Relativity: Lorentz Transformations

Overview

This lecture covers Lorentz transformations in special relativity, explaining how space and time coordinates transform between moving reference frames. The professor derives the consequences of these transformations, including velocity addition, time dilation, length contraction, and addresses several famous paradoxes.

Coordinate Transformations Between Reference Frames

• Two observers: S (stationary) and S′ (moving at velocity u relative to S)
• At initial crossing, both set clocks to zero and synchronized their coordinate origins
• Event: something happening at a specific place and time, described by coordinates (x, t) or (x′, t′)
• Pre-Einstein: x′ = x − ut and x = x′ + ut; time was absolute (t = t′)
• Post-Einstein: space and time coordinates mix when transforming between frames
• The "fudge factor" γ accounts for disagreements about lengths and time intervals

Deriving the Lorentz Factor

• Consider an event triggered by a light pulse emitted when observers meet
• Light travels at velocity c for both observers: x/t = c and x′/t′ = c
• Combining these constraints with transformation equations yields γ = 1/√(1 − u²/c²)
• The Lorentz transformations include this factor in both space and time equations
• All relativistic effects (E = mc², time dilation, Twin Paradox) follow from these equations

Velocity Addition Formula

• Classical expectation: velocities simply add (w = v − u for opposite directions)
• Example: bullet velocity v in frame S becomes w in frame S′ moving at velocity u
• Relativistic formula: w = (v − u)/(1 − uv/c²)
• The denominator prevents any velocity from exceeding c
• Example: train and bullet each at ¾c yields 24c/25, still less than c
• For light itself (v = c), the formula correctly gives w = c

Relativity of Simultaneity

• Two events occurring at same time for one observer (Δt = 0) are not simultaneous for moving observer
• Even when Δt = 0, Δt′ = −uΔx/c² ÷ √(1 − u²/c²) ≠ 0
• Train example: light pulse sent from center to front and back of train
• Observer on train: pulses reach both ends simultaneously
• Observer on ground: back wall rushes toward pulse, front wall retreats; events not simultaneous
• This is not a trivial time zone difference but fundamental disagreement about simultaneity

Time Dilation

• Clocks run fastest in their own rest frame
• Consider clock ticking: Event 1 at (x = 0, t = 0), Event 2 at (x = 0, t = τ₀)
• For stationary clock (Δx = 0), moving observer measures Δt′ = τ₀/√(1 − u²/c²)
• Moving clocks appear to run slow by factor √(1 − u²/c²)
• Light clock explanation: vertical light path in rest frame becomes zigzag in moving frame
• All clocks must behave identically or velocity could be detected, violating relativity postulate
• Biological aging also slows at high speeds

Twin Paradox

• Twin travels at high speed, returns younger than Earth-bound twin
• Paradox: each twin could claim the other is moving and should be younger
• Resolution: traveling twin undergoes acceleration/deceleration during takeoff, turnaround, landing
• Only Earth-bound twin remains in inertial frame throughout journey
• Symmetry breaks during acceleration periods; only one twin can claim to be stationary
• If Δt = 3000 years on Earth and √(1 − u²/c²) = 1/60, traveler ages only 50 years
• Effect confirmed with accelerated particles in labs like Fermilab

Length Contraction

• Rods appear longest in their rest frame
• Contraction factor: L = L₀√(1 − u²/c²)
• To measure moving rod length: find both ends at same time (Δt = 0)
• Events at rod's ends separated by L₀ in rod's frame, by L in observer's frame
• Example: meter stick moving where √(1 − u²/c²) = 0.5 appears half-meter long
• Muons from upper atmosphere survive journey despite short lifetime because atmosphere appears contracted in their frame

Paradoxes and Resolutions

• Hole paradox: half-meter hole in table, meter stick contracted to half-meter falls in
• From stick's perspective: hole is quarter-meter, stick is one meter
• Resolution: front end enters hole first, back end enters later (events not simultaneous)
• Garage paradox: 14-meter car fits in 12-meter garage at sufficient velocity
• Car occupant: entire car was inside garage at one instant
• Parent perspective: front end entered and broke rear wall before back end entered
• All paradoxes resolved through relativity of simultaneity and length measurements

Key Terms & Definitions

• Event: Something happening at specific place and time, described by coordinates
• Lorentz transformation: Equations relating space-time coordinates between moving frames
• γ (gamma factor): 1/√(1 − u²/c²), appears in all relativistic transformation equations
• Proper time (τ₀): Time interval measured in object's rest frame
• Proper length (L₀): Length measured in object's rest frame
• Light clock: Conceptual clock using light pulse bouncing between mirrors
• Inertial frame: Reference frame experiencing no acceleration, where Newton's laws hold