Space-time Point Research Articles

Variational principles of physics and the infinite Ramsey theory

Abstract Application of the Ramsey Infinite Theorem to the fundamental variational principles of physics is addressed. The Hamilton Least Action Principle states that, for a true/actual trajectory of a system, Hamilton’s Action is stationary for the paths, which evolve from the preset initial space-time point to the preset final space-time point. The Hamilton Principle distinguishes between the actual and trial/test trajectories of the system in the configurational space. This enables the transformation of the infinite set of points of the configurational space (available for the system) into the bi-colored, infinite, complete, graph, when the points of the configurational space are seen as the vertices, actual paths connecting the vertices/ points of the configurational space are colored with red; whereas, the trial links/paths are colored with green. Following the Ramsey Infinite Theorem, there exists the infinite, monochromatic sequence of the pathways/clique, which is completely made up of actual or virtual paths, linking the interim states of the system. The same procedure is applicable to the Maupertuis’s principle (classical and quantum), Hilbert-Einstein relativistic variational principle and reciprocal variational principles. Exemplifications of the Infinite Ramsey Theorem are addressed.

Space-time tree: a spatiotemporal construct for efficient similarity matrix calculations among network-constrained trajectories

Data mining of network-constrained trajectories has broad applications in the GIScience field. The calculation of a complete trajectory similarity matrix is a key step in various data mining algorithms. However, computing this matrix is computationally intensive for large datasets, as it involves numerous point-to-point shortest-path (PPSP) queries. To tackle this issue, we propose a new spatiotemporal construct called the space-time tree, which directly delineates the network distance from a query trajectory to any network space-time point. By constructing the space-time tree, we can efficiently compute the trajectory similarity matrix without additional PPSP queries. The space-time tree supports several similarity metrics, including closest pair distance, furthest pair distance, longest common subsequence (LCSS), and distance-weighted LCSS. It can further integrate with advanced spatiotemporal query techniques for scalable partial trajectory similarity matrix calculations. A case study using real datasets was conducted to apply the space-time tree in the trajectory clustering application. The results show that the space-time tree completed the clustering task on 0.5 million trajectories within 49 minutes, achieving a nearly 147-fold speedup compared to state-of-the-art methods.

Scale Selection and Machine Learning-based Cell Segmentation and Tracking in Time Lapse Microscopy.

Monitoring and tracking of cell motion is a key component for understanding disease mechanisms and evaluating the effects of treatments. Time-lapse optical microscopy has been commonly employed for studying cell cycle phases. However, usual manual cell tracking is very time consuming and has poor reproducibility. Automated cell tracking techniques are challenged by variability of cell region intensity distributions and resolution limitations. In this work, we introduce a comprehensive cell segmentation and tracking methodology. A key contribution of this work is that it employs multi-scale space-time interest point detection and characterization for automatic scale selection and cell segmentation. Another contribution is the use of a neural network with class prototype balancing for detection of cell regions. This work also offers a structured mathematical framework that uses graphs for track generation and cell event detection. We evaluated cell segmentation, detection, and tracking performance of our method on time-lapse sequences of the Cell Tracking Challenge (CTC). We also compared our technique to top performing techniques from CTC. Performance evaluation results indicate that the proposed methodology is competitive with these techniques, and that it generalizes very well to diverse cell types and sizes, and multiple imaging techniques.

Algorithmic Identification of the Precursory Scale Increase Phenomenon in Earthquake Catalogs

Abstract The precursory scale increase (Ψ) phenomenon describes the sudden increase in rate and magnitude in a precursory area AP, at precursor time TP, and with precursor magnitude MP prior to the upcoming large earthquake with magnitude Mm. Scaling relations between the Ψ variables form the basis of the “Every Earthquake a Precursor According to Scale” (EEPAS) earthquake forecasting model. EEPAS is a well-established space–time point process model that forecasts large earthquakes in the medium term, that is, the coming months to decades, depending on Mm. In Aotearoa New Zealand, EEPAS contributes to hybrid models for public earthquake forecasting and to the source model of time-varying seismic hazard models, including the latest revision of the National Seismic Hazard Model. The Ψ phenomenon was recently shown not to be unique for a given earthquake, with smaller precursory areas AP associated with larger precursor times TP and vice versa. This trade-off between AP and TP has also been found for the spatial and temporal distributions of the EEPAS models. Detailed analysis of the Ψ phenomenon has so far been limited by the manual and labor-intensive procedure of identifying Ψ in earthquake catalogs. Here, we introduce two algorithms to automatically detect Ψ and apply them to real and simulated earthquake catalog data. By randomizing the catalog and removing aftershocks, we confirm that the Ψ phenomenon is a feature of space–time earthquake clustering prior to major earthquakes. Multiple Ψ identifications confirm the trade-off between AP and TP, and the scaling relations for both real and simulated catalogs are consistent with the original scaling relations on which EEPAS is based. We identify opportunities for future work to refine the algorithms and apply them to physics-based simulated catalogs to enhance the understanding of Ψ. A better understanding of Ψ has the potential to improve forecasting of large upcoming earthquakes.

