Hans-Jürgen Treder

The Weyl-Cartan Space Problem in Purely Affine Theory

According to Poincaré, only the “epistemological sum of geometry and physics is measurable”. Of course, there are requirements of measurement to be imposed on geometry because otherwise the theory resting on this geometry cannot be physically interpreted. In particular, the Weyl-Cartan space problem must be solved, i.e., it must be guaranteed that the comparison of distances is compatible with the Levi-Civita transport. In the present paper, we discuss these requirements of measurement and show that in the (purely affine) Einstein-Schrödinger unified field theory the solution of the Weyl-Cartan space problem simultaneously determines the matter via Einsteins equations. Here the affine field Γikl represents Poincarés sum, and the solution of the space problem means its splitting in a metrical space and in matter fields, where the latter are given by the torsion tensor Γi[kl].

Mach-Einstein doctrine and general relativity

It is argued that, under the assumption that the strong principle of equivalence holds, the theoretical realization of the Mach principle (in the version of the Mach-Einstein doctrine) and of the principle of general relativity are alternative programs. That means only the former or the latter can be realized—at least as long as only field equations of second order are considered. To demonstrate this we discuss two sufficiently wide classes of theories (Einstein-Grossmann and Einstein-Mayer theories, respectively) both embracing Einsteins theory of general relativity (GRT). GRT is shown to be just that “degenerate case” of the two classes which satisfies the principle of general relativity but not the Mach-Einstein doctrine; in all the other cases one finds an opposite situation.These considerations lead to an interesting “complementarity” between general relativity and Mach-Einstein doctine. In GRT, via Einsteins equations, the covariant and Lorentz-invariant Riemann-Einstein structure of the space-time defines the dynamics of matter: The symmetric matter tensor Ttk is given by variation of the Lorentz-invariant scalar densityLmat, and the dynamical equations satisfied by Tik result as a consequence of the Bianchi identities valid for the left-hand side of Einsteins equations. Otherwise, in all other cases, i.e., for the “Mach-Einstein theories” here under consideration, the matter determines the coordinate or reference systems via gravity. In Einstein-Grossmann theories using a holonomic representation of the space-time structure, the coordinates are determined up to affine (i.e. linear) transformations, and in Einstein-Mayer theories based on an anholonomic representation the reference systems (the tetrads) are specified up to global Lorentz transformations. The corresponding conditions on the coordinate and reference systems result from the postulate that the gravitational field is compatible with the strong equivalence of inertial and gravitational masses.

Continuum and discretum―Unified field theory and elementary constants

Unitary field theories and “SUPER-GUT” theories work with an universal continuum, the structured spacetime of R. Descartes, B. Spinoza, B. Riemann, and A. Einstein, or a (Machian(1–3)) structured vacuum according the quantum theory of unitary fields (Dirac,(4,5) and Heisenberg(6–8)). The atomistic aspect of the substantial world is represented by the fundamental constants which are invariant against “all transformations” and which “depend on nothings” (Planck(9–11)). A satisfactory unitary theory has to involve these constants like the mathematical numbers. Today, Plancks conception of the three elementary constants ħ, c, and G may be the key to general relativistic quantum field theory like unitary theory. However, the elementary constants are a question of measurement-theory, also.According to Poppers theory(12–16) of induction, such unitary theories are “universal explaining theories.” The fundamental constants involve the complementarity between the universal statements in unitary theory and the “basic statements” in the language of classical observables.

On general-relativistic and gauge field theories

The fundamental open questions of general relativity theory are the unification of the gravitational field with other fields, aiming at a unified geometrization of physics, as well as the renormalization of relativistic gravitational theory in order to obtain their self-consistent solutions. These solutions are to furnish field-theoretic particle models—a problem first discussed by Einstein. In addition, we are confronted with the issue of a coupling between gravitational and matter fields determined (not only) by Einsteins principle of equivalence, and also with the question of the geometric meaning of a gravitational quantum theory. In our view, all these problems are so closely related that they warrant a general solution. We treat mainly the concepts suggested by Einstein and Weyl.

