Energy and momentum are not two things but one — P = (E/c, pₓ, pₔ, pₕ), a single arrow in spacetime. Its length is frame-independent: the invariant mass. Boost your frame and E and p mix into each other exactly as time and space do, yet (E/c)² − |p|² never changes, and in any collision the total arrow is conserved.
SOURCE Minkowski, Raum und Zeit (Cologne lecture, 1908) — Phys. Z. 10, 104–111; energy–mass groundwork in Planck, Verh. Dtsch. Phys. Ges. 8 (1906). No single stable primary link → AMBER. Rendered, not quoted.
A free body of mass m moving with velocity v carries a four-momentum built from γ = 1/√(1−β²), β = v/c:
The Minkowski length of this arrow is the invariant mass:
Squared time-part minus squared space-part — the same minus sign that governs the interval (c t)² − x². Momentum lives in the space-part; energy in the time-part.
One arrow made of two neighbours. The time-part is the-mass-energy-equivalence (E = γmc², so at rest E = mc²); the space-part is the-relativistic-momentum (p = γmv).
Minkowski (1908) welded them: a body's energy and momentum are the four components of one spacetime vector whose norm (mc)² is invariant and conserved in every interaction. That weld is this sphere.
Re-runs the full proof on demand and re-reads the invariant across a boost. Confirms green when the mass is frame-independent; flips red the instant window 6 corrupts the metric sign.
A parent mass M = 10 at rest decays into two bodies, m₁ = 3 and m₂ = 4 (natural units, c = 1). Also on the bench: a probe body m = 1 at v = 0.6c, and a photon.
All energies and momenta below are computed from these inputs by the engine — no values are typed in by hand.
Momentum–energy plane. The hyperbola E² − p² = m² is the locus of one mass across all frames; a boost slides a body along it. The two decay products (dots) sit on their own m=3, m=4 shells; the parent's arrow is their sum.
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Total energy of the products equals M c²; total momentum is zero; the invariant mass of the two-body system reconstructs the parent M to 1e-9. The metric sign is what makes this hold in every frame.
"Energy and momentum are separate conserved quantities — bolt them into a vector if you like, but the metric is just bookkeeping."
The wall: the minus sign is not bookkeeping. With a Euclidean (E/c)²+|p|² the length changes under a boost, so different observers would disagree on a body's rest mass — a measurable contradiction. Only the Minkowski sign makes mass invariant. That is the fault line window 6 exploits.
Mass grows with speed as m(v) = γm₀.
→ The invariant mass is fixed; γ belongs to E and p. "Relativistic mass" is a retired bookkeeping habit.
A photon is massless so it has no momentum.
→ p = E/c ≠ 0; its four-momentum is null — length zero, not the zero vector.
Kinetic energy adds to |p|c to give total energy linearly.
→ E² = (pc)² + (mc²)² — a Pythagorean sum on the shell, not a linear one.
Flip the metric to a Euclidean (E/c)² + |p|² (wrong sign). Rest mass stops being frame-independent; the witness in window 7 catches the boost mismatch.