THE TIME DILATION

A moving clock ticks slow. dt = γ·dτ — the coordinate time between two ticks of a clock whose own proper time is dτ. Because γ>1 for any motion, the muon born high in the atmosphere lives long enough, in ground time, to reach the ground. Rendered, not quoted.

source Einstein, Zur Elektrodynamik bewegter Körper (1905), Ann. Phys. 17, 891–921 — doi:10.1002/andp.19053221004; moving-clock test: Hafele & Keating, Science 177 (1972). the correctness lives in pure functions, not this canvas
Blue Team · builds & defends
3
The Model

Two events on one worldline — tick, then tock — are separated by proper time dτ in the clock's own rest frame (same place, so only time differs). In a frame where that clock moves at speed v, the two events are at different places, and the coordinate time between them is

β = v/c   γ = 1/√(1−β²)   dt = γ·dτ

Since γ>1 whenever v>0, the moving clock's dτ is squeezed into a larger ground interval dt: fewer ticks per second of ground time — it runs slow. At v=0, γ=1 and dt=dτ.

5
The Lineage

Moving clocks run slow — Einstein 1905, §4. dt = γ dτ is the time component of the-lorentz-transformation, reciprocal with the-length-contraction (L = L₀/γ), and the reason the-twin-paradox is real once one twin accelerates.

Neighbour sphere: the-lorentz-transformation. This window is that transform read on the time axis alone.

7
The Witness live re-check

Re-runs the pure engine now, in your browser, including the planted tamper below. Green = every relativistic identity holds; it flips red the instant window 6 corrupts the law.

witness idle

The Machine
4
Data In in ↓

A clock's proper time dτ and its speed v (as β=v/c). Drag the slider to set β.

β = 0.866
0
The Panel LIT
γ = …

Left: a light clock — a photon bouncing between mirrors. Boosted, its path lengthens to a zig-zag, so each tick takes longer: the γ stretch, drawn live. Right: γ(β) with your β marked. Nothing here is typed in; every number is computed from γ=1/√(1−β²).

8
Data Out out ↓

Coordinate time per tick, and the muon reach. Proven at boot:

…

Muon, γ=10, proper lifetime τ: ground lifetime = 10τ; distance travelled ≈ 9.9499 τc — far enough to reach sea level, which at rest it never could.

Red Team · attacks & breaks
1
The Adversary wall

"If it's all relative, the OTHER frame sees YOUR clock slow too — so who is really slow? The whole thing is a contradiction."

Answer: both are right, and there is no contradiction. Dilation is symmetric because it depends on relative velocity only. The apparent paradox is resolved only by adding a real, frame-independent event — a reunion, which requires acceleration (the-twin-paradox). Symmetry of the effect ≠ symmetry of the histories.

2
The Graveyard
  • "Time dilation makes the moving clock run fast." No — slow. dt=γdτ with γ>1 means ground time dt exceeds the clock's own dτ: fewer ticks per ground second.
  • "Only clocks are affected; real physical decay isn't." Muon lifetime is a clock. Their measured survival to sea level is the effect, tested daily.
  • "The effect scales with v, so +v and −v differ." It depends on β² — identical for +v and −v. Direction is irrelevant.
6
The Tamper

Planted void, disclosed. This swaps the law to dt = dτ/γ — a moving clock that runs fast and ends up ahead. The Witness (7) catches it live.

law intact