Relativity Calculator

Time dilation says a moving clock runs slow by a factor γ = 1/√(1 − v²/c²). At 0.1c, γ ≈ 1.005 — only a half percent stretch. At 0.5c, γ ≈ 1.155. At 0.9c, γ ≈ 2.29. At 0.99c, γ ≈ 7.09. Plug in your velocity as a fraction of c, and the calculator returns γ. So a clock moving at 0.99c through a 1-second interval (stationary frame) ticks only 1/γ ≈ 0.141 seconds in its own frame. This isn't science fiction — GPS satellites must correct for it constantly to stay accurate.

γ = 1/√(1 − v²/c²) where v is speed and c is speed of light, ≈ 3×10^8 m/s. Easier to write β = v/c, so γ = 1/√(1 − β²). At β = 0, γ = 1 (no relativistic effects). As β approaches 1, γ explodes toward infinity. The growth is very gentle for everyday speeds: at v = 1000 m/s (β ≈ 3.3×10^-6), γ ≈ 1 + 5×10¹². You'd need atomic-clock precision to detect this. Real relativistic effects only become noticeable above β ≈ 0.1, which is a tenth of light speed — way beyond chemical rockets.

Lengths along the direction of motion contract by 1/γ in the observer's frame: L = L/γ. Here L is the proper length, measured in the object's own frame. So a 100 m spaceship moving at 0.6c (γ = 1.25) appears 80 m long to a stationary observer. At 0.99c, the same ship would look only 14 m long. Note that lengths perpendicular to motion don't contract. Combined with time dilation, this is what keeps everyone agreeing on the speed of light — different observers measure different distances and times that conspire to give the same c.

Classical KE = ½mv² fails at relativistic speeds. The correct expression is KE = (γ − 1)mc². At low speeds, this reduces to the classical formula via Taylor expansion. At high speeds, KE grows much faster than ½mv² would suggest. So a 1 kg object at 0.9c has γ = 2.29, giving KE = 1.29 × c² × 1 kg ≈ 1.16×10^14 J — about 28 megatons of TNT. This is why accelerating anything to relativistic speeds takes absurd amounts of energy: KE diverges as v approaches c. The calculator handles γ for you.

Speeds in relativity get cleaner expressed as fractions or percentages of c, often called β = v/c. So 30000 km/s = 0.1c = 10% c. Light speed itself is c ≈ 3×10^8 m/s ≈ 1.08×10 km/h. Calculators convert between m/s and percentage of c. Useful when problems give speeds in unusual units or as multiples of c. β > 1 is impossible for massive objects — that's a built-in physics constraint, not a calculator limitation. Once you've got β, plug it into γ formulas, and all relativistic effects follow systematically.

For everyday human standards, time dilation only becomes noticeable above about 10% of c (β ≈ 0.1), where γ reaches roughly 1.005 — half a percent. Astronauts on the ISS, moving at 7.7 km/s (β ≈ 2.6×10^-6), age about 0.007 seconds slower per six months. Nothing dramatic. Particle physics experiments at Fermilab or CERN routinely see particles with γ in the thousands, so their lifetimes look thousands of times longer in the lab frame. For a science-fiction style time dilation, you'd need β > 0.5 or so — beyond the reach of any current technology.

In the classic version, one twin stays on Earth while the other travels at relativistic speed and returns. The travelling twin ages less by a factor of γ during the high-speed legs. So a round trip at 0.8c (γ = 1.667) lasting 10 years on Earth corresponds to about 6 years for the traveller. The asymmetry comes from the traveller's acceleration during turnaround, which breaks the symmetry between the two frames. The calculator computes both elapsed times once you specify journey speed and Earth-frame duration. It's not a paradox so much as a counterintuitive consequence of how time works at high speeds.