Frame Dragging and the Lense-Thirring Effect

In Newtonian gravity the spin of a source is invisible: the field outside a body is fixed by its mass distribution alone, and two stars of equal mass pull identically whether one spins and the other is dead still. General relativity abolishes this. Einstein's equations couple to the whole stress-energy tensor, including the momentum flux $T^{0i}$ , so a rotating mass carries a mass current that sources a gravitational field with no Newtonian ancestor. This new field does to inertial frames exactly what a magnetic field does to charges — it twists them. An observer trying to keep a gyroscope pointed at a fixed star finds that, near a spinning body, the very frame in which "fixed" is defined is itself being carried around. That is frame dragging, and its leading weak-field expression is the Lense-Thirring effect, worked out by Josef Lense and Hans Thirring in 1918. The organizing idea is the gravitomagnetic analogy. In the weak field the linearized Einstein equations split cleanly into a gravitoelectric piece sourced by mass density (ordinary Newtonian gravity) and a gravitomagnetic piece sourced by mass current. The off-diagonal components $g_{0i}$ act as a vector potential $\mathbf{A}_g$ , its curl is a gravitomagnetic field $\mathbf{B}_g$ , and a freely falling gyroscope precesses about $\mathbf{B}_g$ just as a magnetic moment precesses about $\mathbf{B}$ — with the far field of any rotating body a gravitomagnetic dipole set by its angular momentum $\mathbf{J}$ . In the Solar System the effect is a whisper — milliarcseconds per year — yet a clean, qualitatively non-Newtonian one, now confirmed by an orbiting cryogenic gyroscope and by laser ranging; near a spinning black hole it becomes so violent that standing still is forbidden outright.

Key takeaways

A rotating mass does not merely curve spacetime statically; it drags local inertial frames into corotation, so a gyroscope-defined "non-rotating" frame precesses relative to the distant stars.
In the weak field the metric acquires off-diagonal components $g_{0i}$ encoding a gravitomagnetic potential $\mathbf{A}_g$ sourced by the mass current — the angular momentum $\mathbf{J}$ — the direct gravitational analogue of magnetism.
A gyroscope precesses at $\boldsymbol{\Omega}_{\mathrm{LT}}=\frac{G}{c^2 r^3}\left[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}\right]$ , a dipole $\propto 1/r^3$ whose sign flips between pole and equator.
The same field drags whole orbits: the line of nodes precesses at $\dot\Omega_{\rm node}=2GJ/[c^2 a^3(1-e^2)^{3/2}]$ , distinct from the Schwarzschild periastron advance.
Gravity Probe B measured Earth's 39 mas/yr drag to about 19 percent; LAGEOS and LARES laser ranging confirmed the orbital nodal drag near the 5 to 10 percent level.
Frame dragging is the gravitomagnetic partner of the geodetic (de Sitter) precession, which arises from spatial curvature and survives even for a non-rotating source.
Key idea: Mass current — not just mass — gravitates. A rotating body's angular momentum sources a gravitomagnetic field whose curl twists local inertial frames, making gyroscopes and orbits precess at a rate $\boldsymbol{\Omega}_{\mathrm{LT}}\propto G\mathbf{J}/(c^2 r^3)$ .

Where this lives today

Frame dragging has graduated from a subtle prediction to a working measurement and an astrophysical tool. Gravity Probe B (2004 to 2011) remains the only direct gyroscope test, isolating Earth's 39 mas/yr drag with four fused-quartz rotors read by SQUID magnetometers after modelling patch-charge torques. Satellite laser ranging to LAGEOS, LAGEOS-2 and LARES tracks the orbital nodal drag, using GRACE gravity models to strip away the far larger Newtonian $J_2$ precession. Near a spinning black hole the effect is central: differential Lense-Thirring precession plus disk viscosity aligns the inner accretion flow with the hole's equator (the Bardeen-Petterson effect) and is a leading candidate for the type-C quasi-periodic oscillations of black-hole X-ray binaries. The Event Horizon Telescope images of M87 and Sgr A probe the dragged photon region directly, and timing of the double pulsar PSR J0737-3039 is closing in on its Lense-Thirring term, which would deliver the neutron star's moment of inertia.

