T-20 · Special relativity

Pythagoras with a minus sign.

In spacetime the distance between two events is s² = t² − x², and every observer agrees on it. Take speeds from Pythagorean triples, β = a/c, and the Lorentz factor γ = c/b is a fraction: every boost, interval, Doppler factor and proper time on this sheet is exact. Even E² = p² + m² turns out to be the speed's own triple, scaled.

One event, two frames

The Lorentz transformation t′ = γ(t − βx), x′ = γ(x − βt). With β from a Pythagorean triple, γ is a fraction, so the new coordinates are fractions too.

your frame (t, x)	(5, 3)
moving frame (t′, x′)	(4, 0)

Pythagoras with a minus sign

Distance in spacetime is s² = t² − x², not t² + x². A boost changes t and x but never s²: the event slides along its hyperbola. For a timelike event, s is the time a clock reads travelling from the origin to it.

t² − x²	5² − 3² = 16
t′² − x′²	16 the same, exactly
kind	timelike · proper time 4

The factor γ

β = a/c from a triple a² + b² = c² makes 1 − β² = (b/c)², a perfect square, so γ = 1/√(1 − β²) = c/b exactly. Ordinary Pythagoras is what makes the relativistic one come out in fractions.

β, γ	3/5, 5/4
γ²(1 − β²) = 1	exactly
simultaneity	t′ = 0 is the line t = 3/5x, not t = 0