Relativity for space systems

Draft

Relativity enters spaceflight in exactly one place with operational force: clocks. A satellite navigation system is a distributed clock comparison, and at the precision those systems need, clocks at different altitudes and speeds genuinely disagree. Everything else on this page is smaller, though not all of it is negligible.

Two effects, opposite signs

A clock in orbit is subject to two corrections relative to a clock on the ground, and they pull in opposite directions.

Velocity, from special relativity. A moving clock runs slow. To first order in $v^2/c^2$ , the fractional rate offset is

\left(\frac{\Delta f}{f}\right)_{\text{SR}} = -\frac{v^2}{2c^2}

Gravitational potential, from general relativity. A clock deeper in a gravitational well runs slow. For a weak field,

\left(\frac{\Delta f}{f}\right)_{\text{GR}} = \frac{\Delta\Phi}{c^2} = \frac{\mu}{c^2}\left(\frac{1}{r_{\text{ref}}} - \frac{1}{r_{\text{sat}}}\right)

Higher altitude means a shallower well and a faster clock. Higher altitude also means lower orbital speed, and therefore less velocity slowing. Above a certain altitude the gravitational term wins outright.

The GNSS correction, worked

Take a GPS satellite: semi-major axis $a = 26\,560$ km, essentially circular.

Orbital speed.

v = \sqrt{\frac{\mu}{a}} = \sqrt{\frac{398\,600.44}{26\,560}} = 3.874\ \text{km/s}

Special relativistic term.

-\frac{v^2}{2c^2} = -\frac{(3874)^2}{2(2.998\times10^8)^2} = -8.35\times10^{-11}

Over one day of 86 400 s, that is $-7.2\ \mu\text{s}$ . The satellite clock loses 7.2 microseconds per day from its motion.

General relativistic term. Referencing to Earth’s surface at $R_\oplus = 6371$ km:

\frac{\mu}{c^2}\left(\frac{1}{6.371\times10^6} - \frac{1}{2.656\times10^7}\right) = \frac{3.986\times10^{14} \times 1.193\times10^{-7}}{8.988\times10^{16}} = 5.29\times10^{-10}

Over one day, $+45.7\ \mu\text{s}$ . The satellite clock gains 45.7 microseconds per day from being higher in the well.

Net. The gravitational term is more than six times the velocity term:

Net GPS clock rate offset

\Delta t_{\text{net}} \approx +45.7 - 7.2 = +38.5\ \mu\text{s/day}

The value quoted in the literature is 38.6 µs/day. The small difference is the rotation of the ground reference clock, which the calculation above omits by referencing to a non-rotating point on Earth’s surface; the standard treatment references to the rotating geoid, where the centrifugal potential and the surface velocity both contribute.

What that number costs

Light travels 299.8 m in a microsecond. An uncorrected 38.6 µs/day offset becomes a ranging error accumulating at roughly 11.6 km per day. A GPS receiver would be unusable within minutes and absurd within hours.

The correction is applied in hardware, before launch. GPS satellite clocks are built to a nominal 10.23 MHz but are deliberately offset to 10.229 999 995 43 MHz, a fractional shift of $-4.4647 \times 10^{-10}$ , so that once in orbit they tick at 10.23 MHz as observed from the ground. The residual eccentricity-driven variation is corrected in the receiver’s navigation algorithm.

The altitude where the effects cancel

Setting the two terms equal is a clean piece of algebra with a memorable answer. For a circular orbit $v^2 = \mu/r$ , so the total fractional rate offset relative to a clock at infinity is

-\frac{v^2}{2c^2} - \frac{\mu}{rc^2} = -\frac{3\mu}{2rc^2}

while a static clock at the surface offsets by $-\mu/(R_\oplus c^2)$ . Equating:

\frac{3\mu}{2r} = \frac{\mu}{R_\oplus} \quad\Longrightarrow\quad r = \frac{3}{2}R_\oplus

That is $r = 9557$ km, an altitude of about 3186 km. Below it, orbiting clocks run slow relative to the ground; above it, fast. Every operational GNSS constellation sits well above this crossover, which is why all of them gain time.

