Gravitational Time Dilation

Clocks Deep in Gravity Wells

Hovering Clocks Around a Black Hole

Event horizon (rₛ)Photon ring (1.5 rₛ)Static observer clocks

Key Parameters

Schwarzschild

rₛ = 2GM/c²

Time factor

√(1 − rₛ/r)

Photon sphere

1.5 rₛ

ISCO (Schw.)

3 rₛ

At 2 rₛ

dτ/dt ≈ 0.71

At horizon

dτ/dt → 0

What's Happening

The equivalence principle forces it: an accelerating frame mimics gravity, and an accelerating observer's clock runs slow relative to a freely-falling one. So a clock held stationary deep in a gravity well — which requires acceleration to fight the pull — must tick slow.

For a non-rotating mass, the Schwarzschild metric makes this exact: a static clock at radius r ticks at √(1 − rₛ/r) times the rate of a clock at infinity. As r → rₛ the factor → 0 — the clock appears to freeze.

The locally measured second is unchanged. What differs is how distant observers compare their tick rates. Time dilation is a relationship between frames, not a property of any single clock.

Consequences

Photons climbing out of a well lose energy — gravitational redshift

Photons falling in gain energy — blueshift

Higher altitude = faster-running clock (GPS depends on this)

Infalling matter appears to freeze near the horizon (from outside)

An infaller crosses the horizon in finite proper time

Combined SR + GR gives the full clock-rate correction in orbit

Where We See It

GPS satellites

+38 μs/day

Orbiting at ~20,200 km, satellite clocks run faster than ground clocks by ~45 μs/day from gravity, slower by ~7 μs/day from orbital velocity. The net +38 μs/day correction is essential — without it, navigation would drift by 10+ km per day.

Sagittarius A*

z ≈ 2 × 10⁻⁴

Star S0-2 swung within 120 AU of the Milky Way's central supermassive black hole in 2018. Its light was redshifted by exactly the amount predicted by general relativity — the first detection of gravitational redshift from a star orbiting a SMBH.

Neutron star surface

dτ/dt ≈ 0.77

On the surface of a typical 1.4 M☉ neutron star at ~12 km radius, dτ/dt ≈ 0.77 — clocks run at three-quarters the lab rate. Pulsar timing is precise *internally*, but every observed period must be back-corrected for this gravitational dilation.

Sagittarius A* in ED

ED narrative

Commanders can fly within ~2 km of the supermassive black hole at the galactic core. By GR, time outside that close would tick at a fraction of bubble time — but the FSD's spacetime-warping conceit gives the ship its own quasi-flat frame, so the clock in the cockpit keeps station with home.