Gravitomagnetism

Ask what sources gravity and the textbook answer is "mass". But relativity insists that mass and momentum sit together in one object, the stress-energy tensor, and that no source of a field can announce its presence instantaneously. The same logic that turned Coulomb's static charge into Maxwell's full electrodynamics — where moving charge makes a magnetic field — must apply to gravity too. Gravitomagnetism is the working-out of that parallel. Write the metric as $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ with $|h_{\mu\nu}|\ll1$ , impose the Lorenz gauge, and the linearized Einstein equations collapse into wave equations that look line-for-line like the equations for the electromagnetic potentials. The time-time part of the perturbation plays the role of the electrostatic potential; the time-space part plays the role of the vector potential. Mass density sources a gravitoelectric field — ordinary Newtonian gravity — and mass current sources a gravitomagnetic field with no Newtonian counterpart at all. The physical payoff is frame-dragging: a rotating body twists the very definition of "non-rotating" in the spacetime around it. A gyroscope in orbit slowly precesses, an orbital plane is slowly torqued, and a freely falling observer finds that the local inertial frames themselves are dragged along with the source's spin. This is the conceptual engine behind the Lense-Thirring effect, behind Gravity Probe B, and behind the spin-orbit coupling that shapes the gravitational waveforms LIGO and Virgo record. The analogy is a scaffold, not a foundation — gravity is a spin-2 field and electromagnetism spin-1, so stubborn factors of two and a flipped sign survive — but as scaffolding it is one of the most useful pictures in all of gravitation.

Key takeaways

In the weak-field, slow-source limit the linearized Einstein equations in Lorenz gauge become a Maxwell-like pair: mass density $\rho$ sources a gravitoelectric field $\mathbf{E}_g$ , and mass current $\mathbf{j}=\rho\mathbf{v}$ sources a gravitomagnetic field $\mathbf{B}_g$ .
The trace-reversed perturbation splits as $\bar h_{00}=4\Phi/c^2$ and $\bar h_{0i}=-2A_i/c^2$ , defining a gravitoelectric potential $\Phi$ and a gravitomagnetic vector potential $\mathbf{A}$ in exact analogy with the electromagnetic scalar and vector potentials.
A test mass obeys a Lorentz-like law $\dot{\mathbf{v}}=-\mathbf{E}_g-\mathbf{v}\times\mathbf{B}_g$ , and a freely falling gyroscope precesses at $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ — the frame-dragging that is the Lense-Thirring effect.
The analogy is not exact: the graviton's spin-2 character forces the factor $\tfrac12$ in the precession, flips the sign so that parallel mass currents attract, and confines the whole split to linear order in a fixed gauge.
Gravity Probe B measured Earth's frame-dragging at $37.2\pm7.2$ mas/yr against a GR prediction of $\approx39$ mas/yr, and the geodetic term at $6601.8\pm18.3$ mas/yr against a predicted $\approx6606$ mas/yr; LAGEOS/LARES laser ranging sees the orbital node drift of $\approx31$ mas/yr — both confirming the prediction.
Key idea: in Lorenz gauge the linearized Einstein equations split into a Maxwell-like pair — mass density sources a gravitoelectric field $\mathbf{E}_g=-\nabla\Phi$ and mass current sources a gravitomagnetic field $\mathbf{B}_g=\nabla\times\mathbf{A}$ — so a moving mass feels a Lorentz-like force $\mathbf{F}/m=-\mathbf{E}_g-\mathbf{v}\times\mathbf{B}_g$ and a gyroscope precesses at $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ .

The core insight

In Lorenz gauge the linearized Einstein equations split into a Maxwell-like pair: mass density sources a gravitoelectric field $\mathbf{E}_g=-\nabla\Phi$ and mass current density sources a gravitomagnetic field $\mathbf{B}_g=\nabla\times\mathbf{A}$ , so a moving test mass feels a Lorentz-like force $\mathbf{F}/m=-\mathbf{E}_g-\mathbf{v}\times\mathbf{B}_g$ and a freely falling gyroscope precesses at $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ .

Where this lives today

Gravitomagnetism graduated from curiosity to measured physics in the twenty-first century, and it now works quietly inside several frontiers. Gravity Probe B (2004–2011) flew four of the most perfect spheres ever made as gyroscopes in a polar orbit and separated the small gravitomagnetic frame-dragging (measured $37.2\pm7.2\ \mathrm{mas/yr}$ , predicted $\approx39$ ) from the roughly 170-times-larger gravitoelectric geodetic precession (measured $6601.8\pm18.3\ \mathrm{mas/yr}$ , predicted $\approx6606$ ), confirming the Lense-Thirring prediction to roughly $19\%$ . The LAGEOS and LARES satellites attack the same physics from the orbital side: their nodes drift at tens of milliarcseconds per year, read out by picosecond laser ranging against ever-better models of Earth's lumpy gravity field, now pinning the frame-dragging coefficient at the few-percent level. In strong gravity the effect is dominant, not tiny: the Kerr metric's frame-dragging governs the innermost accretion flows around spinning black holes, is invoked in the Bardeen-Petterson alignment of tilted disks and in Blandford-Znajek jet launching, and — via spin-orbit and spin-spin couplings — imprints itself on the gravitational waveforms LIGO, Virgo and KAGRA decode from merging compact binaries. Every precise template bank for a spinning-binary merger is, in part, a gravitomagnetic calculation.

