Abstract The goal of this article is to examine Riemann manifolds with the help of multiplicative arguments. Using the unique metric of multiplicative analysis (proportional metric), Christoffel symbols, connections and curvature tensor fields are designed on the manifolds built with this metric. Through this approach, we have scrutinized the multiplicative Riemann curvature of the multiplicative Euclidean space, uncovering its divergent structural attributes compared to the conventional Euclidean space. Moreover, with the help of the multiplicative Riemann curvature tensor, it has been revealed that the multiplicative Euclidean space is isomorphic to the traditional Euclidean space. Thus, the solutions of some problems that cannot be solved in the traditional Euclidean space will be obtained in the multiplicative space and with the help of isomorphic transformation, their solutions will be obtained in the Euclidean space. Also, illustrative examples are incorporated to facilitate a deeper comprehension of the discussed concepts.

The quantum geometric tensor and quantum Fisher information have recently been shown to provide a unified geometric description of the linear response of many-body systems. However, a similar geometric description of higher-order perturbative phenomena including nonlinear response in generic quantum systems is lacking. In this work, we develop a general framework for the time-dependent quantum geometry of many-body systems by treating external perturbing fields as coordinates on the space of density matrices. We use the Bures distance between the initial and time-evolved density matrix to define geometric quantities through a perturbative expansion. To lowest order, we derive a time-dependent generalization of the Bures metric related to the spectral density of linear response functions, unifying previous results for the quantum metric in various limits and providing a geometric interpretation of Fermi’s golden rule. At next order in the expansion, we define a time-dependent Bures-Levi-Civita connection for general many-body systems. We show that the connection is the sum of one contribution that is related to a second-order nonlinear response function, and a second contribution that captures the higher geometric structure of first-order perturbation theory. We show that in the quasistatic, zero-temperature limit for noninteracting fermions, this Bures connection reduces to the known expression for band-theoretic Christoffel symbols. Our work provides a systematic framework to explore many-body quantum geometry beyond the quantum metric and highlights how higher-order correlation functions can probe this geometry.

Abstract The work reported in “Granular micromechanics‐based identification of isotropic strain gradient parameters for elastic geometrically nonlinear deformations” misidentified key terms in the grain‐pair objective relative displacement when accounting for the second gradient of placement. In this paper, we correct that oversight by deriving a revised expression for the grain‐pair objective relative displacement within the granular micromechanics framework. The amended terms, which resemble Christoffel symbols expressed in terms of strain gradients, modify the contributions of both the normal and tangential components to the strain energy and, consequently, alter the identified strain‐gradient elastic parameters. Importantly, the identification of the standard (first gradient) elastic tensor remains unchanged. This brief paper presents the corrected derivation, the resulting stiffness tensors for anisotropic strain‐gradient elasticity, and updated analytical expressions for the material parameters in both 2D and 3D isotropic settings.

Quantum geometry provides an intrinsic framework for characterizing the geometric structure of quantum states. It highlights its relevance to various aspects of fundamental physics. However, its direct implications for magnetic phenomena remain largely unexplored. Here, we report the observation of electric-field-induced nonlinear magnetization in the nonmagnetic semimetal WTe$_2$ by using a second-harmonic magneto-optical Kerr effect (SMOKE) spectroscopy. We observe a robust nonlinear SMOKE signal that scales quadratically with current and persists up to 200 K. Theoretical modeling and scaling analysis indicate that this nonlinear magnetization is dominated by the orbital contribution and is intrinsically linked to the quantum Christoffel symbol. Just as the Christoffel symbol is a fundamental quantity encoding spacetime geometry in Einstein's general relativity, our work establishes a direct link between quantum geometry and nonlinear magnetization, and provides a geometric perspective for designing future orbitronic devices.

