Rest, total, and kinetic energy in special relativity
A particle with rest mass m has rest energy mc². Relative to an inertial observer, motion increases total energy by the Lorentz factor γ.
Relativistic kinetic energy is total energy minus rest energy and approaches the classical expression only when speed is much smaller than c.
How to calculate relativistic energy
- Enter rest mass: Supply a non-negative mass in kilograms.
- Enter v/c: Use a value with magnitude less than 1.
- Calculate: Review γ and energies in joules, plus kinetic energy in electronvolts.
Formula and variables
β is speed divided by the exact speed of light and must have magnitude below one.
γ = 1/√(1−β²); E = γmc²; KE = (γ−1)mc²- m — Rest mass
- Invariant mass (kg)
- β — Speed fraction
- v/c
- γ — Lorentz factor
- Relativistic factor
One kilogram at 0.8c
Calculate energy for m = 1 kg and β = 0.8.
- m
- 1 kg
- β
- 0.8
- γ = 1/√(1−0.8²) = 1.6667
- KE = (γ−1)mc²
Result: KE ≈ 5.99 × 10¹⁶ J.
The total energy includes the larger rest-energy contribution.
Understanding your results
Energy rises without bound as speed approaches c
For nonzero rest mass, γ diverges as |v| approaches c, so finite energy cannot accelerate the object to light speed.
- At rest, γ = 1 and KE = 0.
- Direction does not change energy because β is squared.
Assumptions
- Special relativity in an inertial frame.
- Input mass is invariant rest mass.
- Potential energy and interactions are excluded.
Limitations
- Does not calculate momentum, composite-system energy, gravitational effects, or massless-particle energy.
- Floating-point precision degrades extremely close to c.
Common mistakes
- Entering percent instead of a fraction of c.
- Using relativistic mass terminology instead of rest mass and total energy.
- Confusing total energy with kinetic energy.
Practical use cases
Special-relativity coursework
Compare classical and relativistic energy scales.
Frequently asked questions
Why must |v/c| be below 1?
A massive particle cannot reach or exceed c in special relativity, and γ is not finite there.
Is mc² kinetic energy?
No. mc² is rest energy; kinetic energy is (γ−1)mc².
Sources and review
- Relativistic Energy — OpenStax University Physics Volume 3. Accessed 2026-07-13.
Reviewed 2026-07-13.