Special Relativity - Notes

An event is determined by spatio-temporal information together, \(t, \vec{r}\). Absolute nature of physics needs vector algebra and calculus formalism

\[\mathbf{X} = c \,t \,\mathbf{e}_0 + \vec{r} = c \, t \, \mathbf{e}_0 + x^1 \mathbf{e}_1 + x^2 \mathbf{e}_2 + x^3 \mathbf{e}_3 = X^{\alpha} \mathbf{E}_{\alpha} \ ,\]

having used Cartesian coordiantes for the space coordinate.

Minkowski metric reads

\[g_{\alpha \beta} = \mathbf{E}_{\alpha} \cdot \mathbf{E}_{\beta} = \text{diag}\{-1, 1, 1, 1\}\]

The reciprocal basis reads \(\mathbf{E}_{\alpha} \cdot \mathbf{E}^{\beta} = \delta_{\alpha}^{\beta}\), \(\mathbf{E}_{\alpha} = g_{\alpha \beta} \mathbf{E}^{\beta}\), s.t. the elementary interval between two events can be written as

\[d \mathbf{X} = d X^{\alpha} \, \mathbf{E}_{\alpha} = \underbrace{d X^{\alpha} \, g_{\alpha \beta}}_{= dX_{\beta}} \, \mathbf{E}^{\beta} = d X_{\beta} \, \mathbf{E}^{\beta} \ ,\]

having used Cartesian coordinates,

\[\begin{split} \begin{aligned} & X^0 = c t \\ & X_0 = c t \end{aligned} \qquad \begin{aligned} & X^1 = x \\ & X_1 = -x \end{aligned} \qquad \begin{aligned} & X^2 = y \\ & X_2 = -y \end{aligned} \qquad \begin{aligned} & X^3 = z \\ & X_3 = -z \end{aligned} \end{split}\]

Its “length”, or better pseudo-norm with Minkowski metric, is invariant and reads

\[d s^2 = d \mathbf{X} \cdot d \mathbf{X} = \left( dX_{\alpha} \mathbf{E}^{\alpha} \right) \cdot \left( dX^{\beta} \mathbf{E}_{\beta} \right) = c^2 \, d t^2 - (dx^1)^2 - (dx^2)^2 - (dx^3)^2 = c^2 \, dt^2 - |d \vec{r}|^2 \]

Note

\(ds\) is invariant todo prove it. And/or add a section about the role of invariance.

For a co-moving observer, \(d \vec{r}' = \vec{0}\), and \(t'\) is commonly indicated with \(\tau\), and its differential is invariant itself, being the product of a constant (\(c\) is a universal constant in special relativity) and an invariant quantity.

\[d s^2 = c^2 dt'^2 - |d \vec{r}'|^2 = c^2 d \tau^2 \ .\]

Given the invariant nature of \(d s\),

\[d s^2 = c^2 \, d \tau^2 = c^2 \, dt^2 - |d \vec{r}|^2 = c^2 \, dt^2 \left[ 1 - \frac{1}{c^2}\frac{|d\vec{r}|^2}{dt^2} \right] = c^2 \, dt^2 \left[ 1 - \frac{|\vec{v}|^2}{c^2} \right]\]

and thus

\[d s = c \, d \tau = \gamma^{-1}(v/c) \, c \, dt \ ,\]

with \(\gamma(w) = \frac{1}{\sqrt{1 - w^2}}\).

