A clock deeper in a gravity well ticks slower — not by illusion, by the metric itself. The factor is Schwarzschild's dτ/dt = √(1 − rs/r), with rs = 2GM/c². It is why a GPS satellite clock, higher in the well, runs fast by tens of microseconds a day and must be corrected. Down the center, data flows: mass and depth go in, the metric decides, the tick-rate comes out. The blue team builds and defends it; the red team tries to break it.
source A. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Ann. Phys. 49 (1916) 769–822 — wikisource / The Foundation of the Generalised Theory of Relativity; confirmed by Hafele–Keating (1972). Rendered, not quoted.
Around a spherical mass M, proper time τ (a real clock) and coordinate time t (a clock at infinity) are locked by one exact factor:
dτ/dt = √(1 − rs/r), rs = 2GM/c².
It is < 1 everywhere outside the horizon: deeper (smaller r) is always slower. It → 1 far away (r >> rs, no dilation) and → 0 at r = rs (frozen, to the distant observer). For the current body:
| quantity | value |
|---|
Einstein 1916: mass curves spacetime; the same curvature that bends orbits also stretches the tick between clocks. The one factor √(1 − rs/r) is the hinge of a whole family.
At r = rs it hits 0 — that surface is the-schwarzschild-radius, where escape velocity = c. Its inverse, 1/√(1−rs/r) − 1, is the redshift the-gravitational-redshift measures. And its weak-field slope is the correction GPS lives on. Each sphere is the next one's premise.
The blue team's live check: recompute the metric over the whole depth ladder and confirm deeper = slower and GPS runs fast. If red flips the sign, this badge is where it shows.
Two inputs and two constants decide everything: a mass M, a radius r from its center, and the fixed G, c. From them: rs = 2GM/c². Real bodies span the range:
| body | rs | surface dτ/dt |
|---|
The middle term that vanishes from the answer is rs itself — you never quote it, you feel it as slowed time. That is what you feed the panel below.
Move the depth — every number is computed from √(1 − rs/r) on the spot, never looked up. The curve is the factor vs r/rs; the dot is you.
What the metric produces, proven: the Sun's horizon rs = 2.95 km; a GPS clock at 20 200 km runs fast by +45.7 µs/day from gravity alone; and at any depth the factor is < 1, monotone in r, →1 far off, →0 at the horizon. The current reading is above; these totals are the output.
The blue team's witness (left) confirms these live; the red team (right) tries to make time run backwards.
And "dτ/dt" is a ratio to a coordinate clock at infinity, not something you can hold. What two real observers compare is the ratio of their factors — a common mistake is to read the bare number as an absolute rate. The metric is not the whole of gravity; it is the first proof that time has a computable depth.
"Time slows because acceleration/g is stronger down there." Cut. It tracks the potential Φ, not local g. A clock in a deep but flat-potential region (zero local acceleration) still runs slow.
"GPS clocks tick slow, being in orbit." Cut. Gravitationally they tick fast (+45.7 µs/day); even after the −7.2 µs velocity term the net is +38 µs/day fast.
"At the horizon, time literally stops." Kept, corrected. Only in the distant observer's coordinate time. The infalling clock ticks normally straight through r = rs.
The red team's move: flip the sign under the root — use √(1 + rs/r) so clocks deeper in the well run faster. Slip that past and GPS "corrections" go the wrong way. The blue team's witness (window 7) is watching.
Flip the sign and the deeper-is-slower law inverts — the witness recomputes, finds a deeper clock running fast, and turns red. Nothing is faked; the attack is real and it is caught.