Observatory · the clock
Time dilation
Clocks run slower deeper in a gravity well, and slower the faster they move. Not the mechanism, time itself: a clock at sea level loses to a clock on a mountain, the clocks in GPS satellites are built to run slow on the ground so that they keep time in orbit, and next to a black hole the effect stops being a correction and becomes the whole story. Two instruments below: the Schwarzschild factor, √(1 − rs/r), with a real black hole's mass behind it, and the Lorentz factor for speed, which is the other half of the same story.
The clock at the horizon
× 1
the near clock runs at this fraction of the far one
that is from the centre of
While an hour passes on the far clock, the near one shows . Spend a year hovering here and pass at home. The factor is √(1 − rs/r): nothing happens to you, your own seconds feel like seconds, it is the rest of the universe that speeds up. At the horizon itself the factor reaches zero, which is why, from far away, nothing is ever seen to cross it. Hovering this close takes a rocket that could hold you against the pull; falling freely is another story, and a shorter one.
The clock on the train
× 1
the moving clock runs at this fraction · γ = 1
This is the special-relativity half, Einstein 1905, with no gravity in it at all: a clock moving past you runs slow by 1/γ, γ = 1/√(1 − v²/c²). At everyday speeds γ is 1 followed by a dozen zeros; at 0.866 c it is exactly 2, at 0.9999 c it is 71. The moving clock shows for every hour of yours. Both clocks are right, and each sees the other one run slow, which is the part people argue about in pubs; it only becomes a one-way difference when one of them turns around and comes back.
Muons made by cosmic rays 15 km up live two microseconds, long enough to fall 600 metres, yet they reach the ground: at 0.9999 c their two microseconds are our 140. The ISS circles at 7.7 km/s, so an astronaut on a six-month stay comes home about 5 milliseconds younger than if they had stayed, before the gravity part gives a little back. Hafele and Keating flew four caesium clocks around the world on airliners in 1971, east and west, and the clocks disagreed with the ones left at home by the tens of nanoseconds that both effects together predicted.
Where it already matters
GPS. The satellites sit 20,200 km up, where Earth's gravity is weaker, so their clocks run fast by about 45 microseconds a day; their orbital speed slows them by about 7, for a net 38 microseconds a day gained. Left uncorrected, position errors would grow by roughly 10 km every day. The satellite clocks are deliberately set to tick slow before launch so that, in orbit, they agree with the ground. Relativity is not a theory you consult about GPS; it is a line item in its design.
Your head and your feet. In 2010 the American standards institute compared two optical clocks one above the other, 33 centimetres apart, and saw the higher one run faster. The difference was a few parts in 1017, and it was there.
The film. The water planet where one hour cost seven years needs a factor of about 61,000. For a non-spinning hole that means hovering at 1 + 3×10−10 rs, closer than any orbit could hold; the film used a very fast-spinning hole, where stable orbits reach far deeper. The slider here is the simple, non-spinning case, which is why it tops out where it does.
Falling in. The clock is for someone hovering, held up against the pull. Someone falling freely feels no pull at all and crosses the horizon in finite time on their own watch; it is the watchers far away who never see it happen, the faller's light stretching and fading instead. Both are right. That is the strange part, and it is not a paradox, just two clocks.