Part II, forthcoming. In place of the real numbers, by which the concept of manifold has been defined so far, we could take other number fields and thus arrive, e. In this part of the article we do not need to take into account this generalisation. In most of the cases, the additional dimensions were taken to be spacelike; nevertheless, manifolds with more than one direction of time also have been studied. In his letter to Einstein of 11 November , he writes [ ], Doc. Perhaps, you are luckier in the search. I am totally convinced that in the end all field quantities will look alike in essence.
But it is easier to suspect something than to discover it. Various reasons instilled in me strong reservations: […] your other remarks are interesting in themselves and new to me. Ishiwara, and G. The result is contained in Hilbert , p. The hints dropped by you on your postcards bring me to expect the greatest.
According to him, the deviation from the Minkowski metric is due to the electromagnetic field tensor:. He claims to obtain the same value for the perihelion shift of Mercury as Einstein [ ], p. The meeting was amicable. In this context, we must also keep in mind that the generalisation of the metric tensor toward asymmetry or complex values was more or less synchronous with the development of Finsler geometry [ ].
Although Finsler himself did not apply his geometry to physics it soon became used in attempts at the unification of gravitation and electromagnetism [ ]. The idea that they keep together the dispersing electrical charges lies close at hand.
Haramein is there a fractal equation for a rotating dual-torus? The continuous symmetry of rotations in space gives rise to the conservation of angular momentum. They are said to form a kind of space called a Calabi-Yau space after Eugenio Calabi of the University of Pennsylvania and Shing-Tung Yau of the University of California at San Diego ; they may also form a generalnation of such a space called an orbifold. Various subcases of affine spaces will occur, dependent on whether the connection is asymmetric or symmetric , i. Remember that in the original dual-resonance theory the number of spacetime dimensions is 26; in superstring theory 10 dimensions are required.
Thus, the idea of a program for building the extended constituents of matter from the fields the source of which they are, was very much alive around Naturforscherversammlung, 19—25 September [ ] showed that not everybody was a believer in it. He claimed that in bodies smaller than those carrying the elementary charge electrons , an electric field could not be measured.
I wish to see this reason in the fact that it is altogether not permitted to describe the electromagnetic field in the interior of an electron as a continuous space function.
The electrical field is defined as the force on a charged test particle, and if no smaller test particles exist than the electron vice versa the nucleus , the concept of electrical field at a certain point in the interior of the electron — with which all continuum theories are working — seems to be an empty fiction, because there are no arbitrarily small measures. Einstein whether he approves of the opinion that a solution of the problem of matter may be expected only from a modification of our perception of space perhaps also of time and of electricity in the sense of atomism, or whether he thinks that the mentioned reservations are unconvincing and is of the opinion that the fundaments of continuum theory must be upheld.
If, in a certain stage of scientific investigation, it is seen that a concept can no longer be linked with a certain event, there is a choice to let the concept go, or to keep it; in the latter case, we are forced to replace the system of relations among concepts and events by a more complicated one.
The same alternative obtains with respect to the concepts of timeand space-distances. In my opinion, an answer can be given only under the aspect of feasibility; the outcome appears dubious to me. But a more precise reasoning shows that in this way no reasonable world function is obtained. It is to be noted that Weyl, at the end of , already had given up on a possible field theory of matter:.
To me, field physics no longer appears as the key to reality; in contrary, the field, the ether, for me simply is the totally powerless transmitter of causations, yet matter is a reality beyond the field and causes its states. Klein on 28 December , see [ ], p. Yet it retains part of its meaning also with regard to questions concerning the constitution of elementary particles.
Because one may try to ascribe to these field concepts […] a physical meaning even if a description of the electrical elementary particles which constitute matter is to be made.
Only success can decide whether such a procedure finds its justification […]. During the twenties Einstein changed his mind and looked for solutions of his field equations which were everywhere regular to represent matter particles:. Let us move into the field chosen by him without too much surprise to see him apparently follow a road opposed to the one successfully walked by the contemporary physicists. After , Einstein first was busy with extracting mathematical and physical consequences from general relativity Hamiltonian, exact solutions, the energy conservation law, cosmology, gravitational waves.
Thus, while lengths of vectors at different points can be compared without a connection, directions cannot.
Indranu Suhendro. Spin-Curvature and the Unification of Fields in a Twisted Space. Spinn-krökning och föreningen av fält i ett tvistat rum. Swedish physics. PDF | A first foundational sketch of the Generalized Theory of Symplectic- Kinemetric Spin-Curvature Structures, creating a purely geometric.
This seemed too special an assumption to Weyl for a genuine infinitesimal geometry:. A metrical relationship from point to point will only then be infused into [the manifold] if a principle for carrying the unit of length from one point to its infinitesimal neighbours is given.
In contrast to this, Riemann made the much stronger assumption that line elements may be compared not only at the same place but also at two arbitrary places at a finite distance. At a point, Equation 98 induces a local recalibration of lengths l while preserving angles, i. If, as Weyl does, the connection is assumed to be symmetric i. With regard to the gauge transformations 98 , remains invariant. From the 1-form dQ , by exterior derivation a gauge-invariant 2-form with follows.
Let us now look at what happens to parallel transport of a length, e.
If X is taken to be tangent to C , i. The same holds for the angle between two tangent vectors in a point cf.
Yet, also today, the circumstances are such that our trees do not grow into the sky. Due to the additional group of gauge transformations, it is useful to introduce the new concept of gauge-weight within tensor calculus as in Section 2. Weyl did calculate the curvature tensor formed from his connection but did not get the correct result ; it is given by Schouten [ ], p.
His Lagrangian is given by , where the invariants are defined by. Weyl had arranged that the page proofs be sent to Einstein. In communicating this on 1 March , he also stated that. In the most general case, the equations will be of 4th order, though. He then asked whether Einstein would be willing to communicate a paper on this new unified theory to the Berlin Academy [ ], Volume 8B , Document , pp. Einstein was impressed: In April , he wrote four letters and two postcards to Weyl on his new unified field theory — with a tone varying between praise and criticism.
His first response of 6 April on a postcard was enthusiastic:. It is a stroke of genious of first rank. Nevertheless, up to now I was not able to do away with my objection concerning the scale. However, as long as measurements are made with infinitesimally small rigid rulers and clocks, there is no indeterminacy in the metric as Weyl would have it : Proper time can be measured. As a consequence follows: If in nature length and time would depend on the pre-history of the measuring instrument, then no uniquely defined frequencies of the spectral lines of a chemical element could exist, i.
He concluded with the words. Only for a vanishing electromagnetic field does this objection not hold. Only in a static gravitational field, and in the absence of electromagnetic fields, does this hold:. Einstein saw the problem, then unsolved within his general relativity, that Weyl alluded to, i. Presumably, such a theory would have to include microphysics.
But I find: If the ds , as measured by a clock or a ruler , is something independent of pre-history, construction and the material, then this invariant as such must also play a fundamental role in theory. Yet, if the manner in which nature really behaves would be otherwise, then spectral lines and well-defined chemical elements would not exist.
Another famous theoretician who could not side with Weyl was H. However, Weyl still believed in the physical value of his theory. There exists an intensive correspondence between Einstein and Weyl, now completely available in volume 8 of the Collected Papers of Einstein [ ].