Theory overview Concept of Cartan-Einstein Unification Theory.
In this web site, Cartan-Einstein unification theory means the physical theory that emerges from the research program consisting in assuming Finslerian teleparallelism (TP) and letting the postulate of logical homogeneity of theoretical physics and geometry run its course. In other words, it is just a matter of cranking the machine. In the process, we occassionally use mathematical tools (Finsler geometry, Kähler calculus, Kaluza-Klein spaces) which may give the impression that one has additional postulates. Let us set the record straight. We use the most comprehensive and powerful mathematical tools that we can handle. If we use a Finslerian setting, it does not mean that we are postulating that the geometry of spacetime is Finslerian, in the same way as the use of curvilinear coordinates in Euclidean space do not make it Riemannian. If our results discover physical laws by letting the geometry be properly Finslerian, that can hardly beä called a postulate. Also, the fifth dimension of our Kaluza-Klein (KK) space, proper time, is one which, like time before Einstein, was already part of physics; it is not one which curls, or hides from us in a micro-micro-micro cosmos. In addition, both the KK space and the Kähler calculus are just new stages in the natural evolution of geometry and the calculus, respectively (see papers 24 - 25).
Origin of Cartan-Einstein Unification Theory.
This theory started with an attempt by Einstein to transcend general relativity (GR) through the postulate of TP. He failed in his trying to implement this postulate in spite of the correct advice given to him by Élie Cartan. Given the pedigree of the theory, the circumstance of Einstein’s failure and the significant inroads made by the present authors, the Cartan-Einstein Unification theory and program can only be viewed as an incredible physical opportunity which has been overlooked. For a description of the postulates see “Theory > Postulates” in this web page.
Einstein gave up in frustration because he did not establish contact with either electrodynamics or GR. Cartan did not pursue the effort beyond his attempt to help Einstein with the first stages of the implementation of his hypothesis. These authors redirected the Cartan-Einstein effort. In the pre-Finslerian setting, there are not enough degrees of freedom and, more importantly, there is not enough structural richness in the set of differential forms that define a space. The freedom must be such that it should permit one to look at the differential invariants from a perspective which need not always be geometric, so as to allow for classical and quantum physics. Einstein would have been extremely pleased to realize that the contact of there is very strong contact of TP with both GR and electrodynamics. And he would have been pleasantly surprised that there is a way of viewing the equations of quantum mechanics in a way which is fully consistent with the geometric spirit, and with a flavor of continuous differential equations rather than the quasi-mystical flavor of quantum mechanics at the time of the Einstein-Bohr battles.
The Role of the Potentials.
What emerges is a “mother of the physics”, so to speak: different physical theories arise as a function of how one chooses to handle the differential invariants (w0, wi, w0i, wjk, i=1,2,3) that define a Finsler bundle. These are the potentials that unify the different parts of the physics. Whether one should go one or two levels up from the curvature to get the potentials (i.e. whether one views the potentials in gravitation as being the metric or the Christoffel symbols is irrelevant). From the fields to the potentials is the old thinking. Forget the fields, as it is misguided to look at their unification, except when convenient for heuristic purposes. The unification must take place at the highest level: the fundamental differential invariants.
For instance, classical mechanics (in the strict sense of a textbook with that title), is the theory of motion that derives from the absolute integral invariant òdw0, in the sense that w0 becomes the Hilbert integral when the torsion is an exact differential). The unified equations of motion of gravitation and electrodynamics emerge inescapably from the equation of the autoparallels in the Finsler bundle. The homogeneous pair of Maxwell’s equations is contained in the first Bianchi identity. The second pair is phenomenological and will therefore have to be derived from the closed system of fundamental equations defining the structure, specifically from the torsion side of the theory. The wjk are the left-invariant forms of SO(3). The Einstein equations and the electromagnetic energy-momentum emerge most clearly in the Kaluza-Klein space generated by (w0, wi, w0i). These differential forms are in turn the solution for a system of equations consisting of two pieces: gravitational and quantum mechanical. The first one is simply the second equation of structure, i.e. the specification of the affine curvature in the Kaluza-Klein space. The second one is a reformulation of the equations that specify the torsion, which is now given by a Kähler equation. This equation is just a generalization of the Dirac equation, except that it emerges naturally in the calculus of differential forms. In other words, the pillars of the physics are not the Maxwell, Einstein and Dirac equations, but just appropriate generalizations of Einstein and Dirac, united ab initio. Maxwell’s theory will have to be viewed as a corruption of quantum mechanics. We are pleased that this point of view has been advanced also by the eminent present day physicist and technology innovator Carver Mead.
