NASA held a big press conference yesterday to announce that the Gravity Probe B experiment had confirmed a prediction of General Relativity that spacetime near Earth should be “twisted” by the Earth’s rotation. A lot of the coverage has focused on the troubled history of the mission (as did the press conference, apparently), but scientifically it’s very impressive.

The shift measured is very, very small– 0.04 arcseconds over the course of a year, or 0.000011 degrees– but agrees nicely with the predictions of relativity. I’m not sure whether to try to work this into the book-in-progress as I revise it– I suspect it’s too complicated to fit into a sentence or two, and I don’t want to take much more space– but it got me thinking about something that does get said quite a bit, namely relativity’s status as one of the best-tested theories in the history of science.

When you get down to it, there are really only two theories in the running for the title of “The Most Precisely Tested Theory in the History of Science”: relativity and quantum mechanics, specifically quantum electro-dynamics (QED). Both theories predict tiny shifts in quantities that are well known from other theories– the rate of ticking of a clock, or the energy difference between two states of an atom– and in both cases, those predictions have withstood a huge battery of experimental tests. There is no question that both general relativity and QED are correct theories, at least within their well-understood limits. (Somewhat embarrassingly, the two don’t play nice together, so neither works well in contexts where both gravity and quantum effects are important. We can’t access any of those contexts experimentally at present, though.)

So, which of the two is *The* Most Precisely Tested Theory in the History of Science?

It’s a little tough to quantify a title like that, but I think relativity can claim to have tested the smallest effects. Things like the aluminum ion clock experiments showing shifts in the rate of a clock set moving at a few m/s, or raised by a foot, measure relativistic shifts of a few parts in 10^{16}. That is, if one clock ticks 10,000,000,000,000,000 times, the other ticks 9,999,999,999,999,999 times. That’s an impressively tiny effect, but the measured value is in good agreement with the predictions of relativity.

In the end, though, I have to give the nod to QED, because while the absolute effects in relativity may be smaller, the precision of the measurements in QED is more impressive. Experimental tests of relativity measure tiny shifts, but to only a few decimal places. Experimental tests of QED measure small shifts, but to an absurd number of decimal places. The most impressive of these is the “anomalous magnetic moment of the electron,” expressed is terms of a number *g* whose best measured value is:

g/2 = 1.001 159 652 180 73 (28)

Depending on how you want to count it, that’s either 11 or 14 digits of precision (the value you would expect without QED is exactly 1, so in some sense, the shift really starts with the first non-zero decimal place), which is just incredible. And QED correctly predicts all those decimal places (at least to within the measurement uncertainty, given by the two digits in parentheses at the end of that).

The Lamb shift in hydrogen hasn’t been measured quite as well, but is also known to many digits, with a measured value of 1057.864 MHz according to this page, which was the clearest statement a quick Google turned up. And there are some other shifts and corrections that have been measured to similar precision.

Now, admittedly, I’m a little biased, in that the QED experiments are closer to my own areas of physics, but for my money, it’s a more impressive accomplishment to correctly predict a small shift to ten decimal places than a tiny shift to two. So, if I had to name one of the two theories as *The* Most Precisely Tested Theory in the History of Science, Einstein loses to Feynman, Schwinger, and Tomonaga. They’re both amazing accomplishments, though, and nothing else comes close.

(Of course, it should be noted that the two theories are not completely distinct– QED is the result of combining quantum mechanics with special relativity, so Einstein had a hand in both, albeit indirectly. General relativity, though, while it incorporates special relativity, is its own thing, and doesn’t mesh well with QED at all.)

There’s a bit of a caveat on the QED precision. If we had even better experiments, everyone basically expects GR’s precision to *keep going* for another ten or twenty decimal places.

In QED, on the other hand, we’re getting close to the end of the “precise” part. You can do a perfect QED calculation, at tree-level, that says g/2=1. You do the calculation at one-loop level and it says g/2 = 1.0011 or something. You add more and more photon/electron loops and eventually you have a calculation that says g/2 = 1.001 159 652 180 73. But … the next bit of the calculation, at least for the muon g-2, isn’t QED any more. You have to start putting *mesons* into the loop diagrams, and those meson calculations simply aren’t precise. They may never be.

So: maybe QED wins in as the most precise experiment-theory correspondence of 2011. But QED has a big, known wall looming on the *theory* side, whereas GR theory is nowhere near the edge of its expected domain of accuracy.

@BM: There is something I never quite understood about the precision of these g-factor calculations. The calculated value depends on the fine-structure constant, right? But the value of that constant is only known to about 5 significant digits, IIRC. So, how can you calculate from that a value with a precision of 11 digits?

