Lorentz transformations

Lorentz Transformations

  • Lorentz Transformations are mathematical equations derived by Hendrik Lorentz.
  • They are central to the theory of Special Relativity, and they show how measurements of space and time by two observers moving relative to each other are related.
  • The transformations reflect the fact that observers moving relative to each other will measure different distances, times, momenta, and energies.

Working of Lorentz Transformations

  • If one observer is moving at a constant velocity relative to another, both observers will agree on the speed of light but will disagree on the actual measurements of space and time.
  • The transformations answer the question of how space and time measurements change for an observer moving relative to an inertial frame.
  • In other words, they provide the link between what two observers moving relative to each other will measure.

Key Components of Lorentz Transformations

  • Position and Time Coordinates: These transformations include the position of an event (x, y, z) and the time (t) at which the event occurs.
  • Relative Velocity: This is the velocity (v) of one observer relative to the other.
  • Speed of Light: (c) is a constant and fundamental part of these transformations.

Lorentz Transformation Equations

  • The simplest Lorentz transformations are for motion in the x-direction. They can be expressed as:
    • x’ = γ(x - vt)
    • t’ = γ(t - vx/c²)
    • Here x’, t’ are the coordinates in the moving frame, x, t are the coordinates in the stationary frame, and γ is the Lorentz factor defined as γ = 1/√(1 - v²/c²)
  • These formulas show how space and time coordinates transform for observers moving relative to one another.

Importance of Lorentz Transformations

  • Lorentz transformations play a crucial role in both Special and General Relativity.
  • They are fundamental to understanding high-speed particle physics, electromagnetic theory, and quantising fields for particle interpretation.
  • They replace the Galilean transformations of classical mechanics when dealing with high speeds, near the speed of light.