Time Dilation Calculator
Solve the special-relativistic time dilation relation for dilated time, proper time, or velocity — with Lorentz factor, unit conversion, and a full step-by-step solution.
Speed ↑
γ grows slowly at everyday speeds and then explodes as v approaches c — at 0.9 c, γ ≈ 2.29; at 0.999 c, γ ≈ 22.4.
Everyday Speeds
At jet or even orbital speeds, v²/c² is so tiny that dilation is only microseconds per day — measurable with atomic clocks but invisible in daily life.
v = 0
With no relative motion, γ = 1 and both observers measure the same time interval — there is no dilation.
v → c limit
As v approaches c, γ diverges to infinity. This is one way to see why massive objects cannot reach the speed of light.
How to Use This Calculator
Time dilation links three quantities: the relative velocity v, the proper time t measured by the moving clock, and the dilated time t′ measured by the stationary observer. Pick what to solve for, then fill in the other two.
1. Choose what to solve for
Pick t′ (dilated time), t (proper time), or v (velocity) using the three buttons under "Solve for".
2. Enter velocity in any unit
Give v as a fraction of c, a percent of c, or in km/s, m/s, km/h, or mph. Presets cover the ISS, a GPS satellite, 0.8 c, and a cosmic-ray muon.
3. Enter the known time
Type a time in seconds, minutes, hours, days, or years — proper and dilated time can each use a different unit.
4. Read the result and steps
The result card shows the solved value and the Lorentz factor γ; the step-by-step panel shows β = v/c, γ, and the final substitution.
5. Switch to Advanced for more depth
Advanced mode adds +1% sensitivity, the exact time difference t′ − t (accurate even for tiny effects), length-contraction factor, and speed in several units.
6. Copy or reset
Use "Copy result" to grab the solved value, or "Reset" to return to the default 0.8 c, 10-year example.
What This Tool Offers
Built around Einstein's special-relativistic time dilation formula for an observer in uniform relative motion.
Solve any of the three variables
Find dilated time, proper time, or the velocity needed to produce a given dilation.
Flexible units
Velocity in c, % of c, km/s, m/s, km/h, or mph; time in seconds through years, converted automatically.
Real-world presets
One-tap presets for the ISS, GPS satellites, 0.8 c, and cosmic-ray muons.
Transparent step-by-step math
Shows β, the Lorentz factor γ, and the substitution, so every answer can be checked by hand.
Precise at tiny effects
Advanced mode uses a numerically stable form for γ − 1, so microsecond-scale dilation at low speeds isn't lost to rounding.
Reference tables
Included below: β-to-γ values, unit conversions for c and years, and real-world time dilation examples.
What Is Time Dilation?
In special relativity, the time between two events depends on the observer's motion. A clock moving relative to an observer is measured to tick more slowly than a clock at rest in the observer's own frame. This is not a mechanical flaw in the clock — it is a property of spacetime itself, and it has been confirmed by experiments with atomic clocks on aircraft, satellites, and by the extended lifetimes of fast-moving particles.