The physical picture

Set two situations side by side. A loop of electric current wraps a magnetic field around itself, dipolar and falling off as $1/r^3$ , and a compass needle placed nearby precesses in that field at the Larmor rate. Now replace the current of charge with a current of mass — a spinning planet, whose rotation carries mass in circles just as the loop carries charge. Linearized general relativity says this mass current wraps a gravitomagnetic field around the spin axis with exactly the same dipolar shape, and a gyroscope placed nearby precesses in it. The two pictures are not merely similar; to linear order in $h_{\mu\nu}$ they are governed by structurally identical equations. The first figure lays them next to each other so the shared skeleton is unmistakable.

Figure

A spinning mass of angular momentum J wraps a dipolar gravitomagnetic field Bg around itself, exactly as a current loop of magnetic moment m wraps a magnetic field B. A test gyroscope precesses at Ω = −½ Bg, the direct analogue of a magnetic dipole precessing at the Larmor rate −(q/2m) B

Two features distinguish gravity from the electromagnetic template, and both trace to spin. First, the compass needle and the gyroscope do not precess at the same rate for a given field: the gyroscope carries an extra factor of one half, $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ versus the Larmor $-(q/2m)\mathbf{B}$ . That $\tfrac12$ is the fingerprint of a spin-2 field and it cannot be scaled away by any choice of units. Second, the force between currents flips sign: parallel electric currents attract but parallel mass currents also attract, so where electromagnetism has like charges repelling, gravity has mass everywhere pulling — the universal attractiveness of gravity, seen through the gravitomagnetic lens. The whole construction, finally, lives in a chosen gauge and is no more Lorentz-invariant than the split of $\mathbf{E}$ and $\mathbf{B}$ is: a boost mixes gravitoelectric into gravitomagnetic. The second figure turns from structure to numbers — the actual precession rates that spaceborne experiments have wrung out of Earth's spin.

Figure

The measured hierarchy. For Gravity Probe B in a 642 km polar orbit general relativity predicts a gravitoelectric geodetic precession near 6606 mas/yr and a far weaker gravitomagnetic frame-dragging near 39 mas/yr — a ratio close to 170 — and GP-B measured 6601.8 ± 18.3 and 37.2 ± 7.2 mas/yr respectively, while LAGEOS satellite laser ranging reads the orbital Lense-Thirring node drift near 31 mas/yr. Bars on a logarithmic scale in milliarcseconds per year

The core equations

The starting point is the linearized Einstein equation in Lorenz gauge. Writing $g_{\mu\nu}=\eta_{\mu\nu}+h_{\mu\nu}$ , forming the trace-reversed perturbation $\bar h_{\mu\nu}=h_{\mu\nu}-\tfrac12\eta_{\mu\nu}h$ , and imposing $\partial^\mu\bar h_{\mu\nu}=0$ , the full field equations reduce to a single sourced wave equation. It is the gravitational twin of $\Box A^\mu=-\mu_0 j^\mu$ .

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

where: $\bar h_{\mu\nu}$ — trace-reversed metric perturbation; $\Box=\eta^{\alpha\beta}\partial_\alpha\partial_\beta$ — flat-space d'Alembertian; $T_{\mu\nu}$ — stress-energy tensor of the source; $G,c$ — Newton's constant and the speed of light. Reading: every component of the perturbation propagates as a wave at speed $c$ , driven by the corresponding component of the source — this is what makes the electromagnetic analogy possible in the first place. For a slow, weakly stressed source ( $T_{ij}\ll T_{00}$ ) only two blocks of the perturbation matter, and they define the two GEM potentials. The identification is a convention choice; the factors are fixed by demanding that the $00$ equation reproduce Newton.

\bar h_{00}=\frac{4\Phi}{c^2},\qquad \bar h_{0i}=-\frac{2A_i}{c^2},\qquad \bar h_{ij}=O(c^{-4}) \qquad (2)

where: $\Phi$ — gravitoelectric (Newtonian) scalar potential; $A_i$ — gravitomagnetic vector potential; the spatial block is higher order for non-relativistic sources. Reading: the time-time part is old Newtonian gravity; the genuinely new physics hides in the time-space part $\bar h_{0i}$ , which has no static Newtonian analogue. Feeding (2) into (1) for a stationary source turns the two wave equations into Poisson equations — one electrostatic, one magnetostatic in form.

