In some ways, mathematics is like literature. It has its own definitions and grammatical rules – although unfortunately these are the bane of too many students’ lives. Which is a great pity, because when used elegantly and clearly, mathematical language can help readers to see things in entirely new ways. Take analogies, for example. They’re obviously powerful in literature – who doesn’t thrill to a creative, well-aimed metaphor? But they can be even more powerful in mathematical physics.

Making *physical* analogies is fundamental in the process of physics, because it helps physicists to imagine new physical phenomena. We still speak of the “flow” of an electric “current”, using liquid metaphors that physicists coined before they knew that electrons existed. On the other hand, the old concept of “ether” – a hypothetical light-carrying medium analogous to water or air – has long passed its use-by date. Physical analogies can be creative and useful, but sometimes they can lead one astray.

The same is true of *mathematical* analogies applied to physical reality – and especially of the interplay between mathematical and physical analogies. An analogy that has tantalised mathematicians and physicists for a century, and which is still a hot if much-debated topic, is that between Albert Einstein’s equations of gravity and James Clerk Maxwell’s equations of electromagnetism. It’s led to an exciting new field of research called “gravito-electromagnetism” – and to the prediction of a new force, “gravito-magnetism”.

The surprising idea of comparing gravity and electromagnetism – two entirely different kinds of phenomena – began with the intriguing mathematical analogy between the equations of Newtonian gravity and Coulomb’s law of electrostatics. Both sets of equations have exactly the same inverse-square form.

In 1913, Einstein began exploring the much more complex idea of a relativistic gravitational analogue of electromagnetic induction – an idea that was developed by Josef Lense and Hans Thirring in 1918. They used Einstein’s final theory of general relativity (GR), which was published in 1916.

Today this so-called “gravito-electromagnetism”, or GEM for short, is generally treated mathematically via the “weak field” approximation to the full GR equations – simpler versions that work well in weak fields such as that of the earth.

It turns out that the mathematics of weak fields includes quantities satisfying equations that look remarkably similar to Maxwell’s. The “gravito-electric” part can be readily identified with the everyday Newtonian downward force that keeps us anchored to the earth. The “gravito-magnetic” part, however, is something entirely unfamiliar – a new force apparently due to the rotation of the earth (or any large mass).

It’s analogous to the way a spinning electron produces a magnetic field via electromagnetic induction, except that mathematically, a massive spinning object mathematically “induces” a “dragging” of space-time itself – as if space-time were like a viscous fluid that’s dragged around a rotating ball. (Einstein first identified “frame-dragging”, a consequence of general relativity elaborated by Lense and Thirring.)

But how far can such mathematical analogies be pushed? Is “gravito-magnetic induction” real? If it is, it should show up as a tiny wobble in the orbit of satellites, and – thanks also to the “geodetic” effect, the curving of space-time by matter – as a change in the direction of the axis of an orbiting gyroscope. (The latter is analogous to the way a magnetic field generated by an electric current changes the orientation of a magnetic dipole.)

Finally, after a century of speculation, answers are unfolding. Independent results from several satellite missions – notably Gravity Probe B, LAGEOS, LARES, and GRACE – have confirmed the earth’s geodetic and frame-dragging effects to varying degrees of precision. For frame-dragging, the best agreement with GR has been within 0.2%, with an accuracy of 5%, but astronomers expect that a new satellite (LARES 2), to be launched at the end of 2019, will, with data from LAGEOS, give an accuracy of 0.2%.

More accurate results will provide more stringent tests of GR, but astrophysicists have already taken gravito-magnetism on board. For instance, it suggests a mechanism to explain the mysterious jets of gas that have been observed spewing out of quasars and active galactic nuclei. Rotating supermassive black holes at the heart of these cosmic powerhouses would produce enormous frame-dragging and geodetic effects. A resulting gravito-magnetic field analogous to the magnetic field surrounding the two poles of a magnet would explain the alignment of the jets with the source’s north-south axis of rotation.

Making analogies is a tricky business, however, and there are some interpretive anomalies still to unravel. To take just one example, questions remain about the meaning of analogical terms such as gravitational “energy density” and “energy current density”. Things are perhaps even more problematic – or interesting – from the mathematical point of view.

For example, there is another, purely mathematical analogy between Einstein’s and Maxwell’s equations, which gives rise to a very different analogy from the GEM equations. To put it briefly, it’s a comparison between the so-called Bianchi identities in each theory.

The existence of two (and in fact several) such different mathematical analogies between the equations of these two physical phenomena is incredibly suggestive of a deeper connection. At present, though, there are some apparent physical inconsistencies between the “electric” and “magnetic” parts in each mathematical approach.

Still, the formal analogies are useful in helping mathematicians find intuitively familiar ways to think about the formidable equations of GR. And there’s always the tantalising possibility that this approach will prove as physically profound as the prediction of gravito-magnetism.

*This is an abridged version of the story that appears in Cosmos 84. To read the full version, subscribe here. *

### Robyn Arianrhod

Robyn Arianrhod is a senior adjunct research fellow at the School of Mathematical Sciences at Monash University. Her research fields are general relativity and the history of mathematical science.

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