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Walk-through

How to Use This Calculator

3 steps

1

Pick a tab for what you need

Use the Dilated Time tab when you know a proper time and a velocity and want the dilated duration. Use the Lorentz Factor tab when you only need γ itself. Use the Velocity Sweep tab to see how γ climbs across a range of reference speeds.

2

Enter the velocity

Enter velocity as either a fraction of the speed of light (e.g. 0.5 for half the speed of light) or in meters per second, and pick the matching unit from the dropdown. The calculator converts m/s to a fraction of c internally.

3

Read the dilated time and Lorentz factor

The result card shows the dilated time (or γ on the Lorentz Factor tab) alongside the interpretation line, which states how many times slower the moving clock runs compared to a stationary observer's clock.

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Reference

Formula & Methodology

2 formulas

Lorentz Factor

γ = 1 / √(1 − v²/c²)

γ (gamma) is the Lorentz factor, v is the relative velocity, and c is the speed of light in a vacuum (299,792,458 m/s). As v approaches c, the term v²/c² approaches 1, the quantity under the square root approaches 0, and γ grows without bound. At v = 0, γ = 1 (no dilation).

Dilated Time

Δt′ = γ · Δt

Δt is the proper time — the time interval measured by a clock moving with the object (its own rest frame). Δt′ is the dilated time — the longer interval measured by a stationary observer watching that clock. Because γ ≥ 1 always, Δt′ ≥ Δt: the moving clock always appears to run slower (or, equivalently, more time passes for the stationary observer) than the observer expects.

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Glossary

Key Terms Explained

7 terms

Time Dilation The relativistic effect where a clock moving relative to an observer is measured to tick more slowly than a clock at rest with that observer. It is a real, measured effect confirmed by particle-accelerator experiments and atomic clocks flown on aircraft — not an illusion of measurement.

Lorentz Factor The dimensionless quantity γ = 1/√(1 − v²/c²) that scales how strongly relativistic effects — time dilation, length contraction, and relativistic mass increase — apply at a given velocity. γ = 1 at rest and grows toward infinity as v approaches c.

Special Relativity Einstein's 1905 theory describing how measurements of space and time differ for observers moving at constant velocity relative to each other. Its two postulates are that the laws of physics are the same in all inertial frames, and that the speed of light in a vacuum is the same for every observer regardless of their motion.

Proper Time The time interval measured by a clock that travels along with the object being timed — i.e., the time in the object's own rest frame. It is always the shortest time interval any observer will measure for that interval; all other observers measure a dilated (longer) time.

Speed of Light The speed at which light travels in a vacuum, c = 299,792,458 meters per second exactly (by definition of the meter). It is the universal speed limit in special relativity — no object with mass can reach or exceed it, since γ diverges to infinity as v approaches c.

Gamma (γ) The conventional symbol for the Lorentz factor. A gamma of 2 means a moving clock ticks at half the rate seen by a stationary observer — one second of proper time corresponds to two seconds of dilated time.

Reference Frame A coordinate system and set of clocks from which measurements of position and time are made. Time dilation is always relative to a specific reference frame — it describes how a clock moving relative to one observer's frame compares to that observer's own clocks, not an absolute slowing of time itself.

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Scenarios

Real-World Examples

3 worked examples

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Half-Light-Speed Spacecraft

Moderate relativistic speed

Proper time (Δt) 1 s Velocity 0.5c

γ = 1/√(1 − 0.5²) = 1/√0.75 ≈ 1.1547. One second on the spacecraft's clock corresponds to about 1.1547 seconds for a stationary observer — the moving clock runs about 15.5% slower. At this speed the effect is noticeable but modest.

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Near-Light-Speed Particle

Extreme relativistic speed

Proper time (Δt) 1 s Velocity 0.99c

γ = 1/√(1 − 0.99²) ≈ 7.0888. One second of proper time now stretches to about 7.09 seconds for a stationary observer — the moving clock runs more than 7 times slower. This is the regime particle accelerators and cosmic-ray muons actually operate in.

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Commercial Jet

Everyday speed — negligible effect

Proper time (Δt) 3600 s (1 hour) Velocity 250 m/s

At 250 m/s, β ≈ 0.00000083, so γ is 1.0000000000003 — indistinguishable from 1 to ordinary precision. Over a full hour, a jet's clock lags a ground clock by only a few hundred nanoseconds, which is exactly the tiny correction that ultra-precise atomic-clock experiments (like Hafele–Keating) were built to detect.

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Deep Dive

Understanding Time Dilation and the Lorentz Factor

3 chapters · 4 min read

Time dilation is one of the most famous — and most misunderstood — predictions of Einstein's special relativity: a moving clock runs slower than a stationary one, and the faster it moves, the bigger the effect. This calculator turns that idea into a concrete number using the Lorentz factor, so you can see exactly how much a clock's tick rate changes at any given velocity.

