Time dilation calculator
Lorentz factor, time dilation and length contraction.
Fill in the fields and the result will appear here automatically.
For two events on one moving clock, its proper interval τ becomes t=γτ in the inertial frame where the clock travels at speed v. Enter β=v/c and τ in seconds. The output gives t, γ, t−τ and the longitudinal length fraction L/L₀. Gravity, acceleration and the itinerary of a journey are not modelled.
How it works
Formula and logic
γ=1/√((1−β)(1+β)); t=γτ; t−τ=τβ²/[√(1−β²)(1+√(1−β²))]; L/L₀=1/γ. The domain is 0≤β<1 and τ>0.
Example
At half the speed of light one second on the moving clock stretches to 1.155 s.
Fields and units
- Fraction of the speed of light — unitless
- Proper time — s
How to use
- — Enter speed as a fraction of light speed and the proper interval in seconds.0.5 means half light speed;1 is outside the domain.
- — The length-fraction row reports the remaining 100/γ per cent, not the percentage decrease. At β=0.5, the length is 86.6025% of L₀, a decrease of≈13.3975%.
Method and limitations
- Calculation method
- Formula and logic
- Data or methodology source
- OpenStax §5.3: proper interval and t=γτ in an inertial frame OpenStax §5.4: remaining longitudinal length L/L₀=1/γ NIST CODATA 2022: exact h, c, e and measured ε₀
- Limitation
- For two events on one moving clock, its proper interval τ becomes t=γτ in the inertial frame where the clock travels at speed v. Gravity, acceleration and the itinerary of a journey are not modelled.
FAQ
Why is nothing visible at everyday speeds?
For β≪1, γ−1≈β²/2. At 250 m/s over one hour of proper time the difference is≈1.252 nanoseconds. Small effects can be measured; there is no universal visibility threshold.
What is proper time?
A proper interval is measured between two events at one location in the clock’s own frame. In a frame where that clock moves, the same interval is γτ. Neither inertial frame is universally the real one.
Why can the speed of light not be reached?
At one the expression under the root goes to zero and the factor to infinity. Physically that means accelerating a massive body would take infinite energy.
Does length really contract?
Yes, but only along the direction of motion and only from the stationary observer’s point of view. For the moving body nothing changes — it is the surrounding world that contracts.
Why can a small time difference remain nonzero?
At small β, rounded γ may display as 1. The difference uses a separate stable expression: β=10⁻⁹ and τ=10²⁰ s give≈50 s. The rounded factor hides digits, not the physical effect.