Abstract This study concerns an inventory of the gravitational force and tidal field induced by filaments, walls, cluster nodes and voids on Megaparsec scales and how they assemble and shape the Cosmic Web. The study is based on a NPart = 5123 ΛCDM dark matter only N-body simulation in a (300h−1 Mpc)3 box at z = 0. We invoke the density field NEXUS+ multiscale morphological procedure to assign the appropriate morphological feature to each location. We then determine the contribution by each of the cosmic web components to the local gravitational and tidal forces. We find that filaments are, by far, the dominant dynamical component in the interior of filaments, in the majority of underdense void regions and in all wall regions. The gravitational influence of cluster nodes is limited, and they are only dominant in their immediate vicinity. The force field induced by voids is marked by divergent outflowing patterns, yielding the impression of a segmented volume in which voids push matter towards their boundaries. Voids manifest themselves strongly in the tidal field as a cellular tapestry that is closely linked to the multiscale cosmic web. However, even within the interior of voids, the dynamical influence of the surrounding filaments is stronger than the outward push by voids. Therefore, the dynamics of voids cannot be understood without taking into account the influence of the environment. We conclude that filaments constitute the overpowering gravitational agent of the cosmic web, while voids are responsible for the cosmic web’s spatial organisation and hence of its spatial connectivity.

The Christoffel symbol is an essential quantity in Einstein's general theory of relativity. We discover that an electric field can induce a nonlinear magnetization in quantum materials, described by a Christoffel symbol defined in the Hilbert space of quantum states (quantum Christoffel symbol). Quite different from the previous scenarios, this orbital magnetization does not need spin-orbit coupling and inversion symmetry breaking. Through symmetry analysis and first-principles calculations, we identify a number of point groups and 2D material candidates (e.g., BiF_{3}, ZnI_{2}, and Ru_{4}Se_{5}) that host this quantum Christoffel nonlinear magnetization. More importantly, this nonlinear magnetization allows the quantum Christoffel symbol to be probed by optical techniques such as magneto-optical Kerr spectroscopy or transport measurements such as tunneling magnetoresistance. This quantum Christoffel nonlinear magnetization gives a paradigm of how geometry dictates physics.

We discuss the problem of singularity crossing in isotropic and anisotropic universes. First, we consider the so called soft or sudden singularities and, in particular the Big Brake singularity. This singularity was discovered in a particular tachyon cosmological model and it was also shown that this kind of singularity arises in a very simple model, where matter is represented by the anti-Chaplygin gas. At the the encounter with the Big Brake singularity the universe has a finite scale factor, a vanishing expansion velocity and an infinite deceleration. The Christoffel symbols also vanish the geodesics are regular and the universe easily can cross such a singularity. Adding to the anti-Chaplygin gas or to the tachyon matter some amount of dust we see that the Big Brake singularity is substituted by a more general soft singularity, its crossing implies a certain transformation of the properties of matter. The crossing of the Big Bang – Big Crunch singularity is more counter-intuitive. However, we describe it for both Friedmann universe and Bianchi-I universe using the field reparametrization of the variables present in models (a scalar field and the metric). Then we consider the Wheeler-DeWitt equation and show that the probability for the universe to find itself at the soft singularity is different from zero, while the encounter with the Big Bang – Big Crunch singularity is suppressed. We analyze the possibility to construct Fock spaces of quantum particles at the vicinity of different cosmological singularities and see when it is possible and when it is not possible. Finally, we present some attempts to develop general approach to the connection between the field reparametrization and the elimination of singularities.

ABSTRACT High‐order accurate discretizations in space and time are applied to the compressible Euler equations on the rotated cubed‐sphere grid. The proposed methodology combines the Direct Flux Reconstruction (DFR) method for spatial discretization and stiffness‐resilient exponential time integration for temporal evolution. The DFR method is quadrature‐free and offers high‐order accuracy, good conservation properties, and geometric flexibility. Stiffness‐resilient exponential integrators allow for larger time steps than explicit methods while maintaining accuracy and stability. The governing equations are formulated using a space‐time tensor formalism, allowing for a general representation in any curvilinear coordinate system. Discretization of these equations presents a number of challenges, including the numerical evaluation of geometric terms, pressure gradients, gravitational forcing, and aliasing errors. Stabilization techniques—such as filtering, numerically consistent calculation of Christoffel symbols, and vertical logarithmic reconstruction—are proposed to address these issues and compute physically plausible solutions. The numerical schemes are evaluated using a series of standard numerical tests.

