Oriented Lorentzian fields, observers and coordinate conventions

Let $(M,g,N)$ be a smooth, Hausdorff, second-countable four-manifold without boundary, with smooth Lorentzian metric $g$ of signature $(-,+,+,+)$ and smooth time-orientation field $N$, as in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model. Supply an independent orientation $o$ of $M$ in the sense of Oriented smooth manifolds and oriented charts. The metric and time orientation do not select $o$. Write ${\mathrm{v}\mathrm{o}\mathrm{l}}_g$ for the oriented metric volume form of The metric volume form, current flux through 3-faces and finite-chain charges; in any positively oriented chart it has components $\sqrt{\mathrm{\mid }\det (g_{ab})\mathrm{\mid }}d{x}^0\wedge d{x}^1\wedge d{x}^2\wedge d{x}^3$.

A smooth real $k$-form is a smooth section of ${\bigwedge }^k{T}^{\ast }M$, and its space is denoted ${\mathrm{\Omega }}^k(M)$, as in A smooth differential $k$-form. In this page, $F\in {\mathrm{\Omega }}^2(M)$ means an arbitrary smooth real two-form. No equation such as $dF=0$, no global potential, and no physical field law is included in this definition.

An observer at $p\in M$ is a vector $n\in T_{p}M$ with $g_p(n,n)=-1$ and $g_p(n,N_p)<0$; its rest space is $n^{\perp }=\{v:g_p(v,n)=0\}$, as in Observers, rest spaces and the spatial projection at an event. A smooth observer congruence on an open set $U\subseteq M$ is a smooth vector field whose value at every point is an observer. Its observer curves are the maximal integral curves, and a chart is adapted to it when its three spatial coordinate functions are constant along those curves, as in Observer congruences, their observer curves and adapted charts. Such a congruence need not be hypersurface-orthogonal.

A local frame field on $U$ is an ordered smooth basis field of $TM{\mathrm{\mid }}_U$. A coordinate frame is the basis of coordinate vector fields supplied by a chart. A tetrad field is a local frame $(E_0,E_1,E_2,E_3)$ with $g(E_A,E_B)=\mathrm{diag}(-1,1,1,1{)}_{AB}$; it is future-directed when $E_0$ is future-directed, as in Local frame fields, coordinate frames and tetrad fields. These are distinct objects: an observer congruence is a vector field, an adapted chart is a coordinate chart, and a tetrad is a basis field. A tetrad field is not presumed to be a coordinate frame. For every observer congruence $n$, each point has a neighborhood carrying an $n$-adapted tetrad; this is a local assertion only.

A worldline on this Lorentzian manifold means a $C^1$ or piecewise-$C^1$ regular curve with future-timelike tangent and specified interval domain, as defined precisely in Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold. Its parameter is arbitrary; the proper-time and four-velocity constructions of that definition apply under their stated hypotheses. A worldline is a mathematical curve here and does not assert the existence or motion of a physical body.

For coordinate bookkeeping only, when a chart is chosen with all four coordinate values assigned the metre, write $x^0=ct$, where $t$ is assigned seconds and $c>0$ is the corresponding scale in $\mathrm{m}{\mathrm{s}}^{-1}$. Then $g_{ab}$ is recorded as dimensionless. This is a unit convention on a chart, not a claim that the chart is inertial, orthonormal, global, or physically realized; $c$ in this mathematical convention has no asserted measured value.

Use the unique torsion-free metric-compatible connection ${\mathrm{\nabla }}^g$ and the curvature convention of Metric compatibility, torsion freedom and the metric-selected connection class and Tensoriality of the curvature, its coordinate formula and its point-normal metric formula. In a length-coordinate chart, the displayed Christoffel formula differentiates dimensionless metric components once, so ${\mathrm{\Gamma }}^a{}_{bc}$ is recorded in ${\mathrm{m}}^{-1}$. The curvature coordinate formula contains one derivative of $\mathrm{\Gamma }$ or a product of two $\mathrm{\Gamma }$ components, so $R^a{}_{bcd}$ and its Ricci and scalar contractions are recorded in ${\mathrm{m}}^{-2}$. These component assignments apply only in length-coordinate charts; other coordinate units transform by their Jacobians.

Let $(M,g,N,o)$ have the oriented, time-oriented Lorentzian structure and volume form of Oriented Lorentzian fields, observers and coordinate conventions. On decomposable $k$-covectors define the induced exterior pairing by

$⟨{\alpha }_1\wedge \cdots \wedge {\alpha }_k,{\beta }_1\wedge \cdots \wedge {\beta }_k{⟩}_g:=\det (g^{-1}({\alpha }_i,{\beta }_j)),$

and extend it bilinearly. For $0\le k\le 4$, its coordinate expression is the pointwise exterior metric on $k$-covectors

$⟨\alpha ,\beta {⟩}_g:=\frac{1}{k!}{\alpha }_{a_1\cdots a_k}{\beta }^{a_1\cdots a_k},$

where indices are raised by $g^{-1}$. There is a unique smooth bundle map ${\star }_g:{\mathrm{\Omega }}^k(M)\to {\mathrm{\Omega }}^{4-k}(M)$ characterized by

