Lorentz Factor Calculator

γ from a speed, the speed back from a γ you need, and the curve drawn with your own β marked on it — because the only thing that matters about γ is where it stops being 1, and that happens later and then faster than anybody expects.

Last updated 11 Nov 2026

Speed → γ, and a target γ → the speed it needs

γ = 1/√(1 − β²), both directions

Speed of the moving thingRelativity questions are almost always set in fractions of c, because that is the only number the formula cares about. If your question gives a speed in m/s, switch the unit below rather than dividing by 299 792 458 by hand — that division is where a tired reader loses a power of ten. Typing c itself, or anything faster, does not break the page: the velocity ratio is capped just below 1 and the answer comes back with a note saying so, because no object with mass can be given that speed.

The speed above is given asOne conversion worth carrying: 1% of c is 2 997 924.58 m/s, about 2998 km/s. A fighter jet is 0.000 002 c and a fast electron in a school tube is about 0.1 c, which is already a 0.5% gamma.

Also: what β would I need for a γ ofThe reverse question, which is the one accelerator physics actually asks. Rearranged, β = √(1 − 1/γ²). A γ of 2 needs 0.866c, a γ of 10 needs 0.995c, and a γ of 100 needs 0.99995c. Values below 1 are impossible and are clamped to 1, which returns β = 0.

γ against β, with the dashed red wall at β = 1 that nothing with mass crosses. The blue dot is your speed; the green dot is the speed your target γ needs. The curve itself is the function and does not move — what moves is where you are on it. The thing to take away is the SHAPE: flat and boring until about 0.8c, then nearly vertical. Half of all the γ on this plot is earned in the last three per cent of β.

1.250000Example

An electron at 0.6c, and the β needed for a γ of 5

One function, two directions, and the approximation that saves the arithmetic

γ = 1 / √(1 − β²) · β = √(1 − 1/γ²) · γ − 1 ≈ β²/2 for β ≪ 1 · γ ≈ 1/√(2(1 − β)) for β near 1

γ: always at least 1, and equal to 1 only at rest. It multiplies times and energies and divides lengths. It is dimensionless, so it never carries a unit and an answer with one is a mistake
β: v/c, between 0 and 1. Above about 0.99 stop quoting β and start quoting 1 − β, which is where all the information has gone
γ − 1 ≈ β²/2: the low-speed shortcut, good to a part in a thousand below β = 0.05. Every GPS and aircraft-clock number is computed this way, because the exact form needs more digits than a calculator keeps
γ ≈ 1/√(2(1−β)): the high-speed shortcut, from 1 − β² = (1−β)(1+β) ≈ 2(1−β). At 1 − β = 10⁻⁸ it gives γ = 7071, against 7071 exactly

Worked example

An electron at 0.6c, and the β needed for a γ of 5
SQUARE β. 0.6² = 0.36. This is the step at which to resist rounding; at low speeds β² is tiny and at high speeds it is close to 1, and both ends punish a short calculator.
SUBTRACT FROM 1. 1 − 0.36 = 0.64. The fact that this is a perfect square is why 0.6c appears in so many questions.
ROOT. √0.64 = 0.8. Note that this is ALSO the answer to the length-contraction question: a rod at 0.6c is 0.8 of its rest length, because L/L₀ = 1/γ.
INVERT. γ = 1/0.8 = 1.25 exactly. A clock at 0.6c runs at 80% of rate; a second of its time is 1.25 s of ours.
NOW GO BACKWARDS. For γ = 5, β = √(1 − 1/25) = √0.96 = 0.979796, which is 2.937×10⁸ m/s. Look at what that cost: quadrupling γ from 1.25 to 5 took β from 0.6 to 0.98.
CHECK THE APPROXIMATION WHILE YOU ARE HERE. β²/2 = 0.18 against the true γ − 1 = 0.25, so the low-speed shortcut is 28% low at 0.6c. It is a low-speed shortcut and 0.6c is not low. Below 0.05c its error is under 0.1%.