\nabla^2\Phi=4\pi G\rho,\qquad \nabla^2\mathbf{A}=\frac{4\pi G}{c^2}\,\mathbf{j} \qquad (3)

where: $\rho$ — mass density, from $T_{00}=\rho c^2$ ; $\mathbf{j}=\rho\mathbf{v}$ — mass current density, from $T_{0i}=-\rho c\,v_i$ . Reading: mass at rest sources $\Phi$ ; mass in motion sources $\mathbf{A}$ — the exact parallel of charge density and charge current in magnetostatics. The fields are built from the potentials just as in electromagnetism, including the induction term that couples a changing vector potential into the gravitoelectric field.

\mathbf{E}_g=-\nabla\Phi-\frac{1}{c}\frac{\partial\mathbf{A}}{\partial t},\qquad \mathbf{B}_g=\nabla\times\mathbf{A} \qquad (4)

where: $\mathbf{E}_g$ — gravitoelectric field (Newtonian gravity plus induction); $\mathbf{B}_g$ — gravitomagnetic field. Reading: $\mathbf{E}_g$ is what a stationary test mass feels as ordinary weight; $\mathbf{B}_g$ acts only on moving masses. Combining the definitions with the source equations and the gauge condition yields the four gravitoelectromagnetic Maxwell equations. Two signs differ from electromagnetism — the source signs — because gravity attracts.

\begin{aligned}&\nabla\cdot\mathbf{E}_g=-4\pi G\rho,&&\nabla\cdot\mathbf{B}_g=0\\[2pt]&\nabla\times\mathbf{E}_g=-\frac{1}{c}\frac{\partial\mathbf{B}_g}{\partial t},&&\nabla\times\mathbf{B}_g=-\frac{4\pi G}{c}\,\mathbf{j}+\frac{1}{c}\frac{\partial\mathbf{E}_g}{\partial t}\end{aligned} \qquad (5)

where: the minus signs on $-4\pi G\rho$ and $-4\pi G\mathbf{j}/c$ encode that the gravitational "charge" (mass) always attracts. Reading: field lines of $\mathbf{E}_g$ converge on mass, and there is no gravitomagnetic monopole, so $\mathbf{B}_g$ is purely dipolar at leading order. A test particle then obeys a Lorentz-like force law, read off from the slow-motion geodesic equation. This is where frame-dragging enters the orbit of a satellite.

\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=-\mathbf{E}_g-\mathbf{v}\times\mathbf{B}_g \qquad (6)

where: $\mathbf{v}$ — velocity of the test particle; the cross term $-\mathbf{v}\times\mathbf{B}_g$ is the gravitomagnetic force that drags orbital planes. Reading: a particle at rest feels only $-\mathbf{E}_g$ (Newton); set it moving and the gravitomagnetic field bends its path sideways. A spinning test gyroscope, carried along a geodesic, precesses relative to the distant stars at half the local gravitomagnetic field — the extra factor being the spin-2 signature.

\boldsymbol{\Omega}_{\text{prec}}=-\frac{1}{2}\,\mathbf{B}_g \qquad (7)

where: $\boldsymbol{\Omega}_{\text{prec}}$ — precession angular velocity of the gyroscope spin; the $\tfrac12$ distinguishes gravity's spin-2 coupling from the electromagnetic Larmor rate. Reading: this is the quantity Gravity Probe B was built to measure directly. Finally, for a compact rotating body of total angular momentum $\mathbf{J}$ , solving the magnetostatic-analogue problem gives a pure dipole field and the corresponding Lense-Thirring precession rate at a field point.

\mathbf{B}_g=\frac{2G}{c^2 r^3}\big[\,3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}\,\big],\qquad \boldsymbol{\Omega}_{\rm LT}=\frac{G}{c^2 r^3}\big[\,3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}\,\big] \qquad (8)

where: $\mathbf{J}$ — total angular momentum of the source; $\hat{\mathbf{r}}$ — unit vector from source to field point; $r$ — distance. Reading: the field falls off as $1/r^3$ with dipole angular structure, and the physical precession $\boldsymbol{\Omega}_{\rm LT}=-\tfrac12\mathbf{B}_g$ is convention-independent even though the split into " $\mathbf{A}$ " and " $\mathbf{B}_g$ " carries author-dependent factors of two.

Derivation AFrom the linearized Einstein equations to the gravitoelectromagnetic Maxwell equations

The goal is to show that the Lorenz-gauge linearized field equation (1), restricted to a slow, weakly stressed source, is not merely analogous to Maxwell's equations but reduces to them term by term, with a definite dictionary between metric components and GEM potentials and two source signs flipped by the attractiveness of gravity. We start from the trace-reversed perturbation, split it into a scalar and a vector potential, take the $00$ and $0i$ components to fix normalizations against the Newtonian limit, and finally assemble the curls and divergences that constitute the four field equations.

Step 1. Begin with $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}/c^4$ in Lorenz gauge $\partial^\mu\bar h_{\mu\nu}=0$ , the exact analogue of the electromagnetic $\Box A_\nu=-\mu_0 j_\nu$ with $\partial^\mu A_\mu=0$ . Why: the gauge condition is what makes the equations decouple into independent wave equations, one per tensor component, so each component can be matched separately to an electromagnetic counterpart.

