Teoria da gravitação num espaço-tempo de Weyl não-integrável

Abstract

In 1918 the German Hermann Weyl developed a unified theory of gravitation and electro­magnetism becoming geometrical both interactions, that is, he associated the potential elec­tromagnetic a 1-form a, after considering that the length of a vector is not preserved under parallel transport as well as with the direction, this also meant that the covariant derivatives of the metric tensor ceased to be null becoming Vag" = a, The gravitational and electromagnetic field equations are obtained from the action I = f (R2 ± v Ft" ) —gd4x in a gauge any and "natural gauge"R = A = constant taking into account that they, as well as the action, should be both invariant under coordinate transformations as invariant under the gauge transformations introduced, namely, gliv = of gin, and di, = cp.+ hi, actually, the first person to speak in scale invariance in physics was the Weyl himself in his article. It is also found that the solutions to the emptiness of Einstein's field equations are also solutions of the corresponding Weyl's field equations. Finally it is shown that the Weyl affine geo­desic may not come from a variational principle by analysing the Helmholtz conditions for the inverse problem of the calculus of variations and discusses about Einstein's criticism of the theory, on which it is concluded that the even seized an inadequate definition of proper time to give his opinion on the work of Weyl, thus, a problem to be solved was to find a good definition of proper time, which leaves open a final version of the Weyl theory.