| Velocity | γ | Coordinate a (parallel), g | Coordinate a (perpendicular), g |
|---|---|---|---|
| 0.3c | 1.048 | 0.868 | 0.910 |
| 0.7c | 1.400 | 0.364 | 0.510 |
| 0.9c | 2.294 | 0.083 | 0.190 |
Define β = v/c and γ = 1/√(1 − β²).
- a = α / γ³
- α = γ³ a
- F = γ³ m₀ a (force parallel to motion)
- a = α / γ²
- α = γ² a
- F = γ m₀ a (force perpendicular to motion)
This tool assumes special relativity in flat spacetime and instantaneous values. If you need position over time under changing force, use numerical integration.
- Select a calculation mode based on what you know.
- Choose whether acceleration is parallel or perpendicular to velocity.
- Enter velocity, then provide the requested acceleration or force.
- Add rest mass to compute force, momentum, and energy values.
- Adjust decimals or scientific notation for clean reporting.
- Press Calculate, then export your results if needed.
1) What is the difference between proper and coordinate acceleration?
Proper acceleration is what an accelerometer measures on the object. Coordinate acceleration is measured in a chosen lab frame. They differ by factors of γ, especially at high speed, and also depend on whether acceleration is parallel or transverse.
2) Why does lab-frame acceleration shrink near light speed?
As v increases, γ grows. For the same proper acceleration, the lab-frame acceleration scales like 1/γ³ (parallel) or 1/γ² (transverse). So the object can keep “feeling” acceleration while gaining speed more slowly in the lab frame.
3) Can I enter v equal to or greater than c?
No. Special relativity requires |v| < c for massive objects. The calculator blocks values at or above light speed to avoid invalid γ values and meaningless acceleration results.
4) When should I choose parallel vs perpendicular?
Choose parallel for thrust or braking along the motion direction. Choose perpendicular for turning or circular motion where acceleration is sideways relative to velocity. The scaling with γ and the force relationship are different in these cases.
5) Which mass should I use in the force formulas?
Use rest mass m₀. The calculator uses the standard special-relativistic relationships between force, γ, and coordinate acceleration. If you are modeling rockets with changing mass, update m₀ per step externally.
6) Are “g” units safe for extreme values?
Yes. “g” is just a convenient scale (1 g = 9.80665 m/s²). For very large accelerations, consider enabling scientific notation to keep the output readable, especially in CSV or PDF exports.
7) What does the PDF export include?
The PDF is a compact single-page report containing your mode, direction, timestamp, and the computed values shown in the results table. It’s designed for sharing in labs, homework, or project notes.
8) Does this cover gravity and general relativity?
No. This calculator models special relativity in flat spacetime. For strong gravitational fields, curved spacetime, or long-duration trajectories near massive bodies, general relativistic methods are needed.