Law Of Universal Gravitation Research Articles

Relativistic Mechanics in Gravitational Field within Oblate Spheroidal Coordinates Based upon Riemannian Geometry for Rotating Homogeneous Mass Distribution

The emergence of the geometrical theory of gravitation (general relativity) by Albert Einstein in his quest to unite special relativity and the Newtonian law of universal gravitation has led to several Mathematical approaches for the exact and analytical solution for all gravitational fields in nature. The first and the most famous analytical solution was the Schwarzchild’s which can be constructed by finding a mapping where the metric tensor takes a simple form i.e. the vanishing of the non-diagonal elements. In this paper, we construct exact solution of the Einstein geometrical gravitational field equation using Riemannian metric tensor called the golden metric tensor that was first developed by Howusu, in the year 2009, for the rotating homogeneous mass distribution within oblate Spheroidal Coordinates. The equations of motion for test particles in the Oblate Spheroidal Geometry were derived using the coefficient of affine connection. Then the law of conservation of momentum and energy are equivalently formulated using the generalized Lagrangian as compared to the analytical solution of the Schwarzchild’s gravitational field. We also derived the planetary equation of motion in the equatorial plane of the Oblate Spheroidal body for this gravitational field.

Proposing Binary Gravitational Search Algorithm Parameters for Back-calculation of Road Pavement Moduli

A Falling Weight Deflectometer is popular equipment to measure surface deflections under imposed loadings, providing the necessary parameters for back-calculating the elastic moduli of road pavements. There are several back-calculation programs available that accurately back-calculate pavement layer moduli. The Gravitational Search Algorithm (GSA), a metaheuristic optimization algorithm inspired by Newton’s law of universal gravitation, is one such algorithm. The Binary Gravitational Search Algorithm (BGSA) is an enhancement algorithm based on GSA that can be used as an efficient search algorithm for back-calculating pavement moduli based on matching measured and calculated surface deflection of road pavements. Choosing the best BGSA parameters is critical for accurate back-calculation of road pavement moduli. Nevertheless, there has not been much study on selecting the best BGSA parameters for back-calculating road pavement moduli in the literature. Therefore, this study proposes strategies for selecting BGSA parameters based on the least computational effort and the least root mean square error between measured and calculated road pavement surface deflections. In this study, the Burmister theory is discussed and the Richardson extrapolation is adopted to improve the accuracy of the calculated points near the pavement surface; the best parameter of BGSA including the agent size A of 50 and an iteration step T of 300 are suggested to back-calculating the road pavement moduli.

Calculating All Dark Energy and Dark Matter Effects through Dynamic Gravity Theory

Newton’s Universal Law of Gravitation was augmented by Einstein’s General Relativity to include factors such as time dilation from speed and gravity field strength. But this augmentation has proven to be incomplete as the math fails in almost all settings outside our solar system hence the need for Dark Energy and Dark Matter to resolve the math. Dynamic Gravity has new math that augments GR in much the same way GR augments Newton’s Law, and this math has the potential to completely explain the motions of all celestial bodies throughout the entire universe with zero need to correct the math with Dark Matter or Dark Energy. This paper explores the notion that gravity must obey the law of conservation of energy as all other forces in this universe have been shown to do. Explaining exactly what gravity is and how it manifests itself on a sub-atomic level. And explaining the many different implications that would be created from this theory. And finally using the math of Dynamic Gravity to calculate Dark Energy and Dark Matter effects within the Milky Way, and between the Milky Way and Andromeda to explain observations without the need of exotic measures.

Review of the gravity model: origins and critical analysis of its theoretical development

This article presents a bibliographic review of the gravitational model in international trade from when it was first associated with Newton's law of universal gravitation. Firstly, I will introduce the concept of gravity in commerce as originally intended by Isard (Q J Econ 68(2):305–320, 1954) in relation to Tinbergen and Tobler’s approaches. Secondly, I will analyse the theoretical roots of international economics according to several authors, including McCallum, Anderson and van Wincoop, Krugman, Tranos, and Nijkamp. Thirdly, attention will be drawn to the evolution of the ideas of the authors mentioned above upon Isard’s initial approach rooted in physics. Furthermore, I will focus on how the tool rooted in physics can be applied or adapted to international trade. Throughout this article, I will try to keep the econophysical nature of this model.

Relativistic Problem of Two Bodies

This paper presents a developed theory of gravitational interaction of two-point bodies that takes into account the finiteness of the speed of light. The relativistic inertial (or gravitational) mass has been determined. The relativistic force of inertia, whose value is invariant to the Lorentz transformation, has been determined. A generalized form of the law of universal gravitation is presented, such that the gravitating masses in it contain corrective relativistic multipliers. Based on these new definitions, a method similar to that of classical mechanics is used to calculate a relativistic orbit of circulation. A new formula for the angle of displacement of perihelion, which takes into account both the value of the focal parameter and the eccentricity of the orbit, and which for the planet Mercury gives the correct numerical result, has been derived.

