Relativistic Time Dilation Calculator
Calculate time dilation, length contraction, and the Lorentz factor at relativistic speeds using Einstein's special relativity
Step-by-Step Solution
Special Relativity Formulas
Einstein's special theory of relativity (1905) shows that measurements of time and space depend on the relative velocity between observer and object. As an object approaches the speed of light, time passes more slowly for it relative to a stationary observer, and its length contracts in the direction of motion.
Lorentz Factor
γ = 1 / √(1 − v²/c²)
The Lorentz factor γ (gamma) quantifies relativistic effects. At rest γ = 1; it increases without bound as v approaches c.
Time Dilation
Δt = γ × Δt₀
- Δt₀ = proper time (measured by the moving clock)
- Δt = dilated time (measured by the stationary observer)
- Moving clocks tick more slowly: Δt ≥ Δt₀
Length Contraction
L = L₀ / γ = L₀ × √(1 − v²/c²)
- L₀ = proper length (measured at rest)
- L = contracted length (measured by observer)
- Objects moving at relativistic speeds appear shorter in the direction of motion
Relativistic Kinetic Energy
K = (γ − 1) × m₀ × c²
- m₀ = rest mass
- c = speed of light (299,792,458 m/s)
- At low speeds this reduces to the classical ½mv²
Lorentz Factor at Various Speeds
| Speed (v/c) | γ (Lorentz factor) | Time dilation | Length contraction |
|---|---|---|---|
| 0.1c (10%) | 1.005 | +0.5% slower | 99.5% of rest length |
| 0.5c (50%) | 1.155 | +15.5% slower | 86.6% of rest length |
| 0.8c (80%) | 1.667 | +66.7% slower | 60.0% of rest length |
| 0.9c (90%) | 2.294 | 2.29× slower | 43.6% of rest length |
| 0.95c (95%) | 3.203 | 3.20× slower | 31.2% of rest length |
| 0.99c (99%) | 7.089 | 7.09× slower | 14.1% of rest length |
| 0.999c (99.9%) | 22.37 | 22.4× slower | 4.5% of rest length |
| 0.9999c (99.99%) | 70.71 | 70.7× slower | 1.4% of rest length |
Real-World Applications
GPS Satellites
GPS satellites orbit at ~14,000 km/h. Special relativity causes their clocks to lose about 7 microseconds per day. Combined with general relativity effects (+45 μs/day), engineers must apply a net correction of ~38 μs/day for accurate positioning.
Particle Accelerators
At the Large Hadron Collider, protons reach 0.999999991c (γ ≈ 7,454). Muons created in the atmosphere at 0.998c live ~5× longer than at rest, allowing them to reach Earth's surface — a direct confirmation of time dilation.
Twin Paradox
If one twin travels at 0.9c for 10 years (ship time), they return to find their Earth twin has aged ~22.9 years. This thought experiment, proposed by Paul Langevin in 1911, has been verified with atomic clocks on aircraft (Hafele–Keating experiment, 1971).
References
The formulas and data used in this calculator are based on Einstein's special theory of relativity and peer-reviewed physics resources:
- OpenStax University Physics Vol. 3 — Time Dilation
- HyperPhysics — Time Dilation
- NIST — Fundamental Physical Constants (speed of light)
- Hafele, J. C. & Keating, R. E. (1972). "Around-the-World Atomic Clocks: Predicted Relativistic Time Gains." Science, 177(4044), 166–168.
- GPS.gov — GPS Accuracy and Relativistic Corrections
- Einstein, A. (1905). "Zur Elektrodynamik bewegter Körper." Annalen der Physik, 322(10), 891–921.
- Taylor, E. F. & Wheeler, J. A. (1992). Spacetime Physics: Introduction to Special Relativity (2nd ed.). W. H. Freeman.
Related Calculators
Note: This calculator uses the equations of special relativity, which assume inertial (non-accelerating) reference frames and flat spacetime. It does not account for general relativistic effects (gravity). At everyday speeds the relativistic corrections are negligibly small.
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