$\alpha \wedge {\star }_g\beta =⟨\alpha ,\beta {⟩}_g{\mathrm{v}\mathrm{o}\mathrm{l}}_g$

at every point. In a positively oriented $g$-orthonormal coframe $(e^0,e^1,e^2,e^3)$, put ${ϵ}_0=-1$ and ${ϵ}_1={ϵ}_2={ϵ}_3=1$. For an increasing index set $I\subseteq \{0,1,2,3\}$, let $I^c$ be its increasing complement, let $s(I,I^c)$ be the sign of the permutation $(I,I^c)$ to $(0,1,2,3)$, and set ${ϵ}_I:={\prod }_{i\in I}{ϵ}_i$. Then

${\star }_g{e}^I={ϵ}_{I}s(I,I^c)e^{I^c}\mathrm{.}$

Consequently, on $k$-forms,

${\star }_g^2=(-1{)}^{k(4-k)+1}\mathrm{id}\mathrm{.}$

In particular ${\star }_g^2=-\mathrm{id}$ on two-forms. For a two-form $F$, the component formula in a positively oriented chart is

$({\star }_{g}F{)}_{ab}=\frac{1}{2}({\mathrm{v}\mathrm{o}\mathrm{l}}_g{)}_{abcd}{F}^{cd}\mathrm{.}$

The exterior derivative in The exterior derivative by the invariant vector-field formula has the local coordinate formula of The local coordinate formula for the exterior derivative, satisfies $d^2=0$, and obeys the graded product rule of The exterior derivative is a graded derivation.

The positive-definite Hodge formula would omit the extra signature sign. It is not a supplier for this item: the single negative direction contributes ${ϵ}_I{ϵ}_{I^c}=-1$ in step 2.2. No metric field equation, Maxwell postulate, global potential, or physical interpretation is used.

The geometric suppliers Oriented Lorentzian fields, observers and coordinate conventions and Local frame fields, coordinate frames and tetrad fields record ${\mathrm{A}\mathrm{C}}_{\omega }$ as a background axiom: every countable family of nonempty sets has a choice function (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). This is an assumption of their development, not a property of a smooth structure. The finite coframe construction and uniqueness-based patching above require no additional choice principle.

Let $(M,g,N,o)$ be as in Oriented Lorentzian fields, observers and coordinate conventions, let $\mathrm{\Omega }:M\to (0,\mathrm{\infty })$ be a smooth dimensionless function, and set $\tilde{g}={\mathrm{\Omega }}^{2}g$ with the same orientation $o$. For every smooth $k$-form $\beta$, $0\le k\le 4$,

${\star }_{\tilde{g}}\beta ={\mathrm{\Omega }}^{4-2k}{\star }_g\beta \mathrm{.}$

In degree $2$ the Hodge star is unchanged. This is a statement about the Hodge operators of two conformally related Lorentz metrics; it asserts no conformal invariance of any curvature or field equation.

Let $U\subseteq M$ be an open set with smooth observer congruence $n$, let $F$ be a smooth real two-form on $U$, and let $c>0$ be the coordinate conversion parameter of Oriented Lorentzian fields, observers and coordinate conventions. Define covectors

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

Here $c$ is the positive coordinate-conversion parameter of Oriented Lorentzian fields, observers and coordinate conventions, with SI unit $\mathrm{m}{\mathrm{s}}^{-1}$ in its length-coordinate convention. The names electric and magnetic distinguish these two covectors; for arbitrary mathematical $F$ no physical field unit or measurement is assigned.

For each $p\in U$, define the rest covector space by

${\mathcal{R}}_{n,p}^{\ast }:=\{\alpha \in T_p^{\ast }M:\alpha (n_p)=0\}\mathrm{.}$

The musical isomorphism ${\mathrm{♭}}_p$ maps the vector rest space $n_p^{\perp }$ onto ${\mathcal{R}}_{n,p}^{\ast }$: $\alpha \in {\mathcal{R}}_{n,p}^{\ast }$ iff $g_p({\mathrm{♯}}_p\alpha ,n_p)=\alpha (n_p)=0$. Moreover, $g_p^{-1}(v^{\mathrm{♭}},w^{\mathrm{♭}})=g_p(v,w)$, so the restriction of $g_p^{-1}$ to ${\mathcal{R}}_{n,p}^{\ast }$ is positive definite by The observer rest-space decomposition and the spatial projection. The covectors $E_p$ and $B_p$ annihilate $n_p$; define $⟨E,B{⟩}_n(p):=g_p^{-1}(E_p,B_p)$. Then

$F=n^{\mathrm{♭}}\wedge E\mathrm{/}c+{\star }_g(n^{\mathrm{♭}}\wedge B),$

$F_{ab}{F}^{ab}=2(⟨B,B{⟩}_n-⟨E,E{⟩}_n\mathrm{/}{c}^2),F\wedge F=-\frac{2}{c}⟨E,B{⟩}_n{\mathrm{v}\mathrm{o}\mathrm{l}}_g\mathrm{.}$

Equivalently, in any positively oriented future tetrad with $E_0=n$, and with ${ϵ}_{123}=+1$ in its oriented spatial triad, the components satisfy $F_{0i}=-E_i\mathrm{/}c$ and $F_{ij}={ϵ}_{ijk}{B}_k$. These are pointwise mathematical decompositions of an arbitrary two-form; no Maxwell equation or physical observation is assumed.