Sections · Time dilation

Vol. Science · Astronomy · Clocks

Time dilation
and two rates

A moving clock ticks slow. A clock higher in a gravitational well ticks fast. Both are on a GPS satellite, and they do not cancel. Pick an altitude, or type a speed. The circular speed is √(GM / r). The Lorentz factor is γ = 1/√(1 − v²/c²). The velocity term on the card is the exact special-relativistic rate, √(1 − v²/c²) − 1, in microseconds over 86,400 seconds. The gravitational term is the weak-field monopole, GM (1/R − 1/r) / c², against a clock at rest on the equatorial radius. GM = 3.986004418×10¹⁴ m³/s², the TCG-compatible value in IERS Conventions (2010) Table 1.1, which is the value Ashby 2003 prints. R = 6,378,137 m. c = 299,792,458 m/s exactly. The GPS preset uses the IS-GPS-200N radius, 26,559,710 m, and computes −7.21 μs/day from speed, +45.65 μs/day from gravity, and +38.44 μs/day net. NIST quotes 7, 45, and 38. The period of the same circle is the satellites desk. A hovering clock near a black hole is the black holes desk. The plain-language receipt is the gravity desk.

Educational calculator. Simplified weak-field model, not for navigation or engineering. No Earth rotation, no quadrupole, no eccentricity, and no Sagnac effect. The ground clock in the gravitational term is at rest at the equatorial radius, not on the rotating geoid. A day in the microsecond rates is 86,400 SI seconds.

A speed, and one year aboard

What the buttons are

Literacy · the words on the readout

Lorentz factor

γ = 1/√(1 − v²/c²). A moving clock’s proper time is the stay-at-home time divided by γ. One year aboard is γ years at home, at constant speed. At orbital speeds γ − 1 is a few parts in 10¹⁰ or 10¹¹, so the card prints 1 + that difference.

Velocity term

√(1 − v²/c²) − 1, times 86,400 seconds, in microseconds. Negative means slow. Ashby’s GPS bookkeeping uses the second-order piece −v²/(2c²). At the speeds of these orbits the two print the same microseconds. They part company as v becomes an appreciable fraction of c.

Gravitational term

GM (1/R − 1/r) / c². The Newtonian potential is −GM/r. A higher clock is less deep in the well, so this term is positive: fast. R is the equatorial radius, 6,378,137 m. The term is zero at the surface and grows as the radius grows.

Net drift

The gravitational fraction plus the Lorentz fraction, over one day of 86,400 seconds. For GPS the gravity term wins, so the net is fast. For the ISS the velocity term wins, so the net is slow. They balance, on this monopole, at r = 3R/2, which is 9,567.21 km.

What is left out

Earth’s spin, the equatorial bulge, the eccentricity of a real GPS orbit, the Sagnac effect, and the Shapiro delay. Ashby’s Φ₀ is an effective potential on the rotating geoid, L_G = 6.969290134×10⁻¹⁰. This page’s surface is only −GM/(R c²). That is why his printed net, +38.58 μs/day, is 0.14 μs/day faster than the +38.44 this page computes. The factory offset he prints, −4.4647×10⁻¹⁰, is the correction applied so the satellite clock keeps geoid time. This page does not apply it. It reports the drift.

Sources · the constants and the GPS story

Classroom clocks. c is the SI constant. GM is the IERS TCG-compatible 3.986004418×10¹⁴ m³/s², the value Ashby prints. The equatorial radius is 6,378,137 m. Rates are computed. GPS on this page is −7.21, +45.65, and +38.44 μs/day. NIST quotes 7, 45, and 38. Ashby’s printed offset is +38.58 μs in 86,400 seconds. Simplified weak-field model. Not for navigation or engineering. Earthrise preview: NASA / Bill Anders (public domain).

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