Special Relativity — worked problems

Rest frame: pure Coulomb, $f^\mu_* = (0, 0, e^2/(4\pi\varepsilon_0 d^2), 0)$. Boost along $\hat x$: $\hat y$-component is invariant.

$$\boxed{\;f^\mu = (0, 0, e^2/(4\pi\varepsilon_0 d^2), 0)\quad\text{(frame-independent)}.\;}$$

Lab-frame 3-force: $\vec F = e^2/(4\pi\varepsilon_0\gamma d^2)\hat y$ — Coulomb repulsion enhanced by $\gamma$ but the magnetic attraction (parallel currents) cancels by $\beta^2/(1-\beta^2)\cdot\gamma = -\gamma\beta^2$, net effect $1/\gamma$. As $v\to c$, 3-force vanishes — the origin of the pinch effect and self-collimation of relativistic beams.

Perfect-fluid stress-energy with $\vec u\to -\vec v$ in the boosted frame: $T'^{0i} = -(4\rho_0\gamma^2/3c)v^i$. With $\rho' = T'^{00} = (3+\beta^2)\gamma^2\rho_0/3$: $$\boxed{\;\vec\pi = -\frac{4\rho'}{3c^2 + v^2}\,\vec v.\;}$$ Newtonian limit: $\vec\pi \to -(\rho_0 + p_0)\vec v/c^2$ — the inertial density is the enthalpy, not the bare energy. The familiar $4/3$ enthalpy factor of radiation appears as soon as photons acquire a bulk velocity.

Galaxy frame: $f^\mu = (\gamma\rho_0 A v/c, 0, 0, 0)$. Cruiser frame: $f'^\mu = (\gamma^2\rho_0 Av/c,\;-\gamma^2\rho_0 Av^2/c^2,\;0,\;0)$.

Decomposition with $u^\mu = (c,0,0,0)$: $f^\mu_\text{heat} = (\gamma^2\rho_0 Av/c, 0, 0, 0)$ — rate of mass-energy gain; $f^\mu_\text{pure} = (0, -\gamma^2\rho_0 Av^2/c^2, 0, 0)$ — relativistic ram-pressure drag. Both non-zero: the force is neither pure nor heat-like.

$\gamma^2$ enhancement: length contraction densifies dust ($\gamma$) AND each dust particle's 4-momentum is boost-enhanced ($\gamma$).

(a) Setting $T'^{11} = \gamma^2(\beta^2\rho_0 - \sigma) = 0$ gives $\beta^2 = \sigma/\rho_0 < 1$. Yes — at this subluminal boost.

(b) $T'^{00} = (\rho_0 - \beta^2\sigma)/(1-\beta^2)$. Differentiating: $(\rho_0 - \sigma)/(1-\beta^2)^2 > 0$, so $T'^{00}$ is monotone-increasing in $\beta^2$; minimum at $\beta=0$ equals $\rho_0$. No — the rest frame already has the smallest energy density. This is the special-relativistic dominant-energy condition.

Perfect fluid with $p_0 = \rho_0/3$: $$T'^{00} = \rho_0(3+\beta^2)/[3(1-\beta^2)],\quad T'^{0i} = (4\rho_0\gamma^2/3)u^i/c,\quad T'^{ij} = (\rho_0/3)\delta^{ij} + (4\rho_0\gamma^2/3c^2)u^i u^j.$$

Shear. Off-diagonal $T'^{ij}_{i\neq j} = (4\rho_0\gamma^2/3c^2)u^i u^j$ vanishes only when frame axes align with $\vec u$. In other frames "shear" appears, but it's a coordinate artefact — the trace-free symmetric part of $T'^{ij}$ is non-zero whenever $\gamma > 1$, encoding a genuine, frame-independent uniaxial anisotropy along the flow.

$\rho \equiv T'^{00} = (\rho_0 + p)\gamma^2 - p$. So $$\boxed{\;\frac{\rho}{\gamma^2} = \rho_0 + p\beta^2 < \rho_0 + p.\;}$$ The bound is approached as $\beta\to 1$. The enthalpy $\rho_0 + p$ is the natural inertial density — appears in the relativistic Euler equation and the sound-speed formula.

$\rho_\text{obs} = c^2\rho_0[(1+w)\gamma^2 - w]$. Monotone-decreasing in $\gamma^2$ if $1+w < 0$, going to $-\infty$ — violates WEC. So $$\boxed{\;\text{WEC} \iff \rho_0 \ge 0\text{ and }w \ge -1.\;}$$ $w < -1$ is phantom matter — sustained phantom dark energy in cosmology drives a Big Rip.

(a) $\partial\mathcal L/\partial x^\mu = 0$, $\partial\mathcal L/\partial\dot x^\mu = c p_\mu$. EL gives $dp_\mu/d\tau = 0$.

(b) EL equation at $O(q)$ contains $\dot x^\nu(\partial_\mu A_\nu - \partial_\nu A_\mu) = \dot x^\nu F_{\mu\nu}$: $$\boxed{\;dp_\mu/d\tau = (q/c) F_{\mu\nu} u^\nu\;}$$ — covariant Lorentz force. The $O(q^2)$ term $\propto A^\nu\partial_\mu A_\nu$ is gauge-dependent (no $F$ structure), correctly dropped. "Minimal coupling" keeps $q^1$, gives $F$ automatically.