Historical Circumstance in Einstein’s Geometrizations in the 1915-1930 Period.
In 1915, Einstein created GR, using the only non-flat geometry available at the time: “the Riemannian geometry of 1915”. Notice that we have not said simply “Riemannian geometry”. The reason is that, in 1915, the spaces of Riemann were false spaces of Riemann (according to Cartan in his 1924 paper “On the Recent Generalizations of the Notion of Space”). He explained that they ceased to be so in 1917, year when Levi-Civita (LC) increased the contents of Riemannian geometry through the addition of concepts of an affine nature (as is the concept of equality of vectors at different points, which simply meant comparison of vectors along a curve for LC) to the purely metric concepts of the original Riemannian geometry (angles, distances, metrics, and related concepts and statements). LC did not create new mathematical objects to represent the new concepts, but simply took some of the old concepts (called Christoffel symbols) and made them play new roles. By the very early twenties, the Levi-Civita option rather than the not yet known TP option had already entered GR without question, unaware as everybody was of the consequence: it usurped by default the throne of the only alternative that makes sense for physics, TP, which was not yet born. The adoption of the affine relations of LC has not had any positive consequence for the GR of 1915.
Soon after LC, Weyl and Eddington started to think (independently of each other) of separating the affine and metric roles. But it was Cartan who produced a series of three major papers on the general theory of spaces endowed with affine concepts and where the metric relations were viewed in the context of the affine relations. He also produced many other papers on several other connections. In 1922, Cartan was already instructing Einstein on TP at the home of the French mathematician Hadamard, in Paris (See the Cartan-Einstein correspondence, edited by R. Debever).
By the late 1920’s, Einstein had realized the serious negative implications of the LC connection. Unaware of Cartan’s publications, he proposed that physics use an alternative to the new Riemannian geometry (i.e. to the extended Riemannian geometry resulting from LC’s), alternative which he called absolute parallelism, modernly called TP. In his correspondence with Cartan, Einstein would claim in 1929, not to have understood Cartan at all during the former’s visit to Paris of 1922. Einstein became interested in TP because, as he put it, there is no parallelism at a distance, i.e. geometric equality, in Riemannian geometry. Our note: this is the case both, without any affine connection, and with the LC connection, but for different reasons. For practical purposes, postulating TP means using only differentiable manifolds where there is a relation of geometric equality between vectors at different tangent spaces; no more, no less. Einstein failed to understand that TP contains the same metric relations as Riemannian geometry. Given the type of problems that he had been concerned with, it would not have been humanely possible to also be in possession of the required mathematical tools. He failed to realize in particular that the addition to the original Riemannian geometry of the affine relations of TP need not contradict the GR of 1915.
Unlike the effort by Einstein, who postulated TP and charged ahead, the present authors meandered their way into the present version of the theory. It all started with the effort to find a torsion such that the equations of the autoparallels (or lines of constant direction) would yield the equations of the motion with Lorentz force. It was clear that the result obtained for the equations of the motion required the connection to be Finslerian. We were fortunate enough to have the services of the highly underrated differential topologist Yeaton H. Clifton. He viewed Finsler geometry as pertaining to bundles which are refibrations of the same set of frames that constitute the usual bundles of Lorentzian connections. He also approaching that geometry with Cartan’s very intuitive approach -which has virtually disappeared from the literature- since such intuitive approach is crucial in understanding Kähler’s calculus for quantum mechanics and GR.
