Hubert Goenner

On the History of Unified Field Theories

This article is intended to give a review of the history of the classical aspects of unified field theories in the 20th century. It includes brief technical descriptions of the theories suggested, short biographical notes concerning the scientists involved, and an extensive bibliography. The present first installment covers the time span between 1914 and 1933, i.e., when Einstein was living and working in Berlin — with occasional digressions into other periods. Thus, the main theme is the unification of the electromagnetic and gravitational fields augmented by short-lived attempts to include the matter field described by Schrödinger’s or Dirac’s equations. While my focus lies on the conceptual development of the field, by also paying attention to the interaction of various schools of mathematicians with the research done by physicists, some prosopocraphical remarks are included.

Exact solutions of the generalized Lane–Emden equation

A generalized Lane–Emden equation with indices (α,β,ν,n) is discussed, which reduces to the Lane–Emden equation proper for α=2, β=1, ν=1. General properties of the set of solutions of this equation are derived, and exact solutions are given. These include a singular solution without free integration constant for arbitrary n and for particular relations between n, ν, and α. Among the two-parameter solutions nonequivalent families of solutions of the same equation are obtained.

On the possibility of phase transitions in the geometric structure of space-time

Abstract It is shown that space-time may be not only in a state which is described by Riemann geometry but also in states which are described by Finsler geometry. Transitions between various metric states of space-time have the meaning of phase transitions in its geometric structure. These transitions together with the evolution of each of the possible metric states make up the general picture of space-time manifold dynamics.

Concerning the generalized Lorentz symmetry and the generalization of the Dirac equation

The work is devoted to the generalization of the Dirac equation for a flat locally anisotropic, i.e., Finslerian space–time. At first we reproduce the corresponding metric and a group of the generalized Lorentz transformations, which has the meaning of the relativistic symmetry group of such event space. Next, proceeding from the requirement of the generalized Lorentz invariance we find a generalized Dirac equation in its explicit form. An exact solution of the nonlinear generalized Dirac equation is also presented.

Finslerian Spaces Possessing Local Relativistic Symmetry

It is shown that the problem of a possibleviolation of the Lorentz transformations at Lorentzfactors γ > 5 × 1010,indicated by the situation which has developed in thephysics of ultra-high energy cosmic rays (the absence of the GZKcutoff), has a nontrivial solution. Its essence consistsin the discovery of the so-called generalized Lorentztransformations which seem to correctly link the inertial reference frames at any values ofγ. Like the usual Lorentz transformations, thegeneralized ones are linear, possess group propertiesand lead to the Einstein law of addition of3-velocities. However, their geometric meaning turns out tobe different: they serve as relativistic symmetrytransformations of a flat anisotropic Finslerian eventspace rather than of Minkowski space. Consideration is given to two types of Finsler spaces whichgeneralize locally isotropic Riemannian space-time ofrelativity theory, e.g. Finsler spaces with a partiallyand entirely broken local 3D isotropy. The investigation advances arguments for the correspondinggeneralization of the theory of fundamental interactionsand for a specific search for physical effects due tolocal anisotropy of space-time.

Einstein tensor and generalizations of Birkhoff's theorem

The Einstein tensors of metrics having a 3-parameter group of (global) isometries with 2-dimensional non-null orbitsG3(2,s/t) are studied in order to obtainalgebraic conditions guaranteeing an additional normal Killing vector. It is shown that Einstein spaces withG3(2,s/t) allow aG4. A critical review of some of the literature on Birkhoffs theorem and its generalizations is given.

Some remarks on the genesis of scalar-tensor theories

Between 1941 and 1962, scalar-tensor theories of gravitation were suggested four times by different scientists in four different countries. The earliest originator, the Swiss mathematician W. Scherrer, was virtually unknown until now whereas the chronologically latest pair gave their names to a multitude of publications on Brans–Dicke theory. P. Jordan, one of the pioneers of quantum mechanics and quantum field theory, and Y. Thiry, known by his book on celestial mechanics, a student of the mathematician Lichnerowicz, complete the quartet. Diverse motivations for and conceptual interpretations of their theories will be discussed as well as relations among them. Also, external factors like language, citation habits, or closeness to the mainstream are considered. It will become clear why Brans–Dicke theory, although structurally a déjà-vu, superseded all the other approaches.

Einstein Tensor and 3‐Parameter Groups of Isometries with 2‐Dimensional Orbits

The algebraic classification of the Weyl and Ricci tensors and the relation between them in a Riemann space with an isometry group possessing a nontrivial isotropy group are reviewed. All metrics with Minkowski signature, invariant under a 3‐parameter isometry group with 2‐dimensional orbits having nondegenerate metrics, are constructed from the group properties and are shown to have Ricci tensors with a double eigenvalue, and the orbits are shown to be surfaces of constant curvature. The null orbits are shown to have a triply degenerate eigenvalue of the Ricci tensor. The various additionally degenerate metrics are classified in further detail, extending the work of Plebanski and Stachel.

A Class of Anisotropic (Finsler-)Space-time Geometries

A particular Finsler-metric proposed in [1, 2]and describing a geometry with a preferred nulldirection is characterized as belonging to a subclasscontained in a larger class of Finsler-metrics with one or more preferred directions (null, space- or timelike). The metrics are classified according to theirgroup of isometries. These turn out to be isomorphic tosubgroups of the Poincare (Lorentz-) group complemented by the generator of a dilatation.The arising Finsler geometries may be used for theconstruction of relativistic theories testing theisotropy of space. It is shown that the Finsler space with the only preferred null direction is the anisotropic space closest to isotropic Minkowski-spaceof the full class discussed.

The Gravitational Field of a Radiating and Contracting Spherically-Symmetric Body with Heat Flow

A model of a highly idealized spherically symmetric object radiating away its mass with constant luminosity is presented. The body starts at t = −∞ with both infinite mass and radius and contracts to a point at t = 0 without forming an event horizon. Its material particles are moving non-geodesically and shearfree while transporting heat to the surface. Unlike in some radiating star models of a similar type, all physically required conditions are satisfied in this model.