Local Spacetime Research Articles

Splitting the spacetime: a systematic analysis of foliation dependence in cosmic averaging

It is a fundamental unsolved question in general relativity how to unambiguously characterize the effective collective dynamics of an ensemble of fluid elements sourcing the local geometry, in the absence of exact symmetries.In a cosmological context this is sometimes referred to as the averaging problem.At the heart of this problem in relativity is the non-uniqueness of the choice of foliation within which the statistical properties of the local spacetime are quantified,which can lead to ambiguity in the formulated average theory. This has led to debate in the literature on how to best construct and view such a coarse-grained hydrodynamic theory.Here, we address this ambiguity by performing the first quantitative investigation of foliation dependence in cosmological spatial averaging. Starting from the aim of constructing slicing-independent integral functionals (volume, mass, entropy, etc.) as well as average functionals (mean density, average curvature, etc.) defined on spatial volume sections, we investigate infinitesimal foliation variations and derive results on the foliation dependence of functionals and on extremal leaves.Our results show that one may only identify fully foliation-independent integral functionals in special scenarios, requiring the existence of associated conserved currents.We then derive bounds on the foliation dependence of integral functionals for general scalar quantities under finite variations within physically motivated classes of foliations.Our findings provide tools that are useful for quantifying, eliminating or constraining the foliation dependence in cosmological averaging.

About Some Applications of Mathematics

In this paper, a new mass-velocity relation is derived. The orbit of planetary motion is deduced based on this new relation and the Newtonian laws of motion along with the concept of proper time. The derivation does not use the tensor concept, but the relativistic orbit is obtained as a first approximation of the modified Newtonian theory. Next, we discuss the possibility of introducing a field dependent metric for the EM field by considering the observation that E2 - c2 B2 or E2 - B2 with c = 1 is approximately an invariant quantity for EM Field [1-3]. It is possible to extend the metric for the gravitational case also [4]. The discussion of anomalous characteristic of Lorentz Transformation (LT) and the introduction of a non-linear transformation connecting Local Space-time coordinates (ct,x) with proper system is continued.

Field Dependant Metric for Gravitational/EM Fields and a NonLinear Transformation for Local to Proper Space-Time World

In this paper, the introduction of a metric for Gravitational Field is examined based on [4]; this can be extended to EM Fields also by change of the parameters involved (ϵ1 , μ1 ) to (ϵ2 , μ2 ) We know that E2 ‒ c2 B2 or E2 ‒B2 with is an invariant quantity for the EM Fields which can be extended to gravitational fields, as done in [1]. We discuss the metric for the gravitational case and extended to the EM fields as well. The discussion of anomalous characteristic of Lorentz Transformation (LT) and the introduction of a non-linear transformation connecting Local Space-time coordinates (ct,x) with proper system (cτ, xτ ) is continued.

The Hole Argument without the notion of isomorphism

In this paper, I argue that the Hole Argument can be formulated without using the notion of isomorphism, and for this reason it is not threatened by the criticism of Halvorson and Manchak (Br J Philos Sci, 2022. https://doi.org/10.1086/719193). Following Earman and Norton (Br J Philos Sci 38, pp. 515–525, 1987), I divide the Hole Argument into two steps: the proof of the Gauge Theorem and the transition from the Gauge Theorem to the conclusion of radical indeterminism. In the analaysis of the first step, I argue that the Gauge Theorem does not rely on the notion of isomorphism but on the notion of the diffeomorphism-invariance of the equations of local spacetime theories; however, for this approach to work, the definition of local spacetime theories needs certain amendments with respect to Earman and Norton’s formulation. In the analysis of the second step, I postulate that we should use the notion of radical indeterminism instead of indeterminism simpliciter and that we should not decide in advance what kind of maps are to be used in comparing models. Instead, we can tentatively choose some kind of maps for this purpose and check whether a given choice leads to radical indeterminism involving empirically indistinguishable models. In this way, the use of the notion of isomorphism is also avoided in the second step of the Hole Argument. A general picture is that physical equivalence can be established by means of an iterative procedure in which we examine various candidate classes of maps, and, depending on the outcomes, we need to broaden or narrow these classes. The Hole Argument can be viewed as a particular instance of this procedure.

