General Covariance Research Articles

Fitting the AFM force–distance curves the correct way

Abstract Data fitting is an indispensable tool in modern metrology. However, as the models become more and more complex the most popular method, ordinary least squares regression, reaches its limit. As the relative uncertainty in the independent variable increases, we can no longer speak about an exactly known independent variable and an uncertain dependent variable. The increasing complexity of the measurement process may give rise to correlationsFurthermore correlations between data may become non negligible: typical sources are e.g. the use of reference samples or crosstalk between sensors. These problems can be treated with generalized least squares. A new algorithm–Optimum Estimate of Function Parameters by Iterated Linearization (OEFPIL) – has been recently suggested which can handle both a wide class of functions as well as general covariance matrices. We illustrate its application in the analysis of force distance curves in AFM which are used to evaluate the mechanical properties of samples such as the Young’s modulus and adhesion. In this work we apply the new algorithm and compare the results to other methods. The uncertainties obtained by OEFPIL are in good agreement with uncertainties obtained by the Monte Carlo method but can be obtained in a more straightforward way.

A unified framework for the analysis of accuracy and stability of a class of approximate Gaussian filters for the Navier–Stokes equations

Abstract Bayesian state estimation of a dynamical system utilising a stream of noisy measurements is important in many geophysical and engineering applications. In these cases, nonlinearities, high (or infinite) dimensionality of the state space, and sparse observations pose key challenges for deriving efficient and accurate data assimilation (DA) techniques. A number of DA algorithms used commonly in practice, such as the Ensemble Kalman Filter (EnKF) or Ensemble Square Root Kalman Filter (EnSRKF), suffer from serious drawbacks such as catastrophic filter divergence and filter instability. The analysis of stability and accuracy of these DA schemes has thus far focused either on finite-dimensional dynamics, or on complete observations in case of infinite-dimensional dissipative systems. We develop a unified framework for the analysis of several well-known and empirically efficient DA techniques derived from various Gaussian approximations of the Bayesian filtering schemes for geophysical-type dissipative dynamics with quadratic nonlinearities. We establish rigorous results on (time-asymptotic) accuracy and stability of these algorithms with general covariance and observation operators. The accuracy and stability results for EnKF and EnSRKF for dissipative PDEs are, to the best of our knowledge, completely new in this general setting. It turns out that a hitherto unexploited cancellation property involving the ensemble covariance and observation operators and the concept of covariance localization in conjunction with covariance inflation play a pivotal role in the accuracy and stability for EnKF and EnSRKF. Our approach also elucidates the links, via determining functionals, between the approximate-Bayesian and control-theoretic approaches to DA. We consider the ‘model’ dynamics governed by the two-dimensional incompressible Navier–Stokes equations (NSEs) and observations given by noisy measurements of averaged volume elements or spectral/modal observations of the velocity field. In this setup, several continuous-time DA techniques, namely the so-called 3DVar, EnKF and EnSRKF reduce to a stochastically forced NSEs. For the first time, we derive conditions for accuracy and stability of EnKF and EnSRKF. The derived bounds are given for the limit supremum of the expected value of the L 2 norm and of the H 1 Sobolev norm of the difference between the approximating solution and the actual solution as the time tends to infinity. Moreover, our analysis reveals an interplay between the resolution of the observations (roughly, the ‘richness’ of the observation space) associated with the observation operator underlying the DA algorithms and covariance inflation and localization which are employed in practice for improved filter performance.

