arXiv subjects

Erhard Scholz

Publications and source records attributed to Erhard Scholz.

At least 19 recordsLinked to original sources

arXiv2026

Bridging the gap between dark matter and MOND by a relativistc scalar field approach

A Lagrangian model for a general relativistic scalar field, formulated in the framework of integrable Weyl geometry, is studied. Under the present assumptions it modifies the light cone structure and induces MOND-like dynamics in the weak field approximation of the Einstein frame (gauge). The Lagrangian contains a Bekenstein-type (``aquadratic'') term and a second order term generating additional mass energy for the scalar field. Both are switched on only if the the scalar field gradient is spacelike and below a MOND-typical threshold, like in the superfluid model of Berezhiani/Khoury. In the weak field limit the Bekenstein term implies a deep MOND equation for the scalar field and leads to MOND\-ian free fall trajectories. The Lagrangian mass term induces non-negligible energy and pressures of the scalar field with the respective consequences for gravitational light deflection.

gr-qc↗

arXiv2022

Mathematical modernity, goal or problem? The opposing views of Felix Hausdorff and Hermann Weyl

This paper contains a case study of the work and self-definition of two important mathematicians during the rise of modern mathematics: Felx Hausdorff (1868--1942) and Hermann Weyl (1885--1955). The two had strongly diverging positions with regard to basic questions of mathematical methodology, which is reflected in the style and content of their mathematical research. Herbert Mehrtens (1990) describes them as protagonists of what he sees as the two opposing camps of ``modernists'' (Hilbert, Hausdorff et al.) and ``countermodernists'' (Brouwer, Weyl et al.). There is no doubt that Hausdorff may be described as a mathematical ``modernist''', while the qualification of Weyl as ``countermodern'' is rather off the track, once his work is taken into account.

math.HO↗

arXiv2022

From heliocentrism to epicycles: A commentary on pre-Ptolemaic astronomy

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.

physics.hist-ph↗

arXiv2022

H. Weyl's and E. Cartan's proposals for infinitesimal geometry in the early 1920s

In the early phase of general relativity Elie Cartan and Hermann Weyl thought about the question of how the role of transformation groups could be transferred from classical geometry (Erlangen program) to differential geometry. They had different starting points and used different techniques, but both generalized the concept of connection arising from Levi-Civita's interpretation of the classical Christoffel symbols as parallel transfer in curved spaces. Their focus differed and Cartan headed toward a much more general framwork than Weyl (non-holonomous spaces versus scale gauge geometry). But there also was an overlap of topics (space problem) and, at the turn to the 1930s, they arrived at an agreement on how to deal with Cartan's infinitesimal geometric structures.

physics.hist-ph↗

arXiv2022

A Weyl geometric scalar field approach to the dark sector

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 non-minimally 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. {\bf 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.

gr-qc↗

arXiv2022

Integrating Dark Matter, Modified Gravity, and the Humanities

Editorial of a special issue on dark matter & modified gravity, distributed across the journals Studies in History and Philosophy of Modern Physics and Studies in History and Philosophy of Science. Published version of the open access editorial (in SHPS) available here: https://doi.org/10.1016/j.shpsa.2021.08.015. The six papers are collected here: https://www.sciencedirect.com/journal/studies-in-history-and-philosophy-of-science-part-b-studies-in-history-and-philosophy-of-modern-physics/special-issue/10CR71RJLWM.

physics.hist-ph↗

arXiv2021

Back to the roots: A discrete Kermack-McKendrick model adapted to Covid-19

A widely used tool for analysing the Covid-19 pandemic is the standard SIR model. It seems often to be used as a black box, not taking into account that this model was derived as a special case of the seminal Kermack-McKendrick theory from 1927. This is our starting point. We explain the setup of the Kermack-McKendrick theory (passing to a discrete approach) and use medical information for specializing to a model which we call {\em adapted K-McK-model}. This includes effects of vaccination, mass testing and mutants. We demonstrate the use of the model by applying it to the development in Germany. As a striking application we demonstrate that a comparatively mild intervention reducing the time until quarantine by one day leads to a drastic improvement. A similar effect can be obtained by certain mass testings as we will demonstrate. We discuss possibilities to apply the model both for predictions and as an analysis tool. We compare the adapted K-McK-model to the standard SIR model and observe condiderable differences if the contact rates are not constant. Finally we compare the model reproduction rate with the empirical reproduction rate determined by the Robert Koch-Institut.

