Frequency-$\omega$ source-free vacuum phasor pairs and their real-field encoding

Let $\omega >0$, let $D\subseteq {\mathbb{R}}^3$ be a nonempty open set, and let $c>0$, ${\mu }_0>0$, ${\epsilon }_0=({\mu }_0{c}^2{)}^{-1}$ be the vacuum constants of Framework-local SI units, the vacuum constants and the exact/measured status of their values; put $T:=2\pi \mathrm{/}\omega$.

(a) [Phasor encoding.] For every pair of $C^{\mathrm{\infty }}$ maps $\tilde{E},\tilde{B}:D\to {\mathbb{C}}^3$ ($C^k$ Euclidean maps and diffeomorphisms) define the associated real fields $E,B:D\times \mathbb{R}\to {\mathbb{R}}^3$ componentwise by

$E(x,t):=\mathrm{Re}(\tilde{E}(x)e^{-i\omega t}),B(x,t):=\mathrm{Re}(\tilde{B}(x)e^{-i\omega t})((x,t)\in D\times \mathbb{R}),$

with the complex exponential and real part of The complex exponential by its power series and Real and imaginary parts, complex conjugation, and modulus. This is the encoding convention of Complex plane-wave fields: real parts of complex exponentials specialized from plane-wave amplitudes to a general $C^{\mathrm{\infty }}$ amplitude, and the complex fields are not themselves physical fields.

(b) [Source-free class.] The pair $(\tilde{E},\tilde{B})$ is called a frequency-$\omega$ source-free vacuum phasor pair on $D$ when it is $C^{\mathrm{\infty }}$ and satisfies, on $D$,

$\mathrm{div}\tilde{E}=0,\mathrm{div}\tilde{B}=0,\mathrm{curl}\tilde{E}=i\omega \tilde{B},\mathrm{curl}\tilde{B}=-i\omega {\epsilon }_0{\mu }_0\tilde{E}=-i\omega c^{-2}\tilde{E},$

with the divergence and curl of Divergence and curl of a $C^1$ vector field applied componentwise to the complex fields. The word "source-free" records that the displayed equations carry no charge or current term; the associated real fields are read as an electric and a magnetic field of The electric and magnetic fields as fixed-time spatial vector fields, $\omega$ is the angular frequency and $T$ the period.

(c) [Status and scope.] This item fixes the encoding, the four phasor equations and the terminology only. It asserts no equation for the associated real fields, no outgoing or boundary condition, no existence or uniqueness statement, no energy or flux reading and no empirical content. In particular the deduction that the associated real fields satisfy the smooth-branch vacuum Maxwell equations with zero sources is not part of this definition; for the pairs used on the exterior-scattering page it is established by the exterior-scattering theorem recorded there. Only $C^{\mathrm{\infty }}$ pairs on an open set are defined, so no boundary trace, weak or distributional class is introduced here.

Well-definedness and conventions

The componentwise reading of the two complex fields is the one of $C^k$ Euclidean maps and diffeomorphisms: $\tilde{E}$ and $\tilde{B}$ are $C^{\mathrm{\infty }}$ exactly when the twelve real coordinate maps $\mathrm{Re}{\tilde{E}}_k$, $\mathrm{Im}{\tilde{E}}_k$, $\mathrm{Re}{\tilde{B}}_k$, $\mathrm{Im}{\tilde{B}}_k$ ($k=1,2,3$) are $C^{\mathrm{\infty }}$ on the open set $D$. The real and imaginary parts and the complex exponential of Real and imaginary parts, complex conjugation, and modulus and The complex exponential by its power series are total on $\mathbb{C}$, so the associated real fields are defined for every $C^{\mathrm{\infty }}$ pair, with no equation assumed and no regularity asserted of them by clause (a). Clause (b) is the subclass cut out by the four displayed equations: the divergence and curl of a $C^{\mathrm{\infty }}$ complex field are the componentwise extensions of the divergence and curl of Divergence and curl of a $C^1$ vector field to $\mathrm{Re}$ and $\mathrm{Im}$ parts, evaluated on the open set $D$, and the right-hand sides use only complex scalar multiplication by the constants $i\omega$ and $i\omega {\epsilon }_0{\mu }_0$. In particular ${\epsilon }_0{\mu }_0=c^{-2}$ by Framework-local SI units, the vacuum constants and the exact/measured status of their values, a positive scalar parameter; all four equations are pointwise, so no selection of data and no choice principle enters the definition.

The units are inherited from The electric and magnetic fields as fixed-time spatial vector fields and Framework-local SI units, the vacuum constants and the exact/measured status of their values: $\tilde{E}$ and $E$ carry $\mathrm{V}{\mathrm{m}}^{-1}$, $\tilde{B}$ and $B$ carry the tesla $\mathrm{T}$, the wavevector-like spatial variable $x$ carries $\mathrm{m}$, $t$ carries $\mathrm{s}$, and $\omega$ carries ${\mathrm{s}}^{-1}$. The complex fields are auxiliary amplitudes encoding the real fields; clause (b) is a condition on those amplitudes, not a new physical assertion about the real fields.