain admissible, admit every compact metric and connection direction, and require the derivative to depend real-linearly only on the three tangent directions and to split into the three partial variations. For each compact real gauge path and each horizontal-lifted compact base flow, supply the jointly smooth matter-bundle diffeomorphisms, their inverses on a common interval, and the admissible transported sections specified in that lemma. These are part of the selected model data; an arbitrary section bundle has no such transport by default.

If the selected matter model is given by a global physical potential $A\in {\mathrm{\Omega }}^1(M)$, use the trivial principal bundle $P=M\times U(1)$ with its global trivialization and normalized representative $a=A\mathrm{/}{\kappa }_0$; its principal connection is locally $-i{z}^{-1}dz+a$. This is the trivial-bundle global-potential branch, and the model may admit any real coupling when its gauge transformations are specified by compactly supported real lifts. A nontrivial bundle sector instead uses the stated local transition data and the model's compatible matter-field transition rules.

The functional is invariant under compactly supported gauge transformations of the fixed bundle sector and under orientation-preserving diffeomorphisms, with the connection and matter fields transported together. In particular, for every smooth compactly supported real gauge parameter $\chi$ with units Wb, its group map is $g_{\chi }=e^{i\chi \mathrm{/}{\kappa }_0}$; it acts by $a_i\mapsto a_i+{\kappa }_0^{-1}d\chi$ and hence by $A_i\mapsto A_i+d\chi$, together with the model's specified matter-field transformation. For a compactly supported vector field $\xi$, transport by the induced horizontal lift gives the physical connection variation ${\alpha }_a={\xi }^b{F}_{ba}$ used by Gauge/diffeomorphism matter identity and field exchange.

Adopt matter-field stationarity as part of this selected model: ${\delta }_{\mathrm{\Psi }}{S}_m=0$ for every admissible compactly supported matter variation. For compactly supported inverse-metric variations $q^{ab}=\delta g^{ab}$ and fixed-sector compact physical connection variations $\alpha \in {\mathrm{\Omega }}^1(M)$, vary the local potentials by $A_i(s)=A_i+s\alpha {\mathrm{\mid }}_{U_i}$, equivalently the normalized representatives by $a_i(s)=a_i+s{\kappa }_0^{-1}\alpha {\mathrm{\mid }}_{U_i}$. The transition data remain fixed, this is a variation of the same bundle connection sector, and $\delta F=d\alpha$. Require the remaining first variation to have the form

$\delta S_m=-\frac{1}{2c}{\int }_M{T}_{ab}^m{q}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g+\frac{1}{c}{\int }_M{j}^a{\alpha }_a{\mathrm{v}\mathrm{o}\mathrm{l}}_g,$

where $T^m$ is a smooth symmetric covariant two-tensor, the Hilbert stress-energy tensor in the convention of The Hilbert stress-energy tensor of a metric-dependent action, and $j$ is a smooth vector current. Here $\alpha$ is the physical-potential variation, not the normalized representative variation; the factor ${\kappa }_0^{-1}$ above makes the two conventions explicit. These normalization, connection-variation, and invariance hypotheses are exactly the formulation interface used by Gauge/diffeomorphism matter identity and field exchange. That conditional mathematical result does not assert that such an action exists for arbitrary proposed matter.

This postulate selects a differentiable action branch and its variation conventions. It does not determine a unique action, assert a local or global solution of its Euler–Lagrange equations, extend the model to singular sources, or establish empirical validity. The separate equation-based model does not depend on this action assumption.

Use the smooth oriented, time-oriented Lorentzian four-manifold, curvature convention and SI constants of Einstein–Maxwell primitive quantities and SI conventions and Oriented Lorentzian fields, observers and coordinate conventions. Choose the actual matter functional $S_m[g,\omega ,\mathrm{\Psi }]$ and its fixed smooth right principal $U(1)$ bundle $\pi :P\to M$ from Selected invariant matter-action contract. In its fixed normalized connection sector, the local representatives are $a_i$, the physical local potentials are $A_i={\kappa }_0{a}_i$, and their globally defined physical curvature satisfies $F{\mathrm{\mid }}_{U_i}=d{A}_i={\kappa }_{0}d{a}_i$ and $dF=0$. A global real potential is not required. Write $R:=g^{ab}{\mathrm{Ric}}_{ab}$ for the scalar curvature of the metric-selected connection, with the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions. For configurations where the action is finite, or for compactly supported action differences on a noncompact manifold, adopt

