Time Dilation Calculator
Compute the time experienced by a stationary observer for a moving clock that ticks through a given proper time, using special relativity. Returns the dilated time interval in the same units as the proper-time input.
Last updated: May 2026
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About this calculator
The formula is t = τ / √(1 − (v/c)²), where τ is the proper time (the time measured in the rest frame of the moving clock), v is the relative velocity in m/s, and c = 299,792,458 m/s is the speed of light. The denominator is 1/γ, where γ is the Lorentz factor; t = γ·τ. Edge cases: when v = 0, γ = 1 and dilated time equals proper time — no relativistic effect. As v approaches c, (v/c)² approaches 1, the denominator approaches 0, and the dilated time grows without bound — a clock at 99.99% c ticks ~70× slower than at rest, and at 99.9999% c ~700× slower. For v ≥ c the formula produces NaN (imaginary square root); special relativity prohibits massive objects from reaching c. At everyday velocities (cars, airliners) the effect is microscopic: at v = 300 m/s (a passenger jet), γ ≈ 1 + 5 × 10⁻¹³, meaning a 1-hour flight clock falls behind by ~2 nanoseconds relative to a ground clock. Time dilation is real and confirmed experimentally: muons created in the upper atmosphere reach Earth's surface despite having a half-life that would let them decay long before, atomic clocks on aircraft show measurable offsets after long flights (Hafele-Keating experiment 1971), and GPS satellites must correct for both special-relativistic time dilation (clocks running slower due to orbital velocity) and general-relativistic dilation (clocks running faster at altitude due to weaker gravity).
How to use
Example 1 — half-light-speed clock. v = 1.5 × 10⁸ m/s (≈ 0.5c), τ = 1 year. Step 1: v/c = 1.5e8 / 2.998e8 ≈ 0.5004. Step 2: (v/c)² ≈ 0.2504. Step 3: 1 − 0.2504 = 0.7496. Step 4: √0.7496 ≈ 0.8658. Step 5: t = 1 / 0.8658 ≈ 1.155 years. Verify: γ at v = 0.5c is 1/√0.75 ≈ 1.1547 — close to the computed result ✓. After 1 year of the moving clock, a stationary observer sees ~1.155 years pass — a 15.5% time-dilation effect. Example 2 — 99% light speed. v = 2.97 × 10⁸ m/s (0.99c), τ = 1 year. Step 1: v/c ≈ 0.99. Step 2: (v/c)² ≈ 0.9801. Step 3: 1 − 0.9801 = 0.0199. Step 4: √0.0199 ≈ 0.1411. Step 5: t = 1 / 0.1411 ≈ 7.09 years. Verify: γ at 0.99c is 1/√0.0199 ≈ 7.089 — matches ✓. The relativistic effect becomes dramatic: a year of proper time corresponds to over 7 years for the stationary observer. This is the physics behind 'interstellar travel paradoxes' — a crew accelerating to 0.99c could reach a star 10 light-years away aging only about 1.4 years while 10 years pass on Earth.
Frequently asked questions
Why is time dilation symmetric, and how does the twin paradox resolve it?
Time dilation is symmetric in special relativity: each inertial observer sees the other's clock running slow. Twin A sees twin B's moving clock dilated; twin B equally sees twin A's moving clock dilated. Both views are correct in their own frames. The twin paradox seems to break this symmetry — when one twin travels to a distant star at high speed and returns, they are younger than the stay-at-home twin, asymmetrically. The resolution: the travelling twin's frame is not inertial throughout; they must accelerate to turn around, breaking the symmetry. During the turn-around, the travelling twin's notion of 'now' on Earth jumps forward dramatically, accounting for the missing time. Special relativity handles this with detailed accounting through the Lorentz transformations or the relativistic Doppler effect; general relativity offers an alternative formulation via the equivalence principle treating the acceleration as a gravitational field. Both approaches agree that the travelling twin ages less, and the effect has been verified with atomic clocks flown around the Earth (Hafele-Keating 1971).