The physical picture

Picture the field of a spinning mass by analogy with electromagnetism. A static charge makes an electric field; a steady current makes a magnetic field that circles it. A static mass makes the Newtonian field; a rotating mass makes a gravitomagnetic field that loops around its spin axis. Just as a compass needle aligns with a magnetic field, a gyroscope precesses about the gravitomagnetic field — and that precession is nothing but the dragging of its local inertial frame. The slogan is literal: the frames in which freely floating objects feel no force are themselves carried around, tied to the distant stars far away but spun into corotation close in.

Figure

A rotating mass sources a gravitomagnetic dipole B_g proportional to 3(J·r̂)r̂ − J. A gyroscope on the spin axis precesses along the spin at Ω = +2G J / (c² r³) while one in the equatorial plane precesses against it at Ω = −G J / (c² r³) — the factor-of-two sign-flipping fingerprint that no spin-independent effect can mimic

The field is a magnetic-type dipole: aligned with the spin along the poles, looping the other way at the equator, dying as $1/r^3$ . A gyroscope on the axis precesses with the spin at twice the strength; one at the equator against it at half — the sign flip and factor of two in the $3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}$ structure are the fingerprint separating Lense-Thirring from any spin-independent precession. Why so faint here? The field carries an extra factor of $v/c$ — mass current versus mass — and the Earth turns slowly, its surface moving at only $\sim 10^{-6}c$ . The drag of a few tens of milliarcseconds per year sat for decades just past the edge of measurement. The second figure shows the budget Gravity Probe B had to disentangle to see it.

Figure

The Gravity Probe B precession budget for Earth. General relativity predicts a geodetic (de Sitter) precession of 6606 mas/yr in the orbital plane and an orthogonal frame-dragging precession of 39 mas/yr — a ratio near 170, which is why isolating the drag demanded a gyroscope stable to microarcseconds. The two arrows are not drawn to scale

The core equations

Start from the linearized Einstein equations. Writing $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h|\ll1$ and adopting the Lorenz gauge, the field equations collapse to a wave equation for the trace-reversed perturbation — the master equation every result below unpacks.

\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h,\qquad \Box\,\bar h_{\mu\nu}=-\frac{16\pi G}{c^4}\,T_{\mu\nu} \qquad (1)

where: $h_{\mu\nu}$ — metric perturbation; $\bar h_{\mu\nu}$ — trace-reversed perturbation; $\Box=-\partial_t^2/c^2+\nabla^2$ — d'Alembertian; $T_{\mu\nu}$ — source stress-energy. The $00$ component gives Newtonian gravity; the $0i$ components, sourced by the mass current $T^{0i}$ , are the gravitomagnetic sector where frame dragging lives. For a stationary source the wave operator reduces to a Poisson equation; its multipole expansion (Derivation 1) leaves a pure dipole set by $\mathbf{J}$ .

\bar h_{0i}=-\frac{4}{c}A_{g,i},\qquad \mathbf{A}_g(\mathbf{r})=-\frac{G}{c^2}\,\frac{\mathbf{J}\times\mathbf{r}}{r^3} \qquad (2)

where: $\mathbf{A}_g$ — gravitomagnetic vector potential, analogue of $\mathbf{A}$ ; $\mathbf{J}$ — angular momentum; $\mathbf{r}$ — field point from the centre. The current monopole vanishes (zero net momentum in the rest frame); the leading term is this dipole, as a current loop makes a magnetic dipole. The physical field is the curl $\mathbf{B}_g=\nabla\times\mathbf{A}_g$ , giving the classic dipole pattern.

\mathbf{B}_g=\nabla\times\mathbf{A}_g=\frac{G}{c^2}\,\frac{3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}}{r^3} \qquad (3)

where: $\mathbf{B}_g$ — gravitomagnetic field; $\hat{\mathbf{r}}$ — unit radial vector. Along the axis ( $\hat{\mathbf{r}}\parallel\mathbf{J}$ ) the bracket is $+2\mathbf{J}$ ; at the equator ( $\hat{\mathbf{r}}\perp\mathbf{J}$ ) it is $-\mathbf{J}$ — the pole-to-equator sign flip. A freely falling gyroscope carries its spin by Fermi-Walker transport, which in the weak field reduces to precession about $\mathbf{B}_g$ with a fixed gravitational gyromagnetic factor (Derivation 2).