Classical gravity and quantum matter fields in unified field theory

The Einstein-Schrödinger purely affine field theory of the non-symmetric field provides canonical field equations without constraints. These equations imply the Heisenberg-Pauli commutation rules of quantum field theory. In the Schrödinger gauging of the Einstein field coordinatesUkli=Γkli−δliΓkmm, this unified geometric field theory becomes a model of the coupling between a quantized Maxwellian field in a medium and classical gravity. Therefore, independently of the question as to the physical truth of this model, its analysis performed in the present paper demonstrates that, in the framework of a quantized unified field theory, gravity can appear as a genuinely classical field.

Gravitation and universal Fermi coupling in general relativity

The generally covariant Lagrangian densityG = ℛ + 2K ℒmatter of the Hamiltonian principle in general relativity, formulated by Einstein and Hilbert, can be interpreted as a functional of the potentialsgikand φ of the gravitational and matter fields. In this general relativistic interpretation, the Riemann-Christoffel form Γkli =kli for the coefficients гkli of the affine connections is postulated a priori. Alternatively, we can interpret the LagrangianG as a functional of φ, gik, and the coefficients гkli. Then the гkli are determined by the Palatini equations. From these equations and from the symmetry гkli = гlki for all matter fields with δℒ/δΓ=0 the Christoffel symbols again result. However, for Diracs bispinor fields, δℒ/δΓ becomes dependent on the Dirac current, essentially with a coupling factor ∼Khc. In this case, the Palatini equations define a new transport rule for the spinor fields, according to which a second universal interaction results for the Dirac spinors, besides Einsteins gravitation. The generally covariant Dirac wave equations become the general relativistic nonlinear Heisenberg wave equations, and the second universal interaction is given by a Fermi-like interaction term of the V-A type. The geometrically induced Fermi constant is, however, very small and of the order 10−81erg cm3

Reference systems and gravitation

It is shown how the formalism of the tetrad theory of gravitation used by Treder (1967a, b, 1970) follows from the more general fibre bundle formalism. This is of interest in the study of the relations between tetrad theories and the general theory of relativity. In particular, the breaking of the principle of general relativity and the interpretation of tetrad fields as reference systems are considered in greater detail.

Einstein and geophysics: Valuable contributions warrant a second look

All scientists know that Albert Einsteins work transformed the principles of physics in atomistic, quantum mechanics, and relativity theory. Many of Einsteins fundamental investigations proved to be relevant to geophysical research as well, though his contributions are often overlooked. Einsteins General Relativity Theory (GRT) initiated a new chapter not only in astronomy, but also in geophysics. Models of the universe based on Einsteins ideas nowadays underpin the main fields of cosmic geophysics, and his basic work on relativistic astrophysics and related fields of science captured the interest of geophysicists such as Emil Wiechert and Hans Ertel.

On the compatibility of spinor field equations with regard to generalisations of Weyl's lemma

Generalisations of Weyls lemma are discussed. In order to secure the compatibility of the spinor field equations, the generalisations may not be arbitrary. It is shown that a contracted Weyl lemma must be valid. This lemma ‘saves’ the duality between the Lorentz covariant and the Einstein covariant representation of the equation of continuity. The meaning of Weyls lemma and its generalisations is discussed in terms of the fibre bundle theory.

On Matter and Metric in Affine Theory of Gravity

The affine theory was conceived as a geometric model, wherein the connection field is the primary structure of the space-time. According to the program lying on the basis of this theory, metric and some sort of matter are somehow to be deduced from the connection field. In the present paper, we point out classical ways to a realization of this program. It is shown that, even in that case where the introduction of the metric seems to exclude the coupling of gravity to matter, the situation is not so hopeless as one may assume. In particular, for a symmetric Einstein tensor, it is answered the old question as to a self-consistent introduction of a metric and a metrical energy-momentum tensor controversially debated by Einstein, Eddington, and Weyl.