How accurate must velocity be for noticeable time dilation, and where is it observed?
Time dilation grows nonlinearly with velocity, dominated by the √(1 − v²/c²) factor. At v = 0.01c (3 × 10⁶ m/s, fast but achievable for spacecraft) γ ≈ 1.00005 — clocks differ by 50 microseconds per hour. At v = 0.1c, γ ≈ 1.005, a 0.5% effect. At v = 0.5c, γ ≈ 1.155, 15.5% effect. At v = 0.99c, γ ≈ 7.09. Beyond 0.99c the effect grows explosively. In practice, measurable time dilation requires either very high velocities (particle accelerators routinely produce particles with γ > 100) or extremely precise clocks (atomic clocks measure parts-per-trillion offsets at jet speeds). GPS satellites orbit at ~3.9 km/s, giving v²/c² ≈ 1.7 × 10⁻¹⁰; uncorrected, their clocks would drift by ~7 μs/day from this effect alone, plus another ~45 μs/day from gravitational dilation in the opposite direction. Without these corrections, GPS positions would drift by ~10 km/day.
Is the speed of light truly the same in all frames, and why does that imply time dilation?
Yes — the constancy of c in all inertial frames is one of Einstein's two postulates of special relativity (the other being the principle of relativity itself), and it's been confirmed by countless experiments to staggering precision (current limits exceed 10⁻¹⁵ on c's frame-dependence). Time dilation follows from this directly. Consider a 'light clock' that bounces a photon between two mirrors a fixed distance apart. In the clock's rest frame, the photon takes time t₀ = 2d/c per bounce. To an observer watching the clock move past, the photon traces a longer zigzag path (combination of vertical bouncing and horizontal motion), so it takes longer. Since the photon's speed is the same c in both frames, the moving clock's tick must be longer in the observer's frame — that's time dilation. Length contraction is the complementary consequence: applying Lorentz transformations consistently, lengths in the direction of motion contract by the same factor that times dilate. Together with mass-energy equivalence, they form the core of special relativity.
What are the common mistakes when applying time dilation?
The biggest mistake is mixing frames — applying γ relative to one frame's measurements while interpreting them in another frame. Proper time is unique to each observer's rest frame; converting between frames requires consistent Lorentz transformations. The second is using non-relativistic velocity addition: at high speeds, velocities don't add linearly; use u' = (u + v)/(1 + uv/c²). The third is forgetting that the formula is symmetric: A sees B's clock dilated and B sees A's clock dilated — there's no 'real' time. People also confuse special-relativistic time dilation (velocity-based) with general-relativistic gravitational time dilation (potential-based); both apply for GPS but with opposite signs. Using v ≥ c gives nonsense (imaginary results) — the formula breaks down because massive objects cannot reach c. Finally, sign errors with relativistic Doppler or assuming time-dilation applies to all observed phenomena (it doesn't — light travel time, aberration, and relativistic beaming need separate treatment).
When should I not use this calculator?
Do not use it for non-inertial (accelerating) reference frames — these require general relativity or careful instantaneous-rest-frame analysis with proper time integrated along the worldline. The formula assumes constant velocity throughout the time interval. It is not appropriate for gravitational time dilation (clocks at different gravitational potentials), which uses GR's gravitational redshift formula, not the SR Lorentz factor. Do not use it for cosmological time dilation of distant supernovae, which involves the scale factor a(t) and the expansion of the universe, not local SR. It is unsuitable for objects moving in curved spacetime (near black holes, in cosmological contexts) — the SR formula is a flat-spacetime approximation. For v ≥ c the formula produces NaN; physical objects cannot reach light speed. Massless particles (photons) don't have a proper time at all because their worldlines are null. For accuracy beyond ~6 significant figures, use the full CODATA value of c and double-precision arithmetic; single-precision can lose precision when v is close to but not equal to c.