Playground
Gamma vs v/c curve with a moving indicator controlled by a beta slider; numerical gamma shown live.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Lorentz Factoroutput Dimensionless stretch factor for time, length, and momentum | dimensionless | 1 | 1 – 10 | |
| Velocity Relative speed between frames | m/s | LT^-1 | 0 – 299792458 | |
| Speed of Light Speed of light in vacuum (constant) | m/s | LT^-1 | 299792458 – 299792458 |
Deep dive
Derivation
From the invariance of the spacetime interval ds^2 = c^2 dt^2 - dx^2, a moving clock's proper time dtau satisfies c^2 dtau^2 = c^2 dt^2 - dx^2 = c^2 dt^2 (1 - v^2/c^2). Thus dt/dtau = 1/sqrt(1 - v^2/c^2) = gamma.
Experimental verification
Muon lifetime measurements at CERN and cosmic-ray showers confirm gamma to better than 0.1%. GPS satellites apply gamma corrections continuously.
Common misconceptions
- Gamma is not an illusion — the clock physically ticks slower in the lab frame
- Gamma is symmetric: each inertial observer sees the other's clock dilated
- Gamma is not a property of the object, but of the relative velocity
Real-world applications
- GPS relativistic corrections
- Particle accelerator design (LHC operates at gamma ≈ 7000)
- Cosmic-ray muon detection
- Relativistic electron beams in synchrotrons
Worked examples
Muon at 0.99c
Given:
- v:
- 296794533
Find: gamma
Solution
gamma = 1/sqrt(1 - 0.99^2) = 1/sqrt(0.0199) ≈ 7.09
LHC proton
Given:
- v:
- 299792455
Find: gamma
Solution
For v ≈ c - 3 m/s, gamma ≈ 7000 (corresponds to 7 TeV beam energy).
Scenarios
What if…
- scenario:
- What if v = 0.5c?
- answer:
- gamma = 1/sqrt(0.75) ≈ 1.155 — only a 15% effect; relativity is subtle at half light speed.
- scenario:
- What if v exceeded c?
- answer:
- 1 - v^2/c^2 goes negative and gamma becomes imaginary — forbidden for real massive particles.
- scenario:
- What if light itself had gamma?
- answer:
- Photons are massless and travel at c in every frame; gamma is undefined for them and replaced by frequency/momentum relations.
Limiting cases
- condition:
- v → 0
- result:
- gamma → 1
- explanation:
- Classical limit: no relativistic effects at low speed.
- condition:
- v → c
- result:
- gamma → ∞
- explanation:
- The factor diverges — massive objects can never reach the speed of light.
- condition:
- v = 0.866c
- result:
- gamma = 2
- explanation:
- Time runs at half rate and lengths contract by half.
Context
Hendrik Lorentz · 1904
Lorentz introduced the factor to preserve Maxwell's equations under frame transformations; Einstein gave it physical meaning in 1905.
Hook
At what speed does your wristwatch tick half as fast to a stationary observer?
Find v such that gamma = 2. Solve 1/sqrt(1 - v^2/c^2) = 2 to get v = c*sqrt(3)/2 ≈ 2.598e8 m/s.
Dimensions:
- lhs:
- gamma → [1]
- rhs:
- 1/sqrt(1 - [LT^-1]^2/[LT^-1]^2) → [1]
- check:
- Dimensionless on both sides. ✓
Validity: Valid in any inertial frame for v < c = 299792458 m/s. Undefined at v = c.