General covariance for quantum states over time

General covariance for quantum states over time

Spacetime symmetry breaking on nongeodesic leaves and a new form of matter

We examine the permanent damage caused by the historical breakdown of full diffeomorphism invariance induced by a foliation. We focus on the case where the foliation is allowed to be nongeodesic after the interactions with the foliation switch off. Gravity and other forms of matter recover full diffeomorphism invariance only at the expense of introducing a new matterlike component, carrying the nonvanishing Hamiltonian (and momentum, as it turns out) left over from the violating past interactions. This matter form must be stress-free in the preferred frame; this is the only way a matter action can mimic the evolution of the leftover Hamiltonian (and momentum) driven by the Dirac hypersurface deformation algebra. Hence, if the preferred frame is nongeodesic, the equivalent matter component must have energy and a momentum current in this frame, but still no spatial stresses: an unusual form of “matter.” It is equivalent to a fluid with anisotropic stress in some regimes, reducing to dust in others, or even displaying completely new features in extreme situations. Its stress energy tensor is conserved. We provide two examples based on accelerated frames: Rindler space-time and the canonical Schwarzchild frame. Published by the American Physical Society 2024

The influence of relativity on the philosophy of symbolic forms

AbstractIn this paper, I will deal with the analogy between Cassirer’s interpretation of relativity and his philosophy of culture. As to the structure of the paper, it will be divided into six parts. I will start with a brief introduction, after which I will succinctly outline Cassirer’s reading of Einstein’s theory, and in particular of general covariance. I will then focus on the presentation of his project for a “systematic philosophy” in the last chapter of Einstein’s Theory of Relativity and subsequently delve into the lexical references to relativity made in Philosophy of Symbolic Forms. After this, I will tackle Cassirer’s rectorship speech, where relativity and the philosophy of culture are explicitly interwoven, but with the important clarification (already provided in Einstein’s Theory of Relativity and Philosophy of Symbolic Forms) that not all kinds of knowledge align with de-anthropomorphized objectivity. The final section contains the concluding remarks.

Light rings and shadows of static black holes in effective quantum gravity

Recently, two types of static black hole models that retain general covariance have been proposed within the Hamiltonian constraint approach to effective quantum gravity (EQG). We have studied the light rings and shadows of these black holes using the topological method and the backward ray-tracing method, respectively. We demonstrate that these light rings in both types of static black holes are standard and unstable according to the classification of light rings. Subsequently, we checked the position of the light rings using the photon trajectory equation. We found that although the quantum parameters do not affect the light rings of these two types of black holes, they do reduce the size of the first type of static black hole in EQG, making it smaller. However, for the second type of static black hole in EQG, we cannot distinguish it from a Schwarzschild black hole based on the shadow alone. Fortunately, the quantum parameters shrink the lensing rings of both types of black holes in EQG, causing the black hole shadow to occupy a larger proportion within the ring. This can serve as a basis for distinguishing whether the black hole is in EQG or general relativity (GR).

Minimum Profile Hellinger Distance Estimation of General Covariate Models

Covariate models, such as polynomial regression models, generalized linear models, and heteroscedastic models, are widely used in statistical applications. The importance of such models in statistical analysis is abundantly clear by the ever-increasing rate at which articles on covariate models are appearing in the statistical literature. Because of their flexibility, covariate models are increasingly being exploited as a convenient way to model data that consist of both a response variable and one or more covariate variables that affect the outcome of the response variable. Efficient and robust estimates for broadly defined semiparametric covariate models are investigated, and for this purpose the minimum distance approach is employed. In general, minimum distance estimators are automatically robust with respect to the stability of the quantity being estimated. In particular, minimum Hellinger distance estimation for parametric models produces estimators that are asymptotically efficient at the model density and simultaneously possess excellent robustness properties. For semiparametric covariate models, the minimum Hellinger distance method is extended and a minimum profile Hellinger distance estimator is proposed. Its asymptotic properties such as consistency are studied, and its finite-sample performance and robustness are examined by using Monte Carlo simulations and three real data analyses. Additionally, a computing algorithm is developed to ease the computation of the estimator.

