Abstract In quantum theory on curved backgrounds, Heisenberg's uncertainty principle is usually discussed in terms of ensemble variances and flat-space commutators. Here we take a different, preparation-based viewpoint tailored to sharp position measurements on spacelike hypersurfaces in general relativity. A projective localization is modeled as a von Neumann--L"uders projection onto a geodesic ball B(r) of radius r on a Cauchy slice, with the post-measurement state described by Dirichlet data. Using DeWitt-type momentum operators adapted to an orthonormal frame, we construct a geometric, coordinate-invariant momentum standard deviation σ p and show that strict confinement to B(r) enforces an intrinsic kinetic-energy floor. The lower bound is set by the first Dirichlet eigenvalue λ 1 of the Laplace--Beltrami operator on the ball, σ p ≥ √λ 1 , and is manifestly invariant under changes of coordinates and foliation. A variance decomposition separates the contribution of the modulus |ψ| from phase-gradient fluctuations and clarifies how the spectral geometry controls momentum uncertainty. 
 Assuming only minimal geometric information, weak mean-convexity of the boundary yields a universal, scale-invariant Heisenberg-type product bound, σ p r ≥ \pi\hbar/2, depending only on the proper radius r.

Publisher Correction: The Newtonian limit of orthonormal frames in metric theories of gravity

Abstract We extend well-known results on the Newtonian limit of Lorentzian metrics to orthonormal frames. Concretely, we prove that, given a one-parameter family of Lorentzian metrics that in the Newtonian limit converges to a Galilei structure, any family of orthonormal frames for these metrics converges pointwise to a Galilei frame, assuming that the two obvious necessary conditions are satisfied: the spatial frame must not rotate indefinitely as the limit is approached, and the frame’s boost velocity with respect to some fixed reference observer needs to converge.

This paper explores some frame bundles and physical implications of Killing vector fields on the two-sphere S2, culminating in a novel application to Maxwell’s equations in free space. Initially, we investigate the Killing vector fields on S2 (represented by the unit sphere of R3), which generate the isometries of the sphere under the rotation group SO(3). These fields, realized as functions Kv:S2→R3, defined by Kv(q)=v×q for a fixed v∈R3 and any q∈S2, generate a three-dimensional Lie algebra isomorphic to so(3). We establish an isomorphism K:R3→K(S2), mapping vectors v=au (with u∈S2) to scaled Killing vector fields aKu, and analyze its relationship with SO(3) through the exponential map. Subsequently, at a fixed point e∈S2, we construct a smooth orthonormal right-handed tangent frame fe:S2\{e,−e}→T(S2)2, defined as fe(u)=(K^e(u),u×K^e(u)), where K^e is the unit vector field of the Killing field Ke. We verify its smoothness, orthonormality, and right-handedness. We further prove that any smooth orthonormal right-handed frame on S2\{e,−e} is either fe or a rotation thereof by a smooth map ρ:S2\{e,−e}→SO(3), reflecting the triviality of the frame bundle over the parallelizable domain. The paper then pivots to an innovative application, constructing solutions to Maxwell’s equations in free space by combining spherical symmetries with quantum mechanical de Broglie waves in tempered distribution wave space. The deeper scientific significance lies in bringing together differential geometry (via SO(3) symmetries), quantum mechanics (de Broglie waves in Schwartz distribution theory), and electromagnetism (Maxwell’s solutions in Schwartz tempered complex fields on Minkowski space-time), in order to offer a unifying perspective on Maxwell’s electromagnetism and Schrödinger’s picture in relativistic quantum mechanics.

In this study we reintroduce the theory of curves by incorporating local fractional calculus. We elucidate the condition for a naturally parametrized curve to be conformable, we define the orthonormal conformable frame of such a curve at any given point. Then we provide a comprehensive explanation of how these newly derived conformable geometric concepts are related to their classical counterparts. Furthermore, we introduce the concept of a conformable rectifying curve and provide its characterizations in terms of this differentiation with respect to arbitrary order. Some illustrative graphs are provided.

In this study, the W-Bertrand curves lying on the Q3 are examined and the notion of γ−Bertrand curves( and α−Bertrand curves, β−Bertrand curves, y−Bertrand curves, respectively). Also, the Bertrand pair {γ,Γ} in terms of their curvature functions are obtained, and the necessary and sufficient conditions for the W-Bertrand curves are expressed using the asymptotic orthonormal frame in Q3. Furthermore, the helix curve is characterized in terms of curvature according to the condition of being W-Bertrand curve pair.