Ensuring symmetric stiffness in impedance-controlled robots is crucial for physically meaningful and stable interaction in contact-rich manipulation. Conventional approaches neglect the change of basis vectors in curved spaces, leading to an asymmetric joint-space stiffness matrix that violates passivity and conservation principles. In this work, we derive a physically consistent, symmetric joint-space stiffness formulation directly from the task-space stiffness matrix by explicitly incorporating Christoffel symbols. This correction resolves long-standing inconsistencies in stiffness modeling, ensuring energy conservation and stability. We validate our approach experimentally on a robotic system, demonstrating that omitting these correction terms results in significant asymmetric stiffness errors. Our findings bridge theoretical insights with practical control applications, offering a robust framework for stable and interpretable robotic interactions.

ABSTRACT We introduce gradus.jl, an open-source and publicly available general relativistic ray-tracing toolkit for spectral modelling in arbitrary spacetimes. Our software is written in the julia programming language, making use of forward-mode automatic differentiation for computing the Christoffel symbols during geodesic integration, and for propagating derivatives through the entire ray-tracer. Relevant numerical methods are detailed, and our models are validated using a number of tests and comparisons to other codes. The differentiability is used to optimally calculate Cunningham transfer functions – used to efficiently pre-compute relativistic effects in spectral models. A method is described for calculating such transfer functions for disc with non-zero vertical height, including the treatment of self-obscuration. An extension of the transfer function formalism that includes timing information is described, and used to calculate high-resolution reverberation lag spectra for a lamppost corona. The lag – frequency and lag – energy spectra for a Shakura–Sunyaev accretion disc with various lamppost heights and Eddington ratios are calculated, and the general impact of disc thickness in reflection models is discussed.

Traditional methods for solving physical equations in curved spaces, particularly in areas like fluid dynamics and continuum mechanics, often face significant complexity due to the necessity of incorporating Christoffel symbols to account for spatial curvature. These symbols complicate the formulation and numerical implementation. In this paper, we present a novel and flexible methodology that entirely obviates the need for Christoffel symbols by fundamentally changing the approach to problem formulation and solution. The method operates by formulating the physical problem directly within a Euclidean 3D Cartesian space, where differential operators are standard and well-defined. The core of our innovation lies in the combined and systematic application of symbolic calculus to perform both the necessary chain rule transformations between the physical curved space and the embedding Euclidean space, and the subsequent projection operations. This powerful symbolic framework allows us to effectively derive and solve the governing equations on the curved geometry without explicitly computing or using Christoffel symbols or specialized curved-space operators. We demonstrate the robustness, flexibility, and advantages of this approach through several examples, including the derivation of the Navier-Stokes equations in cylindrical coordinates, the modeling of complex flows in bent cylindrical tubes, and the simulation of the breakup of viscoelastic threads. These examples highlight the method's ability to simplify the mathematical formulation and provide a robust framework for complex or evolving geometries. The flexibility in choosing basis representations within the Euclidean space is also shown to offer potential benefits for numerical stability in certain applications.

A method is proposed for changing the law of natural motion of a solid body in the gravitational force field by attaching additional bodies to it. Constraintsare established between the original body and the additional ones introduced. These constraintsare considered as ideal and can be both holonomic and nonholonomic. The practical significance of the proposed method is discussed.

