Space-time Curve Research Articles

Optics equation provides approximation of gravitational lenses for students

Straighforward ray trajectory equation uses trigonometric and hyperbolic functions to model curves in space-time, including Einstein rings and fatal attraction near black holes.

Theory of Everything

Our Theory of Everything is made possible by the discovery of a closed fluid dynamic principle, working in a closed fluid space-time system, (the universe). Through the principle of fluid space-time, almost all of the unexplained mysteries of the universe are revealed. How the universe came into being, with matter and the most basic force, Gravity. Newton’s law of gravity in particular, the inverse square law with distance. (the reason is for appearance of the inverse square law, simultaneously in gravitation and electromagnetism, not a coincidence). Special Relativity is an account of how space contracts, time dilates and mass increases, with velocity. General Relativity is an account of how space-time curves. According to Muhammad’s closed fluid dynamic principle, space actually consists of an extremely fluid substance which we call aether. It is therefore fitting that through the closed fluid dynamic principle, we can arrive at an explanation for one major thorn in the side of Special Relativity, the twin paradox. The problem is, it is the twin who accelerates that gets it wrong. Because his assessment of the age of his brother was done without consideration of the fact that when he turned around, made the return journey, he felt his acceleration but did not employ this in Einstein’s time dilation, which was based on inertial reference frames. The task we are confronted with in this paper is to find out how the time dilation addressed in special relativity comes into proceedings in an accelerated frame. We achieve this.

Crossing Versus Locking: Bit Threads and Continuum Multiflows

Bit threads are curves in holographic spacetimes that manifest boundary entanglement, and are represented mathematically by continuum analogues of network flows or multiflows. Subject to a density bound, the maximum number of threads connecting a boundary region to its complement computes the Ryu–Takayanagi entropy. When considering several regions at the same time, for example in proving entropy inequalities, there are various inequivalent density bounds that can be imposed. We investigate for which choices of bound a given set of boundary regions can be “locked”, in other words can have their entropies computed by a single thread configuration. We show that under the most stringent bound, which requires the threads to be locally parallel, non-crossing regions can in general be locked, but crossing regions cannot, where two regions are said to cross if they partially overlap and do not cover the entire boundary. We also show that, under a certain less stringent density bound, a crossing pair can be locked, and conjecture that any set of regions not containing a pairwise crossing triple can be locked, analogously to the situation for networks.

Gravitational potential and galaxy rotation curves in multi-fractional spacetimes

Multi-fractional theories with integer-order derivatives are models of gravitational and matter fields living in spacetimes with variable Hausdorff and spectral dimension, originally proposed as descriptions of geometries arising in quantum gravity. We derive the Poisson equation and the Newtonian potential of these theories starting from their covariant modified Einstein’s equations. In particular, in the case of the theory Tv with weighted derivatives with small fractional corrections, we find a gravitational potential that grows logarithmically at large radii when the fractional exponent takes the special value α = 4/3. This behaviour is associated with a restoration law for the Hausdorff dimension of spacetime independently found in the dark-energy sector of the same theory. As an application, we check whether this potential can serve as an alternative to dark matter for the galaxies NGC7814, NGC6503 and NGC3741 in the SPARC catalogue. We show that their rotation curves at medium-to-large radii can indeed be explained by purely geometric effects, although the Tully-Fisher relation is not reproduced well. We discuss how to fix the small-radius behaviour by lifting some approximations and how to test the model with other observables and an enlarged galaxy sample.

Darboux Vector in Four-Dimensional Space-Time

As the space-time model of the theory of relativity, four-dimensional Minkowski space is the basis of the theoretical framework for the development of the theory of relativity. In this paper, we introduce Darboux vector fields in four-dimensional Minkowski space. Using these vector fields, we define some new planes and curves. We find that the new planes are the instantaneous rotation planes of rigid body moving in four-dimensional space-time. In addition, according to some characteristics of Darboux vectors in geometry, we define some new space curves in four-dimensional space-time and describe them with curvature functions. Finally, we give some examples.

OPTIMIZING VISUALLY IMPAIRED ABILITY TO READ TACTILE PICTOGRAM THROUGH TEXTURE DESIGN

ABSTRACT. This experiment aims to optimize the visually impaired ability when reading tactile pictograms through texture design. Experiment was motivated by two previous studies results by authors show that the visually impaired need a tool to make the reading process more efficient. Overall, the visual semiotic was used to adapt the Space-Time-Curve (STC), Tabrani's visual language theory, as a texture media design system. The design result produces media textures with size composition differences. The media was tested on 20 visually impaired people directly with quantitative analysis technique. The results show that the design system for media textures is not able to optimize visually impaired reading. In general, it concluded that the visually impaired do not perceive a system without memory regarding its variable mechanism as a whole.

