Electrodynamics 2 Summary & Study Notes

🧮 Index Gymnastics (practical_advice_detailed)

Upper vs lower indices — transformation rules. Contravariant (upper) indices and covariant (lower) indices transform oppositely under Lorentz transformations. In index notation:

• Contravariant: $V'^{\mu}=\Lambda^{\mu}_{\ \nu}V^{\nu}$
• Covariant: $V'${\mu}=(\Lambda^{-1})^{\rho}{\ \mu}V_{\rho}

These opposite transformation behaviours guarantee that scalar products are invariant under Lorentz transformations.

🔁 Contraction rules and why they matter

Only one upper index may be contracted with one lower index. This is the Einstein summation convention. Valid contractions produce coordinate-independent scalars or lower-rank tensors, e.g. $V^{\mu}W_{\mu}$ or $T^{\mu}${\ \nu}A^{\nu}{\ \rho}. Contracting two uppers or two lowers requires inserting the metric (e.g. $\eta_{\mu\nu}V^{\mu}W^{\nu}$).

Practical example: the d'Alembertian on a scalar field

$\Box\phi=\partial_{\mu}\partial^{\mu}\phi=\frac{1}{c^{2}}\frac{\partial^{2}\phi}{\partial t^{2}}-\nabla^{2}\phi$.

Here $\partial^{\mu}=\eta^{\mu\nu}\partial_{\nu}$ raises the derivative index with the Minkowski metric $\eta^{\mu\nu}$.

✳ Free indices must match

Any free (unsummed) index must appear on both sides of an equation in the same position (upper vs lower) and represent the same quantity. Example (correct):

$F^{\mu\nu}=\partial^{\mu}A^{\nu}-\partial^{\nu}A^{\mu}$

Wrong index placement immediately signals an inconsistent tensor equation.

🛠 Index positioning as a calculation tool

Reading index positions turns many checks into mechanical verifications. For Lorentz invariance of the interval $s^{2}=\eta_{\mu\nu}x^{\mu}x^{\nu}$, one requires

$\eta_{\mu\nu}\Lambda^{\mu}${\ \rho}\Lambda^{\nu}{\ \sigma}=\eta_{\rho\sigma},

which is the defining orthogonality condition for Lorentz matrices.

⚠ Common pitfalls and how to avoid them

• Reusing dummy indices: do not use the same summed index multiple times in one term.
• Forgetting the metric: to contract two contravariant indices you must use $\eta_{\mu\nu}$.
• Mismatched free indices: check upper/lower positions and index names on each side.

🔎 Practical transformation example (gradient)

Under a boost in the $x$-direction ($\beta=v/c$, $\gamma=1/\sqrt{1-\beta^{2}}$):

$\begin{aligned} x'^{0}&=\gamma(x^{0}-\beta x^{1})\\ x'^{1}&=\gamma(x^{1}-\beta x^{0}) \end{aligned}$

but the gradient transforms with opposite sign on $\beta$:

$\begin{aligned} \partial'_{0}&=\gamma(\partial_{0}+\beta\partial_{1})\\ \partial'_{1}&=\gamma(\partial_{1}+\beta\partial_{0}). \end{aligned}$

This sign difference preserves scalar invariance, e.g. $\partial_{\mu}x^{\mu}$.

(End of notes adapted from practical_advice_detailed (1).pdf)

📐 Contravariant vs Covariant — the geometric viewpoint (contravariant_covariant_intro)

Why index notation? In relativity space and time mix under Lorentz transforms. Index notation tracks transformation properties component-by-component and enforces covariance.

🔼 Contravariant vectors

A contravariant four-vector has components $x^{\mu}$ with $\mu=0,1,2,3$ (e.g. $x^{0}=ct$, $x^{i}=x^{i}$). Under Lorentz transforms:

$x'^{\mu}=\Lambda^{\mu}_{\ \nu}x^{\nu}$.

Think of contravariant vectors as displacements or arrows in spacetime: examples include the four-position, four-momentum $p^{\mu}=(E/c,\mathbf{p})$, and four-current $j^{\mu}=(c\rho,\mathbf{j})$.

🔽 Covariant vectors and the metric

Covariant components are obtained by lowering indices with the metric:

$x_{\mu}=\eta_{\mu\nu}x^{\nu}$,

where for the mostly-minus signature

$\eta_{\mu\nu}=\mathrm{diag}(1,-1,-1,-1)$.

Thus $x_{0}=ct$ but $x_{i}=-x^{i}$ for spatial components. Covariant objects (one-forms) pair naturally with vectors to give scalars.

✖ Scalars and invariants

A scalar invariant forms by contracting a contravariant and covariant index, e.g.

$s= x^{\mu}x_{\mu}=(ct)^{2}-x^{2}-y^{2}-z^{2}$.

This invariance under Lorentz transforms is guaranteed by index structure.

🧩 Tensors and ranks

A tensor of rank $(m,n)$ carries $m$ contravariant and $n$ covariant indices. Transformation rules follow by applying the basic Lorentz factors for each index. The electromagnetic field tensor $F^{\mu\nu}$ is an important example (antisymmetric, six independent components).