The whole function in one column of numbers

The second column is the useful one above β = 0.99. Once 1 − β is small, γ ≈ 1/√(2(1−β)), so every factor of 100 taken off 1 − β multiplies γ by 10. That is also why a particle physicist never quotes a beam speed: the interesting digits are all in the ninth decimal place, and quoting the energy instead carries the same information with no lost precision.
β	1 − β	γ	What it corresponds to
0.000 001	≈ 1	1.000 000 000 000 5	30 km/s — the Earth’s orbital speed
0.10	0.9	1.00504	about 0.5% — the first band where anyone could notice
0.60	0.4	1.25	the tidy textbook speed; 1/γ = 0.8 exactly
0.866	0.134	1.99982	γ = 2. Clocks at half rate, rods at half length
0.950	0.05	3.2026
0.990	0.01	7.0888
0.995	0.005	10.0125	the cosmic-ray muon that reaches sea level
0.999 9	10⁻⁴	70.71	γ ≈ 1/√(2×10⁻⁴) = 70.71 — the high-speed shortcut, exact to five figures
0.999 999 991	9×10⁻⁹	7454	an LHC proton at 7 TeV

γ is flat until it is not, and the last three per cent does half the work

Every relativity formula is γ in a different coat. Time dilation multiplies by it, length contraction divides by it, relativistic momentum and energy are built from it, and velocity addition is the one place it hides. So the single most useful thing a student can carry into an exam is not a formula but a feel for the shape of this function: it is 1 at rest, it is still only 1.005 at a tenth of light speed, it is 2 at 0.866c, and then it goes vertical. Anything that is going to be surprising happens above β = 0.9.

Why the curve is so flat for so long. For small β, γ − 1 ≈ β²/2, so a hundredth of light speed gives a correction of five parts in a hundred thousand. A fighter jet is β = 2×10⁻⁶ and its γ differs from 1 in the twelfth decimal place. That is the honest answer to “why does nobody notice relativity”: not because it is weak, but because the function is quadratic in β near zero and everything you have ever seen moving has β near zero.

Quote 1 − β, not β, at the top end. Above about 0.99 the number β stops carrying information. Writing “0.999 999 991c” is nine characters of which one matters, and reading it off a graph is impossible — the curve near β = 1 is near-vertical, so a pencil line’s width is a factor of two in γ. Physicists therefore quote 1 − β, or the energy, or γ itself. The shortcut γ ≈ 1/√(2(1 − β)) makes the conversion a mental arithmetic job, and it is already accurate to five figures by 1 − β = 10⁻⁴.

Where this goes wrong. Giving γ a unit — it is a pure number. Forgetting to square β, which gives 1/√(1 − β) and an answer that is wrong in a plausible-looking way. Using β²/2 at a speed where it does not apply: at 0.6c it is 28% low and at 0.9c it is 69% low. Taking β greater than 1 from a classical velocity sum, which this page refuses — see the velocity addition page for why that sum was never allowed. And confusing γ with 1/γ: the clock RATE is 1/γ and the clock INTERVAL is γ, so a clock at γ = 2 runs at half speed and records twice as long an interval, and those two sentences are the same sentence.

Frequently asked questions

What are the units of γ?

None. It is the ratio of two speeds squared, subtracted from one and square-rooted and inverted; every unit cancels. An answer like “γ = 2 m/s” is not slightly wrong, it is a category error, and it usually means β was computed without dividing by c.

What speed do I need for γ = 2, 10 or 100?

0.866c for γ = 2, 0.994987c for γ = 10, and 0.99995c for γ = 100. Type the γ you want into the second box and the page returns β and the speed in m/s. The pattern is worth seeing: each factor of 10 in γ costs a factor of 100 off 1 − β. Getting to γ = 1000 needs 1 − β = 5×10⁻⁷.

When can I use γ ≈ 1 + β²/2?

Below about β = 0.05 it is good to better than a part in a thousand, and that covers every satellite, aircraft and planetary speed you will ever meet. The row in the results above gives the exact percentage error at your speed, so you can decide rather than guess. At 0.1c the error is 0.4%; at 0.3c it is 7%; at 0.6c it is 28%.

Why is the curve vertical at β = 1?

Because 1 − β² goes to zero there, so γ goes to infinity. That is not a drawing artefact and it is not a limitation of the theory — it is the statement that accelerating a massive object to c would take infinite energy, which you can see directly as the kinetic energy curve on the kinetic energy page. Light travels at c precisely because it has no rest mass, so the whole (γ − 1)mc² argument never applies to it.

