Special Relativity Fundamentals

Lorentz Transformation

• A mathematical formula describing how space and time coordinates are affected by relative motion
• Developed by Hendrik Lorentz in 1904
• Describes how observers in different inertial reference frames measure the same event
• Key equations:
  • Time dilation: t' = γ(t - vx/c^2)
  • Length contraction: x' = γ(x - vt)
  • Velocity addition: u' = (u + v) / (1 + uv/c^2)
• Where:
  • γ is the Lorentz factor (1 / sqrt(1 - v^2/c^2))
  • v is the relative velocity between the two frames
  • c is the speed of light
  • t and x are time and position in one frame
  • t' and x' are time and position in the other frame

Michelson-Morley Experiment

• Conducted in 1887 by Albert Michelson and Edward Morley
• Tested the hypothesis that light travels at a constant speed in all directions
• Experiment design:
  • Split a light beam into two perpendicular beams
  • Sent each beam to a mirror and reflected back to a detector
  • Compared the time it took for each beam to complete a round trip
• Results:
  • No significant difference in the speed of light in different directions
  • Challenged the concept of a fixed, absolute ether
  • Led to the development of special relativity by Albert Einstein

Mass-Energy Equivalence

• Concept proposed by Albert Einstein in 1905
• States that mass (m) and energy (E) are interchangeable
• Equation: E = mc^2
• Where:
  • m is the rest mass of an object
  • c is the speed of light
  • E is the total energy of the object
• Implications:
  • A small amount of mass can be converted into a large amount of energy
  • A large amount of energy can be converted into a small amount of mass
  • Has far-reaching implications in nuclear physics and beyond