A. Einstein mystery

December 29, 2016 11:12 AM

Grandfather the mathematician took sabbaticals in Berlin in 1927 and 1934. There he became an acquaintance of Einstein. Years later he mentioned talking science to the fellow one afternoon, and I quote: "about a publication of the previous year - on the 'Fünf-bein' as I recall. 'Das ist alles Mist.'" Google fails me on what what the Fünf-bein might refer to, much less why it is all crap. Any thoughts appreciated.
posted by BWA to Science & Nature (8 answers total)

Response by poster: To be clear, it was Einstein saying it was all Mist.
posted by BWA at 11:14 AM on December 29, 2016

Could it be something to do with evolutionary theories about bipedalism?
posted by KateViolet at 11:43 AM on December 29, 2016

Best answer: My physics knowledge is basic but it appears that it has to do with Riemannian manifolds:

see here

And also The vielbein. The Fünfbein would be a five-dimensional manifold. I think relativity uses 4-dimension manifolds?
posted by dis_integration at 12:19 PM on December 29, 2016

Best answer: I believe it was probably Kaluza-Klein theory:

In physics, Kaluza–Klein theory (KK theory) is a unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the usual four of space and time. It is considered to be an important precursor to string theory.

The five-dimensional theory was developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Einstein in 1919,[1] and published them in 1921.[2] Kaluza's theory was a purely classical extension of general relativity to five dimensions. The five-dimensional metric has 15 components. Ten components are identified with the four-dimensional spacetime metric, four components with the electromagnetic vector potential, and one component with an unidentified scalar field sometimes called the "radion" or the "dilaton". Correspondingly, the five-dimensional Einstein equations yield the four-dimensional Einstein field equations, the Maxwell equations for the electromagnetic field, and an equation for the scalar field. Kaluza also introduced the hypothesis known as the "cylinder condition", that no component of the five-dimensional metric depends on the fifth dimension. Without this assumption, the field equations of five-dimensional relativity are enormously more complex.[clarification needed] Standard four-dimensional physics seems to manifest the cylinder condition. Kaluza also set the scalar field equal to a constant, in which case standard general relativity and electrodynamics are recovered identically.

I didn't see this in the Wikipedia article, but I think at one point Einstein changed his mind about the irrelevance of KK theory.
posted by jamjam at 12:31 PM on December 29, 2016 [8 favorites]

Best answer: Agreed that it was probably something to do with Kaluza-Klein theory. These are models in which one attempts to explain all known forces by explaining an extra dimension of space. This would require a mathematical description of a five-dimensional pseudo-Riemannian manifold, and a "fünfbein" (literally "five-legs") is a way of describing these manifolds using a set of five vectors at each point.

The American Physical society has a brief article on Einstein and Kaluza-Klein theory, if you're interested.
posted by Johnny Assay at 4:48 PM on December 29, 2016 [3 favorites]

Best answer: I believe that the first time that Einstein mentioned the "vierbein" (or rather n-bein) formalism was in 1928 [1], which to me would suggest that 1934 is more plausible as time when he would have mentioned some theory with fünfbein.

In fact, I believe that he could well have been referring to the two works titled Einheitliche Theorie von Gravitation und Elektrizität, co-authored with Walther Mayer, that appeared in 1931 and 1932 [2, 3]. This is somewhat related to Kaluza's and Klein's theory, but with a twist. I think this twist could reasonably be described as a "fünfbein theory," even though this term does not appear in the papers.

I'm going to attempt to give a layman's explanation of this vierbein stuff, in order to explain what I mean.

The initial development of general relativity was in a setting of pure Riemannian (or rather, pseudo-Riemannian, but I think that term is newer) setting -- you have a four-dimensional curved spacetime (a so-called manifold), and a way to measure distances (a metric), and that's that. The metric is fundamental. Using the metric, you can develop notions of "shortest" paths between points (the geodesics) and a way to measure geodesic deviation (how the displacement between two geodesics that start off at an infinitesimal distance from each other changes as you move along them), which leads to a notion of connection. And this mathematical framework is enough to develop general relativity.

In [1], Einstein (following work by Cartan and Weyl) added the so-called vierbeine, or tetrads, leading to the concept that he called "Fern-Parallelismus" (often translated as "teleparallelism"). The idea is to consider the curved spacetime to also have a frame field, a set of a four unit vector fields that are orthogonal to each other at every point.

To some extent, this "decouples" the space of vectors from the manifold itself: Normally, to even talk about the numeric value of a vector, you have to first introduce coordinates (say x, y, z, t) on your manifold; you can then describe a vector V as being (for example) "1 length unit in the x-direction, and 2 length units in the y-direction" (note that this also depends on a way to measure lengths, i.e. the metric). If you had a vierbein field, however (i.e. just four orthogonal vector fields), you could instead just say that V is (for example) "1 times the first 'bein' plus 2 times the second". This way, you can do calculations on vectors and vector fields without referring to a coordinate system in the manifold, or a metric.

