A week and a half ago, when the advent calendar reached Newton’s Law of Universal Gravitation, I said that it was the first equation we had seen that wasn’t completely correct. Having done our quick swing through quantum physics, the time has come to correct that equation:

If you say “Einstein equation” to a random person on the street, odds are they’ll immediatley think of “*E=mc*^{2}.” If you ask a physicist to think of *the* Einstein equation, though, this is the one they’ll think of. This is the Einstein field equation from general relativity, and while it’s not as well known as *E=mc*^{2}, it’s considered a far greater achievement within the field.

As explained in APS News, it also appears under the opening credits in the 2003 animated film The Triplets of Belleville, thanks to the friendship between the director and a physicist in Quebec. So it’s artistically significant as well as important for physics.

It’s also the most horrendously complicated of all the equations we’ve seen.

It may not necessarily look it, but those Greek-letter subscripts are a dead giveaway. The symbols here don’t just stand for numbers, they represent tensors, which can loosely be thought of as 4×4 grids of numbers, with their own special rules for multiplication and division. This is really a compact way of representing ten different equations that need to be solved simultaneously to make any predictions.

We’re not really going to explain all that on a blog, so what are these about on a conceptual level? The whole business of general relativity was rather pithily summed up by the late, great John Archibald Wheeler (a man with a real gift for pithily summing things up) as “Matter tells space how to curve, space tells matter how to move.” The right-hand side of this equation describes the matter in some region of space (though the “stress-energy tensor” *T*), and the left-hand side describes the resulting curvature of spacetime. Any matter in the vicinity will move along geodesic curves through this curved spacetime, which are not necessarily straight lines in space. As a result, to an observer watching a bit of matter moving around, it will appear to experience a force. The force in this case is gravity, and the one symbol in this equation that actually stands for a plain old number is *G*, which is the same gravitational constant from Newton’s equation all that time ago.

The business of general relativity involves solving this equation (or, really, these equations) for the “metric tensor” *g*_{μν}. This is the thing that tells you how to combine space and time measurements to form a spacetime distance between two points, according to an observer near one of those points, and it’s so central that Richard Feynman once found his way to a conference by telling a cab dispatcher to take him to the same place as a bunch of distracted guys wandering around saying “g-mu-nu, g-mu-nu” over and over.

What does this tell us? It tells us that the presence of matter causes a change in the way you measure distance and time, depending on where you are relative to a massive object. This means what one observer sees as some distance in space will appear to another observer at a distant position to be a mixture of distance in both space and time. An observer sitting close to a massive object– on the surface of the Earth, say– will see time passing at a different rate than an observer who is farther away– on a satellite in orbit, say. And a length measured by an observer close to a massive object will not agree with the same distance measured by an observer farther away.

This seems completely bizarre, but is absolutely and unequivocally true, as demonstrated by the Global Positioning System. The satellites making up the GPS system contain atomic clocks which broadcast a time signal, and the rate at which those clocks “tick” has to be adjusted to take general relativity into account. Without the relativistic correction, the clocks would drift by some 38 microseconds a day, corresponding to 11km of position uncertainty. As the system works to give you your position on the earth to within a few meters (most of the time), we know that the relativistic correction works, and thus general relativity is correct.

This equation solves a problem that had existed since Newton’s day, namely what causes gravity. Maxwell’s equations allow you to think of electromagnetic forces as being carried from place to place by electromagnetic radiation, but what carried gravity was not obvious, and Newton famously refused to feign a hypothesis about it. Einstein’s theory of general relativity explains what carries gravity: spacetime itself bends in response to mass, and that bending produces the force we see.

This equation also *creates* a problem, because it doesn’t play nice with quantum mechanics. General relativity is a fantastically successful theory, giving correct predictions for everything from the Earth all the way up t galaxy clusters, but the way it works is fundamentally incompatible with the standard methods used in quantum theory. For almost a hundred years, physicists have been beating their heads against this problem, trying to reconcile the two theories, but nothing they’ve tried has yet produced a quantum theory of gravity.

So, as the advent season draws to a close, take a moment to appreciate Einstein’s greatest accomplishment, both its manifest successes and its challenges for the future. And come back tomorrow for the final equation of our countdown to Newton’s birthday.