Physlib.Electromagnetism.Vacuum.HarmonicWave

Given a free space environment $\mathcal{F}$ with speed of light $c$, a wave number $k \in \mathbb{R}$, an amplitude vector $E_0 = (E_{0,0}, \dots, E_{0,d-1}) \in \mathbb{R}^d$, and a phase vector $\phi = (\phi_0, \dots, \phi_{d-1}) \in \mathbb{R}^d$, the electromagnetic $4$-potential $A^\mu$ at a spacetime point $x$ (with time $t$ and spatial coordinates $(x_1, x_2, \dots, x_{d+1})$) for a monochromatic harmonic wave traveling in the $x_1$-direction is defined as follows: 1. The temporal component (scalar potential) is zero: $A^0(x) = 0$. 2. The longitudinal spatial component (along the direction of propagation) is zero: $A^1(x) = 0$. 3. The transverse spatial components for $i \in \{0, \dots, d-1\}$ are given by: \[ A^{i+2}(x) = -\frac{E_{0,i}}{ck} \sin\left(k(ct - x_1) + \phi_i\right) \] where $x_1$ denotes the first spatial coordinate.

For a monochromatic harmonic wave traveling in the $x$-direction in a free space environment $\mathcal{F}$, with wave number $k$, amplitude vector $E_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, the temporal component of the electromagnetic $4$-potential $A^\mu$ (the scalar potential $A^0$) is zero at every spacetime point $x$. That is, $A^0(x) = 0$.

In a free space environment $\mathcal{F}$, consider a monochromatic harmonic wave traveling in the $x_1$-direction with wave number $k$, amplitude vector $E_0$, and phase vector $\phi$. For any spacetime point $x$, the first spatial component (the longitudinal component along the direction of propagation) of the electromagnetic $4$-potential $A^\mu(x)$ is zero: \[ A^1(x) = 0 \] where $A^1$ denotes the component indexed by the first spatial coordinate (represented by `Sum.inr 0` in the formal statement).

Consider a monochromatic harmonic wave traveling in the $x$-direction within a free space environment $\mathcal{F}$, characterized by a wave number $k \in \mathbb{R}$, an amplitude vector $E_0 \in \mathbb{R}^d$, and a phase vector $\phi \in \mathbb{R}^d$. The electromagnetic 4-potential $A^\mu$ associated with this wave is differentiable as a function of spacetime.

For a monochromatic harmonic wave traveling in the $x$-direction in a free space environment $\mathcal{F}$, characterized by a wave number $k \in \mathbb{R}$, an amplitude vector $E_0 \in \mathbb{R}^d$, and a phase vector $\phi \in \mathbb{R}^d$, the electromagnetic potential $A^\mu$ (defined as `harmonicWaveX`) is $n$-times continuously differentiable ($C^n$) for any $n \in \mathbb{N} \cup \{\infty\}$.

For a monochromatic harmonic wave traveling in the $x$-direction in a free space environment $\mathcal{F}$ (with speed of light $c$), defined by a wave number $k \in \mathbb{R}$, an amplitude vector $E_0 \in \mathbb{R}^d$, and a phase vector $\phi \in \mathbb{R}^d$, the scalar potential $\Phi$ (the temporal component of the electromagnetic 4-potential) is identically zero.

Consider a monochromatic harmonic wave traveling in the $x$-direction in a free space environment $\mathcal{F}$ (with speed of light $c$) characterized by a wave number $k$, an amplitude vector $E_0$, and a phase vector $\phi$. For any time $t$ and spatial position $\mathbf{x}$ in $(d+1)$-dimensional space, the first component of the vector potential $\mathbf{A}(t, \mathbf{x})$ (the longitudinal component along the direction of propagation) is zero: \[ (\mathbf{A}(t, \mathbf{x}))_0 = 0 \]

Consider a monochromatic harmonic wave traveling in the $x_1$-direction in free space with speed of light $c$, wave number $k$, transverse amplitude vector $E_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$. The $(i+1)$-th spatial component (representing the transverse components) of the vector potential $\mathbf{A}$ at time $t$ and spatial position $\mathbf{x}$ is given by \[ A_{i+1}(t, \mathbf{x}) = -\frac{E_{0,i}}{ck} \sin(k(ct - x_1) + \phi_i) \] for $i \in \{0, \dots, d-1\}$, where $x_1$ is the first spatial coordinate of $\mathbf{x}$.

For a monochromatic harmonic wave traveling in the $x_1$-direction in free space with speed of light $c$, wave number $k \in \mathbb{R}$, amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and phase vector $\boldsymbol{\phi} \in \mathbb{R}^d$, the transverse components of the spatial vector potential $\mathbf{A}$ at time $t$ and position $\mathbf{x}$ are given by: \[ A_{i+1}(t, \mathbf{x}) = -\frac{E_{0,i}}{ck} \sin(k(ct - x_1) + \phi_i) \] where $i \in \{0, \dots, d-1\}$ and $x_1$ denotes the first spatial coordinate (the direction of propagation).

