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Lorentz Factor Calculator

Find relativistic time dilation from a speed.

How it is calculated

You enter the speed as a percentage of the speed of light c. The tool converts it to β = v ÷ c = percent ÷ 100 and computes γ = 1 ÷ √(1 − β²). It also reports the contraction figure 1 ÷ γ as a percentage: the length of a moving object along its direction of travel, measured by a stationary observer, as a share of its rest length. γ itself is the factor by which a moving clock appears slower. Valid inputs run from 0 up to, but not including, 100, because γ grows without limit as speed approaches c and no result exists at c. γ is shown with four decimals, β with four, the percentage with two.

Worked example

A spacecraft travels at 90% of the speed of light. Step 1: β = 90 ÷ 100 = 0.9000. Step 2: β² = 0.81, so 1 − β² = 0.19. Step 3: √0.19 = 0.4359, and γ = 1 ÷ 0.4359 = 2.2942. Step 4: 1 ÷ γ = 0.4359, so the ship appears contracted to 43.59% of its rest length, and a clock on board appears to tick about 2.29 times slower.

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Common mistakes

  • Entering the speed as a fraction (0.9) when the field expects a percentage; 0.9 means 0.9% of c and gives γ close to 1.0000.
  • Entering a speed in km/h or m/s instead of a percentage of c.
  • Typing 100 and expecting a very large number: at c the formula has no result.
  • Reading the contraction percentage as the amount lost; 43.59% is the length that remains, not the part removed.
  • Assuming relativistic effects matter at everyday speeds: at 1% of c, γ is only about 1.00005.

About this calculator

This calculator finds the Lorentz factor γ from a speed (as a fraction or percentage of the speed of light): γ = 1/√(1 − v²/c²). This factor is central to special relativity — it determines time dilation (moving clocks run slow by a factor of γ) and length contraction (moving objects shrink along their direction of motion by a factor of 1/γ), and it grows without bound as v approaches c.

Physics students and researchers use it to work through relativistic problems in particle physics (where particles routinely reach speeds close to c in accelerators), astrophysics (relativistic jets, cosmic rays), and to understand why GPS satellite clocks need relativistic corrections. Enter a velocity as a fraction of the speed of light to get γ along with the resulting time dilation and length contraction factors.

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