4-Velocity Given the parametric representation of an event in space-time as a function of its proper time, \(\mathbf{X}(\tau)\) or coordinate \(s\), \(\mathbf{X}(s)\) the derivative w.r.t. this parameter is defined as the 4-velocity of the event in space time. Using Cartesian coordinates inducing constant and uniform basis \(\mathbf{E}_{\alpha}\), as a function of the observer time \(t\), \(c t\), \(x^i(t)\), and the transformation of coordinates \(t(\tau)\), with differential \(d t = \frac{1}{\gamma} \, d \tau\)

\[\mathbf{U}(\tau) := \mathbf{X}'(\tau) = \dfrac{d}{d \tau} \left( X^{\alpha}(\tau) \mathbf{E}_{\alpha} \right) = \dfrac{d t}{d \tau} (c t \mathbf{E}_0 + x^i(t) \mathbf{E}_i) = \gamma(v/c) \left( c \mathbf{E}_0 + \dot{x}^i(t) \mathbf{E}_i \right) = \gamma(v/c) \left( c \mathbf{E}_0 + \vec{v} \right)\]

or

\[\mathbf{U}(s) := \mathbf{X}'(s) = \dfrac{d t}{ ds } \dfrac{d}{dt} \mathbf{X}(t) = \dots = \gamma(v/c) \left( \mathbf{E}_0 + \frac{\vec{v}}{c} \right) \ .\]

Note

Using \(s\) as the parameter, \(\mathbf{U}\) is non-dimensional, and has pseudo-norm = 1,

\[\mathbf{U}(s) \cdot \mathbf{U}(s) = \gamma^2 \underbrace{\left( 1 - \frac{|\vec{v}|^2}{c^2} \right)}_{=\gamma^{-2}} = 1 \ .\]

Using \(\tau\) as the parameter, \(\mathbf{U}\) has physical dimension of a velocity and pseudo-norm = \(c\).

4-acceleration \(\mathbf{X}''(\tau)\) or \(\mathbf{X}''(s)\), todo

Dynamics

4-momentum

\[\mathbf{P} = m \mathbf{U}\]

Using Cartesian coordinates and \(\tau\) as independent variable,

\[\mathbf{P} = m \mathbf{U} = m \frac{d \mathbf{X}}{d \tau} = m \gamma (c, \vec{v}) \ .\]

The spatial component is \(\gamma\) times the 3-dimensional momentum \(\vec{p} = m \vec{v}\); the time component reads

\[P^0 = m \gamma(w) c \ ,\]

and for small ratio \(w:= \frac{v}{c}\) it can be expanded in Taylor series around \(w = 0\) as

\[\gamma(w) \sim \gamma(0) + w \, \gamma'(0) + \frac{1}{2} \, w^2 \gamma''(0) + o(w^2) \ ,\]

with

\[\begin{split}\begin{aligned} \left.\gamma(w) \right|_{w=0} & = \left.\frac{1}{\sqrt{1 - w^2}}\right|_{w=0} = 1 \\ \left.\gamma'(w) \right|_{w=0} & = \left.-\frac{1}{2}(1 - w^2)^{-\frac{3}{2}} (- 2 w)\right|_{w=0} = w (1 - w^2)^{-\frac{3}{2}}= 0 \\ \left.\gamma''(w)\right|_{w=0} & = \left. \left( (1-w^2)^{-\frac{3}{2}} + w \left(-\frac{3}{2} \right)(1-w^2)^{-\frac{5}{2}} (- 2 w) \right)\right|_{w=0} = \\ & = \left. \left( (1-w^2)^{-\frac{3}{2}} + 3 w^2 (1-w^2)^{-\frac{5}{2}} \right)\right|_{w=0} = 1 \\ \end{aligned}\end{split}\]

and thus

\[\gamma(w) = 1 + \frac{1}{2} w^2 + o(w^2)\]

and

\[\gamma(v/c) \, m \, c \sim m \, c \left( 1 + \frac{v^2}{c^2} \right) = \frac{1}{c} \left( mc^2 + \frac{1}{2} m |\vec{v}|^2 \right) \]

Thus, recognizing energy (\(E = \gamma m c^2\)) and 3-momentum (\(\vec{p} = m_3 \vec{v}\), with \(m_3 := \gamma m\)), the 4-momentum can be written as

\[\mathbf{P} = m \mathbf{U} = \gamma m \left( 1, \frac{\vec{v}}{c} \right) =: \frac{1}{c} \left( \frac{E}{c}, \vec{p} \right)\]