Pauli-type coupling of spinors and curved spacetime

Abstract In this study we prove that the Pauli interaction---which is associated with a length parameter---emerges when the minimal coupling recipe is applied to the non-degenerate Dirac Lagrangian.
The conventional Dirac Lagrangian is rendered non-degenerate if supplemented by a particular term quadratic in the derivatives of the spinors.
For dimensional reasons, this non-degenerate Dirac Lagrangian is associated with a length parameter $\ell$.
It yields---just as the conventional degenerate Dirac Lagrangian---the standard free Dirac equation in Minkowski space as the $\ell$ terms cancel when setting up the Euler-Lagrange equations.
However, if the Dirac spinor is minimally coupled to gauge fields, then the length parameter~$\ell$ becomes a physical coupling constant yielding novel interactions.
For the U($1$) symmetry the Pauli coupling of fermions to electromagnetic fields arises, modifying the fermion's magnetic moment.
We discuss the impact of these findings on electrodynamics, and estimate the upper bound of the length parameter~$\ell$ from the yet existing discrepancy between the (SM) theory and measurement of the anomalous magnetic moment of light leptons.
This, and also recent studies of the renormalization theory, suggest that the Pauli coupling of leptons to the electromagnetic field is a necessary ingredient in QED, supporting the notion of a fundamental nature of the non-degenerate Dirac Lagrangian.

In a second step we then investigate how analogous ``Pauli-type'' couplings of gravity and matter arise if fermions are embedded in curved spacetime.
The diffeomorphism and Lorentz invariance of that system leads to an anomalous spin-torsion interaction and a curvature-dependent mass correction.
We calculate the mass correction in the De~Sitter geometry of vacuum with the cosmological constant $\Lambda$.
Possible implications for the existence of effective non-zero rest masses of neutrinos are addressed.
Finally, an outlook on the impact of mass correction on the physics of ``Big Bang'' cosmology, black holes, and of neutron stars is provided.

Curvature correlators in nonperturbative 2D Lorentzian quantum gravity

Correlation functions are ubiquitous tools in quantum field theory from both a fundamental and a practical point of view. However, up to now their use in theories of quantum gravity beyond perturbative and asymptotically flat regimes has been limited, due to difficulties associated with diffeomorphism invariance and the dynamical nature of geometry. We present an analysis of a manifestly diffeomorphism-invariant, nonperturbative two-point curvature correlator in two-dimensional Lorentzian quantum gravity. It is based on the recently introduced quantum Ricci curvature and uses a lattice regularization of the full path integral in terms of causal dynamical triangulations. We discuss some of the subtleties and ambiguities in defining connected correlators in theories of dynamical geometry, and provide strong evidence from Monte Carlo simulations that the connected two-point curvature correlator in 2D Lorentzian quantum gravity vanishes. This work paves the way for an analogous investigation in higher dimensions.

Homological aspects of topological gauge-gravity equivalence

In the works of Achúcarro and Townsend and also by Witten, a duality between three-dimensional Chern–Simons gauge theories and gravity was established. In all cases, the results made use of the field equations. In a previous work, we were capable of generalizing Witten’s work to the off-shell cases, as well as to the four-dimensional Yang–Mills theory with de Sitter gauge symmetry. The price we paid is that curvature and torsion must obey some constraints under the action of the interior derivative. These constraints implied on the partial breaking of diffeomorphism invariance. In this work, we first formalize our early results in terms of fiber bundle theory by establishing the formal aspects of the map between a principal bundle (gauge theory) and a coframe bundle (gravity) with partial breaking of diffeomorphism invariance. Then, we study the effect of the constraints on the homology defined by the interior derivative. The main result is the emergence of a non-trivial homology in Riemann–Cartan manifolds.

Axial anomalies of maximally supersymmetric tensor theories

We revisit anomalies of (4, 0) and (3, 1) maximally supersymmetric tensor theories in d = 6. A (4, 0) on-shell tensor multiplet descends to that of the d = 5 maximal supergravity upon a dimensional reduction, hypothesized to offer a strong-coupled UV completion of the latter in the same sense of (2, 0) theories as the UV completion of d = 5 N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 pure Yang-Mills. The gravitational anomalies, found to be nonvanishing, had been computed, although its relevance in the absence of the d = 6 metric is not obvious. We perform a comprehensive anomaly computation for (4, 0) and (3, 1) tensor supermultiplets, respectively, for Sp(4) and Sp(3) × Sp(1) R-symmetry anomalies and the mixed R-gravitational anomaly thereof, and find that anomalies involving R-symmetries cancel out identically. We close with questions on how to address the anomaly in this class of theories with no general covariance.

