Relativity Calculator
The speed tax: γ = 1/√(1 − β²) prices what a fast ride does to clocks, rulers and kinetic bills — with the light barrier refused by name, not by crash.
Relativity Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Special relativity only — no gravity terms
- Inertial frames, constant velocity
In short: Ride at half the speed of light (β = 0.5) and the Lorentz factor reads γ = 1.154701: your clock ticks 1/γ as fast as the stay-home twin's, so a shipboard day (86,400 s) ends 11,575.405113 s late for everyone else — three hours and change, special relativity alone. Rulers ride the same factor (lengths contract to 1/γ along the motion), and every kilogram aboard carries a kinetic bill of (γ − 1)c² = 1.390e+16 J at this speed. At β = 0.6 the arithmetic is a 3-4-5 triangle: γ = 1.25 exactly.
Formula
γ = 1 / √(1 − β²) · dilated t = γ·t₀ · contracted L = L₀/γ · KE = (γ − 1)m·c²
One factor, four consequences. Gamma is dimensionless and never below 1: at rest it is exactly 1, and it grows without bound as speed approaches c — the barrier is in the formula's denominator, which is WHY c is the cap rather than a traffic law. Time dilation stretches intervals, length contraction shortens rulers along the motion only, and the kinetic bill (gamma minus one times m c-squared) is the honest replacement for the half-m-v-squared that fails at speed.
Worked Example
- Enter your speed as a fraction of light speed.
- Read gamma — the single factor behind every card.
- Read a shipboard day's slip against the stay-home clock.
- Push toward 0.99 and watch why the barrier is arithmetic, not legislation.
Defaults: β = 0.5 → γ = 1.154701; a day slips 11,575.405113 s; kinetic bill 1.390e+16 J per kilogram. β = 0.6 → γ = 1.25 exactly.
Strengths & Limits Of This Model
Where this engine is strong
- One factor drives all four cards
- Light barrier refused by name, honestly
Where it stops
- No general-relativistic clocks
- No twin-paradox trip planner
Practical Use Cases
Teaching
the barrier as arithmetic
Cosmic-ray muons
why they reach the ground
Sci-fi sanity checks
what a fast ride really costs
Methodology & Editorial Standards
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Relativity Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why can nothing outrun light, in this arithmetic?
Because γ = 1/√(1 − β²) has no real value at β = 1 or beyond: the denominator dies, and the factor diverges. The speed of light is not a posted limit — it is where the formula that prices time, length and energy stops producing answers. This page refuses the barrier rather than printing a fake factor.
Does the traveller FEEL time slow?
Never — their own clock, heartbeat and coffee all agree at one second per second. The slip is between frames: the stay-home twin's account of the trip logs γ·t₀ seconds against the traveller's t₀. Both are right; that is the point.
What is proper time?
The time measured in the frame where the two events happen at the same place — the traveller's own elapsed seconds. It is the SHORTEST possible reading; every other frame logs more (multiplied by gamma). The day card here prices the stay-home frame's reading of a proper day.
Does length contraction squeeze everything?
No — only along the direction of motion, and only as seen from the other frame. A ship at β = 0.5 reads 1/γ = 86.6% of its rest length to the platform, while the crew measures the platform shortened the same way. No force squeezes anything; geometry disagrees about distances.
Is the GPS correction really this effect?
Partly — and the full story needs gravity too. Special-relativistic motion SLOWS the satellite clocks a few microseconds a day; general-relativistic altitude SPEEDS them about five times more; the net is roughly +38 microseconds per day, quoted approximate. Uncorrected, positions would drift kilometres daily — relativity is civil infrastructure.
How do muons survive to the ground?
Cosmic-ray muons are born high in the atmosphere with lifetimes far too short for the trip at classical arithmetic. At β ≈ 0.998 their clocks run about 15-fold slow against us (or, in their frame, the atmosphere is 15-fold thin) — both frames agree they arrive. The effect this page prices is raining through your detector right now.
What happened to half-m-v-squared?
It is the slow-speed approximation of the honest bill (γ − 1)m·c². At highway speeds the two agree to fifteen decimals; at β = 0.5 the classical guess underprices by about 15% and at β = 0.99 it is not even wrong. The kinetic card computes the honest bill from YOUR beta.
Why quote beta instead of metres per second?
The factor depends only on the RATIO to c, so beta is the natural dial — 0.5 means the same thing for a proton or a starship. Multiply by 299,792,458 m/s (exact) for the ordinary speed; the wavelength page's wave arithmetic and the frequency page carry that conversion spirit for everyday rides.