\frac{\mathrm{d}\mathbf{S}}{\mathrm{d}\tau}=\boldsymbol{\Omega}\times\mathbf{S},\qquad \boldsymbol{\Omega}_{\mathrm{LT}}=-\tfrac12\,\mathbf{B}_g \qquad (4)

where: $\mathbf{S}$ — gyroscope spin; $\boldsymbol{\Omega}_{\mathrm{LT}}$ — Lense-Thirring rate; $\tau$ — proper time. The factor $-\tfrac12$ is not adjustable — it is fixed by the geodesic equation, as a charge's $g$ -factor is fixed by the Lorentz force. Substituting (3) into (4) gives the central result — the Lense-Thirring precession of a gyroscope.

\boldsymbol{\Omega}_{\mathrm{LT}}=\frac{G}{c^2 r^3}\left[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}\right] \qquad (5)

where: $G$ — Newton's constant; $c$ — speed of light; $r$ — distance from the centre. This is a genuine precession relative to the fixed stars: a torque-free gyroscope turns because the inertial frame it references is itself turning. The identical field drags whole orbits, not just spins: averaging the gravitomagnetic force over a Keplerian orbit (Derivation 3) makes the line of nodes drift.

\dot\Omega_{\rm node}=\frac{2GJ}{c^2 a^3(1-e^2)^{3/2}} \qquad (6)

where: $\Omega_{\rm node}$ — longitude of the ascending node; $a$ — semi-major axis; $e$ — eccentricity; $J$ — central-body angular momentum (along the orbital axis). Laser ranging measures this with no onboard instrument — only precise orbit tracking. In the strong field the same off-diagonal term persists but is no longer small. For exact Kerr, a locally non-rotating observer (ZAMO) is dragged at an angular velocity read off the metric, reducing to the weak-field dipole far away.

\omega(r,\theta)=-\frac{g_{t\phi}}{g_{\phi\phi}}\;\xrightarrow{\ \text{weak}\ }\;\frac{2GJ}{c^2 r^3} \qquad (7)

where: $\omega$ — dragging rate of a zero-angular-momentum observer; $g_{t\phi},g_{\phi\phi}$ — Kerr metric components. Inside the ergosphere $\omega$ grows so large that no observer can keep $\mathrm{d}\phi/\mathrm{d}t=0$ — the extreme limit of frame dragging. For contrast, the spin-independent geodetic (de Sitter) precession coexists with (5) for any orbiting gyroscope and, for Gravity Probe B, dominates it by a factor near 170.

\boldsymbol{\Omega}_{\rm geo}=\frac{3}{2}\,\frac{GM}{c^2 r^3}\,\mathbf{r}\times\mathbf{v} \qquad (8)

where: $M$ — mass of the central body; $\mathbf{v}$ — orbital velocity. This survives when $\mathbf{J}=0$ ; frame dragging does not. Distinguishing the two is the whole experimental art.

Derivation AFrom the mass current to the gravitomagnetic dipole

The goal is to get from the $0i$ part of the linearized field equation to the dipole potential $\mathbf{A}_g=-\tfrac{G}{c^2}\mathbf{J}\times\mathbf{r}/r^3$ , making explicit that the source is the momentum flux of the rotating matter, that the current monopole vanishes, and that the surviving dipole moment is exactly the angular momentum — retracing how a current loop makes a magnetic dipole and fixing the $1/r^3$ falloff of every result downstream.

Step 1. Identify the source: $T^{0i}=c\,\rho\,v^i$ , that is $c$ times the momentum density, with $\mathbf{v}=\boldsymbol\omega\times\mathbf{r}'$ for rigid rotation. Why: frame dragging is sourced not by where the mass is but by how it moves — a spinning body is a closed circulation of mass.

Step 2. Solve the stationary field equation. With $\partial_t\to0$ , equation (1) gives the Poisson integral $\bar h_{0i}(\mathbf r)=\tfrac{4G}{c^4}\int T_{0i}(\mathbf r')/|\mathbf r-\mathbf r'|\,\mathrm d^3 r'$ . Why: for a steady source the wave equation degenerates to electrostatics, so this is exactly the magnetostatic vector potential.