Its pseudo-norm reads

\[m^2 = \mathbf{P} \cdot \mathbf{P} = \frac{1}{c^4} \left( E^2 - c^2 |\vec{p}|^2 \right) \]

and thus the relation between \(E\), \(\vec{p}\), \(m\) and \(c\),

\[E^2 = m^2 c^4 + c^2 |\vec{p}|^2 \ ,\]

from which, for \(\vec{v} = \vec{0} \rightarrow \vec{p} = \vec{0}\),

\[E^2 = m^2 c^4 \ ,\]

and keeping only the solution with positive energy (todo reference to Dirac’s equation and anti-matter?)

\[E = m c^2 \ .\]

Lagrangian approach

Free particle.

\[\mathbf{0} = \frac{d \mathbf{P}}{d s} = \frac{d }{d s} \left( m \mathbf{X}'(s) \right)\]

Weak form

\[\begin{split}\begin{aligned} 0 & = \mathbf{W}(s) \cdot \frac{d }{d s} \left( m \mathbf{X}'(s) \right) = \\ & = \frac{d }{d s} \left[ m \mathbf{W}(s) \cdot \mathbf{X}'(s) \right] - m \mathbf{W}'(s) \cdot \mathbf{X}'(s) = \\ \end{aligned}\end{split}\]

Using generalized coordinates \(q^k(s)\), the event can be written in parametric form as \(\mathbf{X}(q^k(s), s)\), while the velocity reads

\[\mathbf{U}(s) = \mathbf{X}'(s) = \frac{d}{ds} \mathbf{X}(q^k(s), s) = {q^k}'(s) \underbrace{\frac{\partial \mathbf{X}}{\partial q^k}(q^k(s), s)}_{=\frac{\partial \mathbf{X}'}{\partial {q^k}'}} + \frac{\partial \mathbf{X}}{\partial s}(q^k(s), s) = \mathbf{U}({q^k}'(s), q^k(s), s)\]

Choosing \(\mathbf{W} = \frac{\partial \mathbf{X}}{\partial q^k} = \frac{\partial \mathbf{X}'}{\partial {q^k}'}\) in the weak form,

\[\begin{split}\begin{aligned} 0 & = \frac{d }{d s} \left[ m \mathbf{W} \cdot \mathbf{X}' \right] - m \mathbf{W}' \cdot \mathbf{X}' = \\ & = \frac{d }{d s} \left[ m \frac{\partial \mathbf{X}'}{\partial {q^k}'} \cdot \mathbf{X}' \right] - m \dfrac{d}{ds} \frac{\partial \mathbf{X}}{\partial {q^k}} \cdot \mathbf{X}' = \\ & = \frac{1}{2} \left[ \frac{d}{d s} \left( \frac{\partial}{\partial {q^k}'}\left( m \mathbf{X}' \cdot \mathbf{X}' \right) \right) - \frac{\partial}{\partial q^k}\left( m \mathbf{X}' \cdot \mathbf{X}' \right) \right] = \end{aligned}\end{split}\]

Defining

\[ f\left({q^k}'(s), q^k(s), s \right) = - m \mathbf{X}'\left({q^k}'(s), q^k(s), s\right) \cdot \mathbf{X}'\left({q^k}'(s), q^k(s), s\right) = - m \ ,\]

multiplying by a “regular” generic function \(w(s)\), neglecting factor \(\frac{1}{2}\) and integrating by parts

\[\begin{split}\begin{aligned} 0 & = - \int_{s=s_a}^{s_b} w(s) \left[ \frac{d}{ds} \frac{\partial f}{\partial {q^k}'} - \frac{\partial f}{\partial q^k} \right] \, ds = \\ & = - \left.\left[ w(s) \frac{\partial f}{\partial {q^k}'} \right]\right|_{s=s_a}^{s_b} + \int_{s=s_a}^{s_b} \left[ w'(s) \frac{\partial f}{\partial {q^k}'} + \frac{\partial f}{\partial q^k} \right] \, ds = \\ & = - \left.\left[ w(s) \frac{\partial f}{\partial {q^k}'} \right]\right|_{s=s_a}^{s_b} + \delta \int_{s=s_a}^{s_b} f\left( {q^k}'(s), q^k(s), s \right) \, ds \ . \end{aligned}\end{split}\]