Abstract This paper is based upon the observation that the translation operator D in a curved space–time must depend upon the position of the particle and thus one must allow the [D, X] commutator to be a function of position X in a generalized Lie algebra. This work consists of two parts; In a purely mathematical development, we first reframe Riemannian geometry (RG) as a Generalized Lie algebra (GLA) by allowing the structure constants to be functions of an Abelian subalgebra as is necessary when translations in a space of n variables depend upon the position in the space. In the second part we show that Einstein’s equations for General Relativity (GR) can now be written as commutation relations in this GLA framework including relativistic Quantum Theory (QT) and the Standard Model (SM) with novel predictions. We begin with an Abelian Lie algebra of n “position” operators, X, whose simultaneous eigenvalues, y, define a real n-dimensional space R(n) with a Hilbert space representation. Then with n new operators defined as independent functions, X′(X), we define contravariant and covariant tensors in terms of their eigenvalues, y and y′ with Dirac notation. We then define n additional operators, D, whose exponential map is, by definition, to translate X in a noncommutative algebra of operators (observables) where the “structure constants” are shown to be the metric functions of the X operators to allow for spatial curvature. The D operators then have a Hilbert space position-diagonal representation as a generalized differential operator plus a Christoffel symbol, Γµ (y), an arbitrary vector function Aµ (y), and the derivative of a scalar function gµn∂ϕ(y)/∂yn. One can then express the Christoffel symbols, and the Riemann, Ricci, and other tensors as commutators in this representation thereby framing RG as a GLA. We then show that this GLA provides a more general framework for RG to support GR, QT, the SM with novel predictions.

This work presents a new black hole solution within the framework of a non-commutative gauge theory applied to Kalb-Ramond gravity. Using the method recently proposed in the literature [Nucl.Phys.B 1017 (2025) 116950], we employ the Moyal twist ∂ r ∧∂ θ to implement non-commutativity, being encoded by parameter Θ. We begin by verifying that the resulting black hole no longer possesses spherical symmetry, while the event horizon remains unaffected by non-commutative corrections. The Kretschmann scalar is computed to assess the corresponding regularity. It turns out that the solution is regular, provided that the Christoffel symbols and related quantities are not expanded to second order in Θ. We derive the thermodynamic quantities, including the Hawking temperature T (Θ,ℓ), entropy S (Θ,ℓ), and heat capacity CV (Θ,ℓ). The remnant mass M rem is estimated by imposing T (Θ,ℓ) → 0, although the absence of a physical remnant indicates complete evaporation. Quantum radiation for bosons and fermions is analyzed via the tunneling method, where divergent integrals are treated using the residue theorem. Notably, in the low-frequency regime, the particle number density for bosons surpasses that of fermions (at least within the scope of the methods considered here). The effective potential for a massless scalar field is obtained perturbatively, enabling the computation of quasinormal modes and the time-domain profiles. Finally, further bounds on Θ and ℓ (Lorentz-violating paramter) are derived from solar system tests, including the perihelion precession of Mercury, light deflection, and the Shapiro time delay.

Abstract This work investigates the geometric and dynamical structure of a spacetime governed by the straight spinning string (SSS) metric. The analysis begins with a comprehensive study of the spacetime geometry, including computation of Christoffel symbols, Ricci tensor, and curvature forms in both coordinate and Cartan formalisms. Despite nontrivial global features such as frame-dragging and conical singularities, the spacetime is shown to be locally flat and Ricci-flat. The symmetries of geodesic motion are then explored via the Noether symmetry approach, yielding conserved quantities associated with the metric's isometries. A Hamiltonian formulation is subsequently developed on a four-dimensional configuration space, incorporating the effects of frame-dragging through canonical momenta and enabling a symplectic reduction. The resulting reduced Hamiltonian reveals hidden integrals linked to dynamical and geometric symmetries. The study proceeds to derive an effective potential for radial geodesic motion, elucidating the interplay between geometry, conserved charges, and singularities. Finally, symbolic and numerical analyses of the potential's behavior uncover a smooth profile near critical radii and offer insights into geodesic confinement and causal structure. Altogether, the results contribute to a deeper understanding of relativistic systems with angular-temporal couplings and topological defects.

 All calculations, numerical evaluations, and plots of the effective potential were performed using \texttt{Maple 2020} to ensure accuracy and clarity in visualization.

