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
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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.
More to read
5.5 The Lorentz Transformation
The Lorentz transformation is the dictionary between two inertial frames in standard setup. It mixes space and time: the primed position blends x with v t, and the primed time blends t with v x over c squared, both scaled by gamma. At everyday speeds gamma is nearly 1 and the mix is invisible, which is why Galileo seemed right. Near light speed the mixing dominates everything.
5.6 Relativistic Velocity Transformation
Velocities do not add by simple sum once speeds grow large. The relativistic rule divides the Galilean sum by a correction factor that keeps every result below c. Fire a probe forward from a fast ship and ground control still clocks it under light speed. The formula protects the second postulate algebraically.
28.4 Relativistic Addition of Velocities
Check any relativistic formula by taking the slow limit first. When v is tiny beside c, gamma slides to 1 and the Lorentz equations become the Galilean ones. Velocity addition likewise sheds its denominator and returns to plain addition. Relativity extends Newton rather than erasing it, which is why slow physics never noticed the difference.
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