It turns out that this freedom allows a wider class of connections on the space of vectors. I believe that Einstein hoped that the extra "components" of the connection could be used to describe electromagnetism, thereby unifying electromagnetism with his theory of gravity. He and Mayer wrote about this in several publications in the following years, but ultimately they realized it wouldn't work. A detailed history of this work is given in [4].

Instead, in 1931 and 1932, Einstein and Mayer revisited the five-dimensional idea of Kaluza and Klein, with one difference. The Kaluza-Klein theory assumed a five-dimensional spacetime, with one space dimensions "curled up" like a very small cylinder. Einstein and Mayer worked with an ordinary four-dimensional spacetime, except that they considered vectors on this space that could be five-dimensional. Obviously, in order to do this, you have to be able to think of vectors as independent from the coordinate system on your manifold. To do this, they used a framework quite similar to that of the tetrads, and I which I think "fünfbein" could be referring to.

From a work by Tilman Sauer, describing this five-dimensional approach of Einstein and Mayer in 1931 and 1932 [5]:

Einstein abandoned the distant parallelism approach when he realized that the tetrad formalism also allowed a different and new perspective on the Kaluza-Klein five-dimensional approach. Together with Walther Mayer with whom he had collaborated already during the final stages of the distant parallelism approach, Einstein now explored a variant of the five-dimensional idea that seemed sufficiently new in order to justify again taking up the Kaluza-Klein approach (Einstein and Mayer 1931, 1932a). The novelty of the approach was that it was no longer the space-time manifold which was enlarged by a fifth space-like dimension. Rather Einstein and Mayer constructed a five-dimensional vector space at each point of four-dimensional space-time. The tetrad formalism allowed for an easy generalization to five dimensions, simply by adding another linearly independent vector to the tetrads. The five-dimensional vector spaces obviously could no longer be identified with tangent spaces of the underlying manifold, but Einstein and Mayer gave a projective mapping from the five-dimensional vector spaces to the four-dimensional tangent spaces.

While Einstein and Mayer succeeded to derive the gravitational and electromagnetic field equations from this new five-dimensional approach, they could not account for the structure of matter. In their first paper, they concluded that the existence of charged particles or currents was incompatible with the field equations. They also remarked that an understanding of quantum theory was not yet conceivable in this approach (Einstein and Mayer 1931). In order to allow for the existence of charged material particles, Einstein and Mayer investigated a generalization of their initial framework. The generalization resulted in a new set of field equations. In a subsequent publication, they investigated mathematical properties of these new equations, specifically the problem of compatibity without, however, commenting on a possible physical interpretation of those equations (Einstein and Mayer 1932a).

Since the five-dimensional vector space approach again ran into difficulties, Einstein and Mayer once more tried another approach (van Dongen 2003). [...]

I haven't tried to understand the difficulties of their approach, but it seems like they abandoned the project shortly thereafter, so it seems plausible that Einstein could consider it "alles Mist" by 1934. Einstein kept working on unifying gravity with eletrocmagnetism using other approaches, but I think the modern verdict (with the benefits of heaps of hindsight) is that these attempts were never successful.

[1] Einstein, A. Riemman-Geometric mit Aufrechterhaltung des Begriffes des Fern-Parallelismus. Sitzungsberichte der Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse (1929) 227-221.
[2] Einstein, A. and Mayer, W. (1931) Einheitliche Theorie von Gravitation und Elektrizität. Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, 541-557
[3] Einstein, A. and Mayer, W. (1932) Einheitliche Theorie von Gravitation und Elektrizität, zweite Abhandlung. Sitzungsberichte der Preussischen Akademie der Wissenschaften zu Berlin, 130-137
[4] Sauer, T. Field equations in teleparallel space–time: Einstein's Fernparallelismus approach toward unified field theory. Historia Mathematica 33 (2006) 399-439
[5] Sauer, T. Einstein’s Unified Field Theory Program. Preprint (2007).
posted by water under the bridge at 9:37 PM on December 29, 2016 [8 favorites]

Response by poster: Yet again you all come through, and with far more detail than I could have imagined. Many thanks to all, this is most welcome.

(FYI, their other commonalities had to with progressive politics and world peace. One of the founders of the Iowa Civil Liberties Union, was grandfather.)
posted by BWA at 6:29 AM on December 30, 2016

Response by poster: (Actually, the year was 1931. My bad. Einstein left Berlin the following year. But the answers still hold water under the bridge's water.)
posted by BWA at 8:43 AM on November 13, 2017