Special relativity challenges our everyday notions of time and space. It reveals that as objects approach the speed of light, strange effects occur: time slows down, lengths shrink, and mass increases.

These relativistic effects have profound implications for our understanding of the universe. They lead to the famous equation E=mc², which shows that mass and energy are interchangeable, and forms the basis for nuclear reactions and our modern view of physics.

Relativistic Effects

Time dilation and length contraction

• Time dilation occurs when moving clocks run slower compared to stationary clocks (twin paradox)
  • Proper time ($\Delta \tau$) measured by clock at rest relative to event
  • Dilated time ($\Delta t$) measured by clock moving relative to event
  • Time dilation formula: $\Delta t = \gamma \Delta \tau$, where $\gamma = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}$
• Length contraction causes moving objects to appear shorter along direction of motion compared to proper length (Lorentz contraction)
  • Proper length ($L_0$) measured by observer at rest relative to object
  • Contracted length ($L$) measured by observer moving relative to object
  • Length contraction formula: $L = \frac{L_0}{\gamma}$
• Relativistic momentum increases as velocity approaches speed of light
  • Classical momentum formula ($p = mv$) not valid at relativistic speeds
  • Relativistic momentum formula: $p = \gamma mv$
  • As velocity approaches $c$, momentum approaches infinity
• Relativistic Doppler effect occurs when light sources and observers are in relative motion

Mass-energy equivalence in nuclear reactions

• Mass-energy equivalence equation $E = mc^2$ shows energy and mass are interchangeable
  • $c$ is speed of light in vacuum ($3 \times 10^8$ m/s)
• In nuclear reactions, small amount of mass converted into large amount of energy
  • Energy released can be calculated using mass-energy equivalence equation
  • In nuclear fission, mass of products slightly less than mass of reactants, difference released as energy (atomic bombs)
  • In nuclear fusion, light nuclei combine to form heavier nuclei, releasing energy (stars, fusion reactors)

Comparing Classical and Relativistic Concepts

Classical vs relativistic physics concepts

• Mass
  • Classical physics: Mass is constant property of object, independent of motion
  • Special relativity: Mass increases with velocity, approaching infinity as velocity approaches $c$
  • Relativistic mass formula: $m = \gamma m_0$, where $m_0$ is rest mass
• Energy
  • Classical physics: Kinetic energy given by $KE = \frac{1}{2}mv^2$
  • Special relativity: Total energy given by $E = \gamma m_0 c^2$, includes rest mass energy and kinetic energy
  • Rest mass energy: $E_0 = m_0 c^2$, energy object possesses even when at rest (intrinsic energy)
• Momentum
  • Classical physics: Momentum given by $p = mv$
  • Special relativity: Momentum given by $p = \gamma mv$, approaches infinity as velocity approaches $c$
  • Relativistic momentum formula reduces to classical formula at low velocities (Newtonian limit)

Fundamental Concepts in Special Relativity

• Inertial reference frames are coordinate systems moving at constant velocity relative to each other
• Simultaneity is not absolute; events simultaneous in one frame may not be in another
• Spacetime combines three spatial dimensions and one time dimension into a four-dimensional continuum
• Lorentz transformation equations relate coordinates between different inertial reference frames