Thus, provided that \(w(s_1) = w(s_2) = 0\), equation of motion of free particle implies stationariety of functional

\[\int_{s=s_a}^{s_b} f\left( {q^k}'(s), q^k(s), s \right) \, ds \ ,\]

i.e.

\[\delta \int_{s=s_a}^{s_b} f\left( {q^k}'(s), q^k(s), s \right) \, ds = 0\]

Using \(t\) as independent parameter, \(ds = \gamma^{-1} \, c \, dt\), the functional can be recast as

\[\int_{t=t_a}^{t_b} -m \, c \, \sqrt{1 - \frac{|\vec{v}|^2}{c^2}} \, dt \ ,\]

to find the (3-dimensional) Lagrangian (multiply by \(c\) to get the right physical dimension; check if it’s required and wheter it’s possible to make \(c\) appear before),

\[\mathscr{L} = - \sqrt{1 - \frac{|\vec{v}|^2}{c^2}} \, m \, c^2 \ ,\]

and retrieve 3-momentum as (being \(\vec{v} = \dot{\vec{r}})\)

\[\begin{split}\begin{aligned} \vec{p} & := \frac{\partial \mathscr{L}}{\partial \dot{\vec{r}}} = \\ & = - m c^2 \frac{1}{2} \left(1-\frac{|\vec{v}|^2}{c^2} \right)^{-\frac{1}{2}} \left( - 2 \frac{\vec{v}}{c^2}\right) = \\ & = m \left(1-\frac{|\vec{v}|^2}{c^2} \right)^{-\frac{1}{2}} \vec{v} = \\ & = \gamma \, m \, \vec{v} \ , \end{aligned}\end{split}\]

and energy as

\[\begin{split}\begin{aligned} E & := \vec{p} \cdot \vec{v} - \mathscr{L} = \\ & = \gamma \, m \, |\vec{v}|^2 + \gamma^{-1} \, m \, c^2 = \\ & = \gamma \, m \, c^2 \left( \frac{|\vec{v}|^2}{c^2} + \gamma^{-2} \right) = \\ & = \gamma \, m \, c^2 \left( \frac{|\vec{v}|^2}{c^2} + 1 - \frac{|\vec{v}|^2}{c^2} \right) = \\ & = \gamma \, m \, c^2 \ . \end{aligned}\end{split}\]

Electromagnetism

Classical electromagnetic theory

Maxwell equations

Maxwell equations read

\[\begin{split}\begin{cases} \nabla \cdot \vec{d} = \rho_f \\ \nabla \times \vec{e} + \partial_t \vec{b} = \vec{0} \\ \nabla \cdot \vec{b} = 0 \\ \nabla \times \vec{h} - \partial_t \vec{d} = \vec{j}_f \end{cases}\end{split}\]

or in vacuum, with \(\rho_f = \rho\), \(\vec{j} = \vec{j}_f\), \(\vec{d} = \varepsilon_0 \vec{e}\), \(\vec{b} = \mu_0 \vec{h}\)

\[\begin{split}\begin{cases} \nabla \cdot \vec{e} = \frac{\rho}{\varepsilon_0} \\ \nabla \times \vec{e} + \partial_t \vec{b} = \vec{0} \\ \nabla \cdot \vec{b} = 0 \\ \nabla \times \vec{b} - \mu_0 \varepsilon_0 \partial_t \vec{e} = \mu_0 \vec{j} \end{cases}\end{split}\]