The geometric Einstein equations obtained from the first equation of structure of TP give rise to a term which becomes in essence the standard electromagnetic (EM) energy-momentum term. But there is another term whose integral over all of space is zero, i.e. similar in this regard to the one used to repair the non-symmetric and non-gauge-invariant canonical EM energy-momentum tensor. As Feynman acknowledged in his Lectures on Physics (Vol. 2, Chapter 27, Section 4), one needs a dedicated experiment that measures the gravitational field to reflect how EM energy-momentum is distributed, as a result precisely of terms of such a type. It can be locally huge, relatively speaking, for appropriate values of parameters and region of space. It is also clear that any term of this type must take positive values in some places and negative ones in other places. All this also applies in the standard theory, but it is ignored or swept under the rug. We solved the geometric Einstein equations of this unified theory for the case of a charged point particle. The symmetry of this problem allows the derivation in both cases of a simple relationship between the electric field E and the gravitational field g’ created by the electric field. The value of g’ in this case is (G/2)1/2 E, where G is Newton’s constant and where all quantities are in the cgs system. The reader should be aware of the fact that we have given the result as if the effect depended on the field. Not so: when E is 1/r2, and therefore inhomogeneous, the expression (-1/2)rdE/dr also is 1/r2 and thus equal to E. But the most important experimental implication of the Cartan-Einstein unified theory will not likely be in the generation of gravitation by electrical means (because it comes at the price of having huge inhomogeneous fields), but rather in the generation of gravity by high magnetic fields. The reason is obvious: nobody is killed by those fields. Of course, there may always be surprising effects in such a formidable theory.
The theoretical implications are two numerous to be mentioned here, and extend to the realm of quantum mechanics (see panes on Publications and on Preprints). We shall expand here on gravitation, since this is the field of physics more directly related to the postulate of TP. The left hand side of the by-Einstein-postulated field equations of 1915 consists of a geometric object which is common to the old and the new (i.e. with LC enhancement) Riemannian geometries, and also to the TP enhancement of the old Riemannian geometry. In 1915, Einstein chose the right hand side to be constituted by, in principle, a pot-pourri of non-gravitational energy-momentum tensors, coming from all branches of physics. As he reportedly said, the left hand of his equations was made of diamond and the right hand side was made of base wood. In contrast, it is a direct consequence of TP that the right hand side of Einstein’s equations is already given by the geometry. The problem now is to identify the physical quantities represented in the geometric terms of the right hand of the Einstein’s equations of TP. Only the very modern mathematics permits the physical interpretation of the right hand side.
Most important for the long term success of this unified theory is the emergence in it of quantum physics, which was completely absent from the work of both Cartan and Einstein. This relation was achieved in a deep way with the work of the great mathematician Erich Kaehler in 1960-1962, who generalized Cartan’s exterior calculus like he had generalized in 1934 Cartan’s theory of exterior systems (which is in essence the theory of systems of partial differential equations).
Kaehler took his calculus far enough to show the geometric roots of the Dirac equation, and thus of Quantum Mechanics. His calculus, however, also had to be extended beyond Kaehler’s own work in more than one way13 (beyond the Levi-Civita connection, beyond tensor-valued clifforms into Clifford-valued clifforms, beyond the usual connections into the realm of the Kaluza-Klein connections associated with Finsler bundles). In this way the Käehler calculus, in addition to providing a language for dealing with both GR and quantum mechanics, brings a geometrized EM field to the Dirac equation. It further enhances the quasi-geometric nature of his Dirac equations.
A word about Kaluza-Klein connections. The canonical Kaluza-Klein theory of Finslerian teleparallelism amounts to a more “dualistic” presentation of geometry than standard differential geometry is, dualistic meaning “of particles in fields”. Fields are now represented in the spacetime part of the Kaluza-Klein space and particles are represented in the fifth dimension. There will be, of course, other ways to approach particles which are even more important than this, namely springing from the geometric structure as “spacetime condensates”. The details as to how this will take place still constitutes an open question.
Jose Vargas and Doug Torr will make every effort to respond to questions on their papers and the subject of Cartan-Kähler Unification.