An active learning workflow for predicting hydrogen atom adsorption energies on binary oxides based on local electronic transfer features

Machine learning combined with density functional theory (DFT) enables rapid exploration of catalyst descriptors space such as adsorption energy, facilitating rapid and effective catalyst screening. However, there is still a lack of models for predicting adsorption energies on oxides, due to the complexity of elemental species and the ambiguous coordination environment. This work proposes an active learning workflow (LeNN) founded on local electronic transfer features (e) and the principle of coordinate rotation invariance. By accurately characterizing the electron transfer to adsorption site atoms and their surrounding geometric structures, LeNN mitigates abrupt feature changes due to different element types and clarifies coordination environments. As a result, it enables the prediction of ∗H adsorption energy on binary oxide surfaces with a mean absolute error (MAE) below 0.18 eV. Moreover, we incorporate local coverage (θl) and leverage neutral network ensemble to establish an active learning workflow, attaining a prediction MAE below 0.2 eV for 5419 multi-∗H adsorption structures. These findings validate the universality and capability of the proposed features in predicting ∗H adsorption energy on binary oxide surfaces.

Dark matter and spacetime symmetry restoration

We examine local physics in the presence of variables: variables associated with the whole of the spacelike surfaces of a foliation. These could be the (pseudo)constants of nature and their conjugate times, but our statements are more general. Interactions between the local and the global (for example, dependence of the local action on global times dual to constants) degrades full space-time diffeomorphism invariance down to spatial diffeomorphism invariance, and so an extra degree of freedom appears. When these presumably primordial global interactions switch off, the local action recovers full invariance and so the usual two gravitons, but a legacy matter component is left over, bearing the extra degree of freedom. Under the assumption that the preferred foliation is geodesic, this component behaves like dark matter, except that 3 of its 4 local degrees of freedom are frozen, forcing its rest frame to coincide with the preferred foliation. The nonfrozen degree of freedom (the number density of the effective fluid) is the survivor of the extra “graviton” present in the initial theory and keeps memory of all the past global interactions that took place in a given location in the preferred foliation. Such “painted-on” dark matter is best distinguished from the conventional one in situations where the preferred frame would be preposterous if all 4 degrees of freedom of dark matter were available. We provide one example: an outflowing halo of legacy matter with exact escape speed at each point and a very specific profile, surrounding a condensed structure made of normal matter. Published by the American Physical Society 2024

Classical gravitational anomalies of Liouville theory

We show that for classical Liouville field theory, diffeomorphism invariance, Weyl invariance and locality cannot hold together. This is due to a genuine Virasoro center, present in the theory, that leads to an energy-momentum tensor with non-tensorial conformal transformations, in flat space, and with a non-vanishing trace, in curved space. Our focus is on a field-independent term, proportional to the square of the Weyl gauge field, WμWμ, that makes the action Weyl-invariant and was disregarded in previous investigations of Weyl and conformal symmetry. We show this term to be related to the classical center of the Virasoro algebra. The mechanism uncovered here is a classical version of the quantum anomalous phenomenon: the generalization to curved space only allows to keep one of the two symmetries enjoyed by the flat space theory, either Lorentz (diffeomorphism) or conformal invariance.