Magnitudes across the regimes

System	Altitude	SR term	GR term	Net
ISS	400 km	−28.3 µs/day	+3.6 µs/day	−24.7 µs/day (slow)
Crossover	3186 km	−20.0 µs/day	+20.0 µs/day	0
GPS (MEO)	20 180 km	−7.2 µs/day	+45.7 µs/day	+38.5 µs/day (fast)
Geostationary	35 786 km	−4.5 µs/day	+51.1 µs/day	+46.5 µs/day (fast)

The ISS figure is the one people find counterintuitive: astronauts on the station age very slightly more slowly than people on the ground, by about 0.01 seconds per year, because at that altitude velocity still dominates.

Shapiro delay

A signal passing near a mass takes longer than the straight-line light time would suggest, because coordinate time runs differently in the curved region. For a signal grazing a body of mass parameter $\mu$ at impact parameter $b$ , the one-way excess is

\Delta t = \frac{2\mu}{c^3}\ln\left(\frac{4 r_1 r_2}{b^2}\right)

For a spacecraft at superior conjunction, with the radio path grazing the solar limb, the round-trip excess reaches approximately 250 microseconds, or about 75 km of apparent range. Deep-space navigation cannot ignore this: it is included in the standard light-time model used by the Deep Space Network.

The Cassini measurement of this delay in 2002 constrained the post-Newtonian parameter $\gamma$ to $1 + (2.1 \pm 2.3) \times 10^{-5}$ , which remains among the tightest tests of general relativity in the solar system.

Perihelion precession

The relativistic correction to the inverse-square law adds a small term that breaks the conservation of the eccentricity vector, so the apsidal line precesses:

\Delta\varpi = \frac{6\pi\mu}{c^2 a(1-e^2)} \ \text{per orbit}

For Mercury this gives 43 arcseconds per century, the residual that Newtonian perturbation theory could not account for and that general relativity explained in 1915. For an Earth satellite the same expression gives a few arcseconds per year, which is below the noise for most missions but is included in precise orbit determination for geodetic satellites such as LAGEOS.

Frame dragging

A rotating mass drags the local inertial frames around with it. The Lense-Thirring precession for a satellite orbit is tiny but has been measured.

Gravity Probe B geodetic: Predicted 6606.1 mas/yr; measured 6601.8 ± 18.3 mas/yr
Gravity Probe B frame dragging: Predicted 39.2 mas/yr; measured 37.2 ± 7.2 mas/yr
LAGEOS/LARES node precession: Predicted ~31 mas/yr; measured to a few per cent
Engineering relevance: None for mission design. Relevant only to geodesy and fundamental physics tests.

What to include, and when

Effect	Include when	Ignore when
Clock rate offsets	Any timing or navigation system at nanosecond precision	Attitude control, thermal, power
Shapiro delay	Deep-space ranging, especially near conjunction	Earth-orbiting missions
Relativistic acceleration	Precise orbit determination, geodetic satellites	Manoeuvre planning, coarse propagation
Frame dragging	Fundamental physics experiments	Everything operational
Sagnac correction	Any Earth-fixed frame at nanosecond timing, including GNSS receivers	Inertial-frame work

The Sagnac term deserves a note, because it is routinely confused with the effects above. It is a consequence of describing signal propagation in a rotating frame rather than a relativistic effect on the clocks themselves, but for a GNSS receiver it reaches tens of nanoseconds, corresponding to tens of metres, and must be applied.

References

Ashby, N. “Relativity in the Global Positioning System”, Living Reviews in Relativity, 6:1, 2003. The definitive treatment of the GNSS case.
Will, C. M. “The Confrontation between General Relativity and Experiment”, Living Reviews in Relativity, 17:4, 2014.
Bertotti, B., Iess, L. and Tortora, P. “A test of general relativity using radio links with the Cassini spacecraft”, Nature, 425:374, 2003.
Everitt, C. W. F. et al. “Gravity Probe B: Final Results of a Space Experiment to Test General Relativity”, Physical Review Letters, 106:221101, 2011.
Petit, G. and Luzum, B. (eds.) IERS Conventions (2010), IERS Technical Note 36. The applied relativistic models used in precise navigation.