Chapter 1: Four-Vectors
Four-vectors are the natural objects in spacetime. They transform simply under Lorentz transformations, have invariant "lengths," and provide a unified framework for relativistic physics.
Four-Position
\( x^\mu = (ct, x, y, z) = (ct, \vec{x}) \)
The four-position combines time and space into a single 4-component vector. The index μ runs from 0 to 3, with x⁰ = ct. Under Lorentz transformations: \( x'^\mu = \Lambda^\mu_{\;\nu} x^\nu \)
Four-Velocity
\( u^\mu = \frac{dx^\mu}{d\tau} = \gamma(c, \vec{v}) \)
The derivative with respect to proper time τ (not coordinate time t). This ensures u^μ transforms as a four-vector. Note: \( u_\mu u^\mu = c^2 \) (invariant magnitude).
The Minkowski Metric
\( \eta_{\mu\nu} = \text{diag}(1, -1, -1, -1) \)
The metric is used to raise/lower indices and compute inner products:
- • \( x_\mu = \eta_{\mu\nu} x^\nu = (ct, -x, -y, -z) \)
- • \( A \cdot B = A_\mu B^\mu = A^0 B^0 - \vec{A} \cdot \vec{B} \)
- • \( x_\mu x^\mu = c^2t^2 - x^2 - y^2 - z^2 \) (invariant interval)