Holomorphic gravity and its regularization of Locally Signed Coordinate Invariance

Holomorphic gravity and its regularization of Locally Signed Coordinate Invariance

Irreversible vierbein postulate: Emergence of spacetime from quantum phase transition

We formulate a model for quantum gravity based on the local Lorentz symmetry and general-coordinate invariance. A key idea is the irreversible vierbein postulate that a tree-level action for the model at a certain energy scale does not contain an inverse vierbein. Under this postulate, only the spinor becomes a dynamical field, and no gravitational background field is introduced in the tree-level action. In this paper, after explaining the transformation rules of the local Lorentz and general-coordinate transformations in detail, a tree-level action is defined. We show that fermionic fluctuations can induce a nonvanishing gravitational background field. Published by the American Physical Society 2024

On the General Covariance of Natural Laws

This paper explores the concept of general covariance in natural laws through the lens of projective geometry and tensor algebra. By introducing the notions of covariance and contravariance using intuitive examples from projections and the scalar product, we illustrate how the covariance of natural laws ensures their universality and objectivity. We also discuss the role of symmetries and conservation principles in relation to the covariant nature of physical equations, highlighting the deep interplay between the mathematical structure of physical theories and the fundamental principles of nature.

Holographic RG from an exact RG: Locality and general coordinate invariance in the bulk

In earlier papers it was shown that the correct kinetic term for scalar, vector gauge field and the spin two field in AdSD+1 space is obtained starting from the exact renormalization group (ERG) equation for a CFTD perturbed by scalar composite, conserved vector current, and conserved traceless energy momentum tensor respectively. In this paper interactions are studied, and it is shown that a flipped version of the Polchinski ERG equation that evolves toward the UV can be written down and is useful for making contact with the usual AdS/CFT prescriptions for correlation function calculations. The scalar-scalar-spin-2 interaction in the bulk is derived from the ERG equation in the large N semiclassical approximation. It is also shown that after mapping to AdS the interaction is local on a scale of the bare cutoff rather than the moving cutoff (which would have corresponded to the anti–de Sitter scale). The map to AdSD+1 plays a crucial role in this locality. The local nature of the coupling ensures that this interaction term in the bulk action is obtained by gauge fixing a general coordinate invariant scalar kinetic term in the bulk action. A wave function renormalization of the scalar field is found to be required for a mutually consistent map of the two fields to AdSD+1. Published by the American Physical Society 2024

Tuning-free sparse clustering via alternating hard-thresholding

Model-based clustering is a commonly-used technique to partition heterogeneous data into homogeneous groups. When the analysis is to be conducted with a large number of features, analysts face simultaneous challenges in model interpretability, clustering accuracy, and computational efficiency. Several Bayesian and penalization methods have been proposed to select important features for model-based clustering. However, the performance of those methods relies on a careful algorithmic tuning, which can be time-consuming for high-dimensional cases. In this paper, we propose a new sparse clustering method based on alternating hard-thresholding. The new method is conceptually simple and tuning-free. With a user-specified sparsity level, it efficiently detects a set of key features by eliminating a large number of features that are less useful for clustering. Based on the selected key features, one can readily obtain an effective clustering of the original high-dimensional data under a general sparse covariance structure. Under mild conditions, we show that the new method leads to clusters with a misclassification rate consistent to the optimal rate as if the underlying true model were used. The promising performance of the new method is supported by both simulated and real data examples.

Cosmological Consequence of Varying Speed of Light and Gravitational Constant

<p>The speed of light is taken to be a constant in a vacuum. This forms the basic tool for the principle of General Covariance, which asserts that all laws of Physics should take the same form in all frames of reference. Without putting inflation into consideration, the theory of varying speed of light (VSL) would solve basic problems of cosmology in the early universe. Furthermore, the Gravitational constant G that occurred in the Friedmann Equations may not have been real constants in the early universe but have some variation with the universe scale factor. Cosmological models with varying physical constants have been of interest in recent years with few works in the literature. A cosmological solution obtained by incorporating variable speed of light and gravitational constant gives a cosmic model which is free from the initial Big Bang singularity and horizon problem. It is also observed here that the early universe was dominated by dark matter, however, as the scale factor increases, the dark energy become dominant.</p>