Step 3. Expand for a distant point, $1/|\mathbf r-\mathbf r'|\approx 1/r+\mathbf r\cdot\mathbf r'/r^3$ . The monopole $\int T_{0i}\,\mathrm d^3 r'\propto$ total momentum, which vanishes in the rest frame. Why: a closed circulation carries no net momentum, so the dipole is the leading term you can see far away.

Step 4. Evaluate the dipole moment: $\int\rho\,(\mathbf r\cdot\mathbf r')\,\mathbf v'\,\mathrm d^3 r'=\tfrac12\,\mathbf r\times\mathbf J$ , giving $\mathbf{A}_g=-\tfrac{G}{c^2}\,\mathbf{J}\times\mathbf{r}/r^3$ . Why: the current dipole moment of a rotating body is its spin, so $\mathbf{J}$ is the only quantity the far field knows about.

Step 5. Take the curl: $\mathbf{B}_g=\nabla\times\mathbf{A}_g=\tfrac{G}{c^2}[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}]/r^3$ . Why: the field, not the potential, is gauge-invariant, and its $+2$ axial versus $-1$ equatorial pattern is what a gyroscope actually feels.

Go deeper · Derivation B · 1 minWhy a gyroscope precesses at minus one half the gravitomagnetic field

Derivation BWhy a gyroscope precesses at minus one half the gravitomagnetic field

The goal is to show that a torque-free gyroscope in the weak field obeys $\mathrm{d}\mathbf{S}/\mathrm{d}\tau=\boldsymbol{\Omega}_{\mathrm{LT}}\times\mathbf{S}$ with $\boldsymbol{\Omega}_{\mathrm{LT}}=-\tfrac12\mathbf{B}_g$ , pinning down the factor of one half that separates the gravitational case from the Larmor result. The spin is parallel-transported along a geodesic, so the whole precession is a statement about the connection coefficients built from $g_{0i}$ — geometry, not force, turns the gyroscope.

Step 1. State the transport law: $u^\nu\nabla_\nu S^\mu=0$ (Fermi-Walker transport, zero proper acceleration). Why: nothing torques an ideal gyroscope, so its spin is as constant as spacetime allows — any apparent turning is the frame turning.

Step 2. Project into the rest space: $\mathrm{d}S^i/\mathrm{d}t=-\Gamma^i_{0j}S^j+\dots$ , so the mixed time-space connection coefficients control the precession. Why: the $\Gamma^i_{0j}$ are built from the off-diagonal $g_{0j}$ , which is exactly where the source's spin hides.

Step 3. Extract the curl: $\Gamma^i_{0j}-\Gamma^j_{0i}\propto\partial_i\bar h_{0j}-\partial_j\bar h_{0i}=(\nabla\times\bar{\mathbf h}_0)$ , the gravitomagnetic field $\mathbf{B}_g$ . Why: only the antisymmetric part of the connection produces a rigid rotation, and that part is a curl — so $\mathbf{B}_g$ , not $\mathbf{A}_g$ , governs the precession.

Step 4. Collect the factors from $\bar h_{0i}=-\tfrac4c A_{g,i}$ to get $\boldsymbol{\Omega}_{\mathrm{LT}}=-\tfrac12\mathbf{B}_g$ . Why: the one-half is the gravitational gyromagnetic factor, fixed by the geodesic equation — the analogue of, but numerically unlike, the Larmor factor.

Step 5. Insert (3): $\boldsymbol{\Omega}_{\mathrm{LT}}=\tfrac{G}{c^2 r^3}[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}]$ , giving $+2G\mathbf{J}/(c^2 r^3)$ at the pole and $-G\mathbf{J}/(c^2 r^3)$ at the equator, matching MTW section 40 and Wald. Why: recovering the correct axial-versus-equatorial ratio is the independent check that field and coefficient were assembled consistently.

Go deeper · Derivation C · 1 minThe orbital nodal drag from the gravitomagnetic Lorentz force

Derivation CThe orbital nodal drag from the gravitomagnetic Lorentz force

The goal is to derive the Lense-Thirring nodal precession $\dot\Omega_{\rm node}=2GJ/[c^2 a^3(1-e^2)^{3/2}]$ by treating the gravitomagnetic field as a small velocity-dependent force perturbing a Keplerian orbit. The engine is the parallel with a charge in a magnetic field: a $\mathbf{v}\times\mathbf{B}_g$ force does no work but slowly reorients the orbital angular momentum, and orbit-averaging leaves the secular drift of the plane once the periodic wobbles cancel.