Theoretical Calculation of the Gravitational Constant Using The Elementary Charge, Speed of Light, Z Boson Mass, and Relativistic Mechanics

The gravitational constant is an unchanging quantity included in the law of universal gravitation. The value of this constant is known much less accurately than other fundamental physical constants, which is due to the experimental difficulties of measuring it. Here we report a simple method for the theoretical calculation of the gravitational constant using the elementary charge, speed of light, Z boson mass, and relativistic mechanics.

PREFACE

PREFACE

On Hooke’s Rule of Nature

Lectures de Potentia Restitutiva or Of Spring: Explaining the Power of Springing Bodies (1678) is an important book for the history of science. This book is better known for Hooke’s presentation of the law that bears his name. This law, or “Rule of Nature” as the author states, is commonly taught within the context of the analysis of elastic bodies and their deformations. However, the framework in which this law was introduced goes beyond the context in which it is currently taught. Alongside the presentation of Hooke’s experiments with springs, the author established his vibratory theory of matter, in which the concepts of congruence and incongruence, initially presented in his Micrographia (1665), would be defined in greater detail. These concepts aimed to theoretically justify the movements of attraction and repulsion in nature. This paper seeks to study the Lectures de Potentia Restitutiva once again to better understand Hooke’s thoughts about the rule which bears his name and his conception of gravity, which the author considered a force. Here Hooke’s definitions of body and motion will be presented, as well as his actual objective when he formulated the so-called Hooke’s Law. As we will see, Hooke intended to create a “philosophical scale” to measure the gravitational attraction between bodies. By considering his previous publications, such as An attempt to prove the motion of the Earth from Observations or Micrographia: or some Physiological Descriptions of Minute Bodies, or even unpublished works such as On the inflection of a direct motion into a curve by supervening Attractive principle, it becomes clear that Hooke was already opening a path toward an understanding of gravity before Newton’s Principia (1687) were published. By taking into account the controversy between Isaac Newton and Robert Hooke, we also intend to strengthen the idea that Hooke was an indispensable contributor to the elaboration of a law of universal gravitation. In addition to all this, it will also be argued that the conclusions achieved by Hooke in Of Spring may have also anticipated Newton’s third law of motion.

The Laws of Planetary Motion (A Teaching Method Inspired by the History of Physics)

In high school, and also in introductory physics courses in higher levels of schooling, the law of universal gravitation of planets is introduced by postulating Johannes Kepler’s three laws, and later Isaac Newton’s law of the inverse of the square of the distance to the Sun. The justification of the laws is only achieved in advanced courses in mechanics. We present material to teach the law of areas and the law of attraction, which is inspired by Newton’s analysis of motions under central forces, as done in Mathematical Principles of Natural Philosophy. The method is geometric, so that the dynamics behind the laws can be discussed with a ruler, triangles, and compass, or some software, without appeal to equations, and can be used on levels of instruction that dispense with analytic treatment.

Action Minimizing Orbits in the 2-Center Problems with Simple Choreography Constraint

The aim of this paper is to study the motion of a \(2+n\)-body problem where two equal masses are assumed to be fixed. We assume that the value of each fixed mass is equal to \(M>0\) and the remaining n moving particles have equal masses \(m>0\). According to Newton’s second law and the universal gravitation law, the n particles move under the interaction of each other and the affection of the two fixed particles. Also, this motion has a natural variational structure. Under the simple choreography constraint, we show that the Lagrangian action functional attains its absolute minimum on a uniform circular motion.

An Image Segmentation Method Based on the Law of Universal Gravitation and Adaptive Affinity Propagation Spectrum Clustering

There are many problems of division in natural and social sciences, and with the development of science and technology, the requirement for division is also increasing. It is difficult to divide accurately by experience and expertise alone, and the most important research branch of the division problem is the clustering algorithm. It is to group similar samples into one class and divide the elements with large differences into different classes. Due to the simplicity and efficiency of the clustering algorithm, it is widely used in image segmentation. The conventional Spectral Clustering (SC) algorithm cannot recognize nonconvex data and has the disadvantage of strong dependence on biased parameter values. To address this problem, the Gravity-based Adaptive Spectral Clustering (GASC) algorithm is proposed in this study. Based on the conventional SC algorithm, the algorithm uses gravity to calculate the similarity (gravity) between data, and uses information entropy and adaptive enhancement (AdaBoost) algorithms to obtain the weights of correct cluster sampling points and wrong cluster sampling points in each cluster, so as to reduce the dependence of the algorithm on bias parameters and reduce the number of wrongly divided sample points. Meanwhile, the GASC algorithm is applied to image segmentation. The implementation process includes three stages: image preprocessing, feature extraction, and clustering. The comparison experiments show that the mean values of normalization and accuracy of the GASC algorithm are improved compared with other clustering algorithms, and the segmentation accuracy for images is higher.