Step 2. Define potentials by $\bar h_{00}=4\Phi/c^2$ and $\bar h_{0i}=-2A_i/c^2$ , and drop $\bar h_{ij}=O(c^{-4})$ for a source with $T_{ij}\ll T_{00}$ . Why: for a non-relativistic source the momentum flux (pressure and stresses) is smaller than the energy density by $v^2/c^2$ , so only the time-time and time-space blocks survive at leading order — exactly the blocks that become $\Phi$ and $\mathbf{A}$ .

Step 3. Take the $00$ component. With $T_{00}=\rho c^2$ , equation (1) gives $\Box\Phi=4\pi G\rho$ , which for a static source is $\nabla^2\Phi=4\pi G\rho$ . Why: this is the Newtonian Poisson equation, and demanding it come out with the standard $4\pi G$ is precisely what fixes the factor of $4$ in the definition $\bar h_{00}=4\Phi/c^2$ .

Step 4. Take the $0i$ component. With $T_{0i}=-\rho c\,v_i$ so that $j_i=\rho v_i$ , equation (1) gives $\Box A_i=(4\pi G/c^2)j_i$ , the magnetostatic analogue. Why: the time-space part of the source is momentum density, i.e. mass current, so it is inevitable that it sources a vector potential in the same way charge current sources the magnetic vector potential.

Step 5. Define $\mathbf{E}_g=-\nabla\Phi-\tfrac1c\partial_t\mathbf{A}$ and $\mathbf{B}_g=\nabla\times\mathbf{A}$ . Taking divergences and curls of these, and using Steps 3–4 plus the gauge condition, produces $\nabla\cdot\mathbf{E}_g=-4\pi G\rho$ , $\nabla\cdot\mathbf{B}_g=0$ , $\nabla\times\mathbf{E}_g=-\tfrac1c\partial_t\mathbf{B}_g$ , and $\nabla\times\mathbf{B}_g=-\tfrac{4\pi G}{c}\mathbf{j}+\tfrac1c\partial_t\mathbf{E}_g$ . Why: the identity $\nabla\cdot(\nabla\times\mathbf{A})=0$ gives the no-monopole law for free, while the source equations feed the Gauss and Ampere laws — the minus signs are the whole physical content, marking gravity as attractive.

Step 6. Note the convention ambiguity: MTW, Wald and Mashhoon normalize $\mathbf{A}$ and $\mathbf{B}_g$ with relative factors of two, but every measurable rate — the precession $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ — is convention-independent. Why: the fields themselves are gauge- and convention-dependent bookkeeping; only the geodesic and spin-transport observables they combine into are physical, so the factor-of-two disputes never touch a prediction.

Go deeper · Derivation B · 2 minThe gravitomagnetic dipole and the Lense-Thirring precession

Derivation BThe gravitomagnetic dipole and the Lense-Thirring precession

The goal is to compute, from the vector Poisson equation $\nabla^2\mathbf{A}=(4\pi G/c^2)\mathbf{j}$ , the gravitomagnetic field produced far outside a compact rotating body of total angular momentum $\mathbf{J}$ , and to convert that field into the frame-dragging precession of a gyroscope — recovering the classic 1918 Lense-Thirring rate. The strategy is to borrow the magnetostatic solution wholesale, since the equation is identical in form, then apply the spin-transport rule (7), watching the factor of two in the field cancel against the one-half in the precession.

Step 1. In the far field the solution of $\nabla^2\mathbf{A}=(4\pi G/c^2)\mathbf{j}$ for a localized current is the dipole $\mathbf{A}(\mathbf{r})=-(G/c^2)\,(\mathbf{J}\times\mathbf{r})/r^3$ , the magnetostatic dipole potential with $\boldsymbol{\mu}\to-(G/c^2)\mathbf{J}$ . Why: only the total angular momentum $\mathbf{J}=\int\mathbf{r}\times\mathbf{j}\,\mathrm{d}^3x$ survives at large $r$ , exactly as only the magnetic dipole moment survives outside a current distribution — higher multipoles fall off faster.

Step 2. Take the curl: $\mathbf{B}_g=\nabla\times\mathbf{A}=(2G/c^2 r^3)\,[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}]$ , the standard dipole angular pattern. Why: the curl of a dipole vector potential always produces the $3(\hat{\mathbf{r}}\cdot)\hat{\mathbf{r}}-(\,)$ structure; the GEM normalization is what carries the leading factor of two into $\mathbf{B}_g$ .

Step 3. Apply the gyroscope rule $\boldsymbol{\Omega}_{\rm prec}=-\tfrac12\mathbf{B}_g$ , so the factor of two cancels and $\boldsymbol{\Omega}_{\rm LT}=(G/c^2 r^3)[3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}]$ (up to the overall sign fixing the drag direction). Why: the cancellation is not an accident — the same spin-2 factor that puts the $\tfrac12$ in the precession is tied to the field normalization, so the physical rate is clean and convention-free.