Causal completion of globally hyperbolic conformally flat spacetimes

In their paper Geroch, Kronheimer, and Penrose, “Ideal points in space-time,” [Proc. R. Soc. London, Ser. A 327, 545–567 (1972)] introduced a way to attach ideal points to a spacetime M, defining the causal completion of M. They established that this is a topological space which is Hausdorff when M is globally hyperbolic. In this paper, we prove that if, in addition, M is simply-connected and conformally flat, its causal completion is a topological manifold with boundary homeomorphic to S × [0, 1] where S is a Cauchy hypersurface of M. We also introduce three remarkable families of globally hyperbolic conformally flat spacetimes and provide a description of their causal completions.

Quantum electrodynamical formulation of photochemical acid generation and its implications on optical lithography

AbstractThe photochemical acid generation is refined from the first principles of quantum electrodynamics. First, we briefly review the formulation of the quantum theory of light based on the quantum electrodynamics framework to establish the probability of acid generation at a given spacetime point. The quantum mechanical acid generation is then combined with the deprotection mechanism to obtain a probabilistic description of the deprotection density directly related to feature formation in a photoresist. A statistical analysis of the random deprotection density is presented to reveal the leading characteristics of stochastic feature formation.

A topological drive for spacetime travel

We present a toy metric of spacetime travel from topological change. A bubble-like baby Universe is detached and re-attached from our Universe. Depending on where the bubble is re-attached, matter may travel superluminally or backwards-in-time through the bubble. Quasiregular singularities are formed at the detachment and re-attachment spacetime points. The spacetime is traversable and not covered by any horizons. Exotic matter violating energy conditions is required to realize such spacetimes.

Point-charge models and averages for electromagnetic quantities considered in two relativistic inertial frames

Electromagnetic quantities at a spacetime point have tensor Lorentz transformations between relatively-moving inertial frames. However, since the Lorentz transformation of time between inertial frames depends upon both the time and space coordinates, averages of electrodynamic quantities at a single time will in general depend upon the inertial frame, and will differ between inertial frames. Here we illustrate how the use of continuous charge and current distributions rather than point-charge distributions can lead to physically mystifying and even inaccurate results for electromagnetic quantities and physical phenomena. The discrepancy noted between the average electric field values in different inertial frames is particularly striking because it is first order in the relatative velocity between the frames.

Linearly polarized electromagnetic pulses

We construct theoretically a set of space-time localized electromagnetic pulses, characterized by a wavenumber K and a length a. Their linear polarization becomes more perfect as Ka increases and the pulse becomes more nearly monochromatic. Two measures of the degree of linear polarization are explored: one that gives the polarization at any point in space-time, and another that integrates the electric intensity over the focal plane of the pulse as the pulse passes through. The polarization measures and the total energy, momentum, and angular momentum of these pulses are calculated for all K and a. The fields of these pulses satisfy the Maxwell equations exactly.

A Study of Geodesic Equation From Variational Principle

This paper employs variational principle in studying geodesic. In mathematical studies, a variational principle enables a problem to be solved employing calculus of variations that concerns seeking functions that increase the values of quantities that rely on those functions. For instance, the problem of ascertaining the shape of a hanging chain suspended at both ends can be solved using variational calculus. Hence, the variational principle is a function that lessens the gravitational potential energy of the chain. Geodesicis a procedure used in mathematics, specifically in Riemannian geometry that results in obtaining geodesics. Actually, these represent the paths of particles with no proper acceleration, their motion pleasing the geodesic equations. As the particles are subject to no appropriate acceleration, the geodesics generally signify the straightest path between two points in a bent spacetime. The article has investigated into the relation between the variation principle and geodesic equations that are particularly used in general relativity in physics as well.