q-bio.PE↗

arXiv2021

Studying the course of Covid-19 by a recursive delay approach

In an earlier paper we proposed a recursive model for epidemics; in the present paper we generalize this model to include the asymptomatic or unrecorded symptomatic people, which we call {\em dark people} (dark sector). We call this the SEPAR$_d$-model. A delay differential equation version of the model is added; it allows a better comparison to other models. We carry this out by a comparison with the classical SIR model and indicate why we believe that the SEPAR$_d$ model may work better for Covid-19 than other approaches. In the second part of the paper we explain how to deal with the data provided by the JHU, in particular we explain how to derive central model parameters from the data. Other parameters, like the size of the dark sector, are less accessible and have to be estimated more roughly, at best by results of representative serological studies which are accessible, however, only for a few countries. We start our country studies with Switzerland where such data are available. Then we apply the model to a collection of other countries, three European ones (Germany, France, Sweden), the three most stricken countries from three other continents (USA, Brazil, India). Finally we show that even the aggregated world data can be well represented by our approach. At the end of the paper we discuss the use of the model. Perhaps the most striking application is that it allows a quantitative analysis of the influence of the time until people are sent to quarantine or hospital. This suggests that imposing means to shorten this time is a powerful tool to flatten the curves

q-bio.PE↗

arXiv2021

From Grassmann complements to Hodge-duality

Hodge duality is a central concept of 20th century algebraic and analytic geometry and plays a non-negligible role also in recent mathematical physics. At first sight one might expect that its origins lie in the 1930s when its name-giving protagonist, William V.D. Hodge, started his mathematical research. On the other hand, a close link between Hodge's theory and the Maxwell equation has sometimes been claimed not only from a systematic point of view but also historically. In the transformation of classical electromagnetism to the relativistic Maxwell theory the influence of Grassmann's alternating product and an operation he called "complement" played a certain role. In fact, Grassmann's "complements" were a linear algebraic template of the later Hodge star operation. This paper surveys the development from Grassmann's complements and their appearance in relativistic electrodynamics to Hodge's theory of harmonic forms.

math.HO↗

arXiv2020

Proposal of a recursive compartment model of epidemics and applications to the Covid-19 pandemic

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 {\em 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.

q-bio.PE↗

arXiv2020

A scalar field inducing a non-metrical contribution to gravitational acceleration and a compatible add-on to light deflection

A scalar field model for explaining the anomalous acceleration and light deflection at galactic and cluster scales, without further dark matter, is presented. It is formulated in a scale covariant scalar tensor theory of gravity in the framework of integrable Weyl geometry and presupposes two different phases for the scalar field, like the superfluid approach of Berezhiani/Khoury. In low acceleration regimes of static gravitational fields (in the Einstein frame) with accordingly low values of the scalar field gradient, the scalar field Lagrangian combines a cubic kinetic term similar to the ``a-quadratic'' Lagrangian used in the first covariant generalization of MOND (RAQUAL) (Bekenstein/Milgrom:1984) and a second order derivative term introduced by Novello et al. in the context of a Weyl geometric approach to cosmology (Novello/Oliveira_et al:1993, Oliveira/Salim/Sautu:1997). In varying with regard to $ϕ$ the latter is variationally equivalent to a first order expression. The scalar field equation thus remains of order two. In the Einstein frame it assumes the form of a covariant generalization of the Milgrom equation known from the classical MOND approach. It implies a corresponding ``non-metrical'' contribution to the acceleration of free fall trajectories. The second order derivative term of the Lagrangian leads to a non-negligible contribution to the energy momentum tensor and an {\em add-on to the light deflection potential} in beautiful agreement with the dynamics of low velocity trajectories. -- In higher sectional curvature regions, respectively for higher accelerations in static fields, the scalar field Lagrangian consists of a Jordan-Brans-Dicke term with sufficiently high value of the JBD-constant to satisfy empirical constraints. Here the dynamics agrees effectively with the one of Einstein gravity.

