arXiv subjects

Erhard Scholz

Publications and source records attributed to Erhard Scholz.

36 records · Page 2Linked to original sources

arXiv2015

MOND-like acceleration in integrable Weyl geometric gravity

In this paper a Weyl geometric scalar tensor theory of gravity with scalar field $Φ$ and scale invariant cubic ("aquadratic") kinetic Lagrangian is introduced. Einstein gauge (comparable to Einstein frame in Jordan-Brans-Dicke theory) is most natural for studying trajectories. In it, the Weylian scale connection induces an additional acceleration which in the weak field, static, low velocity limit acquires the deep MOND form of Milgrom/Bekenstein's gravity. The energy momentum of $Φ$ leads to another add on to Newton acceleration. Both additional accelerations together imply a MOND-ian phenomenology of the model. It has unusual transition functions. They imply higher phantom energy density than in the case of the more common MOND models with transition functions $μ_1(x), \, μ_2(x)$. A considerable part of it is due to the scalar field's energy density which, in our model, gives a scale and generally covariant expression for the self-energy of the gravitational field.

gr-qc↗

arXiv2015

Paving the way for transitions --- a case for Weyl geometry

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.

gr-qc↗

arXiv2014

Higgs and gravitational scalar fields together induce Weyl gauge

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 {\em 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.

gr-qc↗

arXiv2014

The problem of space in the light of relativity: the views of H. Weyl and E. Cartan

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).

math.HO↗

arXiv2011

Weyl geometry in late 20th century physics

Weyl's original scale geometry of 1918 ("purely infinitesimal geometry") was withdrawn from physical theory 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 physics (gravity, quantum mechanics), elementary particle physics, and cosmology. Here we survey the last two segments. It seems that Weyl geometry continues to have an open research potential for the foundations of physics after the turn of the century.

math.HO↗

arXiv2011

Weyl geometric gravity and "breaking" of electroweak symmetry

A Weyl geometric scale covariant approach to gravity due to Omote, Dirac, and Utiyama (1971ff) is reconsidered. It can be extended to the electroweak sector of elementary particle fields, taking into account their basic scaling freedom. Already Cheng (1988) indicated that electroweak symmetry breaking, usually attributed to the Higgs field with a boson expected at $0.1 - 0.3 \,TeV$, may be due to a coupling between Weyl geometric gravity and electroweak interactions. Weyl geometry seems to be well suited for treating questions of elementary particle physics, which relate to scale invariance and its "breaking". This setting suggests the existence of a scalar field boson at the surprisingly low energy of $\sim 1\, eV$. That may appear unlikely; but, as a payoff, the acquirement of mass arises as a result of coupling to gravity in agreement with the understanding of mass as the gravitational charge of fields.

hep-th↗

arXiv2008

Cosmological spacetimes balanced by a scale covariant scalar field

A scale invariant, Weyl geometric, Lagrangian approach to cosmology is explored, with a a scalar field phi of (scale) weight -1 as a crucial ingredient besides classical matter \cite{Tann:Diss,Drechsler:Higgs}. For a particularly simple class of Weyl geometric models (called {\em Einstein-Weyl universes}) the Klein-Gordon equation for phi is explicitly solvable. In this case the energy-stress tensor of the scalar field consists of a vacuum-like term Lambda g_{mu nu} with variable coefficient Lambda, depending on matter density and spacetime geometry, and of a dark matter like term. Under certain assumptions on parameter constellations, the energy-stress tensor of the phi-field keeps Einstein-Weyl universes in locally stable equilibrium. A short glance at observational data, in particular supernovae Ia (Riess ea 2007), shows interesting empirical properties of these models.

astro-ph↗

arXiv2007

Einstein universes stabilized

The hypothesis that gravitational self-binding energy may be the source for the vacuum energy term of cosmology is studied in a Newtonian Ansatz. For spherical spaces the attractive force of gravitation and the negative pressure of the vacuum energy term form a self stabilizing system under very reasonable restrictions for the parameters, among them a characteristic coefficient βof self energy. In the Weyl geometric approach to cosmological redshift, Einstein-Weyl universes with observational restrictions of the curvature parameters are dynamically stable, if βis about 40 % smaller than in the exact Newton Ansatz or if the space geometry is elliptical.

gr-qc↗

arXiv2007

Another look at the Pioneer anomaly

The often quoted equ. (15) in (Anderson_ea:2002) defines the observed Pioneer anomaly $a_P$ such that literal reading suggests $ν_{obs} < ν_{mod$ ($ν_{obs}$ the observed (Doppler) frequency, $ν_{mod}$ the one expected from a relativistic orbit model of the spacecraft). Contextualized reading shows, however, that (15) is meant the other way round, $ν_{obs} > ν_{mod}$. Version 1 to 4 of this paper built upon the first reading and led to a correction term for frequency shifts of wrong sign. Thus the proposal therein does not resolve the Pioneer anomaly, although it gives a correction term for frequencies which has the correct absolute value.

