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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):
![\[ \Delta t' = \gamma \Delta t = \frac{\Delta t}{\sqrt{1-\frac{v^2}{c^2}}} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-850c00da12ee2156dcb212e83b5909b8_l3.png "Rendered by QuickLaTeX.com")
Example: At 0.8c (γ=1.67), 1 Earth hour = 1.67 ship hours
Length Contraction
Objects contract along motion direction:
![\[ L' = \frac{L}{\gamma} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-5c7aebbc349b5892b0d7da35ca2c669e_l3.png "Rendered by QuickLaTeX.com")
Mass-Energy Equivalence
![\[ E = \gamma mc^2 = mc^2 + (\gamma - 1)mc^2 \]](https://on22.com/wp-content/ql-cache/quicklatex.com-1bcd3791bcbbe018a4efe3bc1af80f5b_l3.png "Rendered by QuickLaTeX.com")
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:
![\[ \gamma = \frac{1}{\sqrt{1-0.8^2}} = \frac{1}{0.6} ≈ 1.67 \]](https://on22.com/wp-content/ql-cache/quicklatex.com-fec90ee1634df0c8f407e2522acddaf4_l3.png "Rendered by QuickLaTeX.com")
![\[ \Delta t' = 1.67 × 1 \text{hr} = 1 \text{hr} 40 \text{min} \]](https://on22.com/wp-content/ql-cache/quicklatex.com-65311a645873a3581d9120637eb6657d_l3.png "Rendered by QuickLaTeX.com")
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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