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.

Let $\mathrm{\Omega }$ be an open subset of the oriented, time-oriented Lorentzian domain of Smooth formal matter, EM tensors and observer quantities. Let $\rho ,\sigma \in C^1(\mathrm{\Omega },\mathbb{R})$ with $\rho >0$, and let $U\in C^2(\mathrm{\Omega },T\mathrm{\Omega })$ be future timelike with $g(U,U)=-c^2$. Set $T_m^{ab}=\rho U^a{U}^b$, $j^a=\sigma U^a$, and $a^b=U^a{\mathrm{\nabla }}_a{U}^b$. Assume the formal matter-exchange equation ${\mathrm{\nabla }}_a{T}_m^{ab}=F^b{}_c{j}^c$. Then this equation is equivalent on $\mathrm{\Omega }$ to

${\mathrm{\nabla }}_a(\rho U^a)=0,\rho a^b=\sigma F^b{}_c{U}^c\mathrm{.}$

On $\mathrm{\Omega }$, define $\alpha =\sigma \mathrm{/}\rho$. Under the continuity equation, charge-current conservation ${\mathrm{\nabla }}_a{j}^a=0$ is equivalent to $U^a{\mathrm{\nabla }}_a\alpha =0$. Thus the equation constrains the derivative of $\alpha$ along the dust flow; it does not impose spatial constancy of $\alpha$. The claim is a pointwise conditional identity and supplies no coupled-field existence theorem or Maxwell equation.

• The hypotheses do not impose a constant charge-to-mass ratio. Example Advected nonconstant charge ratio without Maxwell closure ↗ gives an explicit nonconstant ratio satisfying this matter identity while failing sourced Maxwell closure.
• On $\rho =0$, the proof makes no division and does not infer a dust velocity or force law from the exchange equation.

Let $\mathrm{\Omega }$ be an open subset of Smooth formal matter, EM tensors and observer quantities. Let $ϵ,p,\sigma \in C^1(\mathrm{\Omega },\mathbb{R})$ and let $U\in C^2(\mathrm{\Omega },T\mathrm{\Omega })$ be future timelike with $g(U,U)=-c^2$. Set

$T_m^{ab}=\frac{ϵ+p}{c^2}{U}^a{U}^b+p{g}^{ab},a^b=U^a{\mathrm{\nabla }}_a{U}^b,D=U^a{\mathrm{\nabla }}_a,\theta ={\mathrm{\nabla }}_a{U}^a,h^b{}_c={\delta }^b{}_c+\frac{U^b{U}_c}{c^2}\mathrm{.}$

For a current $j^a=\sigma U^a+q^a$ with $q^a{U}_a=0$, define $f^b=F^b{}_c{j}^c$ and assume the mathematical exchange equation ${\mathrm{\nabla }}_a{T}_m^{ab}=f^b$. Then it is equivalent to the longitudinal and rest-space equations

$Dϵ+(ϵ+p)\theta =-U_b{f}^b,\frac{ϵ+p}{c^2}{a}^b+h^{ba}{\mathrm{\nabla }}_{a}p=h^b{}_c{f}^c\mathrm{.}$

The rest electric field of $U$ is $E_U^a=F^a{}_b{U}^b$. If $q^a=\chi E_U^a$ for a specified scalar $\chi \ge 0$, then the electromagnetic contribution to the comoving energy balance is $-U_b{f}^b=\chi g(E_U,E_U)\ge 0$. For $q=0$ it is zero. The Ohmic relation is an additional constitutive hypothesis; no equation of state, entropy law, or fluid existence result follows.

• The source-free and charged cases share the same projection identities. A general conduction current is not assumed to obey Ohm's law; Example Explicit local conduction-energy calculation ↗ supplies a local explicit instance of the optional $q=\chi E_U$ branch.
• These equations are algebraic projections of a supplied balance law, not a global fluid evolution theorem.

Use the smooth oriented, time-oriented Lorentzian four-manifold, signature $(-,+,+,+)$, metric $g$, and future cone of Oriented Lorentzian fields, observers and coordinate conventions. Here timelike means $g(v,v)<0$, null means $v\mathrm{\ne }0$ and $g(v,v)=0$, causal means timelike or null, and a nonzero causal vector is future-directed when $g(v,N)<0$, as in that supplier. A future unit observer is a vector $n$ with $g(n,n)=-1$ and $g(n,N)<0$. At a point $p$, let $T_p:T_{p}M\times T_{p}M\to \mathbb{R}$ be a continuous real symmetric bilinear form, represented by the covariant tensor $T_{ab}$. Define its metric trace by ${\mathrm{tr}}_{g}T:=g^{ab}{T}_{ab}$ and its raised mixed tensor by $T^a{}_b:=g^{ac}{T}_{cb}$.

