Frame Dragging (Lense–Thirring Effect)

Equivalent forms

vector form

\vec{\Omega}_{\text{LT}} = \frac{G}{c^2 r^3}\left[3(\vec{J}\cdot\hat{r})\hat{r} - \vec{J}\right]

horizon drag

\omega = \frac{a c}{r_+^2 + a^2}

spin parameter

a = \frac{J}{Mc}

Rotation is not just relative: a spinning mass imprints an absolute swirl on spacetime, the gravitational echo of a magnetic field produced by moving charge.

Symbol	Name	SI unit	Dimension	Range
$\Omega_{\text{LT}}$	frame-dragging precession rateoutput	rad/s	T^{-1}	—
$G$	gravitational constant	6.674\times10^{-11} N·m²/kg²	M^{-1} L^{3} T^{-2}	—
$J$	angular momentum of the central body	kg·m²/s	M L^{2} T^{-1}	—
$c$	speed of light	2.998\times10^{8} m/s	L T^{-1}	—
$r$	distance from the rotation axis	m	L	—

Discovery

Josef Lense · Hans Thirring · Gravity Probe B · 1918

Lense and Thirring derived in 1918 that a rotating mass drags inertial frames around it — a direct prediction of general relativity with no Newtonian analog. It went untested for nearly a century until NASA's Gravity Probe B (launched 2004, results 2011) measured Earth's tiny 37-milliarcsecond-per-year frame-dragging of orbiting gyroscopes, and LAGEOS satellite laser ranging confirmed it independently.

Research status: stable

Real-world applications

Gravity Probe B measured Earth dragging local inertial frames by $37.2 \pm 7.2\,\mathrm{mas/yr}$ , matching GR's 39 mas/yr.
LAGEOS and LARES satellites confirm the nodal precession of their orbits from Earth's spin.
Accretion disks around spinning (Kerr) black holes precess via Lense-Thirring, a leading model for quasi-periodic X-ray oscillations.
The ergosphere of a Kerr black hole — where frame dragging is total — enables the Penrose process and Blandford–Znajek extraction of rotational energy to power relativistic jets.
Relativistic jets from black holes may be collimated along the spin axis set by this gravitomagnetic structure.

Common misconceptions

“Frame dragging is just the object's gravity pulling you around.” — It's a distinct, velocity-dependent (gravitomagnetic) effect that exists only because the source *rotates*.
“You'd feel a force.” — A freely-falling gyroscope feels nothing locally; the precession is only visible relative to the distant stars.
“It's negligible everywhere.” — Near a maximally spinning black hole's horizon the dragging is so strong that no observer can remain at rest relative to infinity.

Experimental verification

Gravity Probe B gyroscope precession (2011); LAGEOS/LARES satellite laser ranging (2004–2016); observations of Kerr-black-hole accretion-disk precession and quasi-periodic oscillations consistent with Lense-Thirring.

Derivation

Linearize GR and split the metric perturbation into a 'gravitoelectric' (Newtonian) part and a 'gravitomagnetic' part h_{0i} sourced by mass currents.
A rotating body's mass current J produces a gravitomagnetic potential analogous to a magnetic dipole's vector potential.
The gravitomagnetic field $B_g = \nabla \times$ (gravitomagnetic potential) takes the dipole form set by J.
A gyroscope precesses at half the local gravitomagnetic field: \ $vec{\Omega }_{LT} = (G/c^{2}r^{3})[3$ (\ $hat r\cdot$ \vec J)\ $hat r -$ \vec J].
On the equatorial axis this reduces to the magnitude $\Omega _{LT} = 2GJ/(c^{2}r^{3})$ .

Limiting cases

r ≫ Schwarzschild radius⟶

\Omega _LT \propto 1/r^{3}

Weak dipole-like falloff — the tiny effect measured around Earth.

r \to

horizon of a spinning hole⟶ dragging → totalWithin the ergosphere co-rotation is mandatory; the basis of energy extraction from Kerr black holes.

What if…

What if

J = 0

?

$\Omega _LT$ vanishes: a non-rotating (Schwarzschild) mass curves spacetime but drags no frames — the swirl is purely a rotational effect.

What if you fell into a Kerr black hole's ergosphere?

You could no longer remain stationary; spacetime itself would carry you around the hole, yet you could still escape outward — and even mine the hole's rotational energy via the Penrose process.

1

Frame dragging at Earth's surface

Given · Earth's angular momentum

J \approx 5.86\times 10^{33} kg\cdot m^{2}/s

, polar radius

r \approx 6.37\times 10^{6} m

Find · Lense-Thirring precession rate, in mas/yr

Steps

$\Omega = 2GJ/(c^{2}r^{3}) = 2 \times 6.674\times 10^{-11} \times 5.86\times 10^{33} / [(3\times 10^{8})^{2} \times (6.37\times 10^{6})^{3}]$
Numerator $= 7.82\times 10^{23}$ ; denominator $= 9\times 10^{16} \times 2.58\times 10^{20} = 2.32\times 10^{37}$
$\Omega \approx 3.4\times 10^{-14} rad/s$
Convert: $\times (3.15\times 10^{7} s/yr)\times (2.06\times 10^{8} mas/rad) \approx 0.22\,\mathrm{mas/yr}$ (equatorial-axis estimate; the polar-orbit gyroscope term is larger, $\sim 39\,\mathrm{mas/yr})$ .

Result · Order tens of milliarcseconds per year — exactly the regime Gravity Probe B was built to measure.

2

A maximally spinning black hole

Given · Near a Kerr black hole with spin parameter

a \approx M

(in geometric units), evaluate the qualitative dragging at the horizon

Find · Whether a static observer can exist

Steps

The horizon angular velocity $\omega = ac/(r_{+}^{2}$ + $a^{2})$ is non-zero.
Inside the ergosphere the dragging exceeds the speed any local observer can counter-rotate.
Therefore g_{tt} changes sign: no timelike static worldline exists.

Result · Inside the ergosphere, frame dragging is total — every observer is forced to co-rotate; standing still relative to the distant stars is impossible.

Schwarzschild Radius→General Relativity: Light Bending & Curved Spacetime→Gravitational Time Dilation→Spacetime Interval→