A posteriori error estimates for the Richards equation

The Richards equation is commonly used to model the flow of water and air through soil, and it serves as a gateway equation for multiphase flows through porous media. It is a nonlinear advection–reaction–diffusion equation that exhibits both parabolic–hyperbolic and parabolic–elliptic kind of degeneracies. In this study, we provide reliable, fully computable, and locally space–time efficient a posteriori error bounds for numerical approximations of the fully degenerate Richards equation. For showing global reliability, a nonlocal-in-time error estimate is derived individually for the time-integrated H 1 ( H − 1 ) H^1(H^{-1}) , L 2 ( L 2 ) L^2(L^2) , and the L 2 ( H 1 ) L^2(H^1) errors. A maximum principle and a degeneracy estimator are employed for the last one. Global and local space–time efficiency error bounds are then obtained in a standard H 1 ( H − 1 ) ∩ L 2 ( H 1 ) H^1(H^{-1})\cap L^2(H^1) norm. The reliability and efficiency norms employed coincide when there is no nonlinearity. Moreover, error contributors such as space discretization, time discretization, quadrature, linearization, and data oscillation are identified and separated. The estimates are also valid in a setting where iterative linearization with inexact solvers is considered. Numerical tests are conducted for nondegenerate and degenerate cases having exact solutions, as well as for a realistic case and a benchmark case. It is shown that the estimators correctly identify the errors up to a factor of the order of unity.

An intuitive and simplified formulation of general relativistic gravity in freely falling frames

General relativistic gravity in freely falling frames is formulated using undergraduate calculus and matrix algebra. The key to the formulation involves retaining Newton’s first law of motion in the local spacetime of freely falling frames. The formulation shows that the relative acceleration of two particles in the freely falling frame does not depend directly on their relative velocity. Also, the 4 × 4 matrix relating the relative acceleration of two particles to their relative spacetime displacement closely resembles the physics of Newtonian gravity, implying that the local gravitational field obeys Laplace’s equation. The results of this paper and a previous publication [L. R. Miller, Phys. Essays 35, 202 (2022)] provide an intuitive approach that enables the use of lower-level mathematics to arrive at Einstein’s theory of gravity in empty space.

Note on episodes in the history of modeling measurements in local spacetime regions using QFT

Note on episodes in the history of modeling measurements in local spacetime regions using QFT

A localized spacetime Penrose inequality and horizon detection with quasi-local mass

A localized spacetime Penrose inequality and horizon detection with quasi-local mass

Space–time method for analyzing transient heat conduction in functionally graded materials

In this study, a novel approach is presented for simulating transient heat conduction in functionally graded materials (FGM) with varying material properties only in the z direction. The proposed collocation method employs a meshless localized space–time radial basis function (LSTRBF), which yields a linear sparse matrix system, offering advantages over the global method in terms of scalability and suitability for multidimensional problems. Additionally, a novel shape parameter strategy, known as variable leave-one-out cross-validation (LOOCV), is introduced to enhance accuracy while overcoming conventional LOOCV limitations. Four benchmark numerical examples are utilized to demonstrate the simplicity, efficiency, accuracy, and stability of the method. Overall, the proposed LSTRBF method presents a promising alternative for efficiently and accurately simulating transient heat conduction in FGMs.

On the rigidity of cosmological space-times

In this paper, we analyse a family of geometrically well-behaved cosmological space-times (V^{n+1},g), which are foliated by intrinsically isotropic space-like hypersurfaces {M_t}_{tin mathbb {R}}, which are orthogonal to a family of co-moving observers defined by a global time-like vector field U. In particular, this implies such space-times satisfy several of the well-known criteria for isotropic cosmological space-times, although, in the family in question, the simultaneity spaces (M_t,g_t) associated with U can have as sectional curvature a sign-changing function k(t). Being this clearly impossible in the FLRW family of standard cosmological space-times, it motivates us to revisit the geometric rigidity consequences of different definitions of isotropy available in the literature. In this analysis, we divide such definition according to whether the isometries involved are taken to be (local) space-time (STI space-times) or (local) space isometries (SI space-times) of (M_t,g_t) for each t. This subtlety will be shown to be critical, proving that only when space-time isometries are considered one obtains the well-known rigidity properties associated with isotropic cosmological space-times. In particular, SI space-times will be shown to be a strictly larger class than the STI ones, allowing a family of basic cosmological curvature change models which are not even locally isometric to any FLRW space-time.

Local space time constant mean curvature and constant expansion foliations

Inspired by the small sphere-limit for quasi-local energy we study local foliations of surfaces with prescribed mean curvature. Following the strategy used by Ye in [23] to study local constant mean curvature foliations we use a Lyapunov Schmidt reduction in an n+1 dimensional manifold equipped with a symmetric 2-tensor to construct the foliations around a point, prove their uniqueness and show their nonexistence conditions. To be specific, we study two foliation conditions. First we consider constant space-time mean curvature surfaces. They are used to characterizing the center of mass in general relativity [4]. Second, we study local foliations of constant expansion surfaces [18].

