Time Dilation Calculator

Special Relativity, developed by Albert Einstein in 1905, describes how space and time behave when objects move at velocities approaching the speed of light. Two of its most counterintuitive predictions — time dilation and length contraction — have been confirmed experimentally to extraordinary precision.

Time Dilation formula: t' = t / √(1 − v²/c²)

Length Contraction formula: L' = L × √(1 − v²/c²)

Lorentz Factor: γ = 1 / √(1 − v²/c²)

What each variable means:

• t: proper time: time elapsed as measured by the moving observer (e.g., on the spaceship clock)
• t’: dilated time: time elapsed as measured by the stationary observer; always greater than t
• L: proper length: the object’s length in its own rest frame
• L’: contracted length: the observed length from the stationary frame; always less than L
• v: velocity of the moving object (m/s)
• c: speed of light = 299,792,458 m/s ≈ 3 × 10⁸ m/s
• γ (gamma), Lorentz factor; equals 1 at rest, approaches infinity as v → c

Worked example — at 90% the speed of light: v = 0.90c → v²/c² = 0.81 → 1 − 0.81 = 0.19 → √0.19 = 0.4359 → γ = 2.294

A spaceship travels for 1 year (ship time) at 0.90c. A stationary observer measures: t’ = 1 year / 0.4359 = 2.294 years have passed on Earth.

A 100-meter spaceship appears contracted to: L’ = 100 × 0.4359 = 43.59 meters

Real-world verification:

• GPS satellites run at ~14,000 km/h, their clocks gain ~45 microseconds/day due to gravitational effects and lose ~7 µs/day due to velocity time dilation; without relativistic corrections, GPS would drift ~10 km per day.
• Muon decay: Cosmic ray muons created at 15 km altitude with a half-life of 1.5 µs should barely reach sea level, but time dilation allows them to reach us, which has been measured experimentally.