This study is about the dual spacelike curves lying on the dual lightlike cone, which can be either symmetric or asymmetric. We first establish the dual associated curve, which is related to the reference curve. Using these curves and the derivative of the reference curve, we derive the dual asymptotic orthonormal frame. Next, we define the dual structure function, curvature function, and Frenet formulae, and express the curvature function in terms of the dual structure function. This leads to a differential equation that characterizes the dual cone curve in relation to its curvature function. Since curves with constant curvature maintain the same curvature at every point, their geometry is more predictable. Therefore, we assume that the dual cone curvature function is constant and examine how this condition affects the behavior and geometric properties of the dual curves. As a result of this investigation, some new results and definitions are obtained.

Rebricking frames and bases

Given an equivariant noncommutative principal bundle, we construct an Atiyah sequence of braided derivations whose splittings give connections on the bundle. Vertical braided derivations act as infinitesimal gauge transformations on connections. In the case of the principal \mathrm{SU}(2) -bundle over the sphere S^{4}_{\theta} an equivariant splitting of the Atiyah sequence recovers the instanton connection. An infinitesimal action of the braided conformal Lie algebra \mathrm{so}_{\theta}(5,1) yields a five parameter family of splittings. On the principal \mathrm{SO}_\theta(2n,\mathbb{R}) -bundle of orthonormal frames over the sphere S^{2n}_\theta , a splitting of the sequence leads to the Levi-Civita connection for the ‘round’ metric on S^{2n}_\theta . The corresponding Riemannian geometry of S^{2n}_{\theta} is worked out.

Abstract This work introduces a geometrical object that generalizes the quantum geometric tensor; we call it N-bein. Analogous to the vielbein (orthonormal frame) used in the Cartan formalism, the N-bein behaves like a ‘square root’ of the quantum geometric tensor. Using it, we present a quantum geometric tensor of two states that measures the possibility of moving from one state to another after two consecutive parameter variations. This new tensor determines the commutativity of such variations through its anti-symmetric part. In addition, we define a connection different from the Berry connection, and combining it with the N-bein allows us to introduce a notion of torsion and curvature à la Cartan that satisfies the Bianchi identities. Moreover, the torsion coincides with the anti-symmetric part of the two-state quantum geometric tensor previously mentioned, and thus, it is related to the commutativity of the parameter variations. We also describe our formalism using differential forms and discuss the possible physical interpretations of the new geometrical objects. Furthermore, we define different gauge invariants constructed from the geometrical quantities introduced in this work, resulting in new physical observables. Finally, we present two examples to illustrate these concepts: a harmonic oscillator and a generalized oscillator, both immersed in an electric field. We found that the new tensors quantify correlations between quantum states that were unavailable by other methods.

In a recent paper (Kim et al 2023 arXiv:2305.08922 [hep-th]), it has been proposed that the endpoint of the Kerr-AdS superradiant instability is a Grey Galaxy. The conjectured solutions are supposed to be made up of a black hole with critical angular velocity in the centre of AdS, surrounded by a large flat disk of thermal bulk gas that revolves around the black hole. In the analysis of the proposed solutions so far, gravitational effects due to the black hole on the thermal gas have been neglected. A way to estimate these effects is via computing tidal forces. With this motivation, we study tidal forces on objects moving in the Kerr-AdS spacetime. To do so, we construct a parallel-transported orthonormal frame along an arbitrary timelike or null geodesic. We then specialise to the class of fast rotating geodesics lying in the equatorial plane, and estimate tidal forces on the gas in the Grey galaxies, modelling it as a collection of particles moving on timelike geodesics. We show that the tidal forces are small (and remain small even in the large mass limit), thereby providing additional support to the idea that the gas is weakly interacting with the black hole.

A Second-Order SO(3)-Preserving and Energy-Stable Scheme for Orthonormal Frame Gradient Flow Model of Biaxial Nematic Liquid Crystals

In this study, the theory of curves is reconstructed with fractional calculus. The condition of a naturally parametrized curve is described, and the orthonormal conformable frame of the naturally parametrized curve at any point is defined. Conformable helix and conformable slant helix curves are defined with the help of conformable frame elements at any point of the conformable curve. The characterizations of these curves are obtained in parallel with the conformable analysis Finally, examples are given for a better understanding of the theories and their drawings are given with the help of Mathematics.

We explore the cosmological implications of the local limit of nonlocal gravity, which is a classical generalization of Einstein’s theory of gravitation within the framework of teleparallelism. An appropriate solution of this theory is the modified Cartesian flat cosmological model. The main purpose of this paper is to study linear perturbations about the orthonormal tetrad frame field adapted to the standard comoving observers in this model. The observational viability of the perturbed model is examined using all available data regarding the cosmic microwave background. The implications of the linearly perturbed modified Cartesian flat model are examined and it is shown that the model is capable of alleviating the H 0 tension.

