The top floor runs fast

What a skyscraper does to your clock, and the height at which the textbook shortcut stops working.

General relativity says a clock higher in a gravitational field ticks faster. Near Earth's surface the fractional rate difference is usually written gh/c², where g is gravitational acceleration, h the height difference and c the speed of light. In 2010 a NIST group measured this effect with aluminium-ion optical clocks raised by just 33 cm, seeing a shift of roughly 4 parts in 10¹⁷ [1][2]. So the effect is real at human scale. How big is it for a person?

Try it

The default is the 828 m official height of the Burj Khalifa [3] and an 80-year stretch, which no one spends at the top, so read it as an upper bound for a lifetime on the top floor against a lifetime at street level.

Two formulas

Shortcut. Treat g as constant: fractional gain = g·h / c², with g = 9.80665 m/s² (the conventional standard value) and c = 299 792 458 m/s (exact by definition).

Potential. Use Newtonian potential difference between radius R and R+h: fractional gain = GM·(1/R − 1/(R+h)) / c², with GM = 3.986004418×10¹⁴ m³/s², the value in the WGS84 and IERS 2010 constants [4], and R = 6 371 km, a mean radius I chose. This is still a weak-field approximation, and it ignores Earth's rotation and the mismatch between sea-level and mean radius, which shift results at the percent level or less for short heights, so treat the digits past the second as decoration.

What comes out

For 828 m over 80 years, both formulas give about 228 microseconds: the top-floor clock gains roughly a quarter of a millisecond. Nothing a human would notice.

The question I wanted to ask is the reverse: how tall would a tower have to be for the occupant to gain one full second over 80 years? The shortcut says h = c² / (g·T), about 3 630 km. The potential formula, solved numerically, says about 8 410 km. The shortcut is off by a factor of 2.3 because by then g has fallen well below its surface value, so the climb is much cheaper than gh/c² assumes. (A tower that tall is of course a fantasy; the point is where the shortcut fails.) The slider above runs to 100 000 km so you can watch the two formulas separate; the readout shows their ratio.

What I did not check

The Chou et al. paper is the source for the 33 cm experiment and its approximate result; I read summaries of it, not the paper's full text, so I quote only "roughly 4×10^-17", the figure reported by Physics World [2]. My shortcut value for 33 cm is 3.6×10^-17, consistent with that.

Sources

1. C. W. Chou, D. B. Hume, T. Rosenband, D. J. Wineland, "Optical clocks and relativity," Science 329, 1630–1633 (2010).
2. Physics World coverage of the experiment, physicsworld.com (searched 2026-10-24).
3. Council on Tall Buildings and Urban Habitat, ratification of the Burj Khalifa's height as 828 m (2010), as reported by Construction Briefing / KHL.
4. IERS Conventions (2010), IERS Technical Note 36, Petit & Luzum (eds.); WGS84 gives the same GM.