Hyperbolic Medicine: General and Local Hyperbolic Curves on Earth, Influencing Human Physiology

The objective of this work is to study the hyperbolic space-time curves that we perceive in our daily lives through images obtained in different geographical locations on Earth. A bibliographic review of scientific works has been made on hyperbolic curves in medicine, electromagnetic fields, circadian rhythms, and space-time perpendicular to the movement of an organ. On the internet, images in conical perspective have been selected, of cities located at different longitudes and latitudes within the geography of the Earth. They have tried to identify hyperbolic space-time curves. They have then been compared with the hyperbolic curves of the Earth’s magnetic field. Finally, we determine its relationship with the hyperbolic curves of human physiology. We have as conclusions: 1. Human vision is hyperbolic because the space in which we live is deformed by “general hyperbolic curves” that exist at any longitude and latitude of the earth’s geography. 2. The lines of force of the Earth’s magnetic field are hyperbolas that follow “local hyperbolic curves” since they can vary in intensity and even reverse their polarity over time. 3. In the physiology of the human body there are hyperbolic curves that are similar to the lines of force of a magnet and the magnetic field of the Earth. Human physiology can be conditioned by general hyperbolic curves and by local hyperbolic curves. There is an adaptation to that hyperbolic deformation of the space in which we live.

Mathematical modeling for 2D light-sheet fluorescence microscopy image reconstruction

We study an inverse problem for light sheet fluorescence microscopy (LSFM), where the density of fluorescent molecules needs to be reconstructed. Our first step is to present a mathematical model to describe the measurements obtained by an optic camera during an LSFM experiment. Two meaningful stages are considered: excitation and fluorescence. We propose a paraxial model to describe the excitation process which is directly related with the Fermi pencil-beam equation. For the fluorescence stage, we use the transport equation to describe the transport of photons towards the detection camera. For the mathematical inverse problem that we obtain after the modeling, we present a uniqueness result, recasting the problem as the recovery of the initial condition for the heat equation in from measurements in a space–time curve. Additionally, we present numerical experiments to recover the density of the fluorescent molecules by discretizing the proposed model and facing this problem as the solution of a large and sparse linear system. Some iterative and regularized methods are used to achieve this objective. The results show that solving the inverse problem achieves better reconstructions than the direct acquisition method that is currently used.

Navigating four dimensions – upper secondary students’ understanding of movement in spacetime

In contrast to classical physics, general relativity (GR) interprets gravity as geometry. Objects follow geodesic curves in four-dimensional spacetime and curvature creates the illusion of a gravitational force. Despite the scientific relevance of this theory, educational research in the domain of GR is scarce. This study contributes to a growing body of knowledge concerning secondary students’ understanding of GR. Based on a thematic analysis of audio records of student discussions of 97 Norwegian physics students (18-19 years), we present results on students’ understanding of geodesic movement in spacetime. Our findings can give guidance to improve learning and instruction of a key concept of GR at the upper secondary school level.

Direct detection of universal expansion by holonomy in the McVittie spacetime

In general relativity the parallel transfer of a vector around a closed curve in spacetime, or along two curves which together form a closed loop, usually results in a nonzero deficit angle between the vector's initial and final positions. We show that such holonomy in the McVittie spacetime, which represents a gravitating object imbedded in an expanding universe, can in principle be used to directly detect the expansion of the universe, for example by measuring changes in the components of a gyroscopic spin axis. Although such changes are of course small, they are large enough (\D S \sim 10^{-7}) that they could conceivably be measured if the real universe behaved like the McVittie spacetime. The real problem is that virialization will lead to domains decoupled from the global expansion on a scale much larger than that of the solar system, making such an experiment infeasible probably even in principle. Nevertheless the effect is of interest in relation to ongoing discussions, dating back at least to Einstein and Straus, which concern the relationship between the expansion of the universe and local systems.

Globally Maximal Timelike Geodesics in Static Spherically Symmetric Spacetimes: Radial Geodesics in Static Spacetimes and Arbitrary Geodesic Curves in Ultrastatic Spacetimes

This work deals with intersection points: conjugate points and cut points, of timelike geodesics emanating from a common initial point in special spacetimes. The paper contains three results. First, it is shown that radial timelike geodesics in static spherically symmetric spacetimes are globally maximal (have no cut points) in adequate domains. Second, in one of ultrastatic spherically symmetric spacetimes, Morris--Thorne wormhole, it is found which geodesics have cut points (and these must coincide with conjugate points) and which ones are globally maximal on their entire segments. This result, concerning all timelike geodesics of the wormhole, is the core of the work. The third outcome deals with the astonishing feature of all ultrastatic spacetimes: they provide a coordinate system which faithfully imitates the dynamical properties of the inertial reference frame. We precisely formulate these similarities.