Electromagnetic potentials

The electromagnetic field can be written in terms of the electromagnetic potentials

\[\begin{split}\begin{cases} \vec{b} = \nabla \times \vec{a} \\ \vec{e} = -\partial_t \vec{a} - \nabla \varphi \\ \end{cases}\end{split}\]

Lorentz force

A particle in motion in a electromagnetic field is subject to Lorentz force. In classical electromagnetism, the expression of Lorentz force reads

\[\vec{F} = q \left( \vec{e} - \vec{b} \times \vec{v} \right) \ ,\]

whose power is

\[\vec{v} \cdot \vec{F} = \vec{v} \cdot q \left( \vec{e} - \vec{b} \times \vec{v} \right) = q \vec{v} \cdot \vec{e} \ . \]

Electromagnetic potential

\[\begin{split}\begin{cases} \vec{b} = \nabla \times \vec{a} \\ \vec{e} = -\partial_t \vec{a} - \nabla \varphi \\ \end{cases}\end{split}\]

\[\mathbf{A} = \mathbf{E}_{\alpha} A^{\alpha} = \frac{\varphi}{c} \mathbf{E}_0 + \vec{a}\]

\[\symbf{\nabla} \mathbf{A} = \left( \mathbf{E}^{\alpha} \frac{\partial}{\partial X^{\alpha}} \right) \left( A^{\beta} \mathbf{E}_{\beta} \right) = \frac{\partial A^{\beta}}{\partial X^{\alpha}} \mathbf{E}^{\alpha} \otimes \mathbf{E}_{\beta} = g_{\alpha \gamma} \frac{\partial A^{\beta}}{\partial X^{\alpha}} \mathbf{E}_{\gamma} \otimes \mathbf{E}_{\beta} \ .\]

whose components may be collected in a 2-dimensional array (first index for rows, second index for columns),

\[\begin{split}(\nabla \mathbf{A})_{\alpha}^{\beta} = \frac{\partial A^{\beta}}{\partial X^{\alpha}} = \begin{bmatrix} c^{-2} \partial_t \varphi & c^{-1}\partial_x \varphi & c^{-1}\partial_y \varphi & c^{-1}\partial_z \varphi \\ c^{-1} \partial_t a_x & \partial_x a_x & \partial_y a_x & \partial_z a_x \\ c^{-1} \partial_t a_y & \partial_x a_y & \partial_y a_y & \partial_z a_y \\ c^{-1} \partial_t a_z & \partial_x a_z & \partial_y a_z & \partial_z a_z \\ \end{bmatrix}\end{split}\]

or covariant-covariant coomponents,

\[\begin{split}(\nabla \mathbf{A})_{\alpha \beta} = \frac{\partial A_{\beta}}{\partial X^{\alpha}} = g_{\beta \gamma} \frac{\partial A^{\gamma}}{\partial X^{\alpha}} = \begin{bmatrix} c^{-2}\partial_t \varphi & c^{-1}\partial_x \varphi & c^{-1}\partial_y \varphi & c^{-1}\partial_z \varphi \\ -c^{-1}\partial_t a_x & -\partial_x a_x & -\partial_y a_x & -\partial_z a_x \\ -c^{-1}\partial_t a_y & -\partial_x a_y & -\partial_y a_y & -\partial_z a_y \\ -c^{-1}\partial_t a_z & -\partial_x a_z & -\partial_y a_z & -\partial_z a_z \\ \end{bmatrix}\end{split}\]

or contravariant-contravariant coomponents,

\[\begin{split}(\nabla \mathbf{A})^{\alpha \beta} = \frac{\partial A^{\beta}}{\partial X_{\alpha}} = g_{\beta \gamma} \frac{\partial A^{\alpha}}{\partial X^{\gamma}} = \begin{bmatrix} c^{-2}\partial_t \varphi &-c^{-1}\partial_x \varphi &-c^{-1}\partial_y \varphi &-c^{-1}\partial_z \varphi \\ c^{-1}\partial_t a_x & -\partial_x a_x & -\partial_y a_x & -\partial_z a_x \\ c^{-1}\partial_t a_y & -\partial_x a_y & -\partial_y a_y & -\partial_z a_y \\ c^{-1}\partial_t a_z & -\partial_x a_z & -\partial_y a_z & -\partial_z a_z \\ \end{bmatrix}\end{split}\]

