Erhard Scholz | Bergische Universität Wuppertal (original) (raw)

Papers by Erhard Scholz

arXiv (Cornell University), Mar 9, 2017

Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was withdrawn by its aut... more Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was withdrawn by its author from physical theorizing in the early 1920s. It had a comeback in the last third of the 20th century in different contexts: scalar tensor theories of gravity, foundations of gravity, foundations of quantum mechanics, elementary particle physics, and cosmology. It seems that Weyl geometry continues to offer an open research potential for the foundations of physics even after the turn to the new millennium.

arXiv (Cornell University), Aug 3, 2022

If one wants to translate the heliocentric picture of planets moving uniformly on circular orbits... more If one wants to translate the heliocentric picture of planets moving uniformly on circular orbits about the sun to the perspective of a terrestrial observer, using classical (ancient) geometric means only, one is naturally led to the investigation of epicyclic constructions. The announcement of the heliocentric hypothesis by Aristarchos of Samos and the invention of the method of epicycles happened during the 3rd and 2nd centuries BC. The latter developed into the central tool of Hellenistic and Ptolemaic astronomy. In the present literature on the history of astronomy the parallel rise of the heliocentric view and the methods of epicycles is usually considered as a pure contingency. Here I explain why I do not find this view convincing.

How can a mathematician outline a fundamentally new vision of a mathematical discipline? He might... more How can a mathematician outline a fundamentally new vision of a mathematical discipline? He might turn to the philosophy of mathematics and speak about mathematics, i.e. on a metalevel, reflecting his own and other mathematicians' work. Or he might try to sketch the architecture of the new mathematical discipline in question. In the latter case he has to introduce concepts, constructions, and theorems as the central technical building blocks of a mathematical theory. Usually he can draw upon a whole network of results of other scientists, which brings his view closer to tradition and attenuates the novelty of his views. Thus, if an epistemological break is intended, at least some elements of the first, more philosophical approach have to be taken up. The occasion of sharp epistemological turns are rare in the history of mathematics. Riemann's contribution to geometry is a most prominent example. As is well known, Riemann organized his approach to geometry around the new conc...

In the following we survey the attempts to find empirical bounds for the curvature of physical sp... more In the following we survey the attempts to find empirical bounds for the curvature of physical space from astronomical data over a period of roughly a century. Our report will be organized in four sections: (1) Lobachevsky, (2) Gauss and his circle, (3) astronomers of the late 19th century, (4) outlook on the first relativistic cosmological models. In the first three passages we indicate how parallax data were used for inferences on a hypothetical curvature of astronomical space by different authors using only slightly different methodologies. In the last phase a new methodological approach to physical geometry was opened by general relativity, and two completely new data sets came into the game, mass density and cosmological redshift. Before we enter the discussion of the astronomical data, one has to notice that astronomical observations were not the only route toward empirical data on space curvature bounds. We have reports, although only scarce direct information, that C.F.Gauss...

I Die Symmetriekonzepte der Kristallographie und ihre Beziehungen zur Algebra des 19. Jahrhundert... more I Die Symmetriekonzepte der Kristallographie und ihre Beziehungen zur Algebra des 19. Jahrhunderts.- Vorbemerkungen.- 1 Von der phanomenologischen Kristallklassifikation zur Einfuhrung der Kristallsysteme und Kristallklassen.- 1.1 Kristallklassifikation im 18. Jahrhundert: Werner und Rome de l'Isle.- 1.2 Beginnende Mathematisierung im atomistischen Programm: R.J. Hauy.- 1.3 Konstituierung eines alternativen Theoretisierungsprogramms unter dem Einfluss der dynamistischen Philosophie.- 1.4 Charakterisierung der Kristallsysteme durch C.S. Weiss.- 1.5 M.L. Frankenheims Entdeckung der 32 Kristallklassen.- 2 Rationale Vektorraume, Punktsymmetrien und Raumgittertypen im dynamistischen Programm.- 2.1 J.G. Grassmanns "Geometrische Combinationslehre".- 2.2 Rationale Vektorraume in der Kristallographie gegen Ende der 1820er Jahre.- 2.3 Hessels Klassifikation der endlichen raumlichen Punktsymmetriesysteme.- 2.4 Hessels Bestimmung der Kristallklassen.- 2.5 Frankenheims Interpretati...

Scientific Reports

Our lives (and deaths) have by now been dominated for two years by COVID-19, a pandemic that has ... more Our lives (and deaths) have by now been dominated for two years by COVID-19, a pandemic that has caused hundreds of millions of disease cases, millions of deaths, trillions in economic costs, and major restrictions on our freedom. Here we suggest a novel tool for controlling the COVID-19 pandemic. The key element is a method for a population-scale PCR-based testing, applied on a systematic and repeated basis. For this we have developed a low cost, highly sensitive virus-genome-based test. Using Germany as an example, we demonstrate by using a mathematical model, how useful this strategy could have been in controlling the pandemic. We show using real-world examples how this might be implemented on a mass scale and discuss the feasibility of this approach.