Space-Time Warp Curve for Synthesizing Multi-character Motions

This paper introduces a new motion-synthesis technique for animating multiple characters. At a high level, we introduce a hub-sub-control-point scheme that automatically generates many different spline curves from a user scribble. Then, each spline curve becomes a trajectory along which a 3D character moves. Based on the given curves, our algorithm synthesizes motions using a cyclic motion. In this process, space-time warp curves, which are time-warp curves, are embedded in the 3D environment to control the speed of the motions. Since the space-time warp curve represents a trajectory over the time domain, it enables us to verify whether the trajectory causes any collisions between characters by simply checking whether two space-time warp curves intersect. In addition, it is possible to edit space-time warp curves at run time to change the speed of the characters. We use several experiments to demonstrate that the proposed algorithm can efficiently synthesize a group of character motions. Our method creates collision-avoiding trajectories ten times faster than those created manually.

The Particle as a Statistical Ensemble of Events in Stueckelberg–Horwitz–Piron Electrodynamics

In classical Maxwell electrodynamics, charged particles following deterministic trajectories are described by currents that induce fields, mediating interactions with other particles. Statistical methods are used when needed to treat complex particle and/or field configurations. In Stueckelberg-Horwitz-Piron (SHP) electrodynamics, the classical trajectories are traced out dynamically, through the evolution of a 4D spacetime event $x^\mu (\tau)$ as $\tau$ grows monotonically. Stueckelberg proposed to formalize the distinction between coordinate time $x^0 = ct$ (measured by laboratory clocks) and chronology $\tau$ (the temporal ordering of event occurrence) in order to describe antiparticles and resolve problems of irreversibility such as grandfather paradoxes. Consequently, in SHP theory, the elementary object is not a particle (a 4D curve in spacetime) but rather an event (a single point along the dynamically evolving curve). Following standard deterministic methods in classical relativistic field theory, one is led to Maxwell-like field equations that are \hbox{$\tau$-dependent} and sourced by a current that represents a statistical ensemble of instantaneous events distributed along the trajectory. The width $\lambda$ of this distribution defines a correlation time for the interactions and a mass spectrum for the photons emitted by particles. As $\lambda$ becomes very large, the photon mass goes to zero and the field equations become $\tau$-independent Maxwell's equations. Maxwell theory thus emerges as an equilibrium limit of SHP, in which $\lambda$ is larger than any other relevant time scale. Thus, statistical mechanics is a fundamental ingredient in SHP electrodynamics, and its insights are required to give meaning to the concept of a particle.

RELIABILITY ESTIMATION OF MATHEMATICAL DESCRIPTION OF MAIN GAS PIPELINE TOUGH LINE WITH 3RD DEGREE SPLINE

A method for the mathematical description of gas pipeline curve position, which is identified with the upper line, describing for radius vector with coordinates approximated with a cubic spline. To confirm the authenticity of given dependencies is satisfied with the decision of the check tasks assessment in calculating the position of sinfulness and the upper forming parameter estimates change of the pipeline spatial position. Applying a method based on a comparison the changes of space-time curves previously known function, built by the coordinates of points, defined in two ways: a true and with given inaccuracy

A variational perspective on accelerated methods in optimization

Accelerated gradient methods play a central role in optimization, achieving optimal rates in many settings. Although many generalizations and extensions of Nesterov's original acceleration method have been proposed, it is not yet clear what is the natural scope of the acceleration concept. In this paper, we study accelerated methods from a continuous-time perspective. We show that there is a Lagrangian functional that we call the Bregman Lagrangian, which generates a large class of accelerated methods in continuous time, including (but not limited to) accelerated gradient descent, its non-Euclidean extension, and accelerated higher-order gradient methods. We show that the continuous-time limit of all of these methods corresponds to traveling the same curve in spacetime at different speeds. From this perspective, Nesterov's technique and many of its generalizations can be viewed as a systematic way to go from the continuous-time curves generated by the Bregman Lagrangian to a family of discrete-time accelerated algorithms.

