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Gravitomagnetism, a theoretical idea around since 1918, is a predicted consequence of general relativity, from which it was derived. Its existence has been experimentally proven, but allegedly only once, and there are certain variants of the effect that are supported by the evidence to greater or lesser degrees. An international team claimed to discover the effect in the mid-90s, based on data from the LAGEOS I and LAGEOS II spacecraft. The measured effect was within 10% of that predicted by general relativity, although some scientists still doubt the validity of these results. In 2004, Stanford physicists launched Gravity Probe B, an extremely delicate gyroscope package, to measure gravitomagnetism in outer space with much greater precision. Its data is currently being analyzed.
After Einstein presented his theory of general relativity, it took decades to work out all of its predicted consequences. The most famous is the fundamental equivalence between matter and energy, demonstrated vividly by the atomic bomb. Lorentz contraction, the increase in mass and decrease in length seen by an outside observer looking at an object moving at relativistic (near-light) speeds, is another, and has been experimentally verified. Time is known to pass more slowly for objects moving at speeds close to that of light, or even significantly less - the effect has been observed in atomic clocks orbiting the earth.
This poorly exposed and tested consequence, gravitomagnetism, refers to the field that is supposedly created when a massive body rotates rapidly. Gravitomagnetism is misleadingly named - it is not magnetic - the force created emerges from gravity, not electromagnetism. But it is called gravitomagnetism because of the mathematical similarity between the equations that describe this effect and the creation of a magnetic field. In the same way that a magnetic field is created when a charged object rotates, a gravitomagnetic field is created when a massive body rotates. The math used to describe the two is functionally similar The effect could just as easily be called a gravitorotational field, a term which might be less misleading.
A very powerful gravitomagnetic field is expected to be observed around supermassive black holes rotating very rapidly. These black holes may have a mass millions of times greater than the sun and be rotating at a furious rate. Here in the solar system, though, the effect is predicted to be very small - on the order of a few parts per trillion in the overall scheme of gravitational interactions - making it difficult to observe without delicate sensors or proximity to massive planets or the sun.
Stanford's Gravity Probe B was extremely delicate. It contained a gyroscope with an object that was spherical to 40 atomic diameters, possessing a near-homogeneous density distribution. Designed to detect gravitomagnetism, the gyroscope was meant to measure "frame-dragging" - the source of the predicted effect is a small twist in spacetime created by the rotating mass. A spinning gyroscope in a vacuum should rotate with almost perfect uniformity, but it is predicted that gravitomagnetism disturbs this slightly. The simple way to visualize frame-dragging is to imagine a ball rotating on a stretched sheet, that creates a slight twist in the sheet at the same time as creating a major depression.
Another predicted effect is that when a satellite orbits the earth in what should be a perfect circle, it actually ends up in a slightly different place, due to the slight vortex created by the spinning earth. A difficulty in measuring gravitomagnetism is that the earth's equatorial bulge creates discrepancies in satellite/gyroscope behavior that must be correctly subtracted from other data in order to measure the magnitude of genuine frame-dragging.
Although a large amount of data has been returned from the Gravity Probe B, the analysis is ongoing. Gravitomagnetism is quite mysterious, and currently poorly understood. Whether the effect will have practical applications or not is something we will probably not know for at least a few decades.