Time Dilation Calculator

In special relativity, time is not absolute. Two observers in relative motion measure different time intervals between the same pair of events. If a clock is at rest in a moving frame (the proper frame), it measures the proper time $$t_0$$. An observer in a different inertial frame sees that clock running slow by the Lorentz factor:

$$t = \gamma\, t_0 = \frac{t_0}{\sqrt{1 - \beta^2}}$$

where $$\beta = v/c$$ is the velocity as a fraction of the speed of light, and $$\gamma = (1 - \beta^2)^{-1/2}$$ is the Lorentz factor. Key properties of time dilation:

• Reciprocal effect: Each observer sees the other's clock running slow. This is consistent because the two observers disagree on simultaneity.
• γ at low speeds: For everyday velocities, $$\gamma \approx 1 + \frac{v^2}{2c^2}$$, so the effect is negligible. At 1000 km/h, γ differs from 1 by only about 4 parts in 10¹³.
• γ diverges near c: As $$v \to c$$, $$\gamma \to \infty$$. At 99% of c, γ ≈ 7.09; at 99.99%, γ ≈ 70.7.
• Proper time is minimum: The proper time (measured by the clock present at both events) is always the shortest time interval. All other observers measure longer intervals.

Experimental confirmations include the observation that cosmic-ray muons (with a rest-frame lifetime of 2.2 μs) survive the trip from the upper atmosphere to Earth's surface because their clocks run slow in our frame. The Hafele-Keating experiment (1971) flew atomic clocks around the world and measured time differences matching special and general relativistic predictions to within 10%.

Note that gravitational time dilation (from general relativity) is a separate effect not included here. Near massive bodies, clocks run slower in stronger gravitational fields.

The Lorentz factor γ tells you how much time is stretched. A γ of 2 means the moving clock ticks at half the rate of the stationary clock — one second of proper time corresponds to two seconds for the observer. The time difference shows the extra elapsed time the stationary observer measures. At low velocities (β ≪ 1), γ is essentially 1 and time dilation is unmeasurable. The effect only becomes significant above about 10% of the speed of light (β > 0.1, γ ≈ 1.005).