Search Results (21)

Three-dimensional images of multiphase natural and engineered materials obtained by X-ray micro-computed tomography require quantitative processing methods that can describe not only phase volume but also connectivity, spatial heterogeneity, and anisotropy. In this article, X-ray micro-computed tomography is abbreviated as X-$\mu$CT. Scalar descriptors such as fractal dimension, Betti numbers, Euler characteristic, and Minkowski functionals provide compact phase-level summaries of segmented X-$\mu$CT data, but they do not encode where structural heterogeneity occurs, whether connectivity is directionally spanning, how finite sample boundaries affect topological measurements, or how surface-normal orientation is distributed. We propose a unified methodological framework that extends scalar topological and Minkowski-functional analysis of segmented multiphase 3D images by adding cut-response analysis, including its boundary-sensitivity interpretation, directional connectivity and orientation descriptors, and the rank-two surface Minkowski tensor $W_1^{0,2}$. The framework is demonstrated on a previously published segmented geological X-$\mu$CT volume used as a benchmark multiphase geometry with four X-ray-density phases and on synthetic validation geometries with analytically known topology. The results show that the proposed extensions reveal spatial sensitivity, boundary-to-boundary connectivity, and surface fabric that are not captured by scalar phase-level invariants alone. The proposed framework can be used to analyze segmented 3D images of multiphase geological, porous, composite, and engineered samples, thereby expanding quantitative knowledge about their internal structure beyond scalar phase-level descriptors. Full article

We put forward the idea that in addition to invariance under coordinate transformations of general relativity (GR) the gravitational interaction is invariant under arbitrary scale deformations of the metric field, as well as other fields. In addition, we assume that the scaling field has an internal symmetry. Because scale invariance is imposed only on gravity, while the standard model (SM) retains its usual non-Weyl-invariant form, the full framework is not merely a rewriting of GR plus the SM. The global charges that are associated with the internal symmetry could potentially source the gravitational field. Throughout, the framework is treated as a classical generalization of GR on sufficiently sub-Planckian energies; accordingly, any discussion of global charges is to be understood within this effective classical domain. In the case that isotropic deformations are considered, the theory reduces to a Weyl-invariant scalar–tensor (WIST) version of GR. In case that the reference metric is chosen to be Minkowski, the metric factorization takes the standard vierbein form, apart from the additional internal structure assumed here. A few other implications of WIST are considered as well. Full article

We develop the geometry of events in a deformable cellular spacetime, extending our earlier cellular-spaces framework from cellular complexes to cellular events complexes. The framework operates within the conformal class of Minkowski space; in four dimensions, this is the vanishing-Weyl-tensor sector, which excludes Schwarzschild, Kerr, and gravitational-wave spacetimes. The framework treats integer counts of cell crossings as the primitive geometric data: spatial separation between events is the shortest count of face-adjacent cells; temporal separation is the cell-crossing count of a reference light pulse. Newton’s universal clock is replaced by an operational one: the temporal count distance is the ratio of cell length to the speed of light through a cell, and because both quantities are invariants of the co-deformation, the temporal count is itself an invariant: temporal separation is operationally measured via light-pulse counts rather than posited as an external coordinate. Under the co-deformation principle, a single positive scalar field $\rho$ (cell density) controls both the rod length and the clock period. We prove six results, all expressed in terms of counts on the cellular events complex, with a smooth conformally flat metric $\tilde{g}=e^{2\phi }\eta$ ($\phi =-\frac{1}{3}ln\rho$) appearing only as the comparison/calibration object for convergence statements. First, the scalar curvature of the smooth comparison metric is the closed-form differential operator $\tilde{R}=2\square \rho /{\rho }^{1/3}-(8/3){(\partial \rho )}^2/{\rho }^{4/3}$. Second, the volume of a small Alexandrov interval admits an explicit asymptotic expansion in the interval height T, with leading correction $Q(m,u)T^2$ involving an anisotropic invariant at the midpoint m. Third, Q is irreducible to scalar and Ricci-directional invariants alone; the explicit decomposition $Q=\frac{1}{45}\tilde{R}+\frac{1}{5}{\tilde{R}}_{uu}+\frac{1}{2}\mathcal{J}$ exhibits a third independent invariant $\mathcal{J}(m,u)={(u·\partial )}^2\phi \left(m\right)$ as new structural content of the Lorentzian diagnostic. Fourth, the discrete-to-continuum convergence of counts on the cellular events complex yields a counts-only curvature estimator with rate $O\left(\sqrt{a}\right)$ at the joint scaling $T\asymp \sqrt{a}$. Fifth, the smooth comparison metric itself is reconstructible from counts on the discrete complex at rate $O\left(a\right)$: the conformally flat Lorentzian geometry is uniquely determined, up to background Minkowski calibration, by the cellular events complex. Sixth, a finite collection of Alexandrov-interval volume measurements at a fixed midpoint suffices to recover the full local curvature data $\{\tilde{R}\left(m\right),{\tilde{R}}_{\mu \nu }\left(m\right),\mathcal{J}(m,u)\}$ at rate $O\left(\sqrt{a}\right)$ (curvature spectroscopy); and the temporal light-tick count $\lambda$ is essential in a precise sense—there exist conformally flat Lorentzian geometries indistinguishable on every spatial slice by the earlier spatial-only diagnostic but distinguished at the origin by the events-space directional invariant. The framework’s scope is the conformal class of Minkowski: flat FLRW in conformal time, leading-order weak-field gravity, and 2D gravity. This paper is a mathematical contribution to discrete-to-continuum geometry on cellular events complexes; it is not a physical theory of gravity. Full article