Step 4. Evaluate on the spin axis, where $\hat{\mathbf{r}}\parallel\mathbf{J}$ and $3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}=2\mathbf{J}$ , giving the polar magnitude $\Omega_{\rm LT}=2GJ/(c^2 r^3)$ . Why: this is the largest and simplest case, the one the worked example evaluates for Earth, and it shows the field is prograde on the axis — the gyroscope is dragged in the direction the body spins.

Step 5. Cross-check against exact solutions: this dipole is precisely the far-field expansion of the slow-rotation Kerr metric's off-diagonal term $g_{0\phi}\approx-2GJ\sin^2\theta/(c^2 r)$ . Why: agreement with the linearized Kerr metric confirms that GEM is not a separate theory but the weak-field face of the full rotating-body solution, and pins the numerical coefficient beyond doubt. *The polar on-axis rate $2GJ/(c^2r^3)$ is the same number evaluated for Earth in the worked example below, and the dipole field agrees with the leading term of the slow-rotation Kerr $g_{0\phi}$ expansion, confirming that GEM is the weak-field face of the exact rotating-body solution.*

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

Derivation CWhy the gyroscope precesses at minus one half the gravitomagnetic field

The goal is to trace the origin of the stubborn factor of one half in $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ — the single number that most sharply separates gravity's spin-2 coupling from the spin-1 Larmor precession of a magnetic dipole — by following the parallel transport of a spin vector through the linearized connection. The route is to write Fermi-Walker transport for a gyroscope on a slow geodesic, extract the connection coefficients built from $\bar h_{0i}$ , and identify the antisymmetric combination that becomes the curl $\mathbf{B}_g$ .

Step 1. A freely falling gyroscope Fermi-Walker transports its spin: $\mathrm{d}S^i/\mathrm{d}\tau=-\Gamma^i_{0j}S^j+\dots$ , where the leading connection coefficients for a slow gyroscope involve spatial derivatives of $\bar h_{0i}$ . Why: parallel transport is how "keeping the axis fixed" is defined in curved spacetime, so any twisting of the local frames shows up as a nonzero $\Gamma^i_{0j}$ rotating the spin.

Step 2. Compute the relevant coefficient. To linear order $\Gamma^i_{0j}\sim\tfrac12(\partial_j A_i-\partial_i A_j)$ , whose antisymmetric part is $\tfrac12(\nabla\times\mathbf{A})=\tfrac12\mathbf{B}_g$ dotted appropriately. Why: the symmetric part of $\partial A$ does not rotate a vector, so only the curl survives — and the connection carries an intrinsic $\tfrac12$ from the metric derivative $\Gamma\sim\tfrac12\partial h$ .

Step 3. Assemble the rotation: the spin precesses at $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ , a factor of two below the naive Larmor guess $\boldsymbol{\Omega}=-\mathbf{B}_g$ one would copy from $-(q/2m)\mathbf{B}$ with unit charge-to-mass. Why: the Larmor rate for a magnetic dipole already contains one factor of $\tfrac12$ ; the gravitational case contains a second, and that doubling of half-factors is the concrete arithmetic of "spin-2 versus spin-1".

Step 4. Confirm the sign of the physics: with $\mathbf{J}$ up, the on-axis $\mathbf{B}_g$ is up, so $\boldsymbol{\Omega}$ points down — the gyroscope is dragged with the source's rotation, the defining sense of frame-dragging. Why: a coherent sign is the difference between a prograde drag (correct, and what GP-B saw) and an unphysical retrograde one, so tracking it through is essential. *The factor $\tfrac12$ in $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ is consistent with the Schiff frame-dragging formula used for Gravity Probe B and with the Kerr-metric gyroscope precession computed to first order in $J$ .*

Limiting cases

The GEM equations are recovered in a nest of approximations, and each boundary of the nest marks a place where the electromagnetic analogy either simplifies or breaks.

Case	Result	Meaning
Static, non-rotating source	$\mathbf{B}_g\to0,\ \ \nabla^2\Phi=4\pi G\rho$	Pure gravitoelectric limit — ordinary Newtonian gravity, $\Phi=-GM/r$ , no frame-dragging at all.
Stationary rotating source	$\mathbf{A}=-\dfrac{G}{c^2}\dfrac{\mathbf{J}\times\mathbf{r}}{r^3}\ \ (r\gg R)$	Dipolar gravitomagnetic field of angular momentum $\mathbf{J}$ ; the source of the Lense-Thirring effect, the exact analogue of a magnetic dipole.
Slow-motion, weak field	$v/c\to0,\ \ \|h\|\to0$	GEM holds to first order in $h$ and lowest order in $v/c$ ; higher orders bring nonlinear graviton self-interaction with no electromagnetic analogue.
On the spin axis	$\Omega_{\rm LT}=2GJ/(c^2 r^3)$	Maximal, prograde drag; the simplest number to evaluate and the one benchmarked for Earth below.
Spin-2 versus spin-1	$\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$	The factor $\tfrac12$ (against the naive Larmor $1$ ) is the unavoidable signature that the mediating field is a tensor, not a vector.