Time Dilation Formula
where γ = 1/√(1 − v²/c²) is the Lorentz factor. Rearranged for the other two variables:
Formula Variables
| Symbol | Meaning | Notes |
|---|---|---|
| t′ | Dilated time, measured by the stationary observer | Always ≥ t |
| t | Proper time, measured on the moving clock | Shortest interval |
| v | Relative velocity between the frames | 0 ≤ v < c |
| c | Speed of light in vacuum | 299,792,458 m/s |
| γ | Lorentz factor, 1/√(1 − v²/c²) | ≥ 1, dimensionless |
Reference Table — Lorentz Factor by Speed
| v / c (β) | γ | 1 s on the moving clock = | Length contraction (1/γ) |
|---|---|---|---|
| 0.1 | 1.005 | 1.005 s | 0.995 |
| 0.5 | 1.155 | 1.155 s | 0.866 |
| 0.6 | 1.250 | 1.250 s | 0.800 |
| 0.8 | 1.667 | 1.667 s | 0.600 |
| 0.9 | 2.294 | 2.294 s | 0.436 |
| 0.99 | 7.089 | 7.089 s | 0.141 |
| 0.999 | 22.37 | 22.37 s | 0.0447 |
| 0.9999 | 70.71 | 70.71 s | 0.0141 |
Conversion Table — Speed of Light and Time Units
| Quantity | Value |
|---|---|
| 1 c in m/s | 299,792,458 (exact) |
| 1 c in km/s | 299,792.458 |
| 1 c in km/h | ≈ 1,079,252,849 |
| 1 c in mph | ≈ 670,616,629 |
| 1 year (Julian) | 365.25 days = 8,766 hours = 31,557,600 s |
| 1 day | 86,400 s |
Real-World Examples
| Scenario | Speed | Special-relativistic effect |
|---|---|---|
| Airliner / jet | ~250 m/s | ≈ 0.03 µs slower per day |
| GPS satellite | ~3.87 km/s | ≈ 7.2 µs slower per day (velocity effect only) |
| International Space Station | ~7.66 km/s | ≈ 28 µs slower per day (velocity effect only) |
| Cosmic-ray muon | ~0.994 c | γ ≈ 9.1, so a 2.2 µs lifetime appears ≈ 20 µs |
Worked Example
β = v/c = 0.8
γ = 1 / √(1 − 0.8²) = 1 / √0.36 = 1 / 0.6 = 1.6667
t′ = γ · t = 1.6667 × 10
t′ = 16.667 years
Reference & Method Note
Frequently Asked Questions
Each inertial observer sees the other's clock running slow — the effect is symmetric for uniform motion. The asymmetry in scenarios like the twin paradox comes from one traveler accelerating and changing frames, which the simple formula doesn't cover on its own.
Proper time is the time interval measured by a clock that is present at both events, i.e. at rest relative to them. It is the minimum time any observer will measure between those two events; all other observers measure a longer, dilated time.
At v = c the formula divides by zero and γ becomes infinite, and beyond c the square root becomes imaginary. For massive objects, v < c is a hard limit of the theory, so the calculator rejects speeds at or above c.
At everyday speeds γ differs from 1 by a tiny amount (about v²/2c²), so the main displayed value barely changes. Switch to Advanced mode to see the exact difference t′ − t, calculated in a numerically stable way.
What it means
Time dilation is a consequence of special relativity: a clock moving relative to an observer runs slower. The formula t' = γ t, where γ = 1/√(1 – v²/c²), relates the time interval t measured by an observer in the moving frame (proper time) to the time interval t' measured by an observer in a different frame. This effect has been verified experimentally with muon decays and atomic clocks on satellites. It is crucial for GPS synchronization, particle accelerators, and understanding high‑speed phenomena. The formula shows that as v approaches c, time slows down drastically. Understanding time dilation is essential for relativistic kinematics and modern physics.
Worked example
Time Dilation – Two Examples
Real‑World| Parameter | Value |
|---|---|
| t | 1 s |
| v | 1.5×10⁸ m/s |
| Parameter | Value |
|---|---|
| v | 2.7×10⁸ m/s |
Common mistakes
- Speed v: Must be less than c – if v=c, the denominator becomes zero (infinite time).
- Time t: Proper time (in the moving frame) – t is the time in the moving frame, tʹ is the time in the stationary frame? Actually, tʹ = t / γ, where γ = 1/√(1−v²/c²). Be careful about which frame is which.
- Units: Both times in same units (seconds).
- Time dilation: Moving clocks run slower – the moving observer measures a longer time.
- Velocity addition: Not the same as Galilean addition – use relativistic velocity addition.
Applications
Time dilation, t' = t / √(1 − v²/c²), is a consequence of special relativity, stating that moving clocks run slow. This effect is real and must be accounted for in GPS satellite timing, particle accelerators, and high‑speed travel. Engineers use it to synchronise global navigation systems, to correct particle lifetimes in accelerators, and to design spacecraft navigation. The formula is also used in high‑energy physics to predict particle decay times. By understanding time dilation, professionals can ensure accurate timing for modern technology and deepen our understanding of the nature of space and time.
- GPS satellite clock synchronisation and correction
- Particle accelerator physics (muon lifetimes)
- Spacecraft mission timing and relativistic corrections
- High‑speed communication and network synchronisation
- Educational demonstration of relativistic effects
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