Special Relativity Calculator
Work out what happens near the speed of light. Get the Lorentz factor, time dilation, length contraction, relativistic energy, velocity addition and the twin paradox, accurately even at speeds a hair below c.
How to Use
- Pick a calculation: Lorentz factor, Time dilation, Length contraction, Mass & energy, Velocity addition or the Twin paradox.
- Enter the speed and choose how you are giving it: as a fraction or percentage of c, in km/s, m/s, km/h or mph, as a Lorentz factor γ, as 1 − v/c for speeds extremely close to light, or as a rapidity.
- For time, length or energy, use the Find menu to choose the unknown; its field turns into the answer and its unit menu converts it.
- For velocity addition, enter the object’s speed inside the moving frame and the frame’s own speed; a minus sign means the object moves backwards.
- Read the γ graph, the clock and ship drawings or the spacetime diagram, and Show Work for every step.
Worked Example
A muon from a cosmic-ray shower. At rest a muon lives 2.197 µs on average. At v = 0.998c, γ = 1 ÷ √(1 − 0.998²) = 1 ÷ √0.003996 = 15.82, so seen from the ground it lasts Δt = γ × Δt₀ = 15.82 × 2.197 = 34.75 µs and covers 0.998 × 299,792,458 × 34.75 × 10⁻⁶ = 10.40 km instead of 657 m.
Adding speeds. A ship at 0.8c fires a probe forward at 0.6c. The probe’s speed seen from Earth is w = (0.6 + 0.8) ÷ (1 + 0.6 × 0.8) = 1.4 ÷ 1.48 = 0.9459c, still below light speed.
The common mistake: dividing by γ instead of multiplying. It is the moving clock that runs slow, so the time we measure for the muon is longer than its own. Writing Δt = Δt₀ ÷ γ gives 2.197 ÷ 15.82 = 0.1389 µs, which would let the muon travel just 42 m. The right answer is Δt = γ × Δt₀ = 34.75 µs: the moving clock’s time Δt₀ is always the shorter one.
Show Work
Formulas
From Michelson–Morley to GPS
In 1887 Albert Michelson and Edward Morley failed to detect any change in the speed of light as the Earth moved through the supposed ether. George FitzGerald (1889) and Hendrik Lorentz (1892) proposed that moving bodies shrink along their motion, and Lorentz completed the transformation that now carries his name in 1904. In June 1905 Albert Einstein’s paper On the Electrodynamics of Moving Bodies derived it all from two postulates: the laws of physics are the same in every steadily moving frame, and light always travels at the same speed c. A short follow-up that September gave the result now written E = mc². Hermann Minkowski recast the theory as four-dimensional spacetime in 1908.
Paul Langevin described the travelling-twin example in 1911. The predictions have since been measured directly: Bruno Rossi and David Hall timed the decay of cosmic-ray muons at different altitudes in 1941; Joseph Hafele and Richard Keating flew caesium clocks around the world in 1971; and every GPS satellite clock is set to run slightly slow before launch so that, once in orbit, special and general relativity together bring it into step with clocks on the ground.
The speed of light has been exactly 299,792,458 m/s since 1983, when the metre was defined from it, so every number here is exact apart from rounding. Particle masses are NIST CODATA 2022 values.
About This Tool
This calculator covers the core of special relativity in one place: the Lorentz factor with a graph of how it grows, time dilation and length contraction solved for any one unknown, relativistic energy and momentum from a speed or a kinetic energy, the relativistic sum of two velocities, and the twin paradox with a spacetime diagram. Speeds can be typed in many forms, and a decimal speed is read exactly, so 0.99999999999999c or a cosmic-ray proton a few parts in 10²⁴ below c still gives the right γ instead of rounding to infinity. Each mode also shows how far Newton’s answer would be off.
It covers special relativity only: steady motion, no gravity, instant turnarounds. Everything runs in your browser; nothing you enter is sent anywhere.
Related tools: Photon Energy & Wavelength Calculator, Rocket Equation (Δv) Calculator, and Doppler Effect & Wave Calculator.
Frequently Asked Questions
What is the Lorentz factor γ?
γ = 1 ÷ √(1 − v²/c²) is how much time stretches and lengths shrink at speed v. It is 1.005 at 0.1c, 1.155 at 0.5c, 2.294 at 0.9c, 7.089 at 0.99c and 22.37 at 0.999c: almost nothing at everyday speeds, then it climbs without limit as v approaches c.
Does GPS really need relativity?
Yes. A GPS satellite moves at about 3.874 km/s, so special relativity slows its clock by 7.214 µs a day; being higher in Earth’s gravity speeds it up by about 45.7 µs a day (general relativity). The net 38.5 µs a day would make positions drift by kilometres within a day if it were not corrected.
How do muons from the upper atmosphere reach the ground?
A muon lives 2.197 µs on average at rest. At 0.998c, γ = 15.82, so in our frame it lasts 34.75 µs and travels about 10.40 km instead of 657 m. In the muon’s own frame the atmosphere is contracted by the same factor of 15.82 instead.
Why can’t two speeds add up to more than c?
Speeds combine as w = (u + v) ÷ (1 + uv/c²). A probe fired forward at 0.6c from a ship doing 0.8c moves at 1.4 ÷ 1.48 = 0.9459c, not 1.4c. Even 0.9999c plus 0.9999c only gives 1 − 5.0 × 10⁻⁹ of c, and adding c to anything gives exactly c.
How much younger is the travelling twin?
At 0.8c (γ = 5/3), a round trip to a point 4 light-years away takes 10 years by Earth clocks but only 6 years for the traveller, who comes back 4 years younger. The trip is not symmetric because only the traveller turns round, changing frames.
How do I use the Special Relativity Calculator?
Simply type your numbers and read the result, which refreshes the instant you change something. There is nothing to submit and nothing to wait for.
Is it free? Does it work without internet?
Yes to both. It is free with no sign-up, and once the page has loaded it keeps working even with no internet.
Where does my data go?
Nowhere — every calculation runs on your own device. Nothing you enter is uploaded, logged, or stored.
Common Use Cases
Particle physics
A 1 MeV electron has γ = 2.957 and moves at 0.9411c; Newton’s ½mv² would give only 22.63% of its true kinetic energy.
Accelerators
A 6.8 TeV LHC proton has γ ≈ 7,247 and falls short of light speed by about 2.85 m/s, which the 1 − v/c mode shows without rounding to c.
Cosmic rays
The 3.2 × 10²⁰ eV proton seen in 1991 had γ = 3.41 × 10¹¹, a speed of 1 − 4.30 × 10⁻²⁴ of c.
Satellite clocks
At 3.874 km/s a clock loses 7.214 µs per day, the special-relativity part of the GPS clock correction.
Science fiction checks
A 100 m ship at 0.99c is 14.11 m long to a watcher, and each hour on board is 7.089 hours outside.
Homework
Find the speed at which a clock runs at half rate: γ = 2 means v = 0.8660c, 259,600 km/s.
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