Common trapthe misconception

The GEM equations are seductive enough that it is tempting to treat them as gravity's own Maxwell theory — exact, covariant, fundamental. They are none of these. The split into $\mathbf{E}_g$ and $\mathbf{B}_g$ exists only in a fixed gauge (Lorenz) and is not Lorentz-invariant: a boost mixes the two, exactly as it mixes $\mathbf{E}$ and $\mathbf{B}$ , so the fields are frame-dependent bookkeeping, not tensors. The equations hold only to linear order in $h_{\mu\nu}$ and lowest order in source velocities; push to the next order and gravity's nonlinearity — the graviton coupling to its own energy — produces terms with no electromagnetic counterpart. And the spin-2 nature leaves permanent scars even at linear order: the gyroscope's $\tfrac12$ , and the fact that parallel mass currents attract where you might expect a magnetic-style repulsion of like objects. Use GEM as a calculational scaffold and an intuition pump. Do not mistake it for a restructuring of general relativity.

Worked examples

1 · Frame-dragging of a gyroscope at Earth's north pole (fully worked)

A gyroscope hovers just above Earth's north pole. Estimate the Lense-Thirring precession rate it experiences, using Earth's moment of inertia $I=0.331\,MR^2$ with $M=5.97\times10^{24}\ \mathrm{kg}$ , $R=6.37\times10^6\ \mathrm{m}$ , and rotation period $T=86164\ \mathrm{s}$ (one sidereal day).

Step 1. Build Earth's spin angular momentum. The moment of inertia is $I=0.331\,MR^2=0.331\times(5.97\times10^{24})\times(6.37\times10^6)^2=8.0\times10^{37}\ \mathrm{kg\,m^2}$ , and the angular speed is $\omega=2\pi/T=7.29\times10^{-5}\ \mathrm{s^{-1}}$ , so $J=I\omega=\mathbf{5.85\times10^{33}\ kg\,m^2\,s^{-1}}$ .

Step 2. On the spin axis the bracket in (8) is $3(\mathbf{J}\cdot\hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{J}=2\mathbf{J}$ , so the precession magnitude is $\Omega_{\rm LT}=2GJ/(c^2 R^3)$ . Assemble the pieces: $GJ/c^2=(6.674\times10^{-11})(5.85\times10^{33})/(8.99\times10^{16})=4.34\times10^{6}\ \mathrm{m^3\,s^{-1}}$ , and $R^3=(6.37\times10^6)^3=2.59\times10^{20}\ \mathrm{m^3}$ .

Step 3. Divide and double: $\Omega_{\rm LT}=2\times(4.34\times10^{6})/(2.59\times10^{20})=3.36\times10^{-14}\ \mathrm{rad\,s^{-1}}$ . Convert to angle per year with $1\ \mathrm{yr}=3.156\times10^{7}\ \mathrm{s}$ and $1\ \mathrm{rad}=2.063\times10^{8}\ \mathrm{mas}$ : $\Omega_{\rm LT}=3.36\times10^{-14}\times3.156\times10^{7}\times2.063\times10^{8}\approx\mathbf{219\ mas/yr}\approx0.22\ \mathrm{arcsec/yr}$ .

Step 4. Interpret. This surface, on-axis figure is several times the $\approx39\ \mathrm{mas/yr}$ that general relativity predicts for Gravity Probe B's orbit (and that GP-B measured as $37.2\pm7.2\ \mathrm{mas/yr}$ ), and the difference is entirely geometry: GP-B flew a polar orbit at $642\ \mathrm{km}$ altitude, and the orbit average of the dipole field over a full revolution is smaller than the peak on-axis surface value by a factor of a few. The order of magnitude — tens to a couple hundred milliarcseconds per year for Earth — is the honest scale of frame-dragging in the Solar System. Ω_LT ≈ 219 mas/yr on the polar axis at Earth's surface; J ≈ 5.85×10³³ kg m² s⁻¹.

2 · The LAGEOS orbital node drift

The LAGEOS geodetic satellites orbit at semi-major axis $a=1.227\times10^7\ \mathrm{m}$ with small eccentricity. The Lense-Thirring drift of the ascending node of a satellite orbit is $\dot\Omega_{\rm node}=2GJ/[c^2 a^3(1-e^2)^{3/2}]$ . Estimate it for LAGEOS ( $e\approx0$ ) and compare with the quoted $\approx31\ \mathrm{mas/yr}$ .

Step 1. Reuse $2GJ/c^2=2\times4.34\times10^{6}=8.68\times10^{6}\ \mathrm{m^3\,s^{-1}}$ from Example 1, and compute $a^3=(1.227\times10^7)^3=1.85\times10^{21}\ \mathrm{m^3}$ .