Shock waves in an ideal gas with variable density, the radiative and conductive heat fluxes in the presence of gravitational force and magnetic field via the Lie group technique

Abstract We introduce Mahakala, a Python-based, modular, radiative ray-tracing code for curved spacetimes. We employ Google’s JAX framework for accelerated automatic differentiation, which can efficiently compute Christoffel symbols directly from the metric, allowing the user to easily and quickly simulate photon trajectories through non-Kerr spacetimes. JAX also enables Mahakala to run in parallel on both CPUs and GPUs. Mahakala natively uses the Cartesian Kerr–Schild coordinate system, which avoids numerical issues caused by the pole in spherical coordinate systems. We demonstrate Mahakala’s capabilities by simulating 1.3 mm wavelength images (the wavelength of Event Horizon Telescope observations) of general relativistic magnetohydrodynamic simulations of low-accretion rate supermassive black holes. The modular nature of Mahakala allows us to quantitatively explore how different regions of the flow influence different image features. We show that most of the emission seen in 1.3 mm images originates close to the black hole and peaks near the photon orbit. We also quantify the relative contribution of the disk, forward jet, and counterjet to 1.3 mm images.

In this work, we introduce a new fractional derivative that modifies the conventional Riemann-Liouville operator to obtain a set of fractional Einstein field equations within a 2+1 dimensional spacetime by assuming a static and circularly symmetric metric. The main reason for introducing this new derivative stems from addressing the divergence encountered during the construction of Christoffel symbols when using the Caputo operator and the appearance of unwanted terms when using the Riemann-Liouville derivative because of the well-known fact that its action on constants does not vanish, as expected. The key innovation of the new operator ensures that the derivative of a constant is zero. As a particular application, we explore whether the Bañados-Teitelboim-Zanelli black hole metric is a solution to fractional Einstein equations. Our results reveal that for values of the fractional parameter close to one, the effective matter sector corresponds to a charged BTZ solution with an anisotropic cosmological constant.

Abstract The Quantum Natural Gradient (QNG) method enhances optimization in variational quantum algorithms (VQAs) by incorporating geometric insights from the quantum state space through the Fubini-Study metric. In this work, we extend QNG by introducing higher-order integrators and geodesic corrections using the Riemannian Euler update rule and geodesic equations, deriving an updated rule for the Quantum Natural Gradient with Geodesic Correction (QNGGC). We also develop an efficient method for computing the Christoffel symbols necessary for these corrections, leveraging the parameter-shift rule to enable direct measurement from quantum circuits. Through theoretical analysis and practical examples, we demonstrate that QNGGC significantly improves convergence rates over standard QNG, highlighting the benefits of integrating geodesic corrections into quantum optimization processes. Our approach paves the way for more efficient quantum algorithms, leveraging the advantages of geometric methods.

The article focuses on the development and implementation of an effective methodology for measuring high-precision parts on coordinate measuring machines (CMM). The proposed approach addresses the challenges associated with complex geometry measurements under variable environmental conditions by combining advanced mathematical modeling techniques with adaptive error compensation algorithms. The mathematical foundation is based on the application of tensor formalism in Riemannian space, which allows for more precise model-ing of geometric errors using Christoffel symbols and covariant derivatives of the error po-tential. This approach significantly improves the accuracy of spatial positioning. A refined stochastic error model, incorporating Stratonovich integrals and fractional Brownian motion, provides a more accurate description of random processes occurring in the measurement sys-tem. To enhance overall measurement accuracy, an adaptive correction algorithm is pro-posed, based on Itô stochastic differential equations with Fourier-Bessel series expansion, which ensures efficient compensation of systematic errors. In addition, the measurement tra-jectory optimization is formulated as a variational problem, taking into account holonomic and non-holonomic constraints, enabling the optimal positioning strategy. A thermoelastic deformation model based on sixth-rank tensors and Green’s functions is developed to account for temperature-induced deformations, ensuring the reliability of measurements under ther-mal instability. Experimental verification confirmed the effectiveness of the proposed method-ology, demonstrating a 15-20% reduction in systematic errors, a 10-15% decrease in random errors, and an approximately 20% improvement in the accuracy of measurement uncertainty estimation. The proposed methodology combines theoretical advancements with practical so-lutions, providing a robust tool for improving the accuracy and reliability of coordinate measurements in industrial metrology applications.