The electromagnetic field tensor is defined as the anti-symmetric part of the gradient of the 4-electromagnetic potential,

\[\mathbf{F} = \left[ \symbf{\nabla} \mathbf{A} - \left( \symbf{\nabla} \mathbf{A} \right)^T \right]\]

whose components may be collected in a 2-dimensional array (first index for rows, second index for columns),

\[\begin{split}F^{\alpha \beta} = \begin{bmatrix} 0 & -\frac{\underline{e}^T}{c} \\ \frac{\underline{e}}{c} & \underline{b}_{\times} \end{bmatrix} \qquad , \qquad F_{\alpha \beta} = \begin{bmatrix} 0 & \frac{\underline{e}^T}{c} \\ -\frac{\underline{e}}{c} & \underline{b}_{\times} \end{bmatrix} \end{split}\]

Electromagnetic field and electromagnetic field equations

The pair of Maxwell equations

\[\begin{split}\begin{cases} \rho_f = \nabla \cdot \vec{d} \\ \vec{j}_f = - \partial_t \vec{d} + \nabla \times \vec{h} \\ \end{cases}\end{split}\]

can be re-written in \(4\)-formalism, using \(4\)-gradient in Cartesian coordinates

\[\symbf{\nabla} = \mathbf{E}^{\alpha} \frac{\partial}{\partial X^{\alpha}} = \mathbf{E}_0 \, \frac{\partial }{c \partial t} + \mathbf{E}_i \frac{\partial}{\partial x^i} = \mathbf{E}_0 \, \frac{\partial }{c \partial t} + \nabla \ ,\]

and the definition of the 4-current density vector

\[\mathbf{J} = J^{\alpha} \mathbf{E}_{\alpha} = c \rho \, \mathbf{E}_0 + \vec{j}\]

so that

\[\begin{aligned} c \rho \mathbf{E}_0 + \vec{j} = \symbf{\nabla} \cdot \mathbf{F} = \symbf{\nabla} \cdot [ & ( 0\, \mathbf{E}_0 + c \vec{d} ) \otimes \mathbf{E}_0 + ( - \mathbf{E}_0 c \vec{d} + \vec{h}_{\times} ) ] \ , \end{aligned}\]

with the displacement field tensor,

\[\mathbf{D} = D^{\alpha \beta} \, \mathbf{E}_{\alpha} \, \mathbf{E}_{\beta} \ , \]

with components (rows for the first index, columns for the second index)

\[\begin{split}D^{\alpha \beta} = \begin{bmatrix} 0 & -c d_x & - c d_y & - c d_z \\ c d_x & 0 & - h_z & h_y \\ c d_y & h_z & 0 & - h_x \\ c d_z & - h_y & h_x & 0 \\ \end{bmatrix} = \begin{bmatrix} 0 & - c \underline{d}^T \\ c \underline{d} & \underline{h}_{\times} \ . \end{bmatrix} \end{split}\]

The pair of Maxwell equations

\[\begin{split}\begin{cases} \nabla \cdot \vec{b} = 0 \\ \partial_t \vec{b} + \nabla \times \vec{e} = \vec{0} \\ \end{cases}\end{split}\]

can be re-written in \(4\)-formalism as

\[0 = \partial_{\mu} F_{\eta \xi} + \partial_{\eta} F_{\xi \mu} + \partial_{\xi} F_{\mu \eta} \]

Among these \(64 = 4^3\) equations, there are only 4 independent equations.