astro-ph↗

arXiv2007

Scale covariant gravity and equilibrium cosmologies

Causal structure, inertial path structure and compatibility with quantum mechanics demand no full Lorentz metric, but only an integrable Weyl geometry for space time (Ehlers/Pirani/Schild 1972, Audretsch e.a. 1984). A proposal of (Tann 1998,,Drechsler/Tann 1999) for a minimal coupling of the Hilbert-Einstein action to a scale covariant scalar vacuum field $ϕ$ (weight -1) plus (among others) a Klein-Gordon action term opens the access to a scale covariant formulation of gravity. The ensuing scale covariant K-G equation specifies a natural scale gauge vacuum gauge). Adding other natural assumptions for gauge conditions (in particular Newton gauge, with unchanging Newton constant) the chosen Ansatz leads to a class of Weyl geometric Robertson-Walker solutions of the Einstein equation, satisfying $a''a+a'^2= const$, analogous to the Friedmann-Lemaitre equation but with completely different dynamical properties ($a$ the warp function in Riemann gauge). The class has an asymptotically attracting 1-parameter subfamily of extremely simple space-time geometries with an isotropic Robertson-Walker fluid as source of the Einstein equation, discussed as Weyl universes elsewhere (Scholz 2005). Under the assumption of a heuristic gravitational self energy binding Ansatz for the fluid, equilibrium solutions arise, in stark contrast to classical (semi-Riemannian) cosmology. Weyl universes agree very well with a variety of empirical data from observational cosmology, in particular supernovae luminosities and quasar data.

gr-qc↗

arXiv2005

On the geometry of cosmological model building

This article analyzes the present anomalies of cosmology from the point of view of integrable Weyl geometry. It uses P.A.M. Dirac's proposal for a weak extension of general relativity, with some small adaptations. Simple models with interesting geometrical and physical properties, not belonging to the Friedmann-Lema\^ıtre class, are studied in this frame. Those with positive spatial curvature (Einstein-Weyl universes) go well together with observed mass density $Ω_m$, CMB, supernovae Ia data, and quasar frequencies. They suggest a physical role for an equilibrium state of the Maxwell field proposed by I.E. Segal in the 1980s (Segal background) and for a time invariant balancing condition of vacuum energy density. The latter leads to a surprising agreement with the BF-theoretical calculation proposed by C. Castro (2002).

gr-qc↗

arXiv2005

Introducing Groups into Quantum Theory (1926 -- 1930)

In the second half of the 1920s, physicists and mathematicians introduced group theoretic methods into the recently invented ``new'' quantum mechanics. Group representations turned out to be a highly useful tool in spectroscopy and in giving quantum mechanical explanations of chemical bonds. H. Weyl explored the possibilities of a group theoretic approach towards quantization. In his second version of a gauge theory for electromagnetism, he even started to build a bridge between quantum theoretic symmetries and differential geometry. Until the early 1930s, an active group of young quantum physicists and mathematicians contributed to this new challenging field. But around the turn to the 1930s, opposition against the new methods in physics grew. This article focusses on the work of those physicists and mathematicians who introduced group theoretic methods into quantum physics.

math.HO↗

arXiv2004

Philosophy as a cultural resource and medium of reflection for Hermann Weyl

Here we review a kind of post-World-War-II "Nachtrag" to H. Weyl's philosophical comments on mathematics and the natural sciences published in the middle of the 1920s. In a talk given at Zürich in the late 1940s, Weyl discussed F.Gonseth's dialectical epistemology and considered it as being restricted too strictly to aspects of historical change. His own experiences with post-Kantian dialectical philosophy, in particular J.G. Fichte's derivation of the concept of space and matter, had been a stronger dialectical background for his own 1918 studies in purely infintitesimal geometry and the early geometrically unified field theory of matter (extending the Mie-Hilbert program). Although now Weyl distantiated himself from the speculative features of his youthful philosophizing and in particular from his earlier enthusiasm for Fichte, he again had deep doubts as to the cultural foundations of modern mathematical sciences and its role in material culture of high modernity. For Weyl, philosophical "reflection" was a cultural necessity; he now turned towards K. Jasper's and M. Heidegger's existentialism to find deeper grounds, similar to his turn towards Fichte's philosophy after World War I.

math.HO↗

arXiv2004

Local spinor structures in V. Fock's and H. Weyl's work on the Dirac equation (1929)

In early 1929, V. Fock (initially in collaboration with D. Iwanenko) and H. Weyl developed independently from each other a general relativistic generalization of the Dirac equation. In the core, they arrived at the same theory by the introduction of a local (topologically trivial) spinor structures and a lifting of the Levi-Civita connection of underlying space-time. They both observed, in slightly different settings, a characteristic underdetermination of the spin connection by a complex phase factor, which gave the symbolical possibility for a reformulation of Weyl's old (1918) idea to characterize the electromagnetic potential by a differential form transforming as a gauge field. Weyl and Fock realized the common mathematical core of their respective approaches in summer 1929, but insisted on differences in perspective. An interesting difference was discussed by Weyl in his Rouse Ball lecture in 1930,. He contrasted the new type of unification strongly to the earlier geometrically unified field theories (including his own). He was quite explicit that he now considered his earlier ideas on geometrization of ``all of physics'' as premature and declared that the new, more empirically based approach would have to go a long way before it could be considered as a true "geometrization" of matter structures.

