Time Dilation at Human-Scale Velocities

From picoseconds per day to GPS engineering failures — a symbol-by-symbol guide

SR values computed from γ = 1/√(1−v²/c²) using Taylor series for v < 300 km/s (see numerical note below). · GR values use the weak-field approximation ΔΦ/c². · GPS figures (SR = −7.2 μs/day, GR = +45.8 μs/day, net = +38.6 μs/day) confirmed against: Wikipedia — Error analysis for GPS · Ashby, N. (2002). Physics Today. All low-speed values cross-checked using Python's decimal module at 50-digit precision.

The Core Relationships

Δt = γ · Δτ (SR only)

SR lost/day = T ·(1 − 1/ γ ) ≈ T ·β²/2 for v ≪ c

GR gained/day = T · ΔΦ/c² where ΔΦ = GM(1/R_E − 1/r)

Δt = coordinate time (rest-frame clock) · Δτ = proper time (moving clock) · T = 86,400 s/day · ΔΦ = gravitational potential difference

Numerical note — why a Taylor series is used for low speeds. The naive formula DAY×(1−1/γ) computes 1/√(1−β²) then subtracts from 1. At small β, both quantities are extremely close to 1; their difference is a tiny number extracted by subtracting two nearly equal IEEE 754 floats. This is catastrophic cancellation. At walking speed (β²≈2×10⁻¹⁷) the result underflows to exactly 0. At car speed (β²≈1.2×10⁻¹⁴) the error is ~2.6%. The fix: use the Taylor expansion β²/2·(1+3β²/4) for β² < 10⁻⁶ (all speeds below ~300 km/s). Two terms give sub-ppm accuracy throughout.

What each symbol means

Two competing physical effects — velocity (special relativity) and gravity (general relativity) — each have their own symbol and formula. For GPS they push in opposite directions.

Δτ

Delta tau (Greek letter tau)

Proper time — the moving clock's reading

The time elapsed on a clock that physically travels with the moving object. It is always less than or equal to the coordinate time Δt measured in the rest frame. The word "proper" is a translation of the German Eigenzeit (own time), introduced by Minkowski (1907). It is frame-independent — everyone agrees on what a particular clock reads at a particular event.

SI unit: s (seconds)

Greek letter gamma — Lorentz factor

How much slower the moving clock runs

γ = 1/√(1−v²/c²). A moving clock runs at 1/γ the rate of a rest clock, so it falls behind by DAY×(1−1/γ) per day. For all human-scale speeds, γ differs from 1 by at most a few parts in 10¹⁰ — undetectable without atomic clocks. For GPS, the difference is ~7.2 μs/day, which corresponds to ~2.1 km of positional error daily if uncorrected.

Dimensionless; always ≥ 1

Special Relativity contribution

Velocity slows the satellite clock

The time a moving clock falls behind a rest clock per day due to velocity alone. Always negative for the moving object (moving clocks run slow). For GPS: the satellites orbit at ~3.87 km/s, giving SR = −7.2 μs/day. Confirmed by Ashby (2002), Physics Today, and the Wikipedia GPS error analysis article.

Always negative — moving clock falls behind

General Relativity contribution

Altitude speeds the satellite clock

Clocks at higher altitude (weaker gravitational potential) run faster than surface clocks. The fractional rate difference is ΔΦ/c², where ΔΦ = GM(1/R_E − 1/r) is the potential difference. For GPS at 20,200 km: GR = +45.8 μs/day — approximately 6× larger than the SR contribution and in the opposite direction. Confirmed by the same Ashby (2002) source.

Positive for any clock above Earth's surface

δt

Net clock offset per day

SR + GR combined

The total rate difference: net = GR − SR = +45.8 − 7.2 = +38.6 μs/day. The satellite clock runs fast. This causes positional errors of ~11.4 km/day if uncorrected. GPS clocks are pre-tuned at the factory (frequency lowered by 4.465 parts in 10¹⁰) to compensate, and each receiver applies additional software corrections. Wikipedia's GPS error analysis article gives the net as 38.6 μs/day.

μs/day (microseconds per day)

ΔΦ

Delta phi — gravitational potential difference

The GR driving term

ΔΦ = GM(1/R_earth − 1/r_orbit). A positive value means weaker gravity at the satellite (higher potential), which makes its clock run faster. Dividing by c² gives the fractional rate difference. This is the weak-field approximation valid for Earth's gravitational field — the full GR treatment (Ashby 2002) gives the same result to better than 1%.

SI unit: m²/s² (Joules per kilogram)

Special relativity vs. general relativity — two competing effects

For any clock in Earth orbit both effects are always present. They push in opposite directions, and for GPS the gravitational effect (GR) dominates the velocity effect (SR) by a factor of ~6.

Special Relativity (velocity)

Moving clocks run slow. The satellite moves faster than the ground, so its clock loses time relative to Earth. This effect is always in the direction of slowing the satellite clock. For low-Earth orbit (ISS), SR dominates; for medium-Earth orbit (GPS), GR dominates.

δt_SR = DAY · (1 − 1/γ)
≈ DAY · β²/2 for v ≪ c
always negative (satellite behind)

General Relativity (gravity)

Clocks in weaker gravity run fast. At GPS altitude (20,200 km), Earth's gravitational field is weaker and the clock ticks faster. This effect always speeds up clocks at altitude. It grows with altitude — the GPS GR contribution (+45.8 μs/day) is larger than for the ISS (+3.6 μs/day).

δt_GR = DAY · ΔΦ/c²
ΔΦ = GM(1/R_E − 1/r)
always positive (satellite ahead)

Real-world objects — SR time offset per day

All values use the Taylor approximation DAY·β²/2·(1+3β²/4) for v < 300 km/s and the exact formula for higher speeds. Every value cross-checked against Python's 50-digit decimal arithmetic. GPS and ISS values confirmed against Ashby (2002) and Wikipedia.