• If 2 indices are the same, the corresponding equation is the identity \(0 = 0\). As an example, if \(\mu = \eta\)

  \[0 = \partial_{\mu} F_{\mu \xi} + \partial_{\mu} \underbrace{F_{\xi \mu}}_{-F_{\mu xi}} + \partial_{\xi} \underbrace{F_{\mu \mu}}_{=0} = 0 \ , \]

  thus only combinations with different indices may provide some information.

• Given the ordered set of indices \((\mu, \eta, \xi)\), switching a pair of indices provides the same equation. As an example, switching \(\mu\) and \(\eta\)

  \[\begin{split}\begin{aligned} 0 & = \partial_{\eta} F_{\mu \xi} + \partial_{\mu} F_{\xi \eta} + \partial_{\xi} F_{\eta \mu} = \\ & = \partial_{\eta} ( - F_{\xi \mu} ) + \partial_{\mu} ( - F_{\eta \xi} ) + \partial_{\xi} ( -F_{\mu \eta} ) \ . \end{aligned}\end{split}\]

• Thus, only 4 combination of different indices, without taking order into account, provide independent information

  \[\begin{split}\begin{aligned} (1,2,3): & \quad 0 = \partial_{1} F_{23} + \partial_{2} F_{31} + \partial_{3} F_{12} = \partial_x (-b_x) + \partial_y (-b_y) + \partial_z (-b_z) \\ (2,3,0): & \quad 0 = \partial_{2} F_{30} + \partial_{3} F_{02} + \partial_{0} F_{23} = \partial_y \left(-\frac{e_z}{c} \right) + \partial_z \left( \frac{e_y}{c} \right) + \partial_{ct} (-b_x) \\ (3,0,1): & \quad 0 = \partial_{3} F_{01} + \partial_{0} F_{13} + \partial_{1} F_{30} = \partial_z \left(\frac{e_x}{c}\right) + \partial_{ct} (-b_y) + \partial_x \left(-\frac{e_z}{c}\right) \\ (0,1,2): & \quad 0 = \partial_{0} F_{12} + \partial_{1} F_{20} + \partial_{2} F_{01} = \partial_{ct} (-b_z) + \partial_x \left(-\frac{e_y}{c}\right) + \partial_y \left(\frac{e_x}{c}\right) \\ \end{aligned}\end{split}\]

  i.e.

  \[\begin{split}\begin{aligned} (1,2,3): & \quad 0 = - \nabla \cdot \vec{b} \\ (2,3,0): & \quad 0 = -\frac{1}{c} \left[ \partial_t b_x + (\partial_y e_z - \partial_z e_y ) \right] \\ (3,0,1): & \quad 0 = -\frac{1}{c} \left[ \partial_t b_y + (\partial_z e_x - \partial_x e_z ) \right] \\ (0,1,2): & \quad 0 = -\frac{1}{c} \left[ \partial_t b_z + (\partial_x e_y - \partial_y e_x ) \right] \\ \end{aligned}\end{split}\]

  i.e.

  \[\begin{split}\begin{cases} 0 = \nabla \cdot \vec{b} \\ \vec{0} = \partial_t \vec{b} + \nabla \times \vec{e} \\ \end{cases}\end{split}\]

Point particle in electromagnetic field

Lorentz 4-force acting on a point charge of electric charge charge \(q\) reads

\[\mathbf{f} = \mathbf{F} \cdot \mathbf{J} = q \, \mathbf{F} \cdot \mathbf{U} \ .\]

so that the dynamical equation reads

\[m \, \mathbf{X}'' = q \, \mathbf{F} \cdot \mathbf{X}'\]

Energy balance

\[\begin{split}\begin{aligned} \frac{\partial u }{\partial t} & = \\ \frac{\partial \vec{s}}{\partial t} & = \\ \end{aligned}\end{split}\]

…

\[\symbf{\nabla} \cdot \mathbf{T} = - \mathbf{F} \cdot \mathbf{J}\]