Relativity: Time Dilation, Mass-Energy Equivalence, and Modern Applications

Fundamentals of Special Relativity

Einstein’s 1905 theory revolutionized our understanding of space-time by introducing two postulates:

1. The laws of physics are identical in all inertial frames
2. The speed of light (c ≈ 3×10⁸ m/s) is constant in all frames

Core Relativistic Effects

Time Dilation

Moving clocks run slower by factor γ (Lorentz factor):
\[
\Delta t’ = \gamma \Delta t = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}}
\]
Example: At 0.8c (γ=1.67), 1 Earth hour = 1.67 ship hours

Length Contraction

Objects contract along motion direction:
\[
L’ = \frac{L}{\gamma}
\]

Mass-Energy Equivalence

\[
E = \gamma mc^2 = mc^2 + (\gamma – 1)mc^2
\]
Components:

• Rest energy: E₀=mc²
• Kinetic energy: K=(γ-1)mc²

Practical Applications

Global Positioning System

• Satellite clocks run 38μs/day faster due to velocity (SR)
• 45μs/day slower from gravitational time dilation (GR)
• Net correction: -7μs/day

Particle Physics

• LHC protons reach 0.999999991c (γ=7,000)
• Mass increases to 6,500× rest mass

Astrophysics

• Muons (τ=2.2μs) reach Earth’s surface due to γ≈9 at 0.994c
• Black hole event horizons demonstrate extreme spacetime curvature

Worked Example

Time Dilation at 0.8c:
\[
\gamma = \frac{1}{\sqrt{1-0.8^2}} = \frac{1}{0.6} ≈ 1.67
\]
\[
\Delta t’ = 1.67 × 1 \text{hr} = 1 \text{hr} 40 \text{min}
\]

Common Errors

1. Using Newtonian kinetic energy (½mv²) at relativistic speeds
2. Confusing proper time (Δt) and dilated time (Δt’)
3. Misapplying length contraction perpendicular to motion

Practice Problems

1. Calculate γ and time dilation for:
   • 0.6c (γ=1.25)
   • 0.99c (γ≈7.09)
2. Compute the energy equivalent of 1μg mass (E≈90kJ)
3. Explain how GPS requires both special and general relativity.