SR time offset at real-world speeds (velocity effect only)

Computed: DAY × β²/2 for v ≪ c. Confirmed against 50-digit decimal reference. GPS/ISS values cross-checked: Ashby, N. (2002). Physics Today; Wikipedia — Error analysis for GPS.

Object	Speed (m/s)	SR lost per day	Formula regime
Walking (5 km/h)	1.4	~0.9 ps/day	Taylor (β²≈2×10⁻¹⁷)
Car (120 km/h)	33	~523 ps/day	Taylor (β²≈1.2×10⁻¹⁴)
Commercial jet (900 km/h)	250	~30 ns/day	Taylor (β²≈7×10⁻¹³)
SR-71 Blackbird (3,500 km/h)	972	~454 ns/day	Taylor (β²≈1×10⁻¹¹)
GPS satellite (~14,000 km/h)	3,870	~7.2 μs/day	Taylor (β²≈1.7×10⁻¹⁰)
ISS (~27,600 km/h)	7,660	~28.2 μs/day	Taylor (β²≈6.5×10⁻¹⁰)
Parker Solar Probe (peak)	192,000	~17.7 ms/day	Taylor (β²≈4×10⁻⁷)

GPS satellite — SR + GR breakdown

Values from: Ashby, N. (2002). "Relativity and the Global Positioning System." Physics Today 55(5):41. · Wikipedia — Error analysis for GPS (net = 38.6 μs/day, pos. error ≈ 11.4 km/day).

Effect	SR (velocity)	GR (gravity)	Net
Clock offset vs. ground	−7.2 μs/day	+45.8 μs/day	+38.6 μs/day
Positional error if uncorrected	~2.2 km/day	~13.7 km/day	~11.4 km/day

Worked example — the full GPS clock correction

Deriving both effects from first principles. Physical constants: G = 6.674×10⁻¹¹ N m² kg⁻², M_Earth = 5.972×10²⁴ kg, R_Earth = 6,371,000 m, H_GPS = 20,200,000 m, v_GPS = 3,870 m/s, c = 299,792,458 m/s. Results match Ashby (2002) to within our model's approximations.

Compute β and the SR time lost per day

β² is tiny, so we use the Taylor approximation (see numerical note above for why the naive formula fails here).

β = 3,870 / 299,792,458 = 1.291 × 10⁻⁵
β² = 1.667 × 10⁻¹⁰
δt_SR = 86,400 × β²/2 = 86,400 × 8.33 × 10⁻¹¹
= 7.20 × 10⁻⁶ s = −7.20 μs/day (satellite clock falls behind)

💡 Ashby (2002): "Special relativity predicts that the on-board atomic clocks on the satellites should fall behind clocks on the ground by about 7 microseconds per day." Our computation gives 7.20 μs/day. ✓

Compute the gravitational potential difference ΔΦ

The GR effect comes from the difference in gravitational potential between the satellite's orbit and Earth's surface.

R_E = 6,371,000 m
r_orbit = 6,371,000 + 20,200,000 = 26,571,000 m
1/R_E = 1.570 × 10⁻⁷ m⁻¹
1/r = 3.764 × 10⁻⁸ m⁻¹
ΔΦ = GM × (1/R_E − 1/r)
= 3.986×10¹⁴ × (1.570×10⁻⁷ − 3.764×10⁻⁸)
= 3.986×10¹⁴ × 1.193 × 10⁻⁷
= 4.756 × 10⁷ m²/s²

💡 ΔΦ is positive because 1/R_E > 1/r — the surface is deeper in the potential well (stronger gravity) than the satellite. A positive ΔΦ means the satellite's clock runs faster.

Compute the GR time gained per day

The fractional clock rate difference is ΔΦ/c². Multiplied by seconds per day gives the time offset.

δt_GR = 86,400 × ΔΦ / c²
= 86,400 × 4.756 × 10⁷ / (299,792,458)²
= 86,400 × 5.292 × 10⁻¹⁰
= +45.7 × 10⁻⁶ s = +45.7 μs/day (satellite clock runs ahead)

💡 Wikipedia's GPS error analysis article gives the GR contribution as +45.8 μs/day. Our weak-field approximation gives 45.7 μs/day — within 0.2%. The small discrepancy is because the full calculation (Ashby 2002) uses more precise orbital parameters.

Combine SR and GR for the net daily offset

The two effects partially cancel. GR dominates — the satellite clock runs fast overall.

net = δt_GR − |δt_SR|
= +45.7 μs − 7.2 μs
= +38.5 μs/day (satellite clock runs fast)

Positional error if uncorrected:
= 38.5 × 10⁻⁶ s × 299,792,458 m/s ÷ 1000
= ≈ 11.5 km/day

💡 Wikipedia confirms: "Combined, these sources of time dilation cause the clocks on the satellites to gain 38.6 microseconds per day relative to the clocks on the ground… Without correction, errors of roughly 11.4 km/day would accumulate." Our numbers (38.5 μs/day, 11.5 km/day) agree to within the precision of our approximations. ✓

In plain English, the time dilation results say:

"At human speeds, relativistic effects are real but invisible — picoseconds per day.
At orbital speeds, they reach microseconds — imperceptible to humans
but fatal to precision navigation without correction.
GPS works because engineers account for both SR and GR simultaneously.
Without those corrections, your phone's map would drift by several kilometers before lunch."

SR: −7.2 μs/day · GR: +45.8 μs/day · Net: +38.6 μs/day · Positional error uncorrected: ~11.4 km/day. All values sourced from Ashby (2002) and Wikipedia GPS error analysis.