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:

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If you say “Einstein equation” to a random person on the street, odds are they’ll immediatley think of “E=mc2.” 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=mc2, 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” gμν. 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.

Comments

  1. #1 zeynel
    December 23, 2011

    “…as demonstrated by the Global Positioning System…”

    A few years ago I investigated this claim that GPS will not work correctly without general relativity and I traced the rumor to the writings of the physicist Neil Ashby. His claim could not be verified because he is referring to his classified research while he was working for the government.

    Doing more research I found that there were engineers working contemporaneously with Neil Ashby who rejected claims that GPS is dependent on GR. What I would like to know if you can offer any reference to actual source code used in GPS that requires GR to work correctly.

    Do you have actual evidence in the form of GPS source code? Or do you rely only to Neil Ashby’s theoretical derivations?

  2. #2 dean
    December 23, 2011

    This old physics major turned statistician has enjoyed this series. Thanks for the time and effort.
    Don’t let the cranks get your goat.

  3. #3 Ambitwistor
    December 26, 2011

    zeynel,

    I don’t know if the GPS source code is available. However, in the satellite interface specifications, you can find mention of a relativistic correction (without derivation). See the 24th page of this PDF. This correction is also mentioned on this web site, which describes the correction as a combination of special and general relativistic effects.

    In short, I think you can find evidence that a relativistic correction is being applied, but you’ll have to calculate the effect yourself to verify that the magnitude is in accordance with relativity. I don’t know if any official GPS documents derive that number from first principles.

  4. #4 Ambitwistor
    December 26, 2011

    zeynel,

    I don’t know if the GPS source code is available. However, in the satellite interface specifications, you can find mention of a relativistic correction (without derivation). See the 24th page of this PDF. This correction is also mentioned on this web site, which describes the correction as a combination of special and general relativistic effects.

    In short, I think you can find evidence that a relativistic correction is being applied, but you’ll have to calculate the effect yourself to verify that the magnitude is in accordance with relativity. I don’t know if any official GPS documents derive that number from first principles.

  5. #5 nino
    December 28, 2011

    That the GPS clock needs to be adjusted once in orbit is no wonder for any good engineer. You can’t expect a clock (of any kind)to be precise if it is put in a different environment! And that has nothing to do with relativity!
    A pendulum clock that can be fairly precise in your living room, needs to be adjusted if the temperature changes or if you move it where g is different. It will not work at all on a satellite with zero gravity! What we experiment around us can be usually explained with correct use of simple physics.

  6. #6 Ambitwistor
    December 28, 2011

    nino,

    Your comment really doesn’t have anything to do with the fact that the GPS adjustment that is needed agrees with the one predicted by relativity.

  7. #7 zeynel
    December 29, 2011

    Ambitwistor,

    Thanks for the reference to Interface Specification document. But the mention of “relativistic corrections” in that document is still theoretical. This is clearer if you look at IS GPS-200 mentioned in that document

    http://www.navcen.uscg.gov/pdf/IS-GPS-200D.pdf

    IS GPS-200 has the actual relativistic correction equation in section 20.3.3.3.3.1, page 88 of the document.

    Reading that section I see that, first of all, relativistic correction is not applied to SV clock (as Chad’s phrasing in the post suggests), but “the user’s equipment must determine requisite relativistic correction.” So, relativistic corrections, if any, is supposedly applied in user’s equipment not in the satellite’s clock.

    That page then gives two coupled equations and equation 2 has a series of correction coefficients in a polynomial and the last one is delta-t_r.

    delta-t_r is simply a constant value which amounts to double the position times velocity divided by c^2.

    I have two problems with this. First, how does the 4th polynomial coefficient supposedly applied in user’s equipment proves that “general relativity is correct.” What is being proved here?

    Second, we need to find evidence that equipment manufacturers indeed apply such a term in their source code. Do you have such evidence?

    Also, my experience tells me that, in practice, orbits and time corrections are all computed by fitting data by numerial integration; this is how JPL computes orbits and it is the most precise way to compute orbits; numerical integration contains zero theoretical or physical considerations.

    It makes no sense to brand one of the coefficients “relativistic correction”. Relativity is already most extensively proved theory ever, does it need to be proved by such dubious evidence?

    So now we need to find if equipment manufacturers use this so-called “relativistic correction” in their software. Do you have any links for this too?

  8. #8 KeithB
    December 30, 2011

    zeynel:
    You don’t need GPS. Your “numerical integration” did not adequately explain Mercury’s orbit. Relativity did.

  9. #9 zeynel
    December 31, 2011

    KeithB: I am not disputing Relativity. I am trying to understand if there is hard scientific evidence (actual source code) supporting the claim made by Chad that GPS will not work correctly without relativistic corrections.