@Chad:

I never quite liked statements like these, which essentially say that quantum electro-dynamics is a part of or an application of quantum mechanics. IMO, one should say “quantum theory” here instead of “quantum mechanics”. After all, what is usually known as quantum mechanics (the stuff one learns in basic courses) is essentially the quantization of classical mechanics, whereas QED is the quantization of classical electrodynamics, and quantum field theories in general are quantizations of classical field theories. I think saying “quantum mechanics” when one talks about something which essentially has nothing to do with mechanics is quite misleading. “Quantum theory”, on the other hand, is simply a generic term which encompasses all the quantum “things” mentioned above.

I know that complaining about this to someone who has written a book on quantum mechanics himself sounds quite a bit arrogant – but this point has annoyed me for years now. Your thoughts on this…?

I always say “quantum physics”…

@david: “quantum physics” is also o.k. 😉

There’s a bit of a caveat on the QED precision. If we had even better experiments, everyone basically expects GR’s precision to *keep going* for another ten or twenty decimal places.Right, but it hasn’t been tested there. And I don’t think the current prospects for testing GR to another ten decimal places are any better than the prospects for carrying out QED calculations to higher precision.

The calculated value depends on the fine-structure constant, right? But the value of that constant is only known to about 5 significant digits, IIRC. So, how can you calculate from that a value with a precision of 11 digits?If you follow the link to the arxiv preprint about the g-2 results, you’ll see that they cast it as a measurement of the fine structure constant “with alpha^{-1} = 137.035 999 084 (51) [0.37 ppb], and an uncertainty 20 times smaller than for any independent determination of alpha.” I think there may even be a more recent measurement than that one, which I didn’t link because it only provides a value of alpha, not a clear value of g.

I never quite liked statements like these, which essentially say that quantum electro-dynamics is a part of or an application of quantum mechanics. IMO, one should say “quantum theory” here instead of “quantum mechanics”.I’ve never seen a clear distinction being made between “quantum mechanics,” “quantum theory,” and “quantum physics.” People writing about the subject tend to use them more or less interchangeably as far as I can tell. Maybe there

oughtto be a clear distinction between them, but I’m just following the (lack of) pattern of other people who have written about the subject before me.@Chad: First, thanks for mentioning the preprint. I’ve always heard the results stated for g-2; I was not aware that this is used for determining alpha instead.

Second: yes, you are right – it’s quite usual that people use the term “quantum mechanics” instead of “quantum theory”, there is a consistent pattern. And yes, I think there

oughtto be a distinction. Unfortunately, I don’t think that my rant here (and sometimes in other places) will change anyone’s mind about that…Albert Einstein created General Relativity almost one hundred years ago. I have created a more encompassing theory entitled

Beyond Albert Einstein’s Relativity: UFT PHYSICS (by Peter D Rodgers of Brisbane, Australia). My physics paper gives the same results as General Relativity does in the realms measured. But equations from my paper can predict much more than General Relativity predicts.

My paper is at

http://uftphysics.webs.com/

on the internet

or

http://catalogue.nla.gov.au/Record/4514397

in the Australian National Library.

I hope that you enjoy studying it.

Peter Donald Rodgers (BA Double Major Mathematics UQ)

Most instances of “quantum theory” that I’ve heard recently refer specifically to the theory of quantum information/probablility/statistics/computation. But that might be the circles I hang out in.

Biologists have an argument that we have common descent down to way more accuracy than either one. It’s a different sort of calculation, but out of all possible relationship among species ( which far exceeds the number of particles in the universe) we have it down to a fairly narrow corner.

Hmmm. Apropos your post about lab reports …

One of those measurements is more precise, but is a less accurate test of the theory.

When you say “Somewhat embarrassingly, the two don’t play nice together, so neither works well in contexts where both gravity and quantum effects are important” as many others have done I always feel a little off. That is absolutely correct, but … QED doesn’t work without relativity – special relativity. So many people hear the “Relativity vs QED” thing and don’t get that first point. I think it is important to note that you wouldn’t have your exact prediction in QED without Relativity.

Yes General Relativity is more, well, general and has issues with QM, and is what is being tested in the given experiment, but I think it really misleads as to how our main physical models are connected to keep pounding that “Relativity vs QM” thing.

The QED vs. GR argument here forgets one important fact: without precise measurement standards based on QED, precise measurements of GR would not be possible. Therefore QED is necessarily more accurate.

Note also that we have to qualify the GR measurement with the caveat that our best value of Newton’s constant G is pretty coarse compared to, say, alpha. We do however have a pretty good number for G.M where M is the mass of the Earth, thanks to measurements in orbital mechanics. Neither G nor M is measured to particularly high accuracy, but G.M for Earth (and other planets) can be derived from observation of orbiting objects to a higher degree of accuracy.