Comparative Ontology of Theories of Space and Time

With a few exceptions, physics theories are based in a conception of time and space; our two major theories, general relativity, and quantum field theory, differ in their conceptions. Key issues herein include mathematics, logic, intuition, experiment, and ontology, with emphasis on simultaneity and dimensionality of the world. The treatment is through ontological comparison of two theories, space-time theory (special relativity) and energy-time theory (local absolute space and universal time). These two theories share many of the same equations but have different ontology.

Spatio-Temporal Self-Attention Network for Video Saliency Prediction

3D convolutional neural networks have achieved promising results for video tasks in computer vision, including video saliency prediction that is explored in this paper. However, 3D convolution encodes visual representation merely on fixed local spacetime according to its kernel size, while human attention is always attracted by relational visual features at different time. To overcome this limitation, we propose a novel Spatio-Temporal Self-Attention 3D Network (STSANet) for video saliency prediction, in which multiple Spatio-Temporal Self-Attention (STSA) modules are employed at different levels of 3D convolutional backbone to directly capture long-range relations between spatio-temporal features of different time steps. Besides, we propose an Attentional Multi-Scale Fusion (AMSF) module to integrate multi-level features with the perception of context in semantic and spatio-temporal subspaces. Extensive experiments demonstrate the contributions of key components of our method, and the results on DHF1K, Hollywood-2, UCF, and DIEM benchmark datasets clearly prove the superiority of the proposed model compared with all state-of-the-art models.

The averaging problem on the past null cone in inhomogeneous dust cosmologies

Cosmological models typically neglect the complicated nature of the spacetime manifold at small scales in order to hypothesize idealized general relativistic solutions for describing the average dynamics of the Universe. Although these solutions are remarkably successful in accounting for data, they introduce a number of puzzles in cosmology, and their foundational assumptions are therefore important to test. In this paper, we go beyond the usual assumptions in cosmology and propose a formalism for averaging the local general relativistic spacetime on an observer's past null cone: we formulate average properties of light fronts as they propagate from a cosmological emitter to an observer. The energy-momentum tensor is composed of an irrotational dust source and a cosmological constant -- the same components as in the $\Lambda$CDM model for late cosmic times -- but the metric solution is not \emph{a priori}constrained to be locally homogeneous or isotropic. This generally makes the large-scale dynamics depart from that of a simple Friedmann--Lema\^\i tre--Robertson--Walker solution through \emph{backreaction} effects. Our formalism quantifies such departures through a fully covariant system of area-averaged equations on the light fronts propagating towards an observer, which can be directly applied to analytical and numerical investigations of cosmic observables. For this purpose, we formulate light front averages of observable quantities, including the effective angular diameter distance and the cosmological redshift drift and we also discuss the backreaction effects for these observables.

Noether’s theorems and the energy-momentum tensor in quantum gauge theories

Noether's first and second theorems both imply conserved currents that can be identified as an energy-momentum tensor (EMT). The first theorem identifies the EMT as the conserved current associated with global spacetime translations, while the second theorem identifies it as a conserved current associated with local spacetime translations. This work obtains an EMT for quantum electrodynamics and quantum chromodynamics through the second theorem, which is automatically symmetric in its indices and invariant under the expected symmetries (e.g., BRST invariance) without the need for introducing an ad hoc improvement procedure.

A numerical approach to a 2D porous-medium mathematical model: Application to an atherosclerosis problem

This work deals with the study and implementation of a Local Space-Time DG ADER approach, in the finite volume framework with Weighted Essentially Non-Oscillatory (WENO) reconstruction in space, in the context of 2D porous media problems. The particular application performed concerns a biomedical problem dealing with the first stages of atherosclerosis disease, where the artery is considered as a porous medium, although the methodology described can be applied to other situations. The mathematical model on which this research is based is given by a system of two-dimensional nonlinear reaction-diffusion equations with a nonlinear source term in one of the equations, which is a variant of the model proposed originally in N. El Khatib, S. Genieys & V. Volpert (2007) Math. Model. Nat. Phenom., vol 2(2), pp. 126–141 [1]. The 2D model considered in this work has two main features: (i) the artery is taken as a porous medium and (ii) a nonlinear non-homogeneous Neumann boundary condition is incorporated, aimed to account for the recruitment of immune cells through the upper boundary, as a response to the production of cytokines. Certain theoretical properties of the stationary solutions and the evolution solution are stated and proved. Some of them are also verified with the numerical simulation results.

