I hadn’t thought until recently about the fact that this year is the 100th anniversary of Einstein’s discovery of the field equations of general relativity, so there will be quite a few events taking place commemorating this (for a list of some, see here). This week’s Science magazine has a special issue on the topic. It includes news stories about LIGO and gravitational waves, new tests of the equivalence principle, and possible tests of GR from observations of the black hole at the center of our galaxy.

There’s also a review of a book from a few years ago about Einstein’s search for a unified theory, Einstein’s Unification by Jeroen van Dongen. The review addresses something I mentioned in my recent essay about mathematics and physics, that the development of GR provides a good example of a successful theory coming out of not just experiment and “physical intuition”, but motivated also by the serious use of deep mathematical ideas. According to the review:

Einstein employed two strategies in this search [

for the GR field equations]: either starting from a mathematically attractive candidate and then checking the physics or starting from a physically sensible candidate and then checking the mathematics. Although Einstein scholars disagree about which of these two strategies brought the decisive breakthrough of November 1915, they all acknowledge that both played an essential role in the work leading up to it. In hindsight, however, Einstein maintained that his success with general relativity had been due solely to the mathematical strategy. It is no coincidence that this is the approach he adopted in his search for a unified field theory.

Besides the fact that Einstein said so, other evidence for the primacy of the mathematical strategy in this case is the simultaneously successful work by mathematician David Hilbert, who was definitely pursuing the mathematical strategy.

While I think there’s an excellent argument that a mathematical approach was crucial in Einstein’s discovery of the field equations, the later history this book deals with also shows the dangers this can lead to. Einstein spent much of the rest of his life on a fruitless attempt to get a unified theory by pursuing the same mathematics he had so much success with in the case of GR. It’s a good idea to keep in mind both examples. On the one hand, trying out some new deep mathematical ideas can lead to success, on a time scale of a few years. On the other, if you’ve spent 30 years pursuing a mathematical framework that has gone nowhere, maybe you should do something else. A lesson that Einstein’s successors at the IAS might want to keep in mind…

The story about new tests of the equivalence principle contains the usual nonsense about testing “string theory predictions”:

Using beryllium and titanium, they found gravitational and inertial mass equal to one part in 10 trillion, as they reported in Physical Review Letters in 2008. That’s not quite precise enough to test string theory predictions.

That “string theory predicts violations of the equivalence principle” is what used to be called a “factoid”, something not true repeated so often that it becomes a fact. It seems though that usage has changed, with “factoid” now often being used to refer to something true. A new word is needed.

**Update**: See here for an article by Michel Janssen and Jurgen Renn discussing in detail the question of the “mathematical” versus “physical” strategies in Einstein’s discovery of the GR field equations.

Last Updated on

Experimentalist: “how can I use BRST cohomology to proof that the AdS/CFT duality prediction of genus one string correction to beryllium inertial masses is not an artefact of gauge fixing the superconformal symmetry of the D5-BPS black hole background?”

String theorist: “What is beryllium?”

I wonder how much of the Science special issue will remain available online as full text. It deserves to remain available that way indefinitely.

Based on the above excerpt, I find it interesting that the book review seems to gloss over a historical viewpoint which I thought was more or less standard, namely that Einstein was struggling to find a precise formulation of geodesic deviation, presumably without knowing at the start what that notion was.

To put it another way, if one senses that gravity in some sense breaks the rigid metric structure of spacetime as understood in special relativity, but that that metric structure must be preserved as a limiting case, one needs an appropriate mathematical framework to consider such things. Riemannian geometry turned out to be precisely that framework. The

newphysical realization that objects moving solely under the influence of gravity were following essentially inertial trajectories drove this search for a framework.The problem later on was that Einstein was unable to identify such a

keyphysical realization that could drive and guide a similar search for the mathematical structure of a unified theory. We are arguably in a similar position today, despite having vastly more mathematical resources at our disposal. Some would argue that the age of such key physical realizations has past, and so we can only grope furiously through the mathematical possibilities with some past successes to guide us (we hope).Perhaps what Einstein lacked during that last 30 years was a contact with a physical intuition in his mathematics that was as fruitful as the Equivalence Principle was for GR, eventually leading to Riemannian Geometry as the framework for GR. That may also be why Banesh Hoffman lamented the failure of Weyl’s original 1918 unified field theory to succeed at unifying gravitation and electromagnetism, because gauge invariance is such a seemingly reasonable idea, and leads to beautiful theories as well …

PS on my last comment:I really should have said “…especially withvastly more mathematical resources at our disposal.”. The increased size of the search space for appropriate mathematical structures makes the need for the right kind of physical guidance all the more crucial.(Note, by the way, that the obtuse identification of mathematically formulated physical theory with mathematical theory doesn’t help matters at all. This is about more than just the question of relative mathematical rigor.)