$S_{\mathrm{t}\mathrm{o}\mathrm{t}}[g,\omega ,\mathrm{\Psi }]=\frac{c^3}{16\pi G_N}{\int }_M(R-2\mathrm{\Lambda }){\mathrm{v}\mathrm{o}\mathrm{l}}_g-\frac{1}{4{\mu }_{0}c}{\int }_M{F}_{ab}{F}^{ab}{\mathrm{v}\mathrm{o}\mathrm{l}}_g+S_m[g,\omega ,\mathrm{\Psi }]\mathrm{.}$

Adopt that a configuration is dynamically admissible in this branch exactly when the first variation of this total action vanishes independently for every smooth compactly supported symmetric inverse-metric variation, every smooth compactly supported fixed-sector physical connection variation $\alpha \in {\mathrm{\Omega }}^1(M)$, and every admissible compactly supported matter-field variation. The connection variation changes $A_i$ by $\alpha {\mathrm{\mid }}_{U_i}$, equivalently changes $a_i$ by ${\kappa }_0^{-1}\alpha {\mathrm{\mid }}_{U_i}$, while leaving the bundle and transition data fixed; thus $\delta F=d\alpha$. If $M$ has a physical boundary, only variations vanishing on a neighbourhood of that boundary are included. This makes no claim about a Gibbons–Hawking–York or other boundary action.

The earlier postulate Adopt stationary Maxwell action as alternative physical formulation concerns its stated flat electromagnetic and prescribed-potential particle sectors. It supplies no unified matter–gravity action or coupled-solution theorem; the functional above is an additional, explicitly adopted model. An equation-based Einstein–Maxwell formulation is a separate alternative, not a premise of this action branch.

All displayed coefficients are model parameters with the units and interpretations in Einstein–Maxwell primitive quantities and SI conventions. Compact support is essential: the Einstein–Hilbert and Maxwell integrations by parts have boundary terms in general. The postulate asserts neither that its stationary configurations exist nor that the action describes nature.

Let $(M,g)$, the fixed smooth connection sector, the constants $c,G_N,{\mu }_0>0$, constant $\mathrm{\Lambda }$, and the selected matter functional $S_m$ be exactly those of Interior stationary Einstein–Maxwell–matter action. Write $F$ for the connection curvature, $j$ and $T^m$ for the current and Hilbert tensor in the selected matter variation, and define $G_{ab}:={\mathrm{Ric}}_{ab}-\frac{1}{2}R{g}_{ab}$ using the Ricci and scalar curvature contractions in the convention of Oriented Lorentzian fields, observers and coordinate conventions. Also set

$T_{ab}^{\mathrm{E}\mathrm{M}}:=\frac{1}{{\mu }_0}(F_{ac}{F}_b{}^c-\frac{1}{4}{g}_{ab}{F}_{cd}{F}^{cd}),{\kappa }_E:=\frac{8\pi G_N}{c^4}\mathrm{.}$

The homogeneous identity $dF=0$ holds because $F$ is the curvature of the smooth connection in the fixed sector. The adopted configuration is stationary under every admissible compactly supported metric, connection, and matter variation if and only if all of the following hold:

${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b,G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m),{\delta }_{\mathrm{\Psi }}{S}_m[\zeta ]=0\text{ for every admissible compactly supported }\zeta \mathrm{.}$

The last clause is the selected matter model's Euler–Lagrange condition; an explicit local matter equation cannot be given until $S_m$ is specified. These are model deductions from the adopted action and smooth compact variation hypotheses, not evidence that a coupled solution exists or that the model describes nature.

