Time Dilation Calculator
Inputs
| Proper time | 10 sec |
|---|---|
| Velocity | 240,000,000 m/s |
Time Dilation Calculator
Calculate relativistic time dilation using special relativity. Enter a proper time and velocity to find the Lorentz factor γ and the dilated time measured by a stationary observer. Based on Einstein's special theory of relativity.
Inputs
Results
Enter a value to see results.
Time Dilation
Time dilation is a consequence of Einstein's special theory of relativity: a clock in motion relative to a stationary observer ticks more slowly than an identical clock at rest. The faster the clock moves, the greater the slowing. At ordinary speeds the effect is negligibly small, but as the velocity approaches the speed of light it becomes dramatic.
This calculator takes a proper time (time in the moving frame) and a velocity , and returns the Lorentz factor and the dilated time measured by the stationary observer.
How time dilation works
Imagine a clock on a fast-moving spacecraft. A beam of light bouncing between two mirrors ticks once per round trip. From the ground, the light travels a longer diagonal path for each tick, so the ticks are further apart — the clock appears to run slow. The faster the spacecraft, the longer the diagonal, and the more time appears to stretch.
The effect is not a mechanical defect. Biological processes, atomic vibrations, and radioactive decay rates are all affected equally. A traveller on the spacecraft does not feel anything unusual — from their perspective, it is the ground clocks that run slow.
Formula
| Quantity | Symbol | Definition |
|---|---|---|
| Lorentz factor | ||
| Proper time | Time elapsed in the moving frame | |
| Dilated time | Time elapsed for the stationary observer | |
| Relation | ||
| Speed of light | (exact) |
As , and there is no dilation. As , and the moving clock appears frozen to the stationary observer.
Worked example
A spacecraft travels at . The crew experience a journey of .
Step 1 — Lorentz factor:
γ=1−(0.8)21=1−0.641=0.361=0.61≈1.6667Step 2 — Dilated time:
t=γ⋅t0=1.6667×10≈16.67 sWhile the crew ages by 10 seconds, a ground observer measures 16.67 seconds. Enter these values in the calculator to reproduce the result.
Lorentz factor at different speeds
| Speed (fraction of ) | |
|---|---|
| 0.1c | 1.005 |
| 0.5c | 1.155 |
| 0.8c | 1.667 |
| 0.9c | 2.294 |
| 0.99c | 7.089 |
| 0.999c | 22.37 |
Real-world verification
Time dilation is not merely theoretical. The Hafele–Keating experiment in 1971 flew atomic clocks around the world on commercial aircraft and confirmed that the airborne clocks recorded less time than ground clocks, in agreement with both special and general relativistic predictions. GPS satellites orbit at about 14 000 km/h and experience a velocity-based time dilation of roughly −7 µs/day; without correcting for this (and for gravitational time dilation in the opposite direction), GPS positions would drift by kilometres per day.
Limitation: this calculator covers special relativity only
Special relativistic time dilation applies to inertial (non-accelerating) frames. Additional effects arise from gravity (general relativity): clocks deeper in a gravitational field run slower. For most particle physics and kinematic problems the formula here is sufficient; for precise orbital or cosmological calculations both effects must be combined.
Frequently Asked Questions (FAQ)
What is time dilation?
Time dilation is a prediction of Einstein's special theory of relativity: a clock moving relative to a stationary observer ticks more slowly than an identical clock at rest. The faster the clock moves, the slower it ticks, as measured from the stationary frame. This is not an illusion or a mechanical effect — it is a fundamental property of spacetime. GPS satellites, for example, run slightly fast due to the combination of special-relativistic (velocity) time dilation and general-relativistic (gravitational) time dilation; correcting for both is essential for accurate positioning.
What is the Lorentz factor?
The Lorentz factor γ = 1 / √(1 − v²/c²) quantifies how much time, length, and relativistic mass change at a given speed. At v = 0, γ = 1 and there is no relativistic effect. At v = 0.5c, γ ≈ 1.155. At v = 0.9c, γ ≈ 2.294. At v = 0.99c, γ ≈ 7.089. The factor appears throughout special relativity: time intervals are multiplied by γ (dilation), lengths are divided by γ (contraction), and relativistic momentum is p = γmv.
Does time dilation affect us at everyday speeds?
Yes, but the effect is immeasurably small. A passenger on a commercial aircraft at 900 km/h (250 m/s) has v/c ≈ 8 × 10⁻⁷, giving γ − 1 ≈ 3 × 10⁻¹³. Flying for 8 hours, the passenger's clock falls behind by about 0.9 nanoseconds. For a satellite in low Earth orbit at 7 800 m/s, γ − 1 ≈ 3.4 × 10⁻¹⁰, so clocks lose about 7 µs per day. These tiny differences are real and measurable with modern atomic clocks.
What is the twin paradox?
The twin paradox imagines one twin staying on Earth while the other travels at a high fraction of the speed of light and returns. Special relativity predicts that the travelling twin ages less — consistent with time dilation. The 'paradox' seems to arise because, from the traveller's frame, the Earth twin is moving, so shouldn't they age less? The resolution is that the two situations are not symmetric: the traveller must accelerate to turn around, breaking the symmetry. The traveller genuinely ages less. This has been confirmed experimentally with atomic clocks flown on aircraft (Hafele–Keating experiment, 1971).