Why you can push forever and never reach c: the momentum diverges first. Newton says p = m v — double the push, double the speed, no ceiling. Nature says p = γ m v, and as v climbs toward c the factor γ runs to infinity, so a finite force buys ever-smaller gains in speed. Down the center, data flows: mass and velocity go in, the engine computes momentum and energy, the proven law comes out. The blue team builds and defends it; the red team tries to break it.
source G. N. Lewis & R. C. Tolman, The Principle of Relativity, and Non-Newtonian Mechanics, Phil. Mag. 18, 510–523 (1909); after A. Einstein, Zur Elektrodynamik bewegter Körper, Ann. Phys. 17, 891 (1905). wikisource. Rendered, not quoted.
One extra factor turns Newton into Einstein. With β = v/c and γ = 1/√(1−β²), the momentum is p = γ m v — always larger than the classical m v, because γ > 1 whenever v > 0.
For the current velocity, the two momenta side by side (natural units, c = 1):
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The ratio of relativistic to classical momentum is γ. At v = 0 they agree; near c the relativistic value runs away.
Momentum is not alone. It is the space part of the four-momentum (E/c, p); energy is the time part. Boost between frames and they rotate into each other exactly like (ct, x) — same Lorentz geometry, one rung up.
The invariant length of that four-vector is the rest mass: (E/c)² − p² = (m c)². Push forever, never reach c — that single fact is the-four-momentum feeding the-mass-energy-equivalence. Each sphere is the next one's premise.
The blue team's live check: re-run every law of the engine — the classical limit, the divergence, the force cap at c, collision conservation, and the energy–momentum relation. If red tampers, this badge is where it shows.
Two numbers enter: a rest mass m (how much stuff) and a velocity v as a fraction of light speed β = v/c. Everything else — γ, momentum, energy, the force it takes to accelerate — is derived, never assumed. Set them, feed them to the panel below.
Momentum vs speed: relativistic γ m v (diverges at c) against the classical m v line. The red wall is c.
Move any control — γ, p, E and the energy–momentum check are computed on the spot from p = γ m v and E = γ m c², never looked up.
What the machine produces, proven: momentum is p = γ m v — it exceeds the classical m v for every v > 0 and diverges as v → c; a constant force integrated as dp/dt = F drives v toward c asymptotically but never to or past it; total γ m v is conserved in an elastic collision to 1e-9; and it locks to energy through E² = (p c)² + (m c²)².
The blue team's witness (left) confirms these laws live; the red team (right) tries to make them fail.
It is also a free-particle, 1-D instrument: no fields, no potentials, no rotation, and momentum here is the kinetic canonical momentum. In an electromagnetic field the canonical momentum gains a qA/c term this panel does not carry. The law is exact — its scope is the honest limit.
"The mass grows: m → γ m as you speed up." Cut. Rest mass is invariant. What grows is momentum and energy; "relativistic mass" is a bookkeeping fiction that misreads E = γ m c² as a heavier m.
"At high speed just use p = m v, it's close enough." Cut. At 0.99 c the true momentum is ~7× the classical value — the whole reason a collider needs so much energy for so little extra speed.
"A big enough force can push anything past c." Kept, corrected. Force still works — dp/dt = F never stops. But dp adds to a p that diverges, so v only crawls asymptotically toward c. The ceiling is real.
The red team's move: drop the γ — compute momentum the classical way, p = m v — and hope no one checks it against energy. The blue team's witness (window 7) is watching.
Drop γ and the energy–momentum relation E² = (p c)² + (m c²)² — with the true E = γ m c² — no longer closes: E² − (p c)² stops equalling (m c²)². The witness recomputes, disagrees, and turns red. Nothing is faked; the attack is real and it is caught.