Foundations of Physics, 2008

A Weyl geometric approach to cosmology is explored, with a scalar field φ of (scale) weight −1 as... more A Weyl geometric approach to cosmology is explored, with a scalar field φ of (scale) weight −1 as crucial ingredient besides classical matter. Its relation to Jordan-Brans-Dicke theory is analyzed; overlap and differences are discussed. The energy-stress tensor of the basic state of the scalar field consists of a vacuum-like term Λgµν with Λ depending on the Weylian scale connection and, indirectly, on matter density. For a particularly simple class of Weyl geometric models (called Einstein-Weyl universes) the energy-stress tensor of the φ-field can keep spacetime geometries in equilibrium. A short glance at observational data, in particular supernovae Ia (Riess e.a. 2007), shows encouraging empirical properties of these models.

arXiv (Cornell University), Feb 27, 2022

This paper explores the dark sector (dark matter and dark energy) from the perspective of Weyl ge... more This paper explores the dark sector (dark matter and dark energy) from the perspective of Weyl geometric scalar tensor theory (integrable Weyl geometry). In order to account for the galactic dynamics successfully modelled by MOND ("modified Newtonian dynamics"), the nonminimally coupled scalar field considered here has a Lagrangian with two non-conventional contributions in addition to a standard kinetic term: one is inspired by Bekenstein/Milgrom's RAQUAL ("relativistic a-quadratic Lagrangian") from 1983, the other one by a second order term introduced in cosmological studies by Novello et al. in 1993. See, however, the error warning below. We consider the transition to the Einstein gravity on one hand and to scalar field cosmology in the FRW framework on the other. A bouncing cosmological model is tentatively discussed at the end. Error warning Equ. (86) is wrong; it does not take contributions to δL braz g µν into account, which are due to the covariant derivative of the (scale covariant) gradient of the scalar field φ. Therefore the contribution of L braz to the energy tensor of the scalar field is wrong. This flaw is fatal for the derivations in the Milgrom regime; the whole argumentation of part 2 can no longer be upheld. It is here reproduced for documentary reasons only. Contents Error warning 1

Historia Mathematica, 1984

Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis PoincarC. By Erhard Scholz. Boston (Birk... more Geschichte des Mannigfaltigkeitsbegriffs von Riemann bis PoincarC. By Erhard Scholz. Boston (Birkhauser). 1980. 430 pp.

General Relativity and Gravitation

A Correction to this paper has been published: 10.1007/s10714-020-02693-z

During his whole scientific life Hermann Weyl was fascinated by the interrelation of physical and... more During his whole scientific life Hermann Weyl was fascinated by the interrelation of physical and mathematical theories. From the mid 1920s onward he reflected also on the typical difference between the two epistemic fields and tried to identify it by comparing their respective automorphism structures. In a talk given at the end of the 1940s (ETH, Hs 91a:31) he gave the most detailed and coherent discussion of his thoughts on this topic. This paper presents his arguments in the talk and puts it in the context of the later development of gauge theories.

This paper presents three aspects by which the Weyl geometric generalization of Riemannian geomet... more This paper presents three aspects by which the Weyl geometric generalization of Riemannian geometry, and of Einstein gravity, sheds light on actual questions of physics and its philosophical reflection. After introducing the theory's principles, it explains how Weyl geometric gravity relates to Jordan-Brans-Dicke theory. We then discuss the link between gravity and the electroweak sector of elementary particle physics, as it looks from the Weyl geometric perspective. Weyl's hypothesis of a preferred scale gauge, setting Weyl scalar curvature to a constant, gets new support from the interplay of the gravitational scalar field and the electroweak one (the Higgs field). This has surprising consequences for cosmological models. In particular it leads to a static (Weyl geometric) spacetime with "inbuilt" cosmological redshift. This may be used for putting central features of the present cosmological model into a wider perspective.

Starting from a short review of the "classical" space problem in the sense of the 19th ... more Starting from a short review of the "classical" space problem in the sense of the 19th century (Helmholtz -- Lie -- Klein) it is discussed how the challenges posed by special and general relativity to the classical analysis were taken up by Hermann Weyl and Elie Cartan. Both mathematicians reconsidered the space problem from the point of view of transformations operating in the infinitesimal neighbourhoods of a manifold (spacetime). In a short outlook we survey further developments in mathematics and physics of the second half of the 20th century, in which core ideas of Weyl's and/or Cartan's analysis of the space problem were further investigated (mathematics) or incorporated into basic theories (physics).

This is work in progress. We make it accessible hoping that people might find the idea useful. We... more This is work in progress. We make it accessible hoping that people might find the idea useful. We propose a discrete, recursive 5-compartment model for the spread of epidemics, which we call SEPIR-model. Under mild assumptions which typically are fulfilled for the Covid-19 pandemic it can be used to reproduce the development of an epidemic from a small number of parameters closely related to the data. We demonstrate this at the development in Germany and Switzerland. It also allows model predictions assuming nearly constant reproduction numbers. Thus it might be a useful tool for shedding light on which interventions might be most effective in the future. In future work we will discuss other aspects of the model and more countries.

General Relativity and Gravitation, 2015

A common biquadratic potential for the Higgs field h and an additional scalar field φ, non minima... more A common biquadratic potential for the Higgs field h and an additional scalar field φ, non minimally coupled to gravity, is considered in locally scale symmetric approaches to standard model fields in curved spacetime. A common ground state of the two scalar fields exists and couples both fields to gravity, more precisely to scalar curvature R. In Einstein gauge (φ = const, often called "Einstein frame"), also R is scaled to a constant. This condition makes perfect sense, even in the general case, in the Weyl geometric approach. There it has been called Weyl gauge, because it was first considered by Weyl in the different context of his original scale geometric theory of gravity of 1918. Now it seems to get new meaning as a combined effect of electroweak theory and gravity, and their common influence on atomic frequencies.