Step 2. With $e\approx0$ the factor $(1-e^2)^{3/2}\approx1$ , so $\dot\Omega_{\rm node}=8.68\times10^{6}/1.85\times10^{21}=4.70\times10^{-15}\ \mathrm{rad\,s^{-1}}$ .

Step 3. Convert: $4.70\times10^{-15}\times3.156\times10^{7}\times2.063\times10^{8}\approx\mathbf{31\ mas/yr}$ , in agreement with the value confirmed by satellite laser ranging. Because the effect scales as $J/a^3$ , it dwarfs a gyroscope's on-axis rate only near the source; out at LAGEOS's altitude it is the whole orbital plane, not a spinning rotor, that slowly turns. Ω̇_node ≈ 31 mas/yr, matching the LAGEOS measurement.

Physics of the portrait

Gravitomagnetism has no single author — Heaviside guessed it, Einstein's field equations contain it, and a generation of experimenters chased it — but one calculation gave it teeth. In 1918, working out what a rotating mass does to the spacetime around it, an Austrian physicist turned the analogy into a number.

Origins · 1 minHans Thirring and how this was discovered

Hans Thirring

1888–1976 · frame-dragging and the gravitomagnetic field of a rotating mass

Hans Thirring was a Viennese theoretical physicist who, within three years of Einstein's 1915 field equations, asked a question no one else had pressed: if mass tells spacetime how to curve, what does rotating mass do? In 1918, in two short papers, he computed the linearized field inside a slowly rotating spherical shell and found that it drags local inertial frames around with the shell — the effect now called frame-dragging, or, when applied to orbits and gyroscopes, the Lense-Thirring effect after Thirring and the mathematician Josef Lense, who supplied the orbital analysis. The result was a striking vindication of Mach's principle in miniature: the rotating matter partly determines what "non-rotating" means nearby. Thirring saw clearly that the off-diagonal metric term behaved like a magnetic vector potential, making him one of the earliest to think in explicitly gravitomagnetic terms. His prediction lay unmeasured for over ninety years, until Gravity Probe B and satellite laser ranging finally caught Earth's frame-dragging at the tens-of-milliarcseconds-per-year level his equations had foretold. Thirring later worked in cosmic-ray physics and became a committed pacifist, but the 1918 papers remain his monument.

History

1893

— Heaviside frames a gravitational and electromagnetic analogy in his notebooks

1918

— Lense and Thirring compute the frame-dragging of a rotating mass

1960

— Schiff and Pugh independently propose an orbiting-gyroscope test

1963

— Kerr finds the exact metric of a rotating black hole, containing GEM as its weak-field limit

2011

— Gravity Probe B reports the Lense-Thirring precession of Earth confirmed

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §18.1 and §40.7 (1973) · Wald, General Relativity, §4.4 and §6.6 (1984) · Carroll, Spacetime and Geometry, §7.1 (2004) · Hartle, Gravity, ch. 14 (2003) · Schutz, A First Course in General Relativity, §8.4 (2009) · Mashhoon, "Gravitoelectromagnetism: A Brief Review", arXiv:gr-qc/0311030 (2003). The central results are: the linearized field equation $\Box\bar h_{\mu\nu}=-16\pi G\,T_{\mu\nu}/c^4$ in Lorenz gauge; the Newtonian normalization $\nabla^2\Phi=4\pi G\rho$ ; the dipole vector potential $\mathbf{A}=-(G/c^2)\mathbf{J}\times\mathbf{r}/r^3$ , which matches the slow-rotation Kerr $g_{0\phi}$ expansion; and the worked numbers — Earth $J\approx5.85\times10^{33}\ \mathrm{kg\,m^2\,s^{-1}}$ , polar-axis surface rate $\approx219\ \mathrm{mas/yr}$ , LAGEOS node $\approx31\ \mathrm{mas/yr}$ , GP-B GR predictions geodetic $\approx6606\ \mathrm{mas/yr}$ and frame-dragging $\approx39\ \mathrm{mas/yr}$ , measured $6601.8\pm18.3$ and $37.2\pm7.2\ \mathrm{mas/yr}$ (Everitt et al., PRL 106, 221101, 2011). Convention-dependent factors of two appear in $\mathbf{A}$ and $\mathbf{B}_g$ ; physical precession rates are convention-independent.

Prerequisites. Weak-Field Limit and Linearized Gravity · Newtonian Limit of General Relativity. Leads to. Frame Dragging and the Lense-Thirring Effect · Geodetic Precession (de Sitter Effect) · Parametrized Post-Newtonian (PPN) Formalism.

Ask the tutor: Ask me to walk through the Kerr $g_{0\phi}$ expansion, the Schiff orbit-average that turns 219 into 39 mas/yr, or why spin-2 forces that stubborn factor of one half.

Is gravitomagnetism a real force or just a mathematical analogy?

