Lorentz transformationsequation
Imagine a train traveling almost at the speed of light. A passenger on the train and a person on the platform measure the length of the same carriage and the time on the clocks — and get different results! The Lorentz transformations show how these measurements are related so that the speed of light remains the same for both observers. It is as if space and time compress and stretch, adjusting to the motion.
How it works
In everyday life we do not notice this, but at speeds of tens and hundreds of kilometers per second the effects become noticeable. For example, in particle accelerators, protons are accelerated to near-light speeds, and without relativistic corrections their trajectories cannot be calculated. Even the lifetime of unstable particles (muons) in cosmic rays increases because they move at high speed.
💡 GPS satellites move at high speed, and their clocks tick slightly slower than those on Earth. Without taking the Lorentz transformations into account, the navigator would accumulate an error of several kilometers every day!
\gamma = \frac{1}{\sqrt{1 - v^2 / c^2}}
γ — Lorentz factor (dimensionless); v — relative velocity of inertial frames (m/s); c — speed of light in vacuum, fundamental constant (c = 299,792,458 m/s)
t' = \gamma (t - \frac{v x}{c^2})
t' — time in the moving frame (s); t — time in the stationary frame (s); x — coordinate along the direction of motion in the stationary frame (m); v — relative velocity (m/s); c — speed of light (m/s); γ — Lorentz factor
x' = \gamma (x - v t)
x' — coordinate in the moving frame (m); x — coordinate in the stationary frame (m); t — time in the stationary frame (s); v — relative velocity (m/s); γ — Lorentz factor