Calculator
Formula used
The relativistic Lorentz factor is γ = 1 / √(1 − v²/c²). Here v is speed and c is the speed of light.
- β = v/c
- Time dilation: Δt = γ Δτ
- Length contraction: L = L₀/γ
- Longitudinal Doppler factor: D = √((1+β)/(1−β))
- If mass is provided: E = γ m c², K = (γ−1) m c², p = γ m v
How to use this calculator
- Select an input mode: velocity or beta.
- If using velocity, enter the value and choose a unit.
- If using beta, enter a value from 0 to less than 1.
- Keep c at default, or set a custom value if needed.
- Optionally enter rest mass to compute energy and momentum.
- Press Calculate to display results above the form.
- Use the download buttons to export CSV or PDF.
Example data table
Reference values using c = 299,792,458 m/s.
| Input | β (v/c) | γ | L/L₀ | Δt/Δτ |
|---|---|---|---|---|
| 0.10 c | 0.100000 | 1.005038 | 0.994987 | 1.005038 |
| 0.50 c | 0.500000 | 1.154701 | 0.866025 | 1.154701 |
| 0.90 c | 0.900000 | 2.294157 | 0.435890 | 2.294157 |
| 0.99 c | 0.990000 | 7.088812 | 0.141068 | 7.088812 |
Relativistic gamma in practice
1) Lorentz factor overview
The Lorentz factor γ links measurements between inertial frames moving at constant relative speed. It appears whenever you convert time intervals, lengths, or energies between observers. This calculator evaluates γ from either velocity or β = v/c, then reports related quantities for quick interpretation.
2) Why gamma rises sharply near c
Because γ = 1/√(1−β²), the denominator shrinks as β approaches 1. The growth is modest at low speeds (β=0.10 gives γ≈1.005), but becomes dramatic at high speeds (β=0.99 gives γ≈7.089). Small changes in β near light speed produce large changes in γ.
3) Input ranges and numerical stability
A real-valued γ requires v < c, so the form validates limits to prevent nonphysical inputs. For β input, the safe range is 0 ≤ β < 1. When β is extremely close to 1, rounding can dominate; increasing decimal precision helps, but measurement uncertainty should guide the digits you trust.
4) Time dilation you can quantify
Time dilation is often summarized as Δt = γΔτ. If a spacecraft experiences one hour of proper time (Δτ) at γ=2.294, an Earth observer measures about 2.294 hours. The result panel reports Δt/Δτ directly so you can scale any clock interval immediately.
5) Length contraction context
Length contraction along the direction of motion is L = L₀/γ. At β=0.90, the calculator shows L/L₀≈0.436, meaning a 10 m rest length appears as 4.36 m to the observer for whom the object is moving. Always interpret contraction with a clear statement of which frame measures L₀.
6) Energy and momentum outputs
If you provide a rest mass, the tool adds rest energy m c², total energy γ m c², kinetic energy (γ−1) m c², and momentum γ m v. These outputs are useful for particle beams and accelerator contexts where energy scales with γ rather than with v directly.
7) Doppler factor for longitudinal motion
The longitudinal Doppler factor D = √((1+β)/(1−β)) relates frequency shifts for motion directly toward or away from an observer. For β=0.50, D≈1.732; for β=0.90, D≈4.359. Use it as a quick estimate of how spectra or timing signals transform along the line of travel.
8) Common pitfalls and best checks
Confusing speed units, mixing frames, and entering v ≥ c are the most frequent issues. A practical check is to compare β and γ against the example table. If your computed γ seems too large, verify the unit selection, confirm whether you meant km/s versus km/h, and validate that c matches your scenario.
FAQs
1) What is γ used for?
γ converts time, length, and energy between inertial frames. It sets the scale for time dilation, length contraction, and relativistic energy growth at high speeds.
2) Why can’t I enter v equal to c?
At v = c, the term (1−v²/c²) becomes zero, so γ diverges. Massive objects cannot reach light speed in special relativity, so the calculator blocks that input.
3) What’s the difference between velocity mode and beta mode?
Velocity mode converts common units into m/s and then computes β and γ. Beta mode accepts β directly, which is useful when your speed is already expressed as a fraction of c.
4) Does the calculator handle negative velocities?
It assumes speed (magnitude) and requires nonnegative inputs. Direction is not needed for γ, which depends on v². For redshift versus blueshift, interpret Doppler with direction separately.
5) What units are used for energy outputs?
Energy values are reported in joules (J) using E = γ m c² and K = (γ−1) m c². Momentum is reported in kg·m/s from p = γ m v.
6) How precise should my inputs be?
Use precision that matches your measurement uncertainty. Near β≈1, γ changes rapidly, so extra digits can help numerically, but they do not replace accurate experimental or scenario data.
7) When is the Doppler factor shown here valid?
The listed Doppler factor is for longitudinal (line-of-sight) motion in special relativity. Transverse motion and gravitational effects require different formulas and are not included in this tool.