An Alternative to the Alcubierre Theory: Warp Fields by the Gravitation via Accelerated Particles Assertion

<p class="1Body">A summarization of the Alcubierre metric is given in comparison to a new metric that has been formulated based on the theoretical assertion of a recently published paper entitled “gravitational space-time curve generation via accelerated particles”. The new metric mathematically describes a warp field where particle accelerators can theoretically generate gravitational space-time curves that compress or contract a volume of space-time toward a hypothetical vehicle traveling at a sub-light velocity contingent upon the amount of voltage generated. Einstein’s field equations are derived based on the new metric to show its compatibility to general relativity. The “time slowing” effects of relativistic gravitational time dilation inherent to the gravitational field generated by the particle accelerators is mathematically shown to be counteracted by a gravitational equilibrium point between an arrangement of two equal magnitude particle accelerators. The gravitational equilibrium point produces a volume of flat or linear space-time to which the hypothetical vehicle can traverse the region of contracted space-time without experiencing time slippage. The theoretical warp field possessing these attributes is referred to as the two gravity source warp field which is mathematically described by the new metric.</p>

Study of galactic rotation curves in wormhole spacetime

The spacetime of the galactic halo region is described by a wormhole like line element. We assume violation of Null Energy Condition (NEC) in the galactic halo. The Einstein Field equations are solved for two different conditions of pressure and density to obtain physical parameters like tangential velocity of test particles and parameters related to the wormhole geometry. The theoretical rotation curve of the test particles is plotted and compared the same with an observed rotation curve. We obtain a satisfactory fit between the observed curve and the curve obtained from the present theory for the radial distances in the range 9 Kpc to 100 Kpc.

Gravitational Space-Time Curve Generation via Accelerated Charged Particles

A force with an acceleration that is equal to multiples greater than the speed of light per unit time is exerted on a cloud of charged particles. The particles are resultantly accelerated to within an infinitesimal fraction of the speed of light. As the force or acceleration increases, the particles’ velocity asymptotically approaches but never achieves the speed of light obeying relativity. The asymptotic increase in the particles’ velocity toward the speed of light as acceleration increasingly surpasses the speed of light per unit time does not compensate for the momentum value produced on the particles at sub-light velocities. Hence, the particles’ inertial mass value must increase as acceleration increases. This increase in the particles’ inertial mass as the particles are accelerated produce a gravitational field which is believed to occur in the oscillation of quarks achieving velocities close to the speed of light. The increased inertial mass of the density of accelerated charged particles becomes the source mass (or Big “M”) in Newton’s equation for gravitational force. This implies that a space-time curve is generated by the accelerated particles. Thus, it is shown that the acceleration number (or multiple of the speed of light greater than 1 per unit of time) and the number of charged particles in the cloud density are surjectively mapped to points on a differential manifold or space-time curved surface. Two aspects of Einstein’s field equations are used to describe the correspondence between the gravitational field produced by the accelerated particles and the resultant space-time curve. The two aspects are the Schwarzchild metric and the stress energy tensor. Lastly, the possibility of producing a sufficient acceleration or electromagnetic force on the charged particles to produce a gravitational field is shown through the Lorentz force equation. Moreover, it is shown that a sufficient voltage can be generated to produce an acceleration/force on the particles that is multiples greater than the speed of light per unit time thereby generating gravity.

Space-time sketching of character animation

We present a space-time abstraction for the sketch-based design of character animation. It allows animators to draft a full coordinated motion using a single stroke called the space-time curve (STC). From the STC we compute a dynamic line of action (DLOA) that drives the motion of a 3D character through projective constraints. Our dynamic models for the line's motion are entirely geometric, require no pre-existing data, and allow full artistic control. The resulting DLOA can be refined by over-sketching strokes along the space-time curve, or by composing another DLOA on top leading to control over complex motions with few strokes. Additionally, the resulting dynamic line of action can be applied to arbitrary body parts or characters. To match a 3D character to the 2D line over time, we introduce a robust matching algorithm based on closed-form solutions, yielding a tight match while allowing squash and stretch of the character's skeleton. Our experiments show that space-time sketching has the potential of bringing animation design within the reach of beginners while saving time for skilled artists.

Physical Null Conditions: Diameter of a Black Hole Singularity

A black hole is considered a gravitational one-dimensional point of singularity containing an infinite mass within an infinitely small space, and spacetime curves infinitely. Within this singularity all the laws of physics are no longer valid. The physical domains of the large macroscopic scale are described by general relativity and on the small microscopic scale by quantum effects. It is proposed in this new theory that a black hole singularity collapses to a physically defined diameter that perfectly balances between the duality of relativity and quantum states, and in this unified equilibrium condition; physics continues to be obeyed within this non-zero non-infinite black hole singularity.