The gyroscope a spinning Earth turns

Magnetism is electricity seen sideways found that the magnetic field of a current is what the electric field of moving charges looks like once relativity is taken seriously, and that gravity, which also obeys an inverse-square law, must have a magnetic part of its own — a field made by moving mass and felt by moving mass. That essay noted the catch. Electromagnetism gets its magnetism for free from the cancellation of enormous positive and negative charges, so that a tiny drift of one sign is visible; mass comes in one sign only, and gravity’s magnetic part is never seen except as a correction of order $v^2/c^2$ to an already weak force.

The correction is not zero, and it has a definite pattern. A spinning charged ball makes a magnetic dipole field, the field of the loop that behaves like a needle. A spinning mass makes the gravitational analogue, and what that field does to a test body is not to push it but to turn it: the local frames in which a gyroscope keeps a fixed direction are dragged round with the spinning mass. Josef Lense and Hans Thirring worked out the effect on orbits in 1918. Leonard Schiff worked out its effect on a gyroscope in 1960 and proposed measuring it.

The pattern of the turning #

A gyroscope at position $\mathbf{r}$ from the centre of a body with angular momentum $\mathbf{J}$ has its spin axis turned at the angular velocity

$$\boldsymbol{\Omega} = \frac{G}{c^2 r^3}\left[\,3(\mathbf{J}\cdot\hat{\mathbf{r}})\,\hat{\mathbf{r}} - \mathbf{J}\,\right],$$

which is, term for term, the magnetic field of a magnetic dipole $\mathbf{J}$ with the electromagnetic constants replaced by $G/c^2$. Over either pole, where $\hat{\mathbf r}$ is parallel to $\mathbf J$, the bracket is $2\mathbf J$: the gyroscope is turned in the sense of the body’s spin. Over the equator, where $\hat{\mathbf r}$ is perpendicular to $\mathbf J$, the bracket is $-\mathbf J$: it is turned the opposite way, half as fast.

The minus sign over the equator is the first thing the analogy buys. A naive picture of frame dragging — space swirled round by the spinning Earth like honey round a spoon — predicts that everything is turned in the sense of the spin, everywhere. The dipole pattern says otherwise, and for the same reason that a bar magnet’s field points backwards alongside the magnet: the field lines that leave the north pole must return round the outside. Near the equator, the dragging of space nearer the Earth outruns the dragging further out, and a gyroscope spanning that difference is turned backwards by it.

Forty-one milliarcseconds round a polar orbit #

A satellite in a polar orbit passes over both poles and both sides of the equator every hour and a half, and its gyroscope accumulates the turning from every point.

The rate swings between twice the equatorial value and minus the equatorial value, and its average over the orbit is half the equatorial value, in the sense of the spin. For the Earth, with an angular momentum of $5.9\times10^{33}\ \text{kg m}^2/\text{s}$, at an orbital radius of 7027 kilometres, that average is

$$\langle\Omega\rangle = \frac{GJ}{2c^2a^3} \approx 41\ \text{milliarcseconds per year}.$$

Forty-one milliarcseconds is the angle a human hair subtends at three hundred metres. A gyroscope has to keep its axis fixed to a small fraction of that for a year, and the orbit itself adds a much larger turning in the perpendicular direction: the geodetic precession, 6.6 arcseconds a year, which comes from carrying the gyroscope round a closed path through curved space. The turn that two pushes leave behind found a third of that geodetic drift in flat spacetime, as the Thomas precession of a frame carried round a circle; the other two thirds are the curvature of space itself. Frame dragging is a separate effect again, at right angles to the geodetic one in a polar orbit, which is why a polar orbit was chosen: it sends the two drifts along different axes, so that the small one can be read without being swamped.

The roundest objects ever made #

The satellite was Gravity Probe B, proposed in 1959 and 1960, launched in April 2004, and analysed until 2011. Its gyroscopes were four spheres of fused quartz 38 millimetres across, polished so that no point on any of them departed from a perfect sphere by more than about forty atomic layers. A sphere that is not quite round has a slight torque exerted on it by any field, and over a year any torque would turn the axis by far more than the effect being sought. The spheres floated in vacuum, electrostatically suspended, at a temperature of under two kelvin, spinning at about seventy revolutions a second.

