Recover invariant mass and bound a low-speed approximation

1. Define events, frames and conserved quantities

In c=1 units a massive particle has energy E=γm and momentum p=γmv. Energy is measured in reference mass times c² and momentum in reference mass times c; the numerical c=1 convention does not erase their physical dimensions.

Squaring and subtracting gives E²−p²=γ²m²(1−v²)=m². An observer changes E and p together, while invariant rest mass remains the same for this unchanged particle. Referring to γm as a velocity-dependent rest mass would obscure this invariant relation.

Kinetic energy is K=E−m=(γ−1)m. The expansion γ≈1+v²/2+3v⁴/8 at low speed yields K≈mv²/2 plus a positive correction. The tool compares the fractional underestimate (K−Kclassical)/K rather than treating the approximation as a second exact conservation law.

For numerical stability the same kinetic energy can be written mγ²v²/(γ+1), obtained by rationalizing γ−1. This matters for small speeds where subtracting nearly equal numbers loses precision. Approximation error, floating-point cancellation and uncertainty in a measured speed are different issues and should not be conflated.

2. Reconstruct the original case

For m=2 and v=0.6, γ=1.25, E=2.5 and p=1.5. The invariant is 6.25−2.25=4=m². Exact kinetic energy is 0.5, while the classical value is 0.36, a 28% underestimate relative to the exact kinetic energy.

3. Predict and investigate

Does a larger total energy in a moving frame require a larger invariant rest mass?

Reconstruct the invariant from two independently reported quantities, compare exact and approximate kinetic energies across speed and distinguish approximation error from input uncertainty.

Primary mechanism context. Checked 31 October 2026. Original derivations, records and diagrams; no physical apparatus operation or actual independent clock/particle measurement is claimed. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.