✅ Practical advice (summary)

Remember: upper indices transform one way, lower the opposite way; contract only one upper with one lower; free indices must match on both sides. These rules help ensure equations are manifestly Lorentz-covariant and help avoid algebraic errors.

(End of notes adapted from contravariant_covariant_intro.pdf)

⚡ Relativistic Electrodynamics & Field Tensors (NotesOnElectrodynamics_Relativity)

🔁 Lorentz transformations and four-vectors

Use $x^{\mu}=(ct,\mathbf{x})$ and abbreviations $\beta=V/c$, $\gamma=1/\sqrt{1-\beta^{2}}$. The Lorentz boost along $x$ reads

$\bar{x}^{0}=\gamma(x^{0}-\beta x^{1}),\quad\bar{x}^{1}=\gamma(x^{1}-\beta x^{0}),\quad\bar{x}^{2}=x^{2},\quad\bar{x}^{3}=x^{3}.$

Four-vectors like the four-current $J^{\mu}=(c\rho,\mathbf{J})$ obey $\partial_{\mu}J^{\mu}=0$ (continuity equation) and transform as vectors.

🧲 Electromagnetic field tensor

Electric and magnetic fields combine into the antisymmetric electromagnetic field tensor $F^{\mu\nu}$ (six independent components). In matrix form (upper indices):

$F^{\mu\nu}=\begin{pmatrix}0 & E_{x}/c & E_{y}/c & E_{z}/c \\ -E_{x}/c & 0 & -B_{z} & B_{y} \\ -E_{y}/c & B_{z} & 0 & -B_{x} \\ -E_{z}/c & -B_{y} & B_{x} & 0\end{pmatrix}$

Its dual $G^{\mu\nu}$ interchanges electric and magnetic roles and is useful for the homogeneous Maxwell equations.

📜 Maxwell's equations in covariant form

• Inhomogeneous (sources): $\partial_{\nu}F^{\mu\nu}=\mu_{0}J^{\mu}$
• Homogeneous (no magnetic monopoles / Faraday's law): $\partial_{\nu}G^{\mu\nu}=0$ (or the cyclic form $\partial_{\lambda}F_{\mu\nu}+\partial_{\mu}F_{\nu\lambda}+\partial_{\nu}F_{\lambda\mu}=0$).

Written this way, Maxwell's equations are manifestly Lorentz-covariant: transforming $F^{\mu\nu}$ and $J^{\mu}$ with the appropriate factors leaves the equations unchanged in form.

🔄 Field transformations between frames

Electric and magnetic fields mix under boosts. For a boost with velocity $V$ along $x$ (and $\beta=V/c$):

$\begin{aligned} \bar{E}_{x}&=E_{x},\\ \bar{E}_{y}&=\gamma(E_{y}-V B_{z}),\\ \bar{E}_{z}&=\gamma(E_{z}+V B_{y}), \end{aligned}$

and similarly for $\mathbf{B}$. Grouping $\mathbf{E}$ and $\mathbf{B}$ into $F^{\mu\nu}$ explains these mixing rules cleanly.

⚖ Energy, momentum and stress-energy tensor

Electromagnetic energy and momentum appear in the symmetric stress-energy tensor $T^{\mu\nu}$ whose conservation $\partial_{\mu}T^{\mu\nu}=0$ encodes energy and momentum conservation. For fields, components relate to energy density $u$, Poynting vector $\mathbf{S}$ and Maxwell stress tensor.

(End of notes adapted from NotesOnElectrodynamics_Relativity.pdf)

🌍 Principle of Relativity & Intervals (Landau_relativity)

🧭 Principle of relativity and finite signal speed

The Einstein principle of relativity combines (i) the equivalence of inertial frames and (ii) a finite universal signal velocity $c$ (the speed of light in vacuum). This replaces Galilean invariance and implies no absolute time: simultaneity is frame-dependent.

⛳ Intervals and Minkowski geometry

Define the interval between two events

$S_{12}=c^{2}(t_{2}-t_{1})^{2}-(\mathbf{r}${2}-\mathbf{r}{1})^{2}.

For infinitesimal separations this yields the Minkowski line element

$ds^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}$.

A central result: $ds^{2}$ is invariant under Lorentz transformations. This defines pseudo-Euclidean (Minkowski) geometry.

⏳ Timelike, spacelike and lightlike separations

• Timelike: $ds^{2}>0$ — there exists a frame where the two events occur at the same place; a proper time can be associated.
• Spacelike: $ds^{2}<0$ — there exists a frame where the events are simultaneous; no causal influence can connect them.
• Lightlike (null): $ds^{2}=0$ — separation along the light cone; signals travel at $c$.

These classifications are invariant and determine causal structure.

🔁 Group property and velocity composition

Lorentz transformations form a group: sequential boosts combine according to the relativistic velocity addition law (e.g. for colinear velocities $u$ and $v$, $u\oplus v=(u+v)/(1+uv/c^{2})$). Requiring invariance of $c$ is what necessitates modifying classical velocity addition.

🧭 Physical consequences (brief)

No absolute simultaneity, time dilation, length contraction, maximum signal speed $c$, and the relativity of electric and magnetic fields (magnetism emerges from electrostatics plus relativity when charges move). Minkowski geometry provides the unifying language for these effects.

(End of notes adapted from Landau_relativity (1).pdf)