Let $(M,g)$ be a smooth oriented, time-oriented Lorentzian four-manifold with signature $(-,+,+,+)$, and let $U\subseteq M$ be open. Restrict to $U$ the fixed smooth principal $U(1)$ bundle and connection selected in Selected invariant matter-action contract. In its normalized local representatives $a_i$, the physical local potentials are $A_i={\kappa }_0{a}_i$; their global physical curvature satisfies $F{\mathrm{\mid }}_{U_i}=d{A}_i={\kappa }_{0}d{a}_i$ and $dF=0$. The $F$ and $j$ below are the curvature and current of that same connection and selected action. Let $F$ and $j$ be smooth, and assume the adopted Maxwell equations $dF=0$ and ${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b$ of Smooth curved vacuum Maxwell constitutive adoption. Let $S_m$ be the selected smooth invariant matter functional of Selected invariant matter-action contract, satisfying the section-bundle, real functional-domain, compact-variation splitting, and supplied smooth pathwise-transport hypotheses of Gauge/diffeomorphism matter identity and field exchange for this same connection. Suppose its matter fields are on shell. Define $T^{\mathrm{E}\mathrm{M}}$ by the local Maxwell variation and $T^m$ by the Hilbert variation of $S_m$. Then

${\mathrm{\nabla }}_a{T}_{\mathrm{E}\mathrm{M}}^{ab}=-F^{bc}{j}_c,{\mathrm{\nabla }}_a{T}_m^{ab}=F^{bc}{j}_c,{\mathrm{\nabla }}_a(T_{\mathrm{E}\mathrm{M}}^{ab}+T_m^{ab})=0.$

Use the curvature convention of Oriented Lorentzian fields, observers and coordinate conventions and define the scalar curvature by $R:=S=g^{ab}{\mathrm{Ric}}_{ab}$ and the Einstein tensor by $G_{ab}:={\mathrm{Ric}}_{ab}-\frac{1}{2}R{g}_{ab}$. Set ${\kappa }_E:=8\pi G_N\mathrm{/}{c}^4>0$ using the SI constants of Einstein–Maxwell primitive quantities and SI conventions. If, in addition, the pointwise Einstein equation $G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^m)$ holds with constant $\mathrm{\Lambda }$, the contracted Bianchi identity shows that the total conservation law is its necessary source-compatibility condition. This local identity is not a universal gravitational energy statement and does not imply existence of a solution.

Let $U$ be an open subset of the smooth oriented, time-oriented Lorentzian four-manifold and use the metric, units, and electromagnetic conventions of Einstein–Maxwell primitive quantities and SI conventions. A restricted smooth charged-dust configuration consists of constants $m>0$ in kilograms and $q\in \mathbb{R}$ in coulombs, a smooth future-directed vector field $U^a$ with $g_{ab}{U}^a{U}^b=-c^2$, and a smooth scalar field $\nu :U\to (0,\mathrm{\infty })$. Put $u^a:=U^a\mathrm{/}c$, so $g_{ab}{u}^a{u}^b=-1$, and at each $x\in U$ define its local rest space by $H_x:=\{v\in T_{x}M:g_x(v,u_x)=0\}$. The restriction ${\gamma }_x:=g_x{\mathrm{\mid }}_{H_x}$ is a positive-definite inner product. In any basis $(e_1,e_2,e_3)$ of $H_x$ with linear coordinates $(y^1,y^2,y^3)$, its proper 3-volume density is $d{V}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}=\sqrt{\det ({\gamma }_x(e_i,e_j))}\mathrm{\mid }d{y}^{1}d{y}^{2}d{y}^3\mathrm{\mid }$; this density is independent of the chosen basis. The scalar $\nu (x)$ is the proper number density measured in this rest space: the local particle-number measure has density $\nu d{V}_{\mathrm{r}\mathrm{e}\mathrm{s}\mathrm{t}}$. Thus $[\nu ]={\mathrm{m}}^{-3}$. Define the rest-space projector $h^a{}_b={\delta }^a{}_b+U^a{U}_b\mathrm{/}{c}^2$. The associated four-current and matter stress tensor are

$j^a:=q\nu U^a,T_m^{ab}:=m\nu U^a{U}^b\mathrm{.}$

Their length-coordinate SI units are $[U^a]=\mathrm{m}{\mathrm{s}}^{-1}$, $[j^a]=\mathrm{A}{\mathrm{m}}^{-2}$, and $[T_m^{ab}]=\mathrm{J}{\mathrm{m}}^{-3}$. The single-velocity condition means that at every event in the model domain all dust matter there is assigned the same vector $U_x$; a state with distinct future unit timelike velocities at one event is outside this model. In this item, a caustic or multistream breakdown is a point where the material description can no longer be represented by smooth fields $\nu ,U$ with $0<\nu <\mathrm{\infty }$ and one such velocity. The model domain is restricted to open regions excluding those breakdown points. This is a scope convention, not a claim that caustics are avoided or that the fields exist globally. As the continuum closure of this model, adopt on the smooth solution domain

