Lorentz Factor & Special Relativity Calculator

Calculate Lorentz factor γ, time dilation, length contraction, and relativistic momentum.

Max = 1 (speed of light)
For dilation/contraction calculations
For relativistic energy/momentum
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The Lorentz Factor Measures Relativistic Departure from Newtonian Behavior

The Lorentz factor γ=1/√(1−v2/c2) appears throughout special relativity because space and time intervals mix between inertial frames. At everyday speeds, v/c is tiny and γ is almost exactly 1, so Newtonian mechanics works extremely well. As speed approaches c, γ grows rapidly, affecting time dilation, length contraction, momentum, and total energy.

Proper time is measured by a clock traveling with the event sequence, while a frame that sees that clock moving measures a longer interval Δt=γΔτ. A proper length along the motion direction is measured shorter in a frame where the object moves: L=L0/γ.

γ=1/√(1−β2),   β=v/c
SymbolMeaningWhy it appears / units
γLorentz factorDimensionless, always ≥1 for subluminal speeds.
βSpeed fraction v/cDimensionless and less than 1 for massive objects.
cSpeed of light299,792,458m/s exactly in SI.
ΔτProper timeTime interval measured in the frame where events occur at one place.

There is no finite γ for a massive object at v=c; energy required to accelerate a massive object grows without bound as v approaches c. Relativistic momentum is p=γmv, not directly mv.

The Lorentz factor can never be below 1 for a massive object. At v=0 it equals 1, and it increases without bound as v approaches c from below. If an input at or above c produces a real finite γ, the input validation or formula has been misapplied.

Worked Examples

Example 1: v=0.9c (γ=2.29)
γ=1/√(1-0.81)=1/√0.19
Result: γ=2.294 — 1 year on ship = 2.29 years for observer
Time dilation at 90% of light speed
Example 2: Muon at v=0.995c
γ=1/√(1-0.990)=10.01
Result: Muon lifetime 10× longer
Explains why muons reach Earth's surface
Example 3: Speed 0.8c
β=0.8
Result: γ=1.667
A moving clock interval is stretched by about 66.7% relative to its proper time.
Example 4: Speed 0.99c
β=0.99
Result: γ≈7.09
The factor rises sharply close to light speed.

Common Mistakes

⚠️
Using v instead of v/c inside the formula

The square-root term must contain the dimensionless ratio β=v/c.

⚠️
Applying length contraction perpendicular to motion

Standard Lorentz contraction affects dimensions parallel to relative motion, not transverse dimensions.

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Using classical momentum at relativistic speed

At high speed, p=γmv and total energy=γmc² are required.

Frequently Asked Questions

Why can't anything reach c?
As v→c, γ→∞: energy→∞, time stops, length→0. An infinite amount of energy would be needed to accelerate a massive object to exactly c. Only massless particles (photons) travel at c.
GPS needs relativistic corrections?
Yes — GPS satellites travel at ~14,000 km/h (v/c≈1.3×10⁻⁵), causing time dilation of 7 μs/day. General relativity adds +45 μs/day. Net correction: +38 μs/day. Without it, GPS errors accumulate at ~10 km/day.
At what speed does relativity matter?
There is no sharp threshold. Corrections grow with v/c; around 0.1c they are already measurable, while at ordinary vehicle speeds γ differs from 1 by an extremely tiny amount.
Can γ be less than 1?
Not for ordinary subluminal relative motion. Since 1−β² is between 0 and 1, its square root is at most 1, so γ≥1.
Why can’t a massive object reach c?
As v approaches c, γ and the required relativistic energy grow without bound. No finite amount of added energy can accelerate a massive object to exactly light speed.
Is time dilation an illusion?
No. Proper-time differences are measurable physical effects confirmed by particle lifetimes, atomic clocks, and satellite systems. Different inertial frames consistently relate measurements through Lorentz transformations.
When can relativistic corrections be neglected?
For speeds much smaller than c, γ=1/√(1−β2) is very close to 1. Expanding for small β gives γ≈1+β2/2. Once v becomes a substantial fraction of c, the deviation grows rapidly and Newtonian approximations lose accuracy.

Formula Explorer connections

Interpretation: This relationship connects magnetic fields, moving charge, flux, induction or electromagnetic material response. Assumption: Specify field direction and sign convention. Uniform fields, linear materials, negligible edge effects or sinusoidal steady state may be assumed.

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