In general, string theory does predict a violation of the equivalence principle in some regime, since Newton’s constant couples to a scalar field (the dilaton), which generically mixes with other scalar fields.

If Galileo threw such scalar fields off the tower of pisa with a normal weight, they would not fall at the same rate.

The magnitude of this violation is of course completely undetermined (and things like Moduli Stabilization are supposed to make it as tiny as possible), but it is worth keeping in mind.

lun,

If the magnitude is completely undetermined, there is no prediction. It’s also true that some string theorists, when pressed for an example of a prediction of string theory, have argued for “no violation of the equivalence principle” as a prediction, see for example

http://www.math.columbia.edu/~woit/wordpress/?p=151

http://www.math.columbia.edu/~woit/wordpress/?p=559

This is a typical example of a “string theory prediction”, no actual prediction, with claims from string theorists of evidence for string theory if you see something or if you don’t.

Peter, the “reference” you give in the second link is, to put it very mildly, not up to standard of scientific literature, and the second link has very little to do with the equivalence principle (It is relatively easy to incorporate the variation of the fine structure constant into GR, just add a scalar field coupled to the photon with a VEV varying in time in the FRW metric).

Of course what you are saying about the lack of a quantitative prediction is right, especially since string theorists have invented really ingenious ways of making this prediction irrelevant for any conceivable experiment.

Nevertheless, since the driving principle which drove Einstein to develop his mathematics is the EP, it is worth keeping in mind that in string theory the geometrization of gravity is most likely a property of the EFT in certan vacua, not a fundamental principle of the full theory. Quite a few string theorists never even thought about it.

> A new word is needed.

mystification

@ lun,

“…in string theory the geometrization of gravity is most likely a property of the EFT in certain vacua”

What does EFT stand for ?

Thanks!

“EFT” is a conventional acronym for “Effective Field Theory”.

“The review addresses something I mentioned in my recent essay about mathematics and physics, that the development of GR provides a good example of a successful theory coming out of not just experiment and “physical intuition”, but motivated also by the serious use of deep mathematical ideas.”

Some would say it is the only such example.

I read the article about LIGO and GR waves. It was filled with words like ‘certainty’. Reminds me of SUS and the LHC back in the mid 2000’s. So what happens if nor waves are found by LIGO?

John,

The SUSY story is quite different. There’s strong indirect experimental evidence for gravitational waves (the pulsars), none for SUSY. The theoretical arguments for gravitational waves are much stronger also (it’s hard to come up with a theory without them, whereas the simplest theories are ones without SUSY).

It will be interesting if nothing is seen. I’d guess that at first the suspicion would fall on the astrophysics, that the modeling of production of gravitational waves is wrong, or the frequency of events that generate gravitational wave was wrong. But sooner or later it would become clear if there was a major problem with GR, and that would be quite revolutionary.

> “Some would say it is the only such example.”

Regrettably, “some” got a bad case of Gruppenpest and were never heard of again.

I was hoping somebody would use the anniversary to officially retire the “ball rolling on a rubber sheet” pseudo-explanation which is extremely confusing if you try to think about it at all.

srp,

The usual TIME-LIFE picture of a ball on a curved static rubber sheet is not even a pseudo-explanation. The main effect of gravity is that the rubber sheet is being sucked into energy (mass). The curvature of the rubber sheet is a small correction.

John, there are no guaranteed astrophysical sources in the frequency range of advanced-LIGO/VIRGO. So if nothing is seen, the blame will be on astrophysical sources.

Note that CMB polarization experiments also now on hot on the heal of this

and could be lucky if nature chooses a value of r within sensitivity range of upcoming experiments.

A couple of useful references for a modern study of the fundamental mathematics of General Relativity (as opposed to various black-hole solutions and AdS correspondences):

David B. Malament, Topics in the Foundations of General Relativity and Newtonian Gravitational Theory. University of Chicago Press, 2012

R. K. Sachs and H. Wu, General Relativity for Mathematicians. Springer-Verlag, 1977

These are (copyrighted) books, not papers or summaries. Although versions can be found online, they are worth their purchase price. (Although I was a student of David Malament at UCI, I have no financial ties to any of the above authors).