Consider a monochromatic harmonic wave traveling in the $x_1$-direction in free space $\mathcal{F}$ with speed of light $c$, characterized by wave number $k$, amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and phase vector $\boldsymbol{\phi} \in \mathbb{R}^d$. For any component $A_i$ ($i \in \{1, \dots, d+1\}$) of the spatial vector potential $\mathbf{A}$ and any transverse spatial coordinate $x_m$ ($m \in \{2, \dots, d+1\}$), the partial derivative of the component with respect to the transverse coordinate is zero: \[ \frac{\partial A_i}{\partial x_m} = 0 \]

For a monochromatic harmonic wave traveling in the $x_1$-direction (represented by the first spatial coordinate $x_0$) in free space with speed of light $c$, wave number $k \neq 0$, transverse amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, the partial derivative of the $(i+1)$-th component of the vector potential $A_{i+1}$ with respect to the propagation coordinate $x_1$ is given by $$\frac{\partial A_{i+1}(t, \mathbf{x})}{\partial x_1} = \frac{E_{0,i}}{c} \cos(k(ct - x_1) + \phi_i)$$ where $i \in \{0, \dots, d-1\}$.

Consider a monochromatic harmonic wave traveling in the $x$-direction in a free space environment $\mathcal{F}$ (with speed of light $c$), characterized by a wave number $k \in \mathbb{R}$, an amplitude vector $E_0 \in \mathbb{R}^d$, and a phase vector $\phi \in \mathbb{R}^d$. For any time $t$ and spatial position $\mathbf{x}$ in $(d+1)$-dimensional space, the longitudinal component of the electric field $\mathbf{E}(t, \mathbf{x})$ (the component along the direction of propagation) is zero: \[ (\mathbf{E}(t, \mathbf{x}))_0 = 0 \]

For a monochromatic harmonic wave traveling in the $x_1$-direction in free space $\mathcal{F}$ with speed of light $c$, wave number $k \neq 0$, transverse amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, the $(i+1)$-th component of the electric field $\mathbf{E}$ at time $t$ and spatial position $\mathbf{x}$ is given by \[ E_{i+1}(t, \mathbf{x}) = E_{0,i} \cos(k(ct - x_1) + \phi_i) \] for $i \in \{0, \dots, d-1\}$, where $x_1$ denotes the first spatial coordinate of $\mathbf{x}$.

Consider a monochromatic harmonic wave traveling in the $x_1$-direction in free space $\mathcal{F}$ with speed of light $c$, wave number $k \neq 0$, amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$. For any time $t$ and spatial position $\mathbf{x} = (x_1, \dots, x_{d+1})$, let $E_i$ denote the $i$-th component of the electric field $\mathbf{E}$ (where $i \in \{1, \dots, d+1\}$). Then the partial derivative of the $i$-th component of the electric field with respect to the $i$-th spatial coordinate is zero: \[ \frac{\partial E_i}{\partial x_i} = 0 \] (Note: In the formal Lean indexing for $d+1$ dimensions, this corresponds to coordinates $x_0, \dots, x_d$ and components $E_0, \dots, E_d$).

For a monochromatic harmonic wave traveling in the $x_1$-direction in free space $\mathcal{F}$ with speed of light $c$, wave number $k \neq 0$, transverse amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, the partial derivative with respect to time $t$ of the $(i+1)$-th component of the electric field $\mathbf{E}$ at spatial position $\mathbf{x}$ is given by: \[ \frac{\partial E_{i+1}}{\partial t} = -k c E_{0,i} \sin(k(ct - x_1) + \phi_i) \] for $i \in \{0, \dots, d-1\}$, where $x_1$ denotes the first spatial coordinate of $\mathbf{x}$.

For a monochromatic harmonic wave traveling in the $x_1$-direction in free space $\mathcal{F}$ with speed of light $c$, wave number $k \neq 0$, transverse amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, let $\mathbf{E}(t, \mathbf{x})$ denote the electric field at time $t$ and spatial position $\mathbf{x} \in \mathbb{R}^{d+1}$. The divergence of the electric field with respect to the spatial coordinates is zero: \[ \nabla \cdot \mathbf{E} = 0 \] where $\nabla \cdot \mathbf{E} = \sum_{i=1}^{d+1} \frac{\partial E_i}{\partial x_i}$.

Consider a monochromatic harmonic wave traveling in the $x_1$-direction in free space $\mathcal{F}$ with speed of light $c$, characterized by wave number $k$, transverse amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and phase vector $\boldsymbol{\phi} \in \mathbb{R}^d$. For any time $t$ and spatial position $\mathbf{x} \in \mathbb{R}^{d+1}$, the components of the magnetic field matrix $B$ (representing the magnetic field 2-form) associated with the transverse directions are zero. Specifically, for any indices $i, j \in \{1, \dots, d\}$, the matrix entry satisfies: \[ B_{i, j} = 0 \] where index $0$ corresponds to the longitudinal direction of propagation and indices $1, \dots, d$ correspond to the transverse directions.

