--- title: "The Lorentz Transformation and Velocity Addition" description: "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." canonical: https://lightmysky.com/learn/science/the-lorentz-transformation-and-velocity-addition-mt_uG3_gArw19 source: https://lightmysky.com/learn/science/the-lorentz-transformation-and-velocity-addition-mt_uG3_gArw19.md retrieved: 2026-09-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # 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. Subject: Science · Area: Quantum & Modern Physics · Ages 19 to 20 Page: https://lightmysky.com/learn/science/the-lorentz-transformation-and-velocity-addition-mt_uG3_gArw19 ## Ready when they can - 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 ## Lesson: When speed mixes space and time 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. **Example.** 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. **Tip.** 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. **Recap.** Convert frames with the Lorentz pair, add fast speeds with the fraction, and light speed never changes. ## Practice 8 questions on this page, each with its working shown. ## Needs first - [Matrix Multiplication and What It Represents](https://lightmysky.com/learn/mathematics/matrix-multiplication-and-what-it-represents-mt_0fJOMPlCxr) - [Time Dilation and Length Contraction](https://lightmysky.com/learn/science/time-dilation-and-length-contraction-mt_7tpKiFWeG0) ## Opens up - [Relativistic Momentum and Energy](https://lightmysky.com/learn/science/relativistic-momentum-and-energy-mt_zBq87F-j3B)