Relativistic Electrodynamics — Comprehensive Study Notes Summary & Study Notes

⚡ Special Relativity — Postulates and Lorentz Transformations

Postulates: Special relativity is built on two key ideas. First, the principle of relativity: the laws of physics are the same in all inertial frames. Second, the invariance of the speed of light: the speed of light in vacuum is the same for all inertial observers.

Motivation for new transforms: Galilean transformations conflict with the second postulate. Requiring linear maps between inertial frames that preserve straight lines and satisfy the postulates leads to the Lorentz transformations.

Lorentz boost (1D): Define $\beta = v/c$ and the Lorentz factor $\gamma = \frac{1}{\sqrt{1 - \beta^2}}$. For a boost along the $x$-axis,

$ct' = \gamma,(ct - \beta x)$

$x' = \gamma,(x - v t) = \gamma,(x - \beta ct)$

$y' = y, \quad z' = z$.

Matrix form: Package spacetime as $X^\mu = (ct, x, y, z)^T$. A Lorentz boost is $X'^\mu = \Lambda^\mu{}_{\nu} X^\nu$, with $\Lambda$ a $4\times4$ matrix satisfying $\Lambda^T \eta \Lambda = \eta$.

Invariant interval: The quantity

$s^2 = c^2 t^2 - x^2 - y^2 - z^2$

is invariant under Lorentz transformations. This replaces the absolute Newtonian time and is central to causality and classification of timelike, spacelike and lightlike separations.

Comments: Using $ct$ as the time coordinate is convenient because it makes time and space components have same units and simplifies transform algebra.

🧭 Four‑Vectors, Proper Time, and 4‑Momentum

Proper time: For a particle worldline, the infinitesimal proper time $d\tau$ satisfies

$d\tau^2 = dt^2 - \frac{dx^2 + dy^2 + dz^2}{c^2}$

in units where the metric sign convention is $(+,-,-,-)$ for $ds^2 = c^2 d\tau^2$. Proper time is an invariant and equals the time measured in the particle's rest frame.

4‑velocity: Define the 4‑velocity as

$U^\mu = \frac{dx^\mu}{d\tau} = \gamma (c, \mathbf{v})$,

with $U^\mu U_\mu = c^2$ (invariant).

4‑momentum: The 4‑momentum is

$p^\mu = m U^\mu = (E/c, \mathbf{p})$,

and satisfies the energy–momentum relation

$E^2 = \mathbf{p}^{,2} c^2 + m^2 c^4$.

Covariance: 4‑vectors transform under Lorentz transforms as $V'^\mu = \Lambda^\mu{}_{\nu} V^\nu$. Writing physics in terms of 4‑vectors makes invariance under boosts manifest.

📐 Tensors, Metric, and Index Manipulation

Tensors and index notation: Objects with indices carry transformation laws. A contravariant vector is $X^\mu$ and a covariant vector is $X_\mu$. Use the metric to lower/raise indices: $X_\mu = \eta_{\mu\nu} X^\nu$.

Metric: The Minkowski metric is usually written as

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

and obeys $\eta_{\mu\alpha} \eta^{\alpha}{}${\nu} = \delta{\mu}{}^{\nu}.

Einstein summation: Repeated indices are summed. Keep index positions (up/down) correct; this enforces tensorial consistency.

Tensor transformation: A rank-(1,1) object transforms as $T'^\mu{}${\nu} = \Lambda^\mu{}{\alpha} \Lambda_{\nu}{}^{\beta} T^\alpha{}_{\beta}, ensuring coordinate‑independent statements.

Practical rules: Work only with tensors, and ensure indices match on both sides of equations. This is the clean route to a manifestly Lorentz‑covariant theory.

🔌 Electromagnetism in Covariant Form and Magnetism as a Relativistic Effect

4‑current and continuity: Combine charge density and current into the 4‑current $J^\mu = (c\rho, \mathbf{J})$. The continuity equation becomes the covariant form

$\partial_\mu J^\mu = 0$,

which is Lorentz invariant.

Electromagnetic 4‑potential and field tensor: Introduce the 4‑potential $A^\mu = (\phi, \mathbf{A})$. The electromagnetic field tensor is

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

an antisymmetric tensor encoding $\mathbf{E}$ and $\mathbf{B}$.

Maxwell's equations (covariant): The inhomogeneous equations read

$\partial_\mu F^{\mu\nu} = \mu_0 J^\nu$,

and the homogeneous equations are compactly written as

$\partial_{[\alpha} F_{\beta\gamma]} = 0$ (or equivalently $\partial_\mu \tilde F^{\mu\nu}=0$),

where $\tilde F^{\mu\nu}$ is the dual tensor. These forms make gauge invariance and Lorentz covariance manifest.

Field transformations and magnetism: Under boosts, electric and magnetic fields mix; a pure electric field in one frame can appear as a combination of $\mathbf{E}$ and $\mathbf{B}$ in another. This provides a physical explanation for magnetism: the magnetic force can be viewed as a relativistic correction to Coulomb forces arising from Lorentz contraction of charge densities.

Wire example (qualitative): A neutral current‑carrying wire in the lab frame can become charged in the rest frame of a moving test charge because electron and ion densities transform differently under boosts. The resulting electric field in the test charge frame accounts for the magnetic force observed in the lab frame when transformed back.

Gauge freedom: The potentials are not unique: $A^\mu \to A^\mu + \partial^\mu \chi$ leaves $F^{\mu\nu}$ invariant. Working with covariant potentials simplifies solving Maxwell equations and shows the deep connection between gauge symmetry and conservation laws.