Relativistic Time Dilation Calculator

Enter a velocity as a fraction of the speed of light (or in m/s or km/s) and see how time slows down, lengths shrink, and twin travelers age differently. The interactive gamma curve shows the dramatic asymptotic behavior as you approach light speed.

Lorentz Factor vs. Velocity

Clock Comparison

When 1s passes on the stationary clock, only 0.436s passes on the moving clock.

Parameters

Speed input

Velocity (v/c)

Proper time (t₀)

Proper length (L₀)

Twin Paradox Scenario

One-way trip distance

light-years

Presets

Results

Velocity (v/c)

0.9

Lorentz factor (γ)

2.29416

Dilated time (t')

2.29416 s

Contracted length (L')

0.43589 m

Twin Paradox

Stationary twin ages

22.2222 years

Traveling twin ages

9.68644 years

Age difference

12.5358 years

Step-by-Step Calculation

1. Lorentz factor

\gamma = \frac{1}{\sqrt{1 - \beta^2}} = \frac{1}{\sqrt{1 - (0.9)^2}} = 2.29416

2. Time dilation

t' = \gamma \cdot t_0 = 2.294 \times 1 = 2.29416 \text{ s}

3. Length contraction

L' = \frac{L_0}{\gamma} = \frac{1}{2.294} = 0.43589 \text{ m}

4. Twin paradox (round trip to 10 ly)

t_{\text{stationary}} = \frac{2d}{\beta} = \frac{2 \times 10}{0.9} = 22.2222 \text{ years}

5. Traveling twin's time

t_{\text{traveler}} = \frac{t_{\text{stationary}}}{\gamma} = \frac{22.2222}{2.294} = 9.68644 \text{ years}

6. Age difference

\Delta t = 22.2222 - 9.68644 = 12.5358 \text{ years}

Reference Guide

The Lorentz Factor

The Lorentz factor governs all relativistic effects. It starts at 1 for zero velocity and grows without bound as velocity approaches the speed of light.

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}

At everyday speeds, gamma is essentially 1. Even the ISS at 7,660 m/s only produces $\gamma \approx 1.000000000327$ . At 90% of light speed, $\gamma \approx 2.29$ .

Time Dilation

A moving clock runs slower than a stationary one. If a proper time interval $t_0$ passes on the moving clock, a stationary observer measures a longer interval.

t' = \gamma \cdot t_0

This is not an illusion. GPS satellites must correct for relativistic time dilation (both special and general) or positioning errors would accumulate at about 10 km per day.

Length Contraction

Objects moving at high speed appear shorter along the direction of motion. A 1-meter rod moving at 0.9c would be measured as only about 0.44 meters by a stationary observer.

L' = \frac{L_0}{\gamma}

The contraction only affects the dimension parallel to motion. Perpendicular dimensions are unchanged.

The Twin Paradox

If one twin travels at high speed to a distant star and returns, they will be younger than the twin who stayed home. This is not a paradox but a consequence of the traveling twin undergoing acceleration.

For a round trip to a star 10 light-years away at 0.9c, the stationary twin ages about 22 years while the traveler ages only about 10 years. At 0.99c the difference is even more dramatic.

This effect has been experimentally confirmed with atomic clocks on airplanes (Hafele-Keating experiment, 1971) and with muons created in cosmic ray showers.