physics.hist-ph↗

arXiv2004

The changing concept of matter in H. Weyl's thought, 1918 -1930

During the ``long decade'' of transformation of mathematical physics between 1915 and 1930, H. Weyl interacted with physics in two highly productive phases and contributed to it, among others, by his widely read book on {\em Space - Time - Matter (Raum Zeit - Materie)} (1918 -- 1923) and on {\em Group Theory and Quantum Mechanics (Gruppentheorie und Quantenmechanik)} (1928 - 1931). In this time Weyl's understanding of the constitution of matter and its mathematical description changed considerably. At the beginning of the period he started from a "dynamistic" and geometrical conception of matter, following and extending the Mie-Hilbert approch, which he gave up during the year 1920. After transitional experiments with a singularity (and in this sense topological) approach in 1921/22, he developed an open perspective of what he called an ``agency theory'' of matter. The idea for it was formulated already before the advent of the ``new'' quantum mechanics in 1925/26. It turned out to be well suited to be taken over to the quantum view as a kind of ``heritage'' from the first half of the decade. At the end of the period, Weyl completely renounced his earlier belief in the possibility to ``construct matter'' from a geometrically unified field theory. He now posed the possibility of a geometrization of the mathematical forms underlying the rising quantum physical description of matter as a completely open problem for future research.

math.HO↗

arXiv2004

C.F. Gauss' Präzisionsmessungen terrestrischer Dreiecke und seine Überlegungen zur empirischen Fundierung der Geometrie in den 1820er Jahren (C.F. Gauss' high precion measurements of terrestrial triangles and his thoughts on the empirical foundations of geometry in the 1820s)

In the historical literature there has been an extended discussion on the question, whether the report of Sartorius von Waltershausen about C. F. Gauss checking the largest triangle of the geodetical measurement campaign in the kingdom of Hannover as a kind of ``test'' for the Euclididean nature of physical space can be taken seriously or not. Among others, it was argued that it even was logically impossible for Gauss to do so in the early 1820s (i.e. in particular, before J. Bolyai's, N.I. Lobachevsy's, or even B. Riemann's works). This article shows, in which sense Gauss's methodology of curvature of surfaces, although logically developed in all clarity only for surfaces embedded in Euclidean 3-space, could very well be used already in the early 1820 to investigate the above mentioned question in the sense of physical geometry. Although we do not have a definitive proof of the respective calculations by Gauss's own hand, the latter's account of the situation to contemporaries (correspondents, friend and students) changed clearly between the early and the late 1820s. That speaks very much in favour of a basic correctness of Sartorius von Waltershausen's report on this topic.

math.HO↗

arXiv2004

An extended frame for cosmology by integrable Weyl geometry

A conservative extension of general relativity by integrable Weyl geometry is formulated, and a new class of cosmological models ({\em Weyl universes}) is introduced and studied. A short discussion of how these new models behave in the light of observational evidence follows. It turns out that the subclass of Weyl universes with positive ex-ante curvature of spatial sections ({\em Einstein-Weyl universes}) forms an empirically contentful conceptual frame for cosmology. Recent high precision data (supernovae Ia, quasar frequencies, microwave background) are well represented in this type of Weyl geometry and several of the anomalies of the standard approach are avoided or become meaningless.

astro-ph↗

arXiv2004

An outline of Weyl geometric models in cosmology

Already the simplest examples of Weyl geometry, the static space-time models of general relativity modified by an additional time-homogeneous Weylian length connection lead to beautiful cosmological models (Weyl universes) . The magnitude-redshift relation of recent supernovae Ia measurements is in perfect agreement with the prediction of decrease of energy flux in the Weyl models. These data allow to estimate the (ex-ante) spacelike curvature of Weyl universes. Quasar frequency data from the SDSS provide strong evidence of a positive ex-ante curvature. Thus an Einstein-Weyl universe, i.e., an Einstein universe endowed with a Weylian length connection, is in good agreement with supernovae and quasar data. The relative mass-energy density with respect to the critical density of the standard approach, and the relative contribution of the ``vacuum term'' are time-independent in Weyl gauge. Thus the time-evolution anomaly of vacuum energy does not arise. The intervals given in the literature for the dynamically determined matter density parameter are consistent with other data in the Weyl geometric approach. An explanation of the cosmic microwave background by a maximal entropy state of the quantized Maxwell equation, according to a proposal by I.E. Segal for the conformal Einstein universe, translates directly to the Weyl approach. It leads to quantitative estimations for the temperature of the Planck radiation between 2 and 5.3 deg K. Moreover, the anisotropy angles, expected to arise in an Einstein-Weyl universe agrees with the results of multipole measurements of CMB anisotropies.

astro-ph↗