A Unified Invariant Formulation, by Frames, from General Relativity to the Atomic Scale

The aim of this article is the formulation of the basic laws of Physics by frames, i.e. quadruples of exterior differential one forms. The basic operator is a modification of the Hodge-de Rham Laplacian d*d*+*d*d, where * is the hyperbolic star. In this article it is modified depending on the frame. The modified * is invariant w.r. to any diffeomorphism. Consequently, the modified Laplavian is invariant. The field equation developed in this article is a complete alternative to the field equation of General Relativity in vacuum. The frame-field equation yields a derivation of Newtonian (Einstein) law of attraction without recourse to the geodesic postulate. Coulomb law is also derived. Invariant formulation of Maxwell equations is exhibited. Then first order linear approximation is considered. It is used to derive invariant formulation of Schroedinger equation (classical and relativistic) and Dirac equation all of which are linear. The lhs of the field equation, defined on a four dimensional manifold, is the same for all bodies. Thus hopefully, it may set the foundation for a field theory. The interaction of the particles has to be worked out. The basic equation of this article is motivated by the Einstein equation in nonempty space.Comment: Several changes of the sig

On a class of invariant coframe operators with application to gravity

Let a differential 4D-manifold with a smooth coframe field be given. Consider the operators on it that are linear in the second order derivatives or quadratic in the first order derivatives of the coframe, both with coefficients that depend on the coframe variables. The paper exhibits the class of operators that are invariant under a general change of coordinates, and, also, invariant under the global SO(1,3)-transformation of the coframe. A general class of field equations is constructed. We display two subclasses in it. The subclass of field equations that are derivable from action principles by free variations and the subclass of field equations for which spherical-symmetric solutions, Minkowskian at infinity exist. Then, for the spherical-symmetric solutions, the resulting metric is computed. Invoking the Geodesic Postulate, we find all the equations that are experimentally (by the 3 classical tests) indistinguishable from Einstein field equations. This family includes, of course, also Einstein equations. Moreover, it is shown, explicitly, how to exhibit it. The basic tool employed in the paper is an invariant formulation reminiscent of Cartan's structural equations. The article sheds light on the possibilities and limitations of the coframe gravity. It may also serve as a general procedure to derive covariant field equations