We develop a fully gauge-invariant analysis of gravitational-wave polarizations in metric $f\left(R\right)$ gravity with a particular focus on the modified Starobinsky model $f\left(R\right)=R+\alpha R^2-2\mathsf{\Lambda }$, whose constant-curvature solution $R_d=4\mathsf{\Lambda }$ provides a natural de Sitter background for both early- and late-time cosmology. Linearizing the field equations around this background, we derive the Klein–Gordon equation for the curvature perturbation $\delta R$ and show that the scalar propagating mode acquires a mass $m_{\psi }^2=1/\left(6\alpha \right)$, highlighting how the same scalar degree of freedom governs inflationary dynamics at high curvature and the propagation of gravitational waves in the current accelerating Universe. Using the scalar–vector–tensor decomposition and a decomposition of the perturbed Ricci tensor, we obtain a set of fully gauge-invariant propagation equations that isolate the contributions of the scalar, vector, and tensor modes in the presence of matter. We find that the tensor sector retains the two transverse–traceless polarizations of General Relativity, while the scalar sector contains an additional massive scalar propagating degree of freedom, which manifests through breathing and longitudinal tidal responses depending on the wave regime and detector frame. Through the geodesic deviation equation—computed both in a local Minkowski patch and in fully covariant de Sitter form—we independently recover the same polarization content and identify its tidal signatures. The resulting framework connects the extra scalar polarization to cosmological observables: the massive scalar propagating mode sets the range of the fifth force, influences the time evolution of gravitational potentials, and affects the propagation and dispersion of gravitational waves on cosmological scales. This provides a unified, gauge-invariant link between gravitational-wave phenomenology and the cosmological implications of metric $f\left(R\right)$ gravity. Full article

In this work, we investigate tensor perturbations in a de Sitter background within the framework of Chern–Simons modified gravity. We introduce transverse-traceless perturbations and analyze how the Chern–Simons Cotton tensor induces parity-violating modifications to gravitational wave propagation, while the Pontryagin density vanishes at linear order. Using a mode decomposition of the scalar background field, we derive the sub- and super-horizon limits of the wave equations and uncover chiral corrections in the dispersion relations of tensor modes. The resulting birefringence exhibits both amplitude and velocity components, alternating with the phase of the scalar field. Particular solutions sourced by the scalar background show helicity-dependent amplification and a characteristic scaling of the radiated flux that reduces smoothly to the Minkowski limit. The accumulated phase difference between right- and left-handed modes grows quadratically inside the horizon and becomes frozen outside, leaving a permanent parity-violating imprint in the primordial tensor spectrum. Finally, by promoting the Chern–Simons field to a massive dark matter candidate, we demonstrate how its mass-dependent dynamics connect gravitational birefringence to axion-like dark matter phenomenology. Full article

