Introduction
Four-vectors unify space and time into spacetime. Position four-vector x^μ = (ct, x, y, z) transforms simply under Lorentz transformations. Four-vector formalism makes relativistic calculations systematic and reveals deep structure of spacetime physics. This is the modern language of relativity.
Four-Vector Definition
Four-vector has four components transforming like (ct, r) under Lorentz transformations. Contravariant: x^μ = (x0, x¹, x2, x3) = (ct, x, y, z). Covariant: x_μ = (ct, -x, -y, -z). Metric signature (+, -, -, -) or (-, +, +, +). Index placement (upper/lower) is crucial for correct calculations.
Spacetime Interval
Invariant interval: s2 = x^μ x_μ = (ct)2 - r2 = (ct)2 - x2 - y2 - z2. Same in all inertial frames. Defines geometry of Minkowski spacetime. Timelike (s2 > 0, within light cone), spacelike (s2 < 0, outside light cone), lightlike (s2 = 0, on light cone). Classification is invariant.
\nIntroduction
Four-vectors unify space and time into spacetime. Position four-vector x^μ = (ct, x, y, z) transforms simply under Lorentz transformations. Four-vector formalism makes relativistic calculations systematic and reveals deep structure of spacetime physics. This is the modern language of relativity.
Four-Vector Definition
Four-vector has four components transforming like (ct, r) under Lorentz transformations. Contravariant: x^μ = (x0, x¹, x2, x3) = (ct, x, y, z). Covariant: x_μ = (ct, -x, -y, -z). Metric signature (+, -, -, -) or (-, +, +, +). Index placement (upper/lower) is crucial for correct calculations.
Spacetime Interval
Invariant interval: s2 = x^μ x_μ = (ct)2 - r2 = (ct)2 - x2 - y2 - z2. Same in all inertial frames. Defines geometry of Minkowski spacetime. Timelike (s2 > 0, within light cone), spacelike (s2 < 0, outside light cone), lightlike (s2 = 0, on light cone). Classification is invariant.
\nFour-Velocity and Four-Momentum
Four-velocity: u^μ = dx^μ/dτ = γ(c, v) where τ is proper time. Magnitude: u^μ u_μ = c2 (invariant, timelike). Four-momentum: p^μ = mu^μ = (E/c, p). Components: p0 = E/c = γmc, p^i = γmv^i = spatial momentum. Energy and momentum unified into one four-vector.
Lorentz Transformation of Four-Vectors
Matrix form: x'^μ = Λ^μ_ν x^ν where Λ is 4×4 Lorentz transformation matrix. Boost along x: x'0 = γ(x0 - βx¹), x'¹ = γ(x¹ - βx0), x'2 = x2, x'3 = x3. All four-vectors transform identically, making calculations systematic. Tensors extend this to more complex quantities.
Applications
Covariant formulation of electromagnetism (Maxwell's equations in compact form). Relativistic kinematics in particle physics. General Relativity uses curved spacetime with similar four-vector formalism. Standard in high-energy physics calculations. Quantum field theory uses four-vector notation throughout.
\nSolved Example: Four-Vector Calculation
An electron has velocity v = 0.8c in the +x direction. Find its four-velocity and four-momentum. Also find the invariant p^μ p_μ. Solution: First calculate γ: β = 0.8, γ = 1/√(1-0.64) = 1/√0.36 = 1/0.6 = 1.667. Four-velocity: u^μ = γ(c, v, 0, 0) = (1.667c, 1.333c, 0, 0) in units where c is explicit, or u^μ = (5c/3, 4c/3, 0, 0). Check invariance: u^μ u_μ = (5c/3)2 - (4c/3)2 = (25c2/9) - (16c2/9) = 9c2/9 = c2 ✓. Four-momentum: p^μ = mu^μ = (γmc, γmv, 0, 0). For electron: m = 0.511 MeV/c2. p0 = E/c = 1.667 × 0.511 MeV/c = 0.852 MeV/c. p¹ = 1.333 × 0.511 MeV/c = 0.681 MeV/c. p^μ = (0.852, 0.681, 0, 0) MeV/c. Check invariant: p^μ p_μ = (p0)2 - (p¹)2 = (0.852)2 - (0.681)2 = 0.726 - 0.464 = 0.262 (MeV/c)2. This equals (mc)2 = (0.511)2 = 0.261 (MeV/c)2 ✓. Small difference due to rounding. Invariant mass is verified. The four-vector formalism automatically keeps track of the relationship between energy and momentum, ensuring consistency.
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