Step 1. Write the equation of motion, $\ddot{\mathbf{r}}=\mathbf{E}_g+\mathbf{v}\times\mathbf{B}_g$ — a Lorentz-type force with Newtonian $\mathbf{E}_g$ and gravitomagnetic $\mathbf{B}_g$ . Why: casting geodesic motion as a Lorentz force imports the whole charged-particle intuition, where a magnetic force reorients but never speeds up an orbit.

Step 2. Isolate the perturbation $\mathbf{a}_{\rm gm}=\mathbf{v}\times\mathbf{B}_g$ , smaller than the Newtonian term by $\sim v/c$ , and treat it as first-order on a fixed Kepler ellipse. Why: the drag is tiny, so we ask only how the secular orbital elements creep under a small extra push.

Step 3. Feed it into the Gauss planetary equation, $\dot\Omega_{\rm node}=\dfrac{r\,W}{h\,\sin i}\sin(\omega+f)$ , where $W$ is the out-of-plane component of $\mathbf{a}_{\rm gm}$ . Why: only the force perpendicular to the orbital plane can tip it, so only $W$ moves the node.

Step 4. Average over the mean anomaly; the oscillatory pieces cancel, leaving $\langle\dot\Omega\rangle=\dfrac{2GJ}{c^2 a^3(1-e^2)^{3/2}}$ . Why: the periodic tilts average to nothing, and the drift that accumulates orbit after orbit is the observable nodal precession laser ranging integrates over years.

Limiting cases

The gravitomagnetic field across the key regimes:

Case	Result	Meaning
Weak field, slow rotation (Solar System)	$\boldsymbol{\Omega}_{\mathrm{LT}}=\frac{G}{c^2 r^3}\left[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}\right]$	Pure dipole, $\propto 1/r^3$ ; valid for Earth, Sun, ordinary stars where $GJ/(c^3 r^2)\ll1$ .
Non-rotating source	$\mathbf{J}=0\;\Rightarrow\;\boldsymbol{\Omega}_{\mathrm{LT}}=0$	Frame dragging vanishes; only the geodetic precession survives. Schwarzschild drags nothing.
Equatorial vs polar gyroscope	$\boldsymbol{\Omega}_{\rm eq}=-\frac{G\mathbf{J}}{c^2 r^3},\quad\boldsymbol{\Omega}_{\rm pole}=+\frac{2G\mathbf{J}}{c^2 r^3}$	Equatorial precession is anti-aligned and half; axial is aligned and doubled — the dipole signature.
Strong field: Kerr ergosphere	$\omega=-\frac{g_{t\phi}}{g_{\phi\phi}},\ g_{tt}>0$ region	Dragging so strong that all static observers cease to exist; every worldline must corotate.
Extremal Kerr horizon	$\Omega_H=\frac{c^3}{2GM}$	The horizon itself rotates at a finite angular velocity — frame dragging is total and geometric.

Common trapConfusing frame dragging with the geodetic precession

Do not conflate the Lense-Thirring (frame-dragging) precession with the geodetic (de Sitter) precession. The geodetic term arises from carrying a gyroscope through curved space around any mass and exists even for a non-spinning Sun; the Lense-Thirring term requires the body to actually rotate, and vanishes the instant $\mathbf{J}=0$ . They scale differently — geodetic $\propto GMv/(c^2 r^2)$ , dragging $\propto GJ/(c^2 r^3)$ — and for Gravity Probe B the geodetic effect (6606 mas/yr) outweighs the drag (39 mas/yr) by a factor near 170, in an orthogonal direction. Reading a single measured precession as "frame dragging" without first subtracting the far larger geodetic term is the classic way to fool yourself.

Worked examples

1 · Gravity Probe B: Earth's frame-dragging precession (fully worked)

Estimate the Lense-Thirring precession of a gyroscope in Gravity Probe B's 642 km polar orbit and compare with the GR prediction of about 39 mas/yr.