Both, depending on what you mean by "real". The gravitomagnetic field is not a new fundamental interaction — it is a piece of ordinary general relativity, specifically the off-diagonal, time-space part of the metric perturbation from a moving source. But the effects are entirely physical and measured: a gyroscope really precesses, an orbital plane really drifts, and Gravity Probe B and LAGEOS put numbers on it. What is "just an analogy" is the packaging into $\mathbf{E}_g$ and $\mathbf{B}_g$ and Maxwell-like equations. That packaging is a gauge-dependent convenience valid only in the weak field, whereas the underlying frame-dragging is a coordinate-independent feature of the spacetime.

Why does the gyroscope carry a factor of one half that the magnetic compass does not?

Because gravity is a spin-2 field and electromagnetism is spin-1. Concretely, the gyroscope's spin is parallel-transported through the linearized connection, and the connection coefficients that rotate it are built from derivatives of $\bar h_{0i}$ with an intrinsic factor of $\tfrac12$ (since $\Gamma\sim\tfrac12\partial h$ ). When you assemble those into the curl $\mathbf{B}_g$ , that half survives as $\boldsymbol{\Omega}=-\tfrac12\mathbf{B}_g$ . The magnetic Larmor rate $-(q/2m)\mathbf{B}$ already has one factor of two in it; the gravitational case effectively has two such half-factors, and the mismatch is the arithmetic face of "spin-2 versus spin-1". It cannot be removed by any choice of units or normalization.

Why do parallel mass currents attract when parallel electric currents also attract?

The two statements only look like a contradiction if you forget the source signs. In electromagnetism, like charges repel but parallel currents attract — the magnetic force between them wins for co-moving charges. In gravitomagnetism the Gauss law has a flipped sign ( $\nabla\cdot\mathbf{E}_g=-4\pi G\rho$ ) because masses attract, and the Ampere law is flipped to match. The net result is that parallel mass currents attract, consistent with the fact that all gravity is attractive. The clean way to say it: every sign in GEM is chosen so that no configuration of masses ever repels, whereas electromagnetism allows both signs of charge and hence both signs of force.

How is gravitomagnetic frame-dragging different from geodetic precession?

They are two distinct precessions that Gravity Probe B measured separately. Geodetic precession (the de Sitter effect) is gravitoelectric: it arises from a gyroscope moving through the curved space around a mass, and it exists even if the central body does not rotate — it scales with the orbital velocity and the mass. Frame-dragging (the Lense-Thirring effect) is gravitomagnetic: it requires the central body to spin, and it scales with the body's angular momentum $J$ . For GP-B around Earth the geodetic term was about $6606\ \mathrm{mas/yr}$ and the frame-dragging only about $39\ \mathrm{mas/yr}$ — a roughly 170-fold difference, which is exactly why separating them was so hard.

Do I need the full Kerr metric, or is the weak-field picture enough?

For everything in the Solar System, the weak-field GEM picture is plenty: Earth's frame-dragging is a linear-order effect and the dipole formula $\Omega_{\rm LT}=2GJ/(c^2r^3)$ reproduces the measured rates. GEM is in fact the far-field, slow-rotation limit of the exact Kerr metric — the gravitomagnetic dipole is just the leading term of the Kerr $g_{0\phi}$ expansion. You need the full Kerr solution only when the field is strong: near the horizon of a spinning black hole, where frame-dragging becomes so extreme it forces all observers to co-rotate (the ergosphere), the linear approximation fails and there is no shortcut around the exact metric.

Why is there a minus sign in the gravitoelectric Gauss law?

Because gravity only attracts. In electrostatics $\nabla\cdot\mathbf{E}=+\rho/\varepsilon_0$ , and field lines point away from positive charge, which lets like charges repel. In gravitomagnetism $\nabla\cdot\mathbf{E}_g=-4\pi G\rho$ , so the field lines converge onto mass, and a test mass is pulled toward the source. The sign is not a convention you could flip: it is fixed by requiring that the static limit reproduce Newton's attractive force $\mathbf{F}=-m\nabla\Phi$ with $\nabla^2\Phi=+4\pi G\rho$ . The same reasoning flips the sign in the gravitomagnetic Ampere law, and together those two minus signs are what distinguish gravity from electromagnetism at the level of the field equations.

Has frame-dragging actually been detected, and how well?

Yes, by two independent methods. Gravity Probe B (final results 2011) used orbiting gyroscopes and reported Earth's frame-dragging at $37.2\pm7.2\ \mathrm{mas/yr}$ against a prediction near $39\ \mathrm{mas/yr}$ — a confirmation at about the $19\%$ level, limited by unexpected electrostatic patch effects on the rotors. The LAGEOS/LARES program uses satellite laser ranging to track the orbital node drift, and combined with high-resolution models of Earth's static gravity field (from GRACE and GOCE) it claims agreement with general relativity at the few-percent level. Both are hard measurements of a genuinely small effect, and together they leave little doubt that a spinning Earth drags spacetime as Thirring predicted in 1918.