The optical force exerted on a dipole particle can be divided into gradient force, scattering force, and spin–curl force, all of which can be derived from Maxwell’s stress tensor with the dipole approximation. Here, we identify an additional spin–curl force for arbitrary objects beyond the dipole approximation, which is named the generalized spin–curl force in this paper. The generalized spin–curl force originates from the Minkowski force density and depends on the imaginary parts of the permittivity, permeability, and chirality of the object. However, it remains imperceptible in conventional optical force calculations due to its exact cancellation by a compensatory surface force during MST surface integration. The study of the generalized spin–curl force provides critical insights into elucidating the mechanisms underlying optical momentum transfer and internal force distribution within complex media. Furthermore, the generalized spin–curl force offers a novel mechanism for enhancing optical sensors, enabling highly sensitive detection of absorptive or chiral perturbations in systems such as microcavities and metasurfaces. Its ability to manipulate internal force distributions also provides new pathways for advancing optical force probes and chirality-selective sensing at the nanoscale. Full article

We analyse linearised field equations around the Minkowski metric, with its standard flat parallel transport structure, in models of newer GR, which refers to quadratic actions in terms of a nonmetricity tensor. We show that half of the freedom in choosing the model parameters is immediately fixed by asking for reasonable properties of tensors and vectors, defined with respect to spatial rotations, and we accurately describe the much more complicated sector of scalars. In particular, we show that, from the teleparallel viewpoint, the STEGR model with an additional term of a gradient squared of the metric determinant exhibits one and a half new dynamical modes, and not just one new dynamical mode as it was previously claimed. Full article

The paper investigates the vacuum expectation value of the surface energy–momentum tensor (SEMT) for a scalar field with general curvature coupling in the geometry of two branes orthogonal to the boundary of anti-de Sitter (AdS) spacetime. For Robin boundary conditions on the branes, the SEMT is decomposed into the contributions corresponding to the self-energies of the branes and the parts induced by the presence of the second brane. The renormalization is required for the first parts only, and for the corresponding regularization the generalized zeta function method is employed. The induced SEMT is finite and is free from renormalization ambiguities. For an observer living on the brane, the corresponding equation of state is of the cosmological constant type. Depending on the boundary conditions and on the separation between the branes, the surface energy densities can be either positive or negative. The energy density induced on the brane vanishes in special cases of Dirichlet and Neumann boundary conditions on that brane. The effect of gravity on the induced SEMT is essential at separations between the branes of the order or larger than the curvature radius for AdS spacetime. In the considerably large separation limit, the decay of the SEMT, as a function of the proper separation, follows a power law for both massless and massive fields. For parallel plates in Minkowski bulk and for massive fields the fall-off of the corresponding expectation value is exponential. Full article

A model of spacetime is presented. It has an extension to five dimensions, and in five dimensions the geometry is the dual of the Euclidean geometry w.r.t. an arbitrary positive-definite metric. Dually flat geometries are well-known in the context of information geometry. The present work explores their role in describing the geometry of spacetime. It is shown that the positive-definite metric with its flat 5-d connection can coexist with a pseudometric for which the connection is that of Levi–Civita. The 4-d geodesics are characterized by five conserved quantities, one of which can be chosen freely and is taken equal to zero in the present work. An explicit expression for the parallel transport operators is obtained. It is used to construct a pseudometric for spacetime by choosing an arbitrary possibly degenerate inner product in the tangent space of a reference point, for instance, that of Minkowski. By parallel transport, one obtains a pseudometric for spacetime, the metric connection of which extends to a 5-d connection with vanishing curvature tensor. The de Sitter space is considered as an example. Full article

We write the Kerr–Schild tetrads in terms of the flat space–time tetrads and of a (1, 1) tensor $S_{\mu }^{\lambda }$. This tensor can be considered as a projection operator, since it transforms (i) flat space–time tetrads into non-flat tetrads, and vice-versa, and (ii) the Minkowski space–time metric tensor into a non-flat metric tensor, and vice-versa. The $S_{\mu }^{\lambda }$ tensor and its inverse are constructed in terms of the standard null vector field $l_{\mu }$ that defines the Kerr–Schild form of the metric tensor in general relativity, and that yields black holes and non-linear gravitational waves as solutions of the vacuum Einstein’s field equations. We demonstrate that the condition for the vanishing of the Ricci tensor obtained by Kerr and Schild, in empty space–time, is also a condition for the vanishing of the Nijenhuis tensor constructed out of $S_{\mu }^{\lambda }$. Thus, a theory based on the Nijenhuis tensor yields an important class of solutions of the Einstein’s field equations, namely, black holes and non-linear gravitational waves. We also demonstrate that the present mathematical framework can easily admit modifications of the Newtonian potential that may explain the long range gravitational effects related to galaxy rotation curves. Full article

