Special Relativity Review: 4-Vectors & Minkowski Space
Special Relativity Review: 4-Vectors & Minkowski Space
Special relativity unifies space and time into a 4-dimensional pseudo-Riemannian manifold called Minkowski spacetime. The invariant interval replaces the Euclidean distance, and all physical laws must be expressed as tensorial equations under Lorentz transformations. This review establishes the 4-vector calculus that GR generalizes.
Minkowski Metric & Interval
The spacetime interval: \(ds^2 = \eta_{\mu\nu}\,dx^\mu dx^\nu = -c^2 dt^2 + dx^2 + dy^2 + dz^2\). Using signature \((-,+,+,+)\), \(\eta_{\mu\nu} = \text{diag}(-1,1,1,1)\). Indices run \(\mu,\nu = 0,1,2,3\). Einstein summation convention: repeated up/down indices are summed.
Lorentz Transformation & 4-Velocity
Under a boost with \(\beta = v/c\), \(\gamma = (1-\beta^2)^{-1/2}\): \(x'^\mu = \Lambda^\mu{}_\nu x^\nu\). The 4-velocity \(U^\mu = dx^\mu/d\tau = \gamma(c, \mathbf{v})\) satisfies \(\eta_{\mu\nu}U^\mu U^\nu = -c^2\). 4-momentum: \(p^\mu = mU^\mu = (E/c, \mathbf{p})\), with \(p^\mu p_\mu = -m^2c^2\).
Example 1: Invariant Mass
Two photons with energies \(E_1 = E_2 = E\) collide head-on. Find the invariant mass of the system.
Solution: \(p_1^\mu = (E/c, E/c, 0,0)\), \(p_2^\mu = (E/c, -E/c, 0,0)\). Total: \(P^\mu = (2E/c, 0,0,0)\). Invariant: \(P^\mu P_\mu = -(2E/c)^2\), so \(M = 2E/c^2\).
Example 2: Time Dilation via Proper Time
A muon travels at \(\beta = 0.99\). Its rest lifetime is \(\tau_0 = 2.2\,\mu\text{s}\). Find its lab lifetime.
Solution: \(\gamma = 1/\sqrt{1-0.99^2} \approx 7.09\). Lab lifetime: \(t = \gamma\tau_0 \approx 15.6\,\mu\text{s}\). The proper time \(d\tau = ds/c\) is Lorentz-invariant.
Practice
- Show that \(\partial_\mu \partial^\mu = -\partial_t^2/c^2 + \nabla^2\) (the d'Alembertian).
- Prove that \(\eta_{\mu\nu}U^\mu U^\nu = -c^2\) from the definition of proper time.
- A particle has \(E = 5\,\text{GeV}\), \(|\mathbf{p}|c = 4\,\text{GeV}\). Find its rest mass.
- Verify the Lorentz transformation preserves \(ds^2\).
Show Answer Key
1. $\partial_\mu\partial^\mu = \eta^{\mu\nu}\partial_\mu\partial_\nu = \eta^{00}\partial_0^2 + \eta^{11}\partial_1^2 + \eta^{22}\partial_2^2 + \eta^{33}\partial_3^2 = -(1/c^2)\partial_t^2 + \partial_x^2+\partial_y^2+\partial_z^2 = -\partial_t^2/c^2 + \nabla^2$. ✓
2. Proper time: $d\tau^2 = -ds^2/c^2 = dt^2 - d\mathbf{x}^2/c^2$. For massive particles: $U^\mu = dx^\mu/d\tau$. $\eta_{\mu\nu}U^\mu U^\nu = \eta_{\mu\nu}\frac{dx^\mu}{d\tau}\frac{dx^\nu}{d\tau} = \frac{ds^2}{d\tau^2} = \frac{-c^2 d\tau^2}{d\tau^2} = -c^2$. ✓
3. Mass-shell: $E^2 = (pc)^2 + (mc^2)^2$. $E=5$ GeV, $pc=4$ GeV: $mc^2 = \sqrt{25-16} = 3$ GeV. Rest mass $m = 3$ GeV/$c^2$.
4. Lorentz transformation: $x'^\mu = \Lambda^\mu_{\;\nu}x^\nu$ with $\Lambda^T\eta\Lambda = \eta$. $ds'^2 = \eta_{\mu\nu}dx'^\mu dx'^\nu = \eta_{\mu\nu}\Lambda^\mu_{\;\alpha}\Lambda^\nu_{\;\beta}dx^\alpha dx^\beta = \eta_{\alpha\beta}dx^\alpha dx^\beta = ds^2$. The defining property $\Lambda^T\eta\Lambda = \eta$ ensures invariance. ✓