• The weak energy condition (WEC) holds at $p$ when $T_{ab}{v}^a{v}^b\ge 0$ for every future timelike $v\in T_{p}M$.
• The null energy condition (NEC) holds at $p$ when $T_{ab}{\mathrm{\ell }}^a{\mathrm{\ell }}^b\ge 0$ for every nonzero future-directed null $\mathrm{\ell }\in T_{p}M$.
• The strong energy condition (SEC) holds at $p$ when the WEC holds for the trace-reversed tensor $T_{ab}-\frac{1}{2}({\mathrm{tr}}_{g}T)g_{ab}$.
• The dominant energy condition (DEC) holds at $p$ when, for every future-directed timelike $v\in T_{p}M$, the vector $J^a:=-T^a{}_b{v}^b$ is future causal or zero.

These are pointwise algebraic conditions. They are independent of an Einstein equation or an assertion that a physical material realizes $T$. The continuous regularity suffices for the definitions; any differentiability needed for a conservation identity is a separate hypothesis.

For a future unit observer $n$, observer energy density means $T_{ab}{n}^a{n}^b$. The observer energy-current vector is $-T^a{}_b{n}^b$, and its spatial energy flux is the projection of this vector into the positive rest space $n^{\perp }$.

Tong's lecture notes express SEC through a curvature inequality after invoking the Einstein equation. The definition here instead names the pointwise trace-reversed-tensor inequality itself, so no physical field equation is a prerequisite.

In the length-coordinate SI convention, if $T$ has stress-energy units $\mathrm{J}{\mathrm{m}}^{-3}$, the spatial projection of $-T^a{}_b{n}^b$ is an energy-current component with the same units. The corresponding physical energy-flux vector is $c$ times that component, in $\mathrm{W}{\mathrm{m}}^{-2}$.

Use the energy-condition definitions of Pointwise Maxwell and matter energy-condition vocabulary and the matter and Maxwell tensor definitions of Smooth formal matter, EM tensors and observer quantities. At every point of an oriented time-oriented Lorentzian domain, the Maxwell tensor $T_{\mathrm{E}\mathrm{M}}$ of that definition is symmetric, trace-free, and satisfies WEC, NEC, SEC and DEC. Its observer energy density and energy flux in a positively oriented orthonormal rest frame are

$u=\frac{\mathrm{\mid }E{\mathrm{\mid }}^2\mathrm{/}{c}^2+\mathrm{\mid }B{\mathrm{\mid }}^2}{2{\mu }_0},\frac{E\times B}{{\mu }_{0}c},$

and the DEC flux is future causal or zero. A charged-dust tensor with $\rho \ge 0$ satisfies all four conditions. For a perfect-fluid tensor with rest energy density $ϵ$ and pressure $p$, the exact pointwise conditions are

$\mathrm{W}\mathrm{E}\mathrm{C}⟺ϵ\ge 0\text{ and }ϵ+p\ge 0,\mathrm{N}\mathrm{E}\mathrm{C}⟺ϵ+p\ge 0,$

$\mathrm{S}\mathrm{E}\mathrm{C}⟺ϵ+p\ge 0\text{ and }ϵ+3p\ge 0,\mathrm{D}\mathrm{E}\mathrm{C}⟺ϵ\ge \mathrm{\mid }p\mathrm{\mid }\mathrm{.}$

At a point, finite sums of tensors satisfying any one of these conditions still satisfy that condition; in particular $T_{\mathrm{E}\mathrm{M}}+T_m$ satisfies all four for nonnegative dust. No general matter positivity assumption is made, and adding $T_{\mathrm{E}\mathrm{M}}$ to an arbitrary fluid need not repair a violated fluid inequality.

• The strong condition here is defined on the trace-reversed tensor and uses no Einstein field equation. This avoids importing a physical postulate into a mathematical statement.
• The vector $E\times B\mathrm{/}({\mu }_{0}c)$ in the Statement is the spatial component of the observer energy-current, with SI units $\mathrm{J}{\mathrm{m}}^{-3}$. The physical SI energy-flux vector is $E\times B\mathrm{/}{\mu }_0$, in $\mathrm{W}{\mathrm{m}}^{-2}$; it is $c$ times that current component.
• A magnetic field parallel to one spatial axis has principal stresses $(u,u,-u)$; Example Anisotropic magnetic stress and null energy-current vector ↗ verifies the anisotropy and its consequence for perfect-fluid form.

