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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:
- The laws of physics are identical in all inertial frames
- 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:
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
- Using Newtonian kinetic energy (½mv²) at relativistic speeds
- Confusing proper time (Δt) and dilated time (Δt’)
- Misapplying length contraction perpendicular to motion
Practice Problems
- Calculate γ and time dilation for:
- 0.6c (γ=1.25)
- 0.99c (γ≈7.09)
- Compute the energy equivalent of 1μg mass (E≈90kJ)
- Explain how GPS requires both special and general relativity.
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