For a monochromatic harmonic wave traveling in the $x_1$-direction in free space with speed of light $c$, wave number $k \neq 0$, transverse amplitude vector $E_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, the $(0, i+1)$-th component of the magnetic field matrix $B$ at time $t$ and spatial position $\mathbf{x}$ is given by: $$B_{0, i+1}(t, \mathbf{x}) = -\frac{E_{0,i}}{c} \cos(k(ct - x_1) + \phi_i)$$ where $i \in \{0, \dots, d-1\}$ and $x_1$ denotes the first spatial coordinate. In this $(d+1)$-dimensional spatial context, the magnetic field is represented as an antisymmetric matrix.

For a monochromatic harmonic wave traveling in the $x_0$-direction (the first spatial coordinate) in a free space environment $\mathcal{F}$ with speed of light $c$, wave number $k \neq 0$, transverse amplitude vector $E_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, the component $B_{i+1, 0}$ of the magnetic field matrix at time $t$ and spatial position $\mathbf{x}$ is given by: $$B_{i+1, 0}(t, \mathbf{x}) = \frac{E_{0,i}}{c} \cos(k(ct - x_0) + \phi_i)$$ where $i \in \{0, \dots, d-1\}$.

For a monochromatic harmonic wave traveling in the $x_0$-direction in free space with speed of light $c$, wave number $k \neq 0$, transverse amplitude vector $E_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, the partial derivative of any component $B_{i,j}$ of the magnetic field matrix with respect to any transverse spatial coordinate $x_m$ (where $m \in \{1, \dots, d\}$) is zero: \[ \frac{\partial}{\partial x_m} B_{i,j}(t, \mathbf{x}) = 0 \] where $i, j \in \{0, \dots, d\}$ are indices of the magnetic field matrix, $t$ is time, and $\mathbf{x} = (x_0, x_1, \dots, x_d)$ is the spatial position vector.

For a monochromatic harmonic wave traveling in the $x_0$-direction (the first spatial coordinate) in free space with speed of light $c$, wave number $k \neq 0$, transverse amplitude vector $E_0 \in \mathbb{R}^d$, and phase vector $\phi \in \mathbb{R}^d$, the partial derivative of the $(0, i+1)$-th component of the magnetic field matrix $B$ with respect to $x_0$ at time $t$ and spatial position $\mathbf{x}$ is given by: $$\frac{\partial B_{0, i+1}(t, \mathbf{x})}{\partial x_0} = -\frac{E_{0,i} k}{c} \sin(k(ct - x_0) + \phi_i)$$ where $i \in \{0, \dots, d-1\}$.

For a monochromatic harmonic wave traveling in the $x_1$-direction in a free space environment $\mathcal{F}$, characterized by a non-zero wave number $k \in \mathbb{R}$, a transverse amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and a phase vector $\boldsymbol{\phi} \in \mathbb{R}^d$, the resulting electromagnetic $4$-potential $A^\mu$ (defined as `harmonicWaveX`) satisfies Maxwell's equations in the absence of any source charges or currents (i.e., for a zero Lorentz current density $J = 0$).

For a free space environment $\mathcal{F}$ and a monochromatic harmonic wave traveling in the $x_1$-direction—defined by a non-zero wave number $k \in \mathbb{R}$, a transverse amplitude vector $\mathbf{E}_0 \in \mathbb{R}^d$, and a phase vector $\boldsymbol{\phi} \in \mathbb{R}^d$—the resulting electromagnetic potential describes a plane wave propagating in the direction of the first spatial basis vector $\mathbf{e}_1$.

Consider a monochromatic harmonic wave traveling in the $x_1$-direction in a free space environment $\mathcal{F}$ with wave number $k \neq 0$. Let $E_{i+1}(t, \mathbf{x})$ for $i \in \{0, \dots, d-1\}$ denote the $d$ transverse components of the electric field at time $t$ and position $\mathbf{x}$, with corresponding non-zero amplitudes $E_{0,i}$ and phases $\phi_i$. The electric field components satisfy the following polarization ellipse equation: \[ 2d \sum_{i=0}^{d-1} \left( \frac{E_{i+1}(t, \mathbf{x})}{E_{0,i}} \right)^2 - 2 \sum_{i=0}^{d-1} \sum_{j=0}^{d-1} \frac{E_{i+1}(t, \mathbf{x})}{E_{0,i}} \frac{E_{j+1}(t, \mathbf{x})}{E_{0,j}} \cos(\phi_j - \phi_i) = \sum_{i=0}^{d-1} \sum_{j=0}^{d-1} \sin^2(\phi_j - \phi_i). \]