4D supersymmetric gauge theories of spacetime translations

The paper addresses the question whether in four spacetime dimensions, besides standard supergravity theories, field theories exist whose symmetries include local spacetime translations and supersymmetries generated by transformations whose commutators contain infinitesimal local spacetime translations. Several new supersymmetric free field theories are found which may provide theories of that type. It is outlined how these free field theories may be extended to globally supersymmetric theories with the desired properties. One class of such theories is constructed explicitly. These theories are similar to globally supersymmetric Yang-Mills theories in flat spacetime but have supersymmetries generated by transformations whose commutators generate infinitesimal general coordinate transformations with field dependent gauge parameters.

A void in the Hubble tension? The end of the line for the Hubble bubble

The Universe may feature large-scale inhomogeneities beyond the standard paradigm, implying that statistical homogeneity and isotropy may be reached only on much larger scales than the usually assumed ∼100 Mpc. This means that we are not necessarily typical observers and that the Copernican principle could be recovered only on super-Hubble scales. Here, we do not assume the validity of the Copernican principle and let cosmic microwave background, baryon acoustic oscillations, type Ia supernovae, local H 0, cosmic chronometers, Compton y-distortion and kinetic Sunyaev–Zeldovich observations constrain the geometrical degrees of freedom of the local structure, which we parametrize via the ΛLTB model—basically a non-linear radial perturbation of a FLRW metric. In order to quantify if a non-Copernican structure could explain away the Hubble tension, we pay careful attention to computing the Hubble constant in an inhomogeneous Universe, and we adopt model selection via both the Bayes factor and the Akaike information criterion. Our results show that, while the ΛLTB model can successfully explain away the H 0 tension, it is favored with respect to the ΛCDM model only if one solely considers supernovae in the redshift range that is used to fit the Hubble constant, that is, 0.023 < z < 0.15. If one considers all the supernova sample, then the H 0 tension is not solved and the support for the ΛLTB model vanishes. Combined with other data sets, this solution to the Hubble tension barely helps. Finally, we have reconstructed our local spacetime. We have found that data are best fit by a shallow void with δ L ≈ −0.04 and Mpc, which, interestingly, lies on the border of the 95% credible region relative to the standard model expectation.

Algebraic Theory of Patterns as Generalized Symmetries

We generalize the exact predictive regularity of symmetry groups to give an algebraic theory of patterns, building from a core principle of future equivalence. For topological patterns in fully-discrete one-dimensional systems, future equivalence uniquely specifies a minimal semiautomaton. We demonstrate how the latter and its semigroup algebra generalizes translation symmetry to partial and hidden symmetries. This generalization is not as straightforward as previously considered. Here, though, we clarify the underlying challenges. A stochastic form of future equivalence, known as predictive equivalence, captures distinct statistical patterns supported on topological patterns. Finally, we show how local versions of future equivalence can be used to capture patterns in spacetime. As common when moving to higher dimensions, there is not a unique local approach, and we detail two local representations that capture different aspects of spacetime patterns. A previously developed local spacetime variant of future equivalence captures patterns as generalized symmetries in higher dimensions, but we show that this representation is not a faithful generator of its spacetime patterns. This motivates us to introduce a local representation that is a faithful generator, but we demonstrate that it no longer captures generalized spacetime symmetries. Taken altogether, building on future equivalence, the theory defines and quantifies patterns present in a wide range of classical field theories.

The art of building a smooth cosmic distance ladder in a perturbed universe

How does a smooth cosmic distance ladder emerge from observations made from a single location in a lumpy Universe? Distances to the Type Ia supernova (SN1A) in the Hubble flow are anchored on local distance measurements to sources that are very nearby. We described how this configuration could be built in a perturbed universe where lumpiness is described as small perturbations on top of a flat Friedmann-Lemaıtre Robertson-Walker (FLRW) spacetime.We show that there is a non-negligible modification (about 11%) to the background FLRW area distance due to the presence of inhomogeneities in the immediate neighbourhood of an observer. We find that the modification is sourced by the electric part of the Weyl tensor indicating a tidal deformation of the local spacetime of the observer.We show in detail how it could impact the calibration of the SN1A absolute magnitude in the Hubble flow. We show that it could potentially resolve the SN1A absolute magnitude and Hubble tensions simultaneously without the need for early or late dark energy.