Mass is frozen energy. A body at rest already holds E₀ = mc²; set it moving and the energy grows as E = γmc², bound to momentum by E² = (pc)² + (mc²)². The rest energy is the invariant floor every frame agrees on; a massless particle sits at the far limit where E = pc. Rendered, not quoted.
A particle of rest mass m moving at speed v. Write β = v/c and γ = 1/√(1−β²). Then:
Kinetic energy is the excess over rest, K = (γ−1)mc². The invariant mass (E²−(pc)²)/c⁴ is what every observer measures the same — it is the geometric length of the four-momentum.
Mass is frozen energy — Einstein 1905. E = mc² is the v = 0 face of E² = (pc)² + (mc²)², the invariant length of the-four-momentum, built on the-relativistic-momentum p = γmv. Neighbour spheres carry the boost and the momentum; this one supplies the floor they all agree on.
Live re-check. It recomputes E two ways — kinematically as γmc² and from the energy–momentum relation — and confirms they agree to 1e-9. If the RED tamper flips the sign in that relation, this badge catches it instantly and turns red.
Rest mass m and speed β = v/c (natural units, c = 1). Rest energy is the anchor; the total energy and momentum are computed live from β.
Total energy E = γmc² (amber) rises without bound as β→1; the rest energy mc² (dashed) is the floor. The gap is the kinetic energy.
E² = (pc)² + (mc²)² holds to 1e-9 across all β; the recovered invariant mass equals the input m in every boosted frame; and the massless limit collapses to E = pc.
Second push: “relativistic mass increases.” That older bookkeeping folds γ into m; here m is invariant and γ sits with the energy. Same physics, honest labels.
E = mc² is the whole story.
→ Only for v = 0. Moving bodies obey E = γmc²; the full invariant is E² = (pc)² + (mc²)².
K = ½mv² at every speed.
→ Only the low-β limit. (γ−1)mc² adds a (3/8) m v⁴/c² term and higher orders.
A photon has zero energy because m = 0.
→ m = 0 gives E = pc, not zero. Light carries energy and momentum with no rest mass.
The disclosed planted void. Flip the sign in the energy–momentum relation to E² = (pc)² − (mc²)². The rest-energy limit then wants √(−m²c⁴) — the WITNESS (7) catches the mismatch live.