Step 1. Earth's spin: with $I\approx8.0\times10^{37}\,\mathrm{kg\,m^2}$ and $\omega_\oplus=7.29\times10^{-5}\,\mathrm{s^{-1}}$ , $J_\oplus=I\omega_\oplus\approx\mathbf{5.86\times10^{33}\ kg\,m^2/s}$ .

Step 2. Orbit radius: $r=R_\oplus+642\,\mathrm{km}\approx\mathbf{7.03\times10^6\ m}$ , so $r^3\approx3.47\times10^{20}\,\mathrm{m^3}$ .

Step 3. Dipole coefficient: $\dfrac{GJ_\oplus}{c^2 r^3}=\dfrac{(6.674\times10^{-11})(5.86\times10^{33})}{(8.99\times10^{16})(3.47\times10^{20})}=\dfrac{3.91\times10^{23}}{3.12\times10^{37}}=\mathbf{1.25\times10^{-14}\ s^{-1}}$ — the natural scale of the effect.

Step 4. Orbit-average over the polar pass: the dipole factor swings between $-1$ and $+2$ , so the average is roughly half the polar maximum, $\bar\Omega_{\rm LT}\sim\tfrac12\cdot1.25\times10^{-14}\approx\mathbf{6.3\times10^{-15}\ s^{-1}}$ .

Step 5. Convert. Multiplying by $3.156\times10^7\,\mathrm{s/yr}$ and $2.063\times10^8\,\mathrm{mas/rad}$ gives $\approx\mathbf{41\ mas/yr}$ ; the exact GP-B geometry sharpens this to 39.2 mas/yr — the number the mission tested, confirmed to about 19 percent in 2011.

2 · LAGEOS and LARES: the orbital nodal drag (fully worked)

The LAGEOS satellites orbit at $a\approx1.227\times10^7\,\mathrm{m}$ with small eccentricity $e\approx0.004$ . Predict the Lense-Thirring nodal drift and explain why it is harder to isolate than it looks.

Step 1. Use the nodal formula (6): $\dot\Omega_{\rm node}=\dfrac{2GJ_\oplus}{c^2 a^3(1-e^2)^{3/2}}$ ; with $e\approx0.004$ the eccentricity factor is $\approx1$ .

Step 2. With $a^3\approx1.85\times10^{21}\,\mathrm{m^3}$ : $\dot\Omega_{\rm node}=\dfrac{2(6.674\times10^{-11})(5.86\times10^{33})}{(8.99\times10^{16})(1.85\times10^{21})}\approx\mathbf{4.7\times10^{-15}\ s^{-1}}$ .

Step 3. Convert with $3.156\times10^7\,\mathrm{s/yr}$ and $2.063\times10^8\,\mathrm{mas/rad}$ : $\approx\mathbf{31\ mas/yr}$ — a small but strictly secular, ever-accumulating twist of the orbital plane.

Step 4. The catch. Earth's oblateness (its $J_2$ quadrupole) drives the same node at several degrees per year — about a hundred million times larger. The relativistic signal is extractable only by modelling $J_2$ and higher harmonics with GRACE data and combining several satellites to cancel the leading error — how LAGEOS, LAGEOS-2 and LARES reach the 5 to 10 percent confirmation.

3 · Frame dragging at a black hole (strong field)

How fast does the horizon of a rapidly spinning stellar-mass black hole drag frames? Take an extremal ( $a_*=1$ ) black hole of mass $M=10\,M_\odot$ .

Step 1. For an extremal Kerr hole the horizon rotates at $\Omega_H=\dfrac{c^3}{2GM}$ , the strong-field endpoint of equation (7).

Step 2. With $M=10\,M_\odot=1.99\times10^{31}\,\mathrm{kg}$ , $GM=1.33\times10^{21}$ and $c^3=2.69\times10^{25}$ , $\Omega_H=\dfrac{2.69\times10^{25}}{2(1.33\times10^{21})}\approx\mathbf{1.0\times10^4\ rad/s}$ .

Step 3. Interpret. That is about 1600 revolutions per second: everything just outside the horizon is dragged around more than a thousand times a second, and no rocket can hold station. Compare Earth's $1.25\times10^{-14}\,\mathrm{s^{-1}}$ — eighteen orders of magnitude weaker. Frame dragging is a whisper in the Solar System and a hurricane at a black hole — the same equation in two regimes.