astro-ph.GA↗

arXiv2020

Light cone and Weyl compatibility of conformal and projective structures

In the literature different concepts of compatibility between a projective structure and a conformal structure on a differentiable manifold are used. In particular compatibility in the sense of Weyl geometry is slightly more general than compatibility in the Riemannian sense. An often cited paper [Ehlers-Pirani-Schild:1972] introduces still another criterion which is natural from the physical point of view: every light like geodesics of of the conformal structure is a geodesics of the projective structure. Their claim that this type of compatibility is sufficient for introducing a Weylian metric has recently been questioned in [Trautman:2012] and [Scholz:2019]. Here it is proved that the conjecture of EPS is correct.

math.DG↗

arXiv2019

Gauging the spacetime metric -- looking back and forth a century later

H. Weyl's proposal of 1918 for generalizing Riemannian geometry by local scale gauge (later called {\em Weyl geometry}) was motivated by mathematical, philosophical and physical considerations. It was the starting point of his unified field theory of electromagnetism and gravity. After getting disillusioned with this research program and after the rise of a convincing alternative for the gauge idea by translating it to the phase of wave functions and spinor fields in quantum mechanics, Weyl no longer considered the original scale gauge as physically relevant. About the middle of the last century the question of conformal and/or local scale gauge transformation were reconsidered by different authors in high energy physics (Bopp, Wess, et al.) and, independently, in gravitation theory (Jordan, Fierz, Brans, Dicke). In this context Weyl geometry attracted new interest among different groups of physicists (Omote/Utiyama/Kugo, Dirac/Canuto/Maeder, Ehlers/Pirani/Schild and others), often by hypothesizing a new scalar field linked to gravity and/or high energy physics. Although not crowned by immediate success, this ``retake'' of Weyl geometrical methods lives on and has been extended a century after Weyl's first proposal of his basic geometrical structure. It finds new interest in present day studies of elementary particle physics, cosmology, and philosophy of physics.

physics.hist-ph↗

arXiv2018

E. Cartan's attempt at bridge-building between Einstein and the Cosserats -- or how translational curvature became to be known as {\em torsion}

Élie Cartan's "généralisation de la notion de courbure" (1922) arose from a creative evaluation of the geometrical structures underlying both, Einstein's theory of gravity and the Cosserat brothers generalized theory of elasticity. In both theories groups operating in the infinitesimal played a crucial role. To judge from his publications in 1922--24, Cartan developed his concept of generalized spaces with the dual context of general relativity and non-standard elasticity in mind. In this context it seemed natural to express the translational curvature of his new spaces by a rotational quantity (via a kind of Grassmann dualization). So Cartan called his translational curvature "torsion" and coupled it to a hypothetical rotational momentum of matter several years before spin was encountered in quantum mechanics.

math.HO↗

arXiv2017

The unexpected resurgence of Weyl geometry in late 20-th century physics

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.

math.HO↗

arXiv2016

Clusters of galaxies in a Weyl geometric approach to gravity

A model for the dark halos of galaxy clusters, based on the Weyl geometric scalar tensor theory of gravity (WST) with a MOND-like approximation, is proposed. It is uniquely determined by the baryonic mass distribution of hot gas and stars. A first heuristic check against empirical data for 19 clusters (2 of which are outliers), taken from the literature, shows encouraging results. Modulo a caveat resulting from different background theories (Einstein gravity plus $ΛCDM$ versus WST), the total mass for 15 of the outlier reduced ensemble of 17 clusters seems to be predicted correctly (in the sense of overlapping $1\,σ$ error intervals).

astro-ph.CO↗

arXiv2016

The changing faces of the Problem of Space in the work of Hermann Weyl

During his life Weyl approached the problem of space (PoS) from various sides. Two aspects stand out as permanent features of his different approaches: the {\em unique determination of an affine connection} (i.e., without torsion in the terminology of Cartan) and the question {\em which type of group} characterizes physical space. The first feature came up in 1919 (commentaries to Riemann's inaugural lecture) and played a crucial role in Weyl's work on the PoS in the early 1920s. He defended the central role of affine connections even in the light of Cartan's more general framework of connections with torsion. In later years, after the rise of the Dirac field, it could have become problematic, but Weyl saw the challenge posed to Einstein gravity by spin coupling primarily in the possibility to allow for non-metric affine connections. Only after Weyl's death Cartan's approach to infinitesimal homogeneity and torsion became revitalized in gravity theories.

math.HO↗

arXiv2016

Weyl's search for a difference between `physical' and `mathematical' automorphisms

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.

physics.hist-ph↗