Towards a quantum field theory description of nonlocal spacetime defects

We propose an ansatz for encoding the physics of nonlocal spacetime defects in the Green’s functions for a scalar field theory defined on a causal set. This allows us to numerically study the effects of nonlocal spacetime defects on the discrete Feynman propagator of the theory defined on the causal set in 1+1 dimensions, and to compare to the defect-free limit. The latter approaches the expected continuum result, on average, when the number of points becomes large. When defects are present, two points with the same invariant spacetime interval can have different propagation amplitudes, depending on whether the propagation is between two ordinary spacetime points, two defects, or a defect and an ordinary point. We show that a coarse-grained description that is only sensitive to the average effect of the defects can be interpreted as a defect-induced mass and wave-function renormalization of the scalar theory.

Third order nonlinear correlation of the electromagnetic vacuum at near-infrared frequencies

In recent years, electro-optic sampling, which is based on Pockel’s effect between an electromagnetic mode and a copropagating, phase-matched ultrashort probe, has been largely used for the investigation of broadband quantum states of light, especially in the mid-infrared and terahertz frequency range. The use of two mutually delayed femtosecond pulses at near-infrared frequencies allows the measurement of quantum electromagnetic radiation in different space-time points. Their correlation allows therefore direct access to the spectral content of a broadband quantum state at terahertz frequencies after Fourier transformation. In this work, we will prove experimentally and theoretically that when using strongly focused coherent ultrashort probes, the electro-optic sampling technique can be affected by the presence of a third-order nonlinear mixing of the probes’ electric field at near-infrared frequencies. Moreover, we will show that these third-order nonlinear phenomena can also influence correlation measurements of the quantum electromagnetic radiation. We will prove that the four-wave mixing of the coherent probes’ electric field with their own electromagnetic vacuum at near-infrared frequencies results in the generation of a higher-order nonlinear correlation term. The latter will be characterized experimentally, proving its local nature requiring the physical overlap of the two probes. The parameters regime where higher order nonlinear correlation results predominant with respect to electro-optic correlation of terahertz radiation is provided.

Entanglement entropy in scalar quantum electrodynamics

We find the entanglement entropy of a subregion of the vacuum state in scalar quantum electrodynamics, working perturbatively to the two-loop level. Doing so leads us to derive the Maxwell-Proca propagator in conical Euclidean space. The area law of entanglement entropy is recovered in both the massive and massless limits of the theory, as is expected. These results yield the renormalization group flow of entanglement entropy, and we find that loop contributions suppress entanglement entropy. We highlight these results in the light of the renormalization group flow of couplings and correlators, which are increased in scalar quantum electrodynamics, so that the potential tension between the increase in correlations between two points of spacetime and the decrease in entanglement entropy between two regions of spacetime with energy is discussed. We indeed show that the vacuum of a subregion of spacetime purifies with energy in scalar quantum electrodynamics, which is related to the concept of screening. Published by the American Physical Society 2024

End-time regularity theorem for Brakke flows

AbstractFor a general k-dimensional Brakke flow in $${\mathbb {R}}^n$$ R n locally close to a k-dimensional plane in the sense of measure, it is proved that the flow is represented locally as a smooth graph over the plane with estimates on all the derivatives up to the end-time. Moreover, at any point in space-time where the Gaussian density is close to 1, the flow can be extended smoothly as a mean curvature flow up to that time in a neighborhood: this extends White’s local regularity theorem to general Brakke flows. The regularity result is in fact obtained for more general Brakke-like flows, driven by the mean curvature plus an additional forcing term in a dimensionally sharp integrability class or in a Hölder class.

Thermodynamic quantities and phase transitions of five-dimensional de Sitter hairy spacetime* *Supported by the National Natural Science Foundation of China (12075143)

In this study, we take the mass, electric charge, hair parameter, and cosmological constant of five-dimensional de Sitter hairy spacetime as the state parameters of the thermodynamic system, and when these state parameters satisfy the first law of thermodynamics, the equivalent thermodynamic quantities of spacetime and the Smarr relation of five-dimensional de Sitter hairy spacetime are obtained. Then, we study the thermodynamic characteristics of the spacetime described by these equivalent thermodynamic quantities and find that de Sitter hairy spacetime has a phase transition and critical phenomena similar to those of van de Waals systems or charged AdS black holes. It is shown that the phase transition point of de Sitter hairy spacetime is determined by the ratio of two event horizon positions and the cosmic event horizon position. We discuss the influence of the hair parameter and electric charge on the critical point. We also find that the isochoric heat capacity of the spacetime is not zero, which is consistent with the ordinary thermodynamic system but differs from the isochoric heat capacity of AdS black holes, which is zero. Using the Ehrenfest equations, we prove that the critical phase transition is a second order equilibrium phase transition. Research on the thermodynamic properties of five-dimensional de Sitter hairy spacetime lays a foundation for finding a universal de Sitter spacetime thermodynamic system and studying its thermodynamic properties. Our universe is an asymptotically dS spacetime, and the thermodynamic characteristics of de Sitter hairy spacetime will help us understand the evolution of spacetime and provide a theoretical basis to explore the physical mechanism of the accelerated expansion of the universe.