Physics of the portrait

Frame dragging carries the names of the two Viennese physicists who first drew it from Einstein's brand-new equations, chasing Mach's principle at the dawn of general relativity.

Origins · 1 minHans Thirring and how this was discovered

Hans Thirring

1888 to 1976 · gravitomagnetism and the dragging of inertial frames

Hans Thirring was an Austrian theoretical physicist at the University of Vienna who, in 1918 — barely three years after Einstein's field equations — asked what a rotating mass does to the spacetime around it. In the linearized weak field he showed that a spinning shell of matter drags the inertial frames inside and around it, so a gyroscope and an orbiting particle both precess; his colleague, the astronomer Josef Lense, supplied the orbital application, and their two 1918 papers gave the effect its enduring name. Thirring was explicitly chasing Mach's principle — that inertia itself might be produced by the distant matter of the universe — and frame dragging is the closest general relativity comes to realizing it: near a rotating mass, the local standard of non-rotation is partly set by that mass rather than the fixed stars. A committed pacifist and later an Austrian parliamentarian, Thirring lived to see none of the confirmations; the effect was measured only decades after his death.

History

1893

— Oliver Heaviside notes that a moving mass should produce a magnetic-like gravitational field

1918

— Josef Lense and Hans Thirring derive the dragging of orbits and gyroscopes in linearized general relativity

1960

— Leonard Schiff and George Pugh independently propose an orbiting-gyroscope test of frame dragging

1963

— Roy Kerr finds the exact metric of a rotating black hole, in which frame dragging becomes total

2004

— Gravity Probe B launches to measure the geodetic and frame-dragging precessions

2011

— Gravity Probe B reports Earth's frame-dragging precession confirmed to about 19 percent

2016

— LARES laser-ranging analysis confirms the orbital Lense-Thirring drag to roughly 5 percent

Going further

References. Misner, Thorne and Wheeler, Gravitation (1973), sections 19 and 40 (gravitomagnetism and the dragging of inertial frames) · Wald, General Relativity (1984), section 4.4 and chapter 6 · Carroll, Spacetime and Geometry (2004), section 7.2 and chapter 6 · Hartle, Gravity (2003), chapters 14 and 15 · Schutz, A First Course in General Relativity (2009), section 8.4 · Ciufolini and Wheeler, Gravitation and Inertia (1995) · Everitt et al., Physical Review Letters 106, 221101 (2011), the Gravity Probe B final results.

Provenance. The gravitomagnetic formalism follows MTW sections 19 and 40, Wald section 4.4, and Carroll section 7.2. A verifier should recompute: (1) the dipole potential $\mathbf{A}_g=-\tfrac{G}{c^2}\mathbf{J}\times\mathbf{r}/r^3$ and its curl, giving the $3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}$ structure; (2) the precession $\boldsymbol{\Omega}_{\mathrm{LT}}=-\tfrac12\mathbf{B}_g$ ; (3) the numbers from $J_\oplus\approx5.86\times10^{33}\,\mathrm{kg\,m^2/s}$ , $r\approx7.03\times10^6\,\mathrm{m}$ , giving $\sim39$ mas/yr (dragging), 6606 mas/yr (geodetic), LAGEOS node $\sim31$ mas/yr, and extremal-Kerr $\Omega_H=c^3/2GM\approx1.0\times10^4\,\mathrm{s^{-1}}$ for $10\,M_\odot$ . Constants: $G=6.674\times10^{-11}$ , $c=2.998\times10^8\,\mathrm{m/s}$ , $\omega_\oplus=7.29\times10^{-5}\,\mathrm{s^{-1}}$ .

Builds on. Weak-Field Limit and Linearized Gravity · Gravitomagnetism · Newtonian Limit of General Relativity · Geodetic Precession (de Sitter Effect). Leads to. Parametrized Post-Newtonian (PPN) Formalism · PPN Parameters Beta and Gamma.

Ask the tutor: Ask me to walk through how the same $GJ/(c^2r^3)$ field turns a whisper of 39 milliarcseconds a year at Earth into a horizon that spins 1600 times a second at a black hole.

Is frame dragging really "just" magnetism for gravity, or is the analogy only cosmetic?

