Geodetic Precession (Gravity Probe B)

Equivalent forms

magnitude

|\Omega_{\text{geo}}| = \frac{3}{2}\frac{GM}{c^2 r}\,\omega_{orb}

gpb value

\Omega_{\text{geo}} \approx 6.6\ \text{arcsec/yr (GP-B)}

A free gyroscope, touched by nothing, quietly rotates because the geometry it threads is not flat.

Symbol	Name	SI unit	Dimension	Range
$\Omega_{\text{geo}}$	geodetic precession rateoutput	rad/s	T^{-1}	—
$M$	central mass	kg	M	—
$r$	orbital radius	m	L	—
$v$	orbital velocity	m/s	L T^{-1}	—
$c$	speed of light	2.998e8 m/s	L T^{-1}	—

Discovery

Willem de Sitter / Gravity Probe B · 1916

de Sitter predicted in 1916 that the Earth-Moon system's orbital angular momentum would precess in the Sun's field. The pure gyroscopic version waited for Gravity Probe B (launched 2004, results 2011), whose four ultra-precise quartz gyroscopes in polar orbit measured the geodetic effect to 0.3% — a landmark confirmation of spacetime curvature.

Research status: stable

Real-world applications

Gravity Probe B's primary measurement, confirming GR's spatial curvature.
Lunar Laser Ranging detects the Earth-Moon geodetic (de Sitter) precession $of \sim 19$ milliarcsec/yr.
Binary-pulsar spin precession (e.g. PSR B1913+16) is the relativistic geodetic effect in strong fields.

Common misconceptions

'A torque tips the gyroscope.' — No torque acts; the precession is pure geometry (parallel transport).
'Geodetic and frame-dragging are the same.' — Geodetic comes from the mass (curvature); frame-dragging comes from the mass's spin.
'It's a tiny SR effect only.' — Two-thirds of it is genuine spatial curvature, beyond special relativity.

Experimental verification

Gravity Probe B (2011): 6,602 +/- 18 marcsec/yr vs 6,606 predicted; Lunar Laser Ranging confirms the de Sitter term; binary pulsars show geodetic spin precession.

Derivation

Parallel-transport a spin vector along a circular geodesic in the Schwarzschild metric.
The spatial part of the metric is curved; after a full orbit the vector fails to return to its start direction.
The mismatch per orbit is the geodetic angle, giving rate $\Omega = (3/2)(GM/c^2\,\mathrm{r})(v/r) = (3/2)(GM/c^2\,\mathrm{r}) \omega _orb$ .
The factor 3/2 splits into 1/2 from special-relativistic Thomas precession and 1 from spatial curvature.
For Gravity Probe $B (r = R_E + 642\,\mathrm{km})$ this evaluates to 6.6 arcsec/yr.

Limiting cases

Weak field, r large⟶ Omega small,

\sim r^{-5/2}

Nearly flat space precesses negligibly.

Strong field (pulsar)⟶ degrees per yearCompact-binary curvature makes the effect huge and directly observable.

What if…

What if the central body also spins?

Add the much smaller frame-dragging (Lense-Thirring) term — GP-B measured that $\sim 39$ marcsec/yr piece separately.

What if r grows large?

$\Omega \sim 1/r^{5/2}$ (via omega_orb): the effect fades quickly with distance, vanishing in flat space.

1

Gravity Probe B rate

Given · Earth

M = 5.97e24\,\mathrm{kg}

, orbit

r = 7.02e_{6} m

, omega_orb from

a \sim 97 \min

period.

Find · Order of the geodetic precession

Steps

$GM/(c^2\,\mathrm{r}) = 6.674e-11*5.97e24/(9e16*7.02e_{6}) \sim 6.3e-10$ .
$\omega _orb = 2\,\mathrm{pi} / (5820\,\mathrm{s}) \sim 1.08e-3\,\mathrm{rad/s}$ .
$\Omega = 1.5 * 6.3e-10 * 1.08e-3 \sim 1.0e-12\,\mathrm{rad/s} \sim 6.6$ arcsec/yr after unit conversion.

Result ·

\sim 6.6

arcseconds per year — the value Gravity Probe B confirmed.

Frame Dragging (Lense–Thirring Effect)→Schwarzschild Radius→Gravitational Time Dilation→General Relativity: Light Bending & Curved Spacetime→