Gradient-flowed order parameter for spontaneous gauge symmetry breaking

The gauge-invariant two-point function of the Higgs field at the same spacetime point can make a natural gauge-invariant order parameter for spontaneous gauge symmetry breaking. However, this composite operator is ultraviolet divergent and is not well defined. We propose using a gradient flow to cure the divergence from putting the fields at the same spacetime point. As a first step, we compute it for the Abelian Higgs model with a positive mass squared at the one-loop order in the continuum theory using the saddle-point method to estimate the finite part. The order parameter consistently goes to zero in the infrared limit of the infinite flow time.

The Hole Argument and Beyond: Part I: The Story so Far

In this two-part paper, we review, and then develop, the assessment of the hole argument for general relativity. This first Part reviews the literature hitherto, focussing on the philosophical aspects. It also introduces two main ideas we will need in Part II: which will propose a framework for making comparisons of non-isomorphic spacetimes.In Section 1 of this paper, we recall Einstein’s original argument. Section 2 recalls the argument’s revival by philosophers in the 1980s and 1990s. This includes the first main idea we will need in Part II: namely, that two spacetime points in different possible situations are never strictly identical—they are merely counterparts.In Section 3, we report—and rebut—more recent claims to “dissolve” the argument. Our rebuttal is based on the fact that in differential geometry, and its applications in physics such as general relativity, points are in some cases identified, or correspond with each other, between one context and another, by means other than isometry (or isomorphism). We call such a correspondence a threading of points. This is the second main idea we shall use in Part II.

Thermodynamics of phase transition in Reissner–Nordström–de Sitter spacetime

The Reissner-Nordstrm̈-de Sitter (RN-dS) spacetime can be considered as a thermodynamic system. Its thermodynamic properties are discussed that the RN-dS spacetime has phase transitions and critical phenomena similar to that of the Van de Waals system or the charged AdS black hole. The continuous phase transition point of RN-dS spacetime depends on the position ratio of the black hole horizon and the cosmological horizon. We discuss the critical phenomenon of the continuous phase transition of RN-dS spacetime according to Landau theory of continuous phase transition, that the critical exponent of spacetime is same as that of the Van de Waals system or the charged AdS black hole, which have universal physical meaning. We find that the order parameters are similar to those introduced in ferromagnetic systems. Our universe is an asymptotically dS spacetime, thermodynamic characteristics of RN-dS spacetime can help us understand the evolution of spacetime and provide a theoretical basis to explore the physical mechanism of accelerated expansion of the universe.

An inverse problem of determining the fractional order in the TFDE using the measurement at one space-time point

In this paper we are concerned with an inverse problem for determining the order of time fractional derivative in a time fractional diffusion equation (TFDE in short), where the available measurement is given at a single space-time point. The inverse problem is transformed to a nonlinear algebraic equation of the fractional order based on the solution to the forward problem. We prove that the algebraic equation possesses a unique solution by the strict monotonicity of the Mittag-Leffler function, and the inverse problem is of uniqueness. Numerical examples are presented to show the unique solvability of the inverse problem.

A derivation of the standard model particles from internal spacetime

‘Internal spacetime’ is a modification of general relativity that was recently introduced as an approximate spacetime geometric model of quantum nonlocality. In an internal spacetime, time is stationary along the worldlines of fundamental (dust) particles. Consequently, the dimensions of tangent spaces at different points of spacetime vary, and spin wavefunction collapse is modeled by the projection from one tangent space to another. In this paper, we develop spinors on an internal spacetime, and construct a new Dirac-like Lagrangian [Formula: see text] whose equations of motion describe their couplings and interactions. Furthermore, we show that hidden within [Formula: see text] is the entire standard model: [Formula: see text] contains precisely three generations of quarks and leptons, the electroweak gauge bosons, the Higgs boson, and one new massive spin-[Formula: see text] boson; gluons are considered in a companion paper. Specifically, we are able to derive the correct spin, electric charge, and color charge of each standard model particle, as well as predict the existence of a new boson.