The Lorentz Transformation and Velocity Addition
One set of equations converts coordinates between frames and contains dilation and contraction as special cases. Adding velocities with it never gives a result above the speed of light.
What a learner can do afterwards
- Transforms the coordinates of an event between two frames
- Recovers time dilation and length contraction from the transformation
- Adds two large velocities relativistically and checks the low-speed limit
1 · Read
Two frames in relative motion mix space and time when they compare notes. With the motion along x at speed v, the Lorentz change reads x prime equals gamma times x minus v t, and t prime equals gamma times t minus v x over c squared, while y and z stay put. Gamma is 1 over the square root of 1 minus v squared over c squared, and it grows as v nears light speed.
Dilation and contraction hide inside those two lines. For two ticks at the same moving spot, the transform hands you a stretched time: that is time dilation. For a rod whose ends you mark at the same primed time, the length comes out divided by gamma: that is length contraction. Even simultaneity bends: flashes seen together trackside, at the two ends of a 26 m car passing at half light speed, flash at different times for the seated passenger.
Add speeds with u equals v plus u prime over 1 plus v u prime over c squared. A ship at half light speed fires a canister forward at 0.75 c, and Earth sees 1.25 c over 1.375, which is 0.909 c, not 1.25 c. At everyday speeds the bottom of the fraction sits near 1, and the rule melts back into ordinary addition.
Feed light speed into the rule and light speed comes out, from any ship at any speed. Anything built from slower parts stays below c.
Convert frames with the Lorentz pair, add fast speeds with the fraction, and light speed never changes.
2 · Watch
Take it off screen
Where it sits
8 questions wait behind this lesson, each with its answer explained. Every answer feeds the sky: stars light as they are learned, and dim when it is time to come back.