Introduction
Lorentz transformations relate space and time coordinates between inertial frames moving at constant relative velocity. They replace Galilean transformations, preserving the speed of light while mixing space and time coordinates. These are the mathematical foundation of special relativity.
Standard Configuration
Frame S' moves at velocity v along x-axis relative to S. Origins coincide at t = t' = 0. Coordinates perpendicular to motion (y, z) are unchanged. Transformations mix x and t coordinates. This simplified configuration contains all essential physics of Lorentz transformations.
Lorentz Transformation Equations
x' = γ(x - vt), t' = γ(t - vx/c2), y' = y, z' = z where γ = 1/√(1-v2/c2) is Lorentz factor. Inverse: x = γ(x' + vt'), t = γ(t' + vx'/c2). Symmetric between frames (principle of relativity). The mixing of x and t shows space and time are interconnected.
\nIntroduction
Lorentz transformations relate space and time coordinates between inertial frames moving at constant relative velocity. They replace Galilean transformations, preserving the speed of light while mixing space and time coordinates. These are the mathematical foundation of special relativity.
Standard Configuration
Frame S' moves at velocity v along x-axis relative to S. Origins coincide at t = t' = 0. Coordinates perpendicular to motion (y, z) are unchanged. Transformations mix x and t coordinates. This simplified configuration contains all essential physics of Lorentz transformations.
Lorentz Transformation Equations
x' = γ(x - vt), t' = γ(t - vx/c2), y' = y, z' = z where γ = 1/√(1-v2/c2) is Lorentz factor. Inverse: x = γ(x' + vt'), t = γ(t' + vx'/c2). Symmetric between frames (principle of relativity). The mixing of x and t shows space and time are interconnected.
\nLorentz Factor
γ = 1/√(1-β2) where β = v/c. For v << c: γ ≈ 1 + 1/2β2, recovers Newtonian limit. At v = 0.6c: γ = 1.25. At v = 0.8c: γ ≈ 1.67. At v = 0.99c: γ ≈ 7.09. As v → c: γ → ∞. γ ≥ 1 always. This factor determines all relativistic effects magnitude.
Interval Invariance
Spacetime interval s2 = (cΔt)2 - (Δx)2 - (Δy)2 - (Δz)2 is invariant under Lorentz transformations. All inertial observers agree on s2. Classification: timelike (s2 > 0, cause-effect possible), spacelike (s2 < 0, no causal connection), lightlike (s2 = 0, light signal only).
Rapidity
Alternative parameterization: velocity addition is simpler using rapidity φ where tanhφ = v/c. Lorentz transformation becomes hyperbolic rotation in spacetime. Velocity addition: φ_total = φ1 + φ2 (simple addition). γ = coshφ. Useful in particle physics for adding multiple velocities.
\nSolved Example: Lorentz Transformation Calculation
Frame S' moves at v = 0.6c relative to S. In S, an event occurs at x = 100 m, t = 2 μs. Find coordinates in S'. Also find spacetime interval. Solution: First calculate γ: β = 0.6, β2 = 0.36. γ = 1/√(1-0.36) = 1/√0.64 = 1/0.8 = 1.25. x' = γ(x - vt) = 1.25 × (100 - 0.6×3×10^8 × 2×10^-6) = 1.25 × (100 - 360) = 1.25 × (-260) = -325 m. t' = γ(t - vx/c2) = 1.25 × (2×10^-6 - 0.6×3×10^8×100/(9×10^16)) = 1.25 × (2×10^-6 - 2×10^-7) = 1.25 × 1.8×10^-6 = 2.25×10^-6 s = 2.25 μs. In S', the event occurs at x' = -325 m, t' = 2.25 μs. Event is to the left of S' origin and occurs later than in S. Spacetime interval: s2 = (cΔt)2 - (Δx)2 = (3×10^8 × 2×10^-6)2 - (100)2 = (600)2 - 10000 = 360000 - 10000 = 350,000 m2. Same in S': (3×10^8 × 2.25×10^-6)2 - (-325)2 = (675)2 - 105625 = 455625 - 105625 = 350,000 m2. Interval is invariant ✓. Timelike interval (positive), so event could be causally connected to origin.
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