We show that a Minkowski phase space endowed with a bracket relatively to a conformable differential realizes a Poisson algebra, confering a bi-Hamiltonian structure to the resulting manifold. We infer that the related Hamiltonian vector field is an infinitesimal Noether symmetry, and compute the corresponding deformed recursion operator. Besides, using the Hamiltonian–Jacobi separability, we construct recursion operators for Hamiltonian vector fields in conformable Poisson–Schwarzschild and Friedmann–Lemaître–Robertson–Walker (FLRW) manifolds, and derive the related constants of motion, Christoffel symbols, components of Riemann and Ricci tensors, Ricci constant and components of Einstein tensor. We highlight the existence of a hierarchy of bi-Hamiltonian structures in both the manifolds, and compute a family of recursion operators and master symmetries generating the constants of motion. Full article

It is well known that extraordinary photons in hyperbolic metamaterials may be described as living in an effective Minkowski spacetime, which is defined by the peculiar form of the strongly anisotropic dielectric tensor in these metamaterials. Here, we demonstrate that within the scope of this approximation, the sound waves in hyperbolic metamaterials look similar to gravitational waves, and therefore the quantized sound waves (phonons) look similar to gravitons. Such an analogue model of quantum gravity looks especially interesting near the phase transitions in hyperbolic metamaterials where it becomes possible to switch quantum gravity effects on and off as a function of metamaterial temperature. We also predict strong enhancement of sonoluminescence in ferrofluid-based hyperbolic metamaterials, which looks analogous to particle creation in strong gravitational fields. Full article

In this paper, we present a complete proof of the so-called First On-Shell Theorem that determines dynamical content of the unfolded equations for free symmetric massless fields of arbitrary integer spin in any dimension and arbitrary integer or half-integer spin in four dimensions. This is achieved by calculation of the respective ${\sigma }_{-}$ cohomology both in the tensor language in Minkowski space of any dimension and in terms of spinors in $Ad{S}_4$. In the d-dimensional case $H^p\left({\sigma }_{-}\right)$ is computed for any p and interpretation of $H^p\left({\sigma }_{-}\right)$ is given both for the original Fronsdal system and for the associated systems of higher form fields. Full article

We consider a gauge theory of vector fields in $3D$ Minkowski space. At the free level, the dynamical variables are subjected to the extended Chern–Simons (ECS) equations with higher derivatives. If the color index takes n values, the third-order model admits a $2n$-parameter series of second-rank conserved tensors, which includes the canonical energy–momentum. Even though the canonical energy is unbounded, the other representatives in the series have a bounded from below the 00-component. The theory admits consistent self-interactions with the Yang–Mills gauge symmetry. The Lagrangian couplings preserve the energy–momentum tensor that is unbounded from below, and they do not lead to a stable non-linear theory. The non-Lagrangian couplings are consistent with the existence of a conserved tensor with a 00-component bounded from below. These models are stable at the non-linear level. The dynamics of interacting theory admit a constraint Hamiltonian form. The Hamiltonian density is given by the 00-component of the conserved tensor. In the case of stable interactions, the Poisson bracket and Hamiltonian do not follow from the canonical Ostrogradski construction. Particular attention is paid to the “triply massless” ECS theory, which demonstrates instability even at the free level. It is shown that the introduction of extra scalar field, serving as Higgs, can stabilize the dynamics in the vicinity of the local minimum of energy. The equations of motion of the stable model are non-Lagrangian, but they admit the Hamiltonian form of dynamics with a Hamiltonian that is bounded from below. Full article

Without the mass-energy equivalence available on Minkowski spacetime $\mathbb{M}$, it is not possible on 4-dimensional non-relativistic Galilei/Newton spacetime $\mathbb{G}$ to combine 3-momentum and total mass-energy in a single tensor object. However, given a fiducial frame, it is possible to combine 3-momentum and kinetic energy into a linear form (particle) or $(1,1)$ tensor (continuum) in a manner that exhibits increased unity of classical mechanics on flat relativistic and non-relativistic spacetimes $\mathbb{M}$ and $\mathbb{G}$. As on $\mathbb{M}$, for a material continuum on $\mathbb{G}$, the first law of thermodynamics can be considered a consequence of a unified dynamical law for energy-momentum rather than an independent postulate. Full article