A mass curves the spacetime around it. If the mass is spinning, the spacetime around it is also dragged in the direction of the spin.
Lense and Thirring derived the effect from the Einstein equation in . It is called the dragging of inertial frames, or the Lense-Thirring effect.
The axis of a spinning top that feels no outside force, a gyroscope, normally keeps its direction against the distant stars. Near a spinning mass the very standard of what counts as not rotating is dragged, so the axis slowly shifts.
The boundary from which not even light can get out is the horizon of a black hole. The spacetime of a rotating black hole is described by the solution Kerr found in , the Kerr solution, and in it there is a second boundary outside the horizon.
Inside that boundary no amount of rocket thrust can keep you at rest against the distant stars. The dragging is so strong that you have to go round the way the hole turns. The region is called the ergosphere.
The ergosphere lies outside the horizon, so you can go in, be carried round in the direction of the spin, and come back out.
The same dragging happens around the spinning Earth.
Put a gyroscope on a satellite in an orbit km high that passes over both poles, a polar orbit, and its axis shifts by milliarcseconds a year. A milliarcsecond is a millionth of a degree, so milliarcseconds is about a hundred thousandth of a degree.
The same gyroscope also turns because the mass of the Earth curves spacetime. This is called geodetic precession, and at milliarcseconds a year it is some times larger than the dragging.
The two tip the axis in perpendicular directions, so they can be measured apart.
In the satellite Gravity Probe B, carrying four gyroscopes, was launched into a polar orbit. The gyroscopes were quartz spheres cm across, made as close to perfectly round as was then possible.
The results, announced in , gave a geodetic precession of and a frame dragging of milliarcseconds a year. The predictions of and both lie within the error bars.
Laser ranging to the LAGEOS satellites, which follows their orbits by timing laser pulses bounced back to the ground, has also confirmed how far the dragging turns the planes of their orbits.
Inside the ergosphere there are orbits whose energy, as measured from far away, is negative.
Drop something in, split it in two, put one piece onto a negative energy orbit and let the other escape. Energy is conserved over the two pieces, so the escaping piece carries out more energy than went in. Penrose showed this in , and it is called the Penrose process.
A mechanism that carries the spin energy out through the magnetic field threading the black hole was proposed in and is called the Blandford-Znajek process. It is thought to be the source of the jets reaching out from the centres of galaxies.
In either case, the spin of the black hole slows by the amount of energy drawn out.