Let $M$ be a smooth manifold with an affine connection $\mathrm{\nabla }$, let $I\subseteq \mathbb{R}$ be any interval with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and let $z:I\to M$ be continuous. We say that $z$ is $C^2$ on $I$ in this definition when the following chartwise conditions hold. At each interior parameter value ${\tau }_0\in \mathrm{int}I$, there are a chart $x:U\to x(U)\subseteq {\mathbb{R}}^n$ around $z({\tau }_0)$ and an open interval $J\subseteq \mathrm{int}I$ containing ${\tau }_0$, with $z(J)\subseteq U$, such that $x\circ z$ is twice continuously differentiable on $J$. At each included endpoint ${\tau }_0\in I\setminus \mathrm{int}I$, there are such a chart and a relative neighborhood $J=I\cap ({\tau }_0-\delta ,{\tau }_0+\delta )$, with $\delta >0$ and $z(J)\subseteq U$, for which the first and second derivatives on $J\cap \mathrm{int}I$ extend continuously to ${\tau }_0$; their endpoint values are the corresponding one-sided derivatives. Here $n=\dim M$, and differentiability of $x\circ z$ is componentwise as a map of one real variable. Smooth chart transitions and $C^2$ composition make this condition independent of the chosen chart ($C^r$ and smooth maps between smooth manifolds, $C^k$ Euclidean maps are closed under componentwise algebra and composition); at an included endpoint, the first- and second-derivative chain-rule expressions have the corresponding continuous one-sided limits. On a time-oriented Lorentzian four-manifold, this is exactly the curve regularity specified in clause (a) of Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold; that definition's causal conditions are not part of the present regularity convention.

The parameter $\tau$ is an arbitrary real parameter here; no causal or nonzero-velocity condition is imposed. Put $U(\tau ):=z^{\mathrm{\prime }}(\tau )$, using the one-sided derivative at included endpoints. A vector field along $z$ assigns $V(\tau )\in T_{z(\tau )}M$ to each $\tau \in I$; it is continuous when its component functions are continuous in every chart along $z$. In a chart $x=(x^1,\dots ,x^n)$ covering a segment of $z$, write $U^a(\tau ):=\frac{d}{d\tau }(x^a\circ z)(\tau )$ and let ${\mathrm{\Gamma }}^a{}_{bc}$ be the Christoffel symbols of $\mathrm{\nabla }$. The components $U^a$ transform by the tangent-map Jacobian on chart overlaps, by the chain rule, so $U$ is a vector field along $z$. All indices range from $1$ to $n$, and repeated upper/lower indices are summed. Define the covariant acceleration of $z$ on this segment by the component formula

$({\mathrm{\nabla }}_{U}U{)}^a(\tau ):=\frac{d{U}^a}{d\tau }(\tau )+{\mathrm{\Gamma }}^a{}_{bc}(z(\tau ))U^b(\tau )U^c(\tau )\mathrm{.}$

The local $C^2$ regularity makes $U^a$ continuously differentiable, including one-sided endpoint derivatives, so these components are continuous on the relative parameter interval. The name and notation extend the pullback covariant derivative of Covariant derivative along a curve from smooth curves; the coordinate expression is well-defined on overlaps by The covariant acceleration formula is well-defined along a $C^2$ curve ↗.

For every smooth manifold with affine connection, every interval $I\subseteq \mathbb{R}$ with nonempty interior (Intervals of $\mathbb{R}$: the nine order-convex forms, nondegeneracy, and length), and every curve $z:I\to M$ that is $C^2$ in the chartwise sense of Covariant acceleration along a $C^2$ curve, the coordinate formula in that definition transforms as a tangent vector along $z$ on chart overlaps. At included endpoints the derivatives are one-sided. It therefore defines a unique continuous vector field along $z$, including its endpoints.

Let $\mathrm{\Omega }$ be an oriented open subset of the time-oriented Lorentzian domain of Smooth formal matter, EM tensors and observer quantities. Let $F$ and $j$ be smooth, let $T_m^{ab}$ be smooth and symmetric, and suppose

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

Let $T_{\mathrm{E}\mathrm{M}}$ be the tensor of Smooth formal matter, EM tensors and observer quantities and assume the exact conditional identity of Curved electromagnetic stress divergence, ${\mathrm{\nabla }}_a{T}_{\mathrm{E}\mathrm{M}}^{ab}=-F^b{}_c{j}^c$, under its stated smoothness and source-sign hypotheses. For a smooth Killing field $\xi$ on $\mathrm{\Omega }$, meaning ${\mathcal{L}}_{\xi }g=0$, define