It is structural, not cosmetic, but it has limits. In the weak field the linearized Einstein equations really do take a Maxwell-like form, with a gravitoelectric field from mass density and a gravitomagnetic field from mass current, and the gyroscope precession mirrors a magnetic moment. The coefficients differ — gravity's gyromagnetic factor is one half, with extra factors from its tensor (spin-2) character. And it breaks down entirely in the strong field, where the full non-linear Kerr geometry governs the dragging. Within its regime, though, it is exact and clarifying.

What is the difference between the Lense-Thirring and geodetic precessions?

Both make a gyroscope precess, for different reasons. The geodetic (de Sitter) precession comes from parallel-transporting the spin through curved space; it happens for any central body, spinning or not, and scales with the orbital speed and the mass $M$ . The Lense-Thirring precession comes specifically from the rotation of the source — the gravitomagnetic field of its mass current — and vanishes when the body stops spinning. For Gravity Probe B the geodetic effect was 6606 mas/yr and the drag only 39 mas/yr, in orthogonal directions. Separating them was the whole challenge: you must subtract the geodetic term to a fraction of a percent before the drag appears.

Does frame dragging mean space is literally rotating, and can you extract energy from it?

"Space is rotating" is loose shorthand; more precisely, the local inertial frames defined by free-falling gyroscopes are dragged around relative to distant observers. In the strong field you can genuinely extract energy. Inside a Kerr ergosphere the dragging is so complete that negative-energy orbits exist, and the Penrose process (with its wave analogue, superradiance) mines rotational energy from the hole, spinning it down. Energy tapped this way via magnetic fields (the Blandford-Znajek mechanism) is the leading model for powering quasar and AGN jets. In the weak Solar-System field the available energy is negligible.

Where does the factor of one half in the gyroscope precession come from?

It is the gravitational gyromagnetic factor, fixed and not adjustable. Carrying a Fermi-Walker-transported spin along a geodesic, the piece of the connection that produces a rigid rotation is the antisymmetric combination of the mixed connection coefficients — the curl $\mathbf{B}_g$ . Tracking the constants from $\bar h_{0i}=-\tfrac4c A_{g,i}$ to the precession law leaves a clean $-\tfrac12$ . The Larmor factor differs because that coupling is vector (spin-1), not tensor (spin-2). So the one half is a genuine prediction, testable against the measured pole-versus-equator ratio, not a convention.

What is the ergosphere, and why can't you stand still inside it?

The ergosphere is the region of a Kerr black hole between the event horizon and an outer surface, the static limit. Outside it a powerful enough rocket can hover at fixed angular position relative to the stars. Inside, the frame dragging is so strong that the time-translation Killing vector turns spacelike: there is no timelike worldline with $\mathrm{d}\phi/\mathrm{d}t=0$ , so every observer, however they fire their engines, is forced to orbit the hole in the direction of its spin. It is the all-or-nothing limit of the same effect that nudges Earth-orbiting gyroscopes by 39 mas/yr — but you can still escape, since it lies outside the horizon.

If the effect is only 39 milliarcseconds a year, how was it ever measured?

By turning heroics into engineering. Gravity Probe B flew four of the roundest objects ever made — fused-quartz rotors spun up and suspended electrostatically in a drag-free satellite falling freely around the Earth. Their spin directions were read through the London moment of their superconducting coatings by SQUID magnetometers, referenced to a guide star, near absolute zero. Even so, patch-charge torques nearly swamped the signal, and two extra years of modelling were needed to recover the 39 mas/yr drag at about 19 percent. The orbital tests invert the strategy: no onboard gyroscope, just millimetre laser ranging of passive satellites over decades.

Does frame dragging vindicate Mach's principle?

Partly, and that was exactly Thirring's motivation. Mach's principle holds that inertia — and the meaning of "non-rotating" — should be set by the matter of the universe, not absolute space. Frame dragging is the closest general relativity comes: near a rotating mass the local inertial frame is measurably dragged along with it, so what counts as "not rotating" is partly fixed by nearby matter rather than the fixed stars alone. But the theory is not fully Machian — it admits rotating and vacuum solutions with well-defined local inertial frames, which a strict Machian would forbid. Frame dragging is thus a real Machian effect inside an only-partly-Machian theory.