Why Motion Slows Down Clocks

Special relativity starts from a strange but experimentally verified fact: the speed of light is the same for every observer, no matter how fast they're moving. To keep that true, space and time have to bend around each other — an observer watching a clock move past them will measure that clock ticking more slowly than an observer traveling alongside it does. This isn't a trick of light delay or measurement error; it is a real difference in the rate time passes, confirmed by comparing atomic clocks flown on airplanes, by the extended lifetimes of fast-moving muons in the atmosphere, and by the everyday operation of GPS satellites.

Reading the Lorentz Factor

The Lorentz factor γ = 1/√(1 − v²/c²) is the single number that captures how strong the relativistic effect is at a given velocity. At everyday speeds — cars, planes, even orbital spacecraft — γ is so close to 1 that the correction is measured in nanoseconds. It's only as velocity approaches a significant fraction of c that γ climbs noticeably: at 50% of c it's about 1.15, at 90% of c it's about 2.29, and at 99% of c it's about 7.09. Push toward 99.99% of c and γ exceeds 70 — the closer v gets to c, the more explosively γ grows, which is also why no massive object can ever reach c itself (it would require infinite energy to push γ to infinity).

Limits and What This Calculator Assumes

This calculator implements the standard special-relativity time dilation formula for a single, constant relative velocity between two inertial (non-accelerating) observers — it does not account for gravitational time dilation (from general relativity, relevant near massive bodies) or for acceleration effects (relevant to the classic "twin paradox"). It also assumes velocity is entered accurately as a fraction of c or in m/s; garbage in the velocity field, including any value at or above the speed of light, produces an undefined result rather than a false answer, since nothing with mass can travel at or beyond c.

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Questions

Frequently Asked Questions

6 questions

What is the formula for time dilation?+

Δt′ = γ · Δt, where Δt is the proper time (measured by the moving clock itself), Δt′ is the dilated time (measured by a stationary observer), and γ = 1/√(1 − v²/c²) is the Lorentz factor derived from the relative velocity v and the speed of light c.

What is the Lorentz factor at 0.5c?+

At v = 0.5c, γ = 1/√(1 − 0.5²) = 1/√0.75 ≈ 1.1547. A clock moving at half the speed of light runs about 15.5% slower than a stationary observer's clock — one second of proper time becomes about 1.1547 seconds of dilated time.

Does time dilation matter in everyday life?+

Not noticeably. At everyday speeds — walking, driving, flying — the Lorentz factor differs from 1 by only a few parts in a trillion or less, producing time differences measured in nanoseconds over hours of travel. The effect only becomes significant at a meaningful fraction of the speed of light.

How does GPS relate to time dilation?+

GPS satellites experience both special-relativistic time dilation (from their orbital velocity, which slows their clocks slightly) and general-relativistic time dilation (from weaker gravity at altitude, which speeds their clocks up). The net effect is corrected for in the satellite clocks' design — without the correction, GPS position errors would accumulate by several kilometers per day.

What units does this calculator use for velocity?+

You can enter velocity either as a fraction of the speed of light (e.g. 0.5 for half of c) or in meters per second, using the unit dropdown next to the velocity field. The calculator converts m/s to a fraction of c internally before computing the Lorentz factor.

What happens as velocity approaches the speed of light?+

As v approaches c, γ grows without bound and dilated time grows correspondingly large. At v = c exactly (or above), the formula is undefined — 1 − v²/c² becomes zero or negative — which reflects the physical fact that no object with mass can reach or exceed the speed of light. This calculator returns a validation message rather than a number in that case.

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Time Dilation Calculator

Dilated Time — Δt′ = γ·Δt

Lorentz Factor — γ = 1 / √(1 − v²/c²)

Velocity Sweep — γ at reference speeds

Result

How to Use This Calculator

Pick a tab for what you need

Enter the velocity

Read the dilated time and Lorentz factor

Formula & Methodology

Key Terms Explained

Real-World Examples

Half-Light-Speed Spacecraft

Near-Light-Speed Particle

Commercial Jet

Understanding Time Dilation and the Lorentz Factor

Why Motion Slows Down Clocks

Reading the Lorentz Factor

Limits and What This Calculator Assumes

Frequently Asked Questions

Get a branded PDF of your results

Time Dilation Calculator

Dilated Time — Δt′ = γ·Δt

Lorentz Factor — γ = 1 / √(1 − v²/c²)

Velocity Sweep — γ at reference speeds

Result

How to Use This Calculator

Pick a tab for what you need

Enter the velocity

Read the dilated time and Lorentz factor

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding Time Dilation and the Lorentz Factor

Why Motion Slows Down Clocks

Reading the Lorentz Factor

Limits and What This Calculator Assumes

Frequently Asked Questions

Get a branded PDF of your results

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