We analytically extend the 5D Myers–Perry metric through the event and Cauchy horizons by defining Eddington–Finkelstein-type coordinates. Then, we use the orthonormal frame formalism to formulate and perform separation of variables on the massive Dirac equation, and analyse the asymptotic behaviour at the horizons and at infinity of the solutions to the radial ordinary differential equation (ODE) thus obtained. Using the essential self-adjointness result of Finster–Röken and Stone’s formula, we obtain an integral spectral representation of the Dirac propagator for spinors with low masses and suitably bounded frequency spectra in terms of resolvents of the Dirac Hamiltonian, which can in turn be expressed in terms of Green’s functions of the radial ODE.

Abstract In 1946, Dennis Gabor introduced the analytic signal for real‐valued signals f. Here, H is the Hilbert transform. This complexification of functions allows for an analysis of their amplitude and phase information and has ever since given well‐interpretable insight into the properties of the signals over time. The idea of complexification has been reconsidered with regard to many aspects: examples are the dual tree complex wavelet transform, or via the Riesz transform and the monogenic signal, that is, a multi‐dimensional version of the Hilbert transform, which in combination with multi‐resolution approaches leads to Riesz wavelets, and others. In this context, we ask two questions: Which pairs of real orthonormal bases (ONBs), Riesz bases, frames and Parseval frames and can be “rebricked” to complex‐valued ones ? And which real operators A allow for rebricking via the ansatz ? In this short note, we give answers to these questions with regard to a characterization which linear operators A are suitable for rebricking while maintaining the structure of the original real valued family. Surprisingly, the Hilbert transform is not among them.

We consider curves γ:[0,1]→R3\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma : [0, 1]\\rightarrow {\\mathbb {R}}^3$$\\end{document} endowed with an adapted orthonormal frame r:[0,1]→SO(3)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r: [0, 1]\\rightarrow SO(3)$$\\end{document}. We wish to deform such framed curves (γ,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma , r)$$\\end{document} while preserving two contraints: a local constraint prescribing one of its ‘curvatures’ (i.e., off-diagonal elements of r′rT\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$r'r^T$$\\end{document}), and a global constraint prescribing the initial and terminal values of γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} and r. We proceed in two stages. First we deform the frame r in a way that is naturally compatible with the constraints on r, by interpreting the local constraint in terms of parallel transport on the sphere. This provides a link to the differential geometry of surfaces. The deformation of the base curve γ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma $$\\end{document} is achieved in a second step, by means of a suitable reparametrization of the frame. We illustrate this deformation procedure by providing some applications: first, we characterize the boundary conditions on (γ,r)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$(\\gamma , r)$$\\end{document} that are accessible without violating the local constraint; then, we provide a short proof of a smooth approximation result for framed curves satisfying both the differential and the global constraints. Finally, we also apply these ideas to elastic ribbons with nonzero width.

The 4+1 formalism in general relativity expresses the Einstein equations as a manifestly covariant initial value problem, resulting in a pair of first order evolution equations for the metric γμv and intrinsic curvature Kμv of spacetime geometry (μ, v = 0, 1, 2, 3). This approach extends the Stueckelberg-Horwitz-Piron (SHP) framework, a covariant approach to canonical particle mechanics and field theory employing a Lorentz scalar Hamiltonian K and an external chronological parameter τ. The SHP Hamiltonian generates τ-evolution of spacetime events xμ (τ) or ψ (x, τ) in an a priori unconstrained phase space; standard relativistic dynamics can be recovered a posteriori by imposing symmetries that express the usual mass shell constraint for individual particles and fields as conservation laws. As a guide to posing field equations for the evolving metric, we generalize the structure of SHP electrodynamics, with particular attention to O(3,1) covariance. Thus, the 4+1 method first defines a 5D pseudo-spacetime as a direct product of spacetime geometry and chronological evolution, poses 5D field equations whose symmetry must be broken to 4D, and then implements the implied 4+1 foliation to obtain evolution equations. In this paper, we sharpen and clarify the interpretation of this decomposition by introducing a fixed orthonormal quintrad frame and a 5D vielbein field that by construction respects the preferred 4+1 foliation. We show that for any diagonal metric, this procedure enables the evolution equation for the metric to be replaced by an evolution equation for the vielbein field itself, simplifying calculation of the spin connection and curvature.

By analyzing algebraically special solutions of the Maxwell equations in the Kerr space-time, we obtain the exact expressions for the polarization characteristics of electromagnetic waves emitted from the vicinity of a black hole. We revealed the asymmetry of dependence of the ellipticity angle on the polar angle for the fundamental mode and the first harmonics of polarized radiation. This creates a basis for the new method of evaluation of the intrinsic angular momentum of the Kerr black hole. It is shown that the existence of singular points of the solution in a local orthonormal frame is a consequence of the Poincaré–Brouwer theorem.

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