Is 1/γ the same thing as the clock rate?

Yes, and keeping the two straight is worth a mark. A clock moving with γ = 2 TICKS at half rate, so 1/γ = 0.5 is the rate. The INTERVAL it records between two events is γ times smaller than the lab’s, which is the same statement upside down. Length contraction uses 1/γ directly: L/L₀ = 1/γ.

The page refuses my speed. Why?

It does not refuse it; it caps it. Anything at or above c is clamped to β = 0.999 999 999, which gives γ = 22 360.68, and a note says so. The cap exists so that a typo does not blank the page and vanish the diagram, which is what an infinite γ does. The physics behind the cap is that no object with mass can be given the speed c, at any energy.

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References

CODATA Internationally Recommended 2022 Values of the Fundamental Physical Constants, NIST Standard Reference Database 121. Read on 11 November 2026 from the machine-readable table at physics.nist.gov/cuu/Constants/Table/allascii.txt, which prints “speed of light in vacuum 299 792 458 (exact)” — exact because the 1983 definition of the metre fixes it, so c² = 8.9875517873681764×10¹⁶ m²/s² carries no uncertainty either. The same table gives G = 6.674 30(15)×10⁻¹¹ m³kg⁻¹s⁻², which is the least precisely known of the constants on this page by a factor of ten thousand, standard acceleration of gravity 9.806 65 (exact), and the rest energies used here: electron 0.510 998 950 69(16) MeV, proton 938.272 089 43(29) MeV, muon 105.658 3755(23) MeV.
AQA, AS and A-level Physics (7407, 7408) specification, section 3.12.3 Special relativity, inside the option 3.12 Turning points in physics (A-level only). Read from aqa.org.uk on 11 November 2026. It is worth knowing exactly what a board asks for: 3.12.3.3 requires proper time and time dilation in the form t = t₀ / √(1 − v²/c²), with evidence from muon decay; 3.12.3.4 requires length contraction as l = l₀√(1 − v²/c²); and 3.12.3.5 requires mass-energy equivalence, the form E = m₀c²/√(1 − v²/c²), graphs of mass and kinetic energy against speed, and Bertozzi’s experiment as direct evidence. Note what that implies: AQA still teaches relativistic mass. This page cites the specification; it does not claim the board endorses it.
A. Einstein, “Zur Elektrodynamik bewegter Körper”, Annalen der Physik 17 (1905) 891–921 — the paper in which the two postulates, the Lorentz transformation, time dilation and length contraction all appear together, and the companion three-page note “Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?”, Annalen der Physik 18 (1905) 639–641, which adds the mass-energy result. Cited for the result, not for a number: neither paper was read for this build, and nothing on this page rests on a figure taken from them.
Particle Data Group, Review of Particle Physics — Summary Table of Leptons, read on 11 November 2026 from pdg.lbl.gov/2025/tables/rpp2025-sum-leptons.pdf. Muon mean life τ = (2.196 981 1 ± 0.000 002 2)×10⁻⁶ s; muon mass 105.658 3755 ± 0.000 0023 MeV. It is a MEAN life, not a half-life: the half-life is τ ln 2 = 1.5230 µs, and quoting one where the other is meant is the commonest error in a muon question. cτ = 658.64 m, which is the distance a muon would cover in one mean lifetime if there were no dilation at all.
CBSE / NCERT, Physics, Class XII, Curriculum 2025–26, read on 11 November 2026 from the government-hosted syllabus at edustud.nic.in (SYLLABUS_2025_26/12/12_Physics_EM.pdf). The nine units are Electrostatics, Current Electricity, Magnetic Effects of Current and Magnetism, Electromagnetic Induction and Alternating Currents, Electromagnetic Waves, Optics, Dual Nature of Radiation and Matter, Atoms and Nuclei, and Electronic Devices. Special relativity, time dilation, length contraction and the Lorentz factor appear nowhere in it. The only trace of the subject is “mass-energy relation, mass defect” in Chapter 13, Nuclei — and the document does not even write E = mc² out. So if you are on this syllabus, this topic is reading beyond it rather than revision for it, which is a useful thing to know the night before a paper.