${\mathrm{\nabla }}_a(\nu U^a)=0,m{\mathrm{\nabla }}_{U}U=q{F}^{\mathrm{♯}}(U),$

where ${\mathrm{\nabla }}_U{U}^a:=U^b{\mathrm{\nabla }}_b{U}^a$ and $g(F^{\mathrm{♯}}(v),w)=F(w,v)$, as in the external-charge convention of Stationary charged action and Lorentz force. The latter supplies test-body sign and normalization context only; it proves neither the continuum number equation nor a continuum or coupled PDE evolution law.

This definition applies only while $\nu ,U,F,g$ remain smooth and the single-velocity dust description is valid. It makes no claim that such solutions exist globally or avoid caustics, and it does not assign a distributional stress tensor to point particles.

For the smooth charged-dust model of Restricted smooth charged-dust model, with constant $m>0$ and $q$, current $j^a=q\nu U^a$, and stress $T_m^{ab}=m\nu U^a{U}^b$, every configuration satisfying its adopted closure obeys

${\mathrm{\nabla }}_a{j}^a=0,{\mathrm{\nabla }}_a{T}_m^{ab}=F^{bc}{j}_c\mathrm{.}$

This is a model deduction from the continuum equations, not a consequence of the external test-particle theorem and not a claim that a smooth continuum solution exists.

On the smooth Lorentzian geometry and SI convention of Einstein–Maxwell primitive quantities and SI conventions, choose a fixed smooth bundle sector and the classical charged-scalar functional $S_{\phi }$ of Classical complex-scalar connection derivative and action. A configuration of this model is a smooth complex scalar section $\phi$ and the smooth connection and metric fields in that definition. In the action branch, take this actual functional as the selected matter action $S_m:=S_{\phi }$, with the same fixed-sector compact variations. The scalar is classical; the dimensionless field normalization, the positive coefficient $K_{\phi }$, the smooth real potential $V$, and the permitted coupling $\lambda$ are exactly those of Classical complex-scalar connection derivative and action.

In its circle-bundle branch, $\lambda {\kappa }_0=k\in \mathbb{Z}$ for the selected representation; in the separate trivial real-potential branch, $\lambda$ may be any real number. No quantum rest-mass identification, material-response claim, or existence theorem for the coupled equations is part of this definition.

Use the specified model of Specified classical charged complex-scalar model and the adopted total action of Interior stationary Einstein–Maxwell–matter action. Write $P_a:=D_a\phi$, $L_{\phi }:=g^{ab}\overline{P_a}{P}_b+V(\mathrm{\mid }\phi {\mathrm{\mid }}^2)$ and

$E_{\phi }:=D_a{P}^a-V^{\mathrm{\prime }}(\mathrm{\mid }\phi {\mathrm{\mid }}^2)\phi ,j^a:=K_{\phi }\lambda \mathrm{Im}(\bar{\phi }{P}^a),$

$T_{ab}^{\phi }:=K_{\phi }\mathrm{Re}(\overline{P_a}{P}_b)-\frac{K_{\phi }}{2}{g}_{ab}{L}_{\phi }\mathrm{.}$

For smooth fields and the compact fixed-sector variations, stationarity of the selected scalar matter action in every scalar direction is equivalent to $E_{\phi }=0$. Stationarity of the full action in all fields then gives

${\mathrm{\nabla }}_a{F}^{ab}=-{\mu }_0{j}^b,G_{ab}+\mathrm{\Lambda }{g}_{ab}={\kappa }_E(T_{ab}^{\mathrm{E}\mathrm{M}}+T_{ab}^{\phi }),$

with $dF=0$ for the connection curvature. On the scalar matter equation,

${\mathrm{\nabla }}_a{j}^a=0,{\mathrm{\nabla }}_a(T^{\phi }{)}^a{}_b=F_{ba}{j}^a\mathrm{.}$

These are conditional field equations and local balance identities for the specified classical branch. They assert neither existence of a smooth solution nor empirical or quantum validity.