Reading the direction of a featureless spinning ball without touching it is the hard part, and it was done with a piece of physics that the field a spinning superconductor makes of itself described: a superconductor that rotates generates a magnetic moment exactly aligned with its rotation axis. The quartz spheres were coated with niobium, which becomes superconducting at those temperatures, so each spinning gyroscope carried a tiny magnetic moment along its own axis, and a superconducting loop around it detected the orientation of that moment to a small fraction of a milliarcsecond. A telescope on the satellite, locked onto the star IM Pegasi, supplied the fixed reference. And the satellite itself was flown around the gyroscopes rather than the other way round: a proof mass inside it was allowed to fall freely, and thrusters fed by the helium boiling off the cryostat kept the whole spacecraft centred on it, so that air drag and the pressure of sunlight pushed on the spacecraft and never reached the spheres. The gyroscopes were as nearly in free fall as anything built, which is what a measurement of the local inertial frames requires: any force on them would be indistinguishable from a twist of the frames themselves.

The geodetic drift was measured to 0.3 per cent of its predicted value. The frame-dragging drift was measured to 19 per cent — far worse than the original goal of one per cent, because patches of electric charge on the spheres’ surfaces and the inside of their housings exerted torques that the designers had not anticipated, and modelling them out of the data took five years. But 37.2 against a prediction of 39.0, with an uncertainty of 7.2, is a measurement of an effect 170 times smaller than the drift beside it, by an instrument that had to hold its axis steady to a part in a hundred thousand million of a radian per second.

A whole orbit dragged round #

There is a second way to see the effect, which needs no gyroscope at all. An orbit is itself a kind of gyroscope: its plane has an orientation, fixed by the orbiting body’s angular momentum, and frame dragging turns it. The node of an orbit — the line where its plane crosses the equator — is dragged round in the sense of the Earth’s spin at

$$\dot\Omega_{\text{node}} = \frac{2GJ}{c^2 a^3 (1 - e^2)^{3/2}},$$

which for the LAGEOS satellites, at 12,270 kilometres from the Earth’s centre, is about 31 milliarcseconds a year. LAGEOS are dense balls covered in corner reflectors, built for laser ranging, and their orbits are tracked to centimetres. The difficulty is that the Earth’s equatorial bulge turns the same node a million times faster; the trick, used by Ignazio Ciufolini and Erricos Pavlis in 2004, was to combine two satellites with different orbits so that the bulge’s effect cancelled, leaving the relativistic one. Their result agreed with the prediction within about ten per cent, and later work with the LARES satellite claims a few per cent. Gravity Probe B measured the dragging of a spinning ball; the laser-ranged satellites measure the dragging of an orbit; they agree.

Dragging fades faster than the geodetic drift #

Both relativistic drifts weaken with distance, but not at the same rate.

Frame dragging falls as $1/r^3$, like any dipole field, while the geodetic drift falls as $r^{-5/2}$, because it combines the field’s $1/r^2$ with an orbital speed that falls only as $r^{-1/2}$. So the ratio of the two worsens steadily outward, from 162 at the satellite’s radius to over a thousand at the Moon’s, and frame dragging has to be measured close to the Earth or not at all. The Moon’s own orbit does carry a frame-dragging signature, and lunar laser ranging is sensitive to it, but there it is entangled with every other influence on the Moon’s motion.

A pendulum at the pole, and a ring of light on the ground #

On the Earth’s surface the same dragging acts on everything that keeps a direction. A Foucault pendulum at the North Pole keeps its plane of swing fixed relative to the local inertial frame while the Earth turns beneath it once a day — the effect the ellipse a pendulum turns by itself warns can be mimicked by a pendulum’s own imperfections. But the local inertial frame at the pole is itself dragged by the Earth’s spin, at $2GJ/c^2R^3$, about $3 imes10^{-14}$ radians per second — a fifth of an arcsecond a year. The pendulum’s plane turns relative to the distant stars by that much. No pendulum is good enough to see it; the friction of its suspension and the ellipticity of its swing produce drifts millions of times larger.

The instrument that might see it from the ground is not mechanical. A ring laser — light sent both ways round a closed loop several metres across — measures rotation through the difference in the round-trip times of the two beams, and the large ring at Wettzell in Bavaria resolves the Earth’s rotation to a few parts in a billion, enough to see the wobble of its axis and the tides of the solid Earth. The compass that finds the axis the Earth turns about found the mechanical version reading the Earth’s rotation from its gyroscopic torque; a ring laser reads it from light. Frame dragging changes what such an instrument measures by about one part in a billion of the Earth’s rotation rate, and proposals to build a set of rings with different orientations, accurate enough to separate the dragging from the geodetic term, are being studied. If one succeeds, it will have measured gravitomagnetism without leaving the ground.

