The Cartan-Einstein unification theory is based on the Einstein postulates of Postulates(a) Teleparallelism (TP) and (b) Logical Homogeneity of Geometry and Theoretical Physics. Some may argue that we are also assuming that (c) the world is Finslerian.
Here is the key point about this issue. One can view a pre-Finslerian
geometry from a Finslerian perspective. This is similar to computing in
Euclidean space with curvilinear coordinates. If one computes in the
Finsler bundle the autoparallels in the base space, one cannot avoid
identifying the electromagnetic field with a piece of a properly
Finslerian torsion.(a) Teleparallelism. In
1915, GR had been formulated in terms of what is called the curvature
of Riemann. The latter was a purely metric concept, as there was not
yet a concept of comparing vectors at a distance. That came only in
1917 with the Levi-Civita connection (LCC). The Christoffel symbols
existed, but for non-affine purposes. But the LCC, being path
dependent, does not provide a concept of equality of vectors at a
distance. Thus, one cannot integrate vector-valued integrands. In the
late 1920’s, Einstein grabbed the bull by the horns and postulated TP, i.e. that the connection has to provide equality of vectors at a distance.
Einstein did not succeed in implementing TP. For what he did on TP,
see his papers of 1930 (“Theorie unitaire…” and “Auf die
Riemann-Metric…”). Complete reference is given in “Documentation> Bibliography”.The most immediate consequence of TP is that the affine curvature is zero, which in turn implies the form of the metric curvature in terms of the torsion and its derivatives. That was realized by Cartan. By contraction, one obtains that the Einstein tensor equals a combination of geometric terms. But Cartan did not connect the torsion with the electromagnetic field. Suppose that one day TP were to prove its worth and become the new paradigm. The generations trained in the new paradigm would say: did physicists in the 20th century not understand that, with the LCC, one cannot integrate energy-momentum distributions? TP is a logical necessity! GR without any affine connection makes more sense that GR with LCC. (b) Logical Homogeneity of Geometry and Theoretical Physics.
Einstein subscribed during the twenties and thirties to the thesis of
(to use his own words) “Logical Homogeneity of Geometry and Theoretical
Physics”. To explain it, he used the example of the oldest branch of
physics, stating that “The propositions of Euclid contain affirmations
as to the relations of practically rigid bodies”. Many similar
statements can be found in “Geometry and Experience” (1921), “On the
Methods of Theoretical Physics” (1933) and “The Problem of Space, Ether
and the Field in Physics” (1934), specially the first two.The modern version of the postulate of logical homogeneity would be that the “structural equations” of the postulated geometric structure must constitute the field equations of the physics. Einstein
did not have the concept of equations of structure that would have
allowed this formulation of his “thesis”, i.e. of our postulate of
logical homogeneity. Without ever speaking of this postulate, which
Einstein did not mention in his work on TP, Cartan had discovered it on
his own and given a mathematical version of it when he distinguished
between the work on unification by physicists and demiurges. A demiurge
is one whose field equations imply what the geometry of the space is,
and not just take it for granted. Since the geometry is specified by the
equations of structure and Bianchi identities, it is natural to
identity them with the field equations, so that the latter will then
automatically imply what the structure is.In order to succeed, Einstein should have been a demiurge. He was not. He ignored both the affine and metric curvatures in his proposed system of field equations. This was his most serious error, since that very equation fully geometrizes Einstein’s gravitational field equation and reveals the presence of a new, non-quadratic electromagnetic energy momentum term. (c) Finslerian Connections. The
authors of this web site are developing TP in a Finslerian setting.
Finsler geometry is a web of research undertakings by different schools,
which pursue different agendas. It is most commonly defined as the
geometry of spaces where the length of curves of general form (i.e. not
necessarily quadratic) is defined. But, as Cartan said, “the metric does not contain all the geometric reality of the space… one can define the space by its equations of structure.” The
equations of structure are a property of the affine connection, and,
in Cartan’s way of doing geometry, connections live in the frame
bundles of spaces (this is in his 1922 paper, “On the equations of
structure …”).One can have properly Finslerian connections even if the metric of the space is the usual Riemannian metric. Furthermore, the usual metric compatible affine connections with torsion are particular cases of Finslerian connections and can, therefore, always be represented in Finsler bundles. If one obtains the equations of the autoparallels in the Finslerian framework using the connection that results from solving the system constituted by the first equation of structure and the compatibility of metric and connection, one obtains equations with terms that take the form of the combined gravitational and EM equations of motion. The connection happens to be Finslerian after all, but in a deceptively simple way. And one does not even need TP to get this result. One might say that this postulate is not so much a postulate as a consequence of the previous one. Finsler geometry provides a more general geometric framework than Riemann plus torsion, and it allows for greater flexibility in organizing the differential invariants that characterize the spacetime. For instance, those invariants also define a rich Kaluza-Klein space. It suffices to crank the machinery constituted by those three postulates in order to, apparently, get the physics. In those areas where we have reached so far, we reproduce the physics. When there are discrepancies, the new version looks only richer and more nuanced. |
Jose Vargas and Doug Torr will make every effort to respond to questions on their papers and the subject of Cartan-Einstein-Kähler Unification. Authors |