Special Relativity Calculator — Time Dilation, Length

Compute time dilation, length contraction, relativistic energy and momentum.

About this calculator

A special relativity calculator computes time dilation, length contraction, relativistic momentum, mass-energy equivalence (E = mc²), and velocity addition for systems moving at significant fractions of the speed of light. These effects are negligible at everyday speeds but become significant above ~10% c.

Special relativity is one of the best-tested theories in physics. GPS satellites must include both special relativistic time dilation (orbital velocity) and general relativistic time dilation (weaker gravity at altitude) corrections to function correctly — without them, GPS would drift by ~10 km per day.

Formula

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

  • γ — Lorentz factor (≥ 1)
  • v — Relative velocity
  • c — Speed of light = 3 × 10⁸ m/s
  • Δt' — γ × Δt (moving clocks tick slower)

Worked example

Inputs: Spaceship travels at 0.8c for 10 years (Earth time)

  1. γ = 1 / √(1 − 0.64) = 1 / √0.36 = 1 / 0.6 = 1.667
  2. On the spaceship: Δt' = 10 / 1.667 = 6 years experienced by astronauts
  3. Distance contracted: a 10 light-year journey appears 6 ly to the travellers
  4. Rest mass 70 kg human gains relativistic energy: E = γmc² = 1.05 × 10¹⁹ J (≈ 100 times annual world energy use, illustrating why such travel is impractical)

Frequently asked questions

Why can't anything travel faster than light?

As an object accelerates toward c, its relativistic mass (γm) grows without bound — at v = c, γ would be infinite, requiring infinite energy to accelerate further. The cosmic speed limit emerges naturally from the geometry of spacetime as encoded by the Lorentz transformations. Massless particles (photons) MUST travel at c; massive particles can only approach it asymptotically.

Has time dilation been experimentally tested?

Yes, extensively. Atomic clocks flown on commercial airliners have measurably less elapsed time than ground reference clocks. Muons created in the upper atmosphere (with very short proper lifetime) reach the ground in measurable quantities only because of their time dilation. GPS satellites continuously demonstrate both special and general relativistic time dilation. The predictions match observations to 10+ significant figures.

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