Inertia from the rest of the universe #

Thirring’s original interest in 1918 was not the Earth. It was a question Ernst Mach had raised: what decides which frames are inertial — which frames are not rotating — if not the matter in the universe? Newton had said absolute space decides; Mach suspected the distant stars. Thirring computed what a massive spherical shell, rotating, does to the space inside it, and found that the inertial frames inside are dragged round at a fraction of the shell’s rotation, about $4GM/3c^2R$ of it for a shell of mass $M$ and radius $R$.

For any ordinary shell that fraction is tiny. But it grows as the shell becomes more compact, and in 1966 Dieter Brill and Jeffrey Cohen showed that a shell approaching its own horizon size drags the frames inside it almost completely: a gyroscope inside, and the shell, rotate together, and an observer inside has no way to tell that the shell is turning. For a universe whose mass is comparable to what is needed to close it, the same reasoning suggests that the local inertial frames are tied to the average motion of the matter in it — which is what is observed, to within a part in a million million per year in the absence of any rotation of the universe relative to the inertial frames. Whether general relativity fully embodies Mach’s idea is still argued; that it embodies a measurable part of it is what Gravity Probe B confirmed. The same distinction underlies the forces that are not there: a rotating frame’s centrifugal and Coriolis forces are rotation relative to the inertial frames, and frame dragging is the claim that those frames are not fixed in advance, but made by mass in motion.

Where nothing can stand still #

Near the Earth the dragging is a correction of one part in $10^{14}$ per second. Near a spinning black hole it is the dominant feature of the geometry.

Far from the hole the dragging follows the same $2GJ/c^2r^3$ that turns gyroscopes over the Earth’s poles. Close in, the exact Kerr solution departs from it, and at the horizon the dragging rate becomes the hole’s own rotation rate, the angular velocity of the surface that only lets things in. More striking is the region outside the horizon where the dragging is too strong to resist: inside a surface that reaches $2GM/c^2$ on the equator, no observer can remain at rest relative to the distant stars, whatever thrust it has, because staying still would mean moving against the dragged space faster than light. Every observer there is carried round with the hole. That region is the ergosphere, and it is outside the horizon, so things can enter it and come out again.

Roger Penrose showed in 1969 that the ergosphere stores extractable energy: a particle that enters it and splits, with one half falling into the hole on an orbit of negative energy as seen from far away, lets the other half escape with more energy than the original had. The hole spins down by exactly the difference, and the area that is not allowed to shrink sets the limit — up to 29 per cent of a maximally spinning hole’s mass can be extracted. Magnetic fields threading the ergosphere are thought to tap the same energy in the jets from the centres of active galaxies, through the Blandford–Znajek process.

A round Earth, a circular orbit and published numbers #

The pattern and the orbit figures use the weak-field formula for a spinning sphere, which ignores the Earth’s oblateness — its bulge contributes higher multipoles to the dragging, a correction of about a tenth of a per cent — and treats the orbit as circular and exactly polar. The predictions here therefore differ from the experiment’s own by a few tenths of a per cent in the geodetic term, which is why the measured value is compared with the drift computed here rather than with the experiment’s published prediction. The measured values themselves are the published results, typed in, not computed.

The Kerr figure shows the dragging only on the equator and only for a static, non-orbiting observer’s frame; elsewhere the ergosphere is smaller, touching the horizon at the poles. And none of the figures shows the gravitomagnetic force on moving bodies, the analogue of the magnetic force, which deflects a moving test mass sideways; it is real, it modifies orbits, and it is what the node of the LAGEOS orbits responds to, but at these speeds it is far too small to see except through its accumulated effect on a whole orbit.

Still open: how closely the dragging matches near strong fields #

Frame dragging is now measured to a few per cent near the Earth. Near compact objects it is measured indirectly: the precession of orbits in binary pulsars includes a frame-dragging term from the pulsar’s spin, the tilt of a white dwarf’s companion orbit in one system has been attributed to it, and the spins inferred from the gravitational waves of merging black holes are spins whose dragging shaped the last orbits. What has not been done is a measurement in a strong field precise enough to test whether the dragging follows the Kerr solution exactly, or differs from it in the way some alternative theories of gravity predict — differences that would show first in the ergosphere and the innermost orbits, where the dashed rule and the solid curves part company.

The analogy that started this essay is the general lesson. Wherever an inverse-square field is carried by something that moves, a second field appears, with the dipole shape of magnetism; for gravity it turns gyroscopes rather than compass needles, along the spin over the poles and against it over the equator. The Earth drags the frames near it by forty milliarcseconds a year. A black hole drags them so hard that, inside its ergosphere, nothing can stand still.