Inverse-Square Orbits Revisited

Kepler’s laws are an essential part of any mechanics course at the introductory level. They are commonly used to illustrate the power of Newton’s laws at explaining and predicting the motion of planets and comets around the sun. While the derivation of the second and third laws are outlined in every standard textbook, that of the first law (and the shape of orbits in general) is invariably omitted, due mainly to the insufficient proficiency of the targeted students in differential equations. The paper offers a workaround this difficulty. Assuming familiarity with the basic properties of conics and employing Newton’s law of universal gravitation, we manage to obtain, with very little calculus, the classic result: orbits are conic sections whose nature is determined by the body’s total energy.

Law of Gravity Blurred by Perturbed Planetary Orbits for Alien Observers

The Fermi paradox is the contradiction between the high probability for aliens and its lack of evidence. This is perhaps the most significant of all questions in astronomy. Nowadays, thousands of exoplanets have been detected, providing us more information directly from alien planets to revisit Fermi paradox. Over recent years, the detection of habitable exoplanets has grown rapidly, and most of them in multi-planet system with fairly compact structures, which is partly due to our observational techniques. In such systems, planets’ orbits are significantly perturbed by gravitational interactions from each other. To travel within or out of a planetary system, one must understand the law governing planetary motion, but can extraterrestrials deduce the universal law of gravitation by observing planets in their system, as Johannes Kepler and Isaac Newton had done? Here we show that due to the close planetary orbits in systems hosting habitable exoplanets, planetary orbits significantly deviate from the Keplerian orbit. In our N-body simulation over a period of 300 days, out of the ten planetary systems which all include habitable exoplanets in our sample, the mean anomaly deviates from a Keplerian orbit by more than 5 degrees while the system remains stable as the semi-major axis remains relatively constant. This would significantly complicate what the observers on those planets observe as the motion in the system would not appear to have a pattern periodically. Although planets indeed affect each other in all systems, the compact nature of these systems have magnified the effect and provide no way for the extraterrestrials inhabiting those planets to discover the universal law of gravitation. Extraterrestrials lacking the law of gravity definitely cannot facilitate traveling beyond their planets. That could be one possible resolution of Fermi paradox, reducing the population of high-developed civilization in Universe.

Methodology to determine the probalistic demand of mass attraction centers based on the classification within the framework of macrosystems theory

Introduction. The calculation of correspondence was described at the beginning of the 20 century in the form of a gravitational model and is based on analogy with the law of universal gravitation. The development of this model can be called J. Wilson’s approach, in which correspondence calculations are performed using an entropy model. The entropy approach operates with various expressions for the entropy of a macroscopic system. At the same time, its equilibrium is achieved at the maximum value of the selected entropy function. The purpose of this work is to develop a methodology for determining the probabilistic demand of mass attraction centers - Shopping centers and to demonstrate the results of its application on the example of shopping centers located on the territory of the city of Tula. Probabilistic demand is necessary to obtain the so-called ‘a priori probabilities’ in the expression of the entropy of the transport macrosystem.Methods and Materials. For the development of the methodology, as well as its further use, the most convenient and promising scientific platform is the theory of transport macrosystems, which is a special case of the general theory of macrosystems. Developed in the works of mainly domestic scientists, it allows you to perform various tasks specific to transport systems.Conclusions. The method of determining the probabilistic demand of the mass attraction centers -Shopping centers, which consists in obtaining a priori probabilities of being in them and their capacities for solving problems of finding equilibrium distributions of visitors, was developed with the aim of further developing the macroscopic approach in the study of Shopping centers. The main purpose of the technique is to use the results obtained in solving problems about the equilibrium states of the drains of transport and the road network in the framework of the theory of transport macrosystems.

Does quantum entanglement turn Rowlands’ ‘principle of duality’ into a ‘law’?

Physical laws are permanently observed. They are universal but not proven. Newton’s universal law of gravitation and three laws of motion, derived from the observations and theories of Copernicus, Kepler, Galileo and others, were used to take astronauts to the moon and return them safely to earth. Theories about the origin of gravity and whether gravitational forces are fundamental, quantum, entropic, induced, emergent, curved spacetime or fields like electromagnetism are still being probed. General relativity is a geometric theory, not a law, about gravity. In a book called Zero to Infinity: The Foundations of Physics Peter Rowlands asserts that physics is “founded entirely on the principle of duality.” Wave-particle duality is one of the most fundamental representations of quantum objects. Photons, electrons, neutrinos and even molecules have been shown to be in superposition and entangled. This paper will examine the role that quantum entanglement plays in the fermion/boson relationship of Rowlands’ nilpotent Dirac equation. Many other entangled physical dualities, identified as being foundational to physics, strongly suggest that his ‘principle of duality’ should be promoted into a physical law.

IMPACT OF THE INTERACTION OF MOVING PLANETS ON THEIR ORBITS

The research raises the problem of determining and measuring the interaction of planets on the trajectory of their orbits around the Sun. Its solution becomes possible only based on an adequate mathematical model. For this purpose, Newton's law of universal gravitation was adapted to the case of moving masses in a wide range of velocities in flat Euclidean space and physical time. The finite speed of propagating the gravitational field is considered. Differential equations of planets’ motion are obtained. Transients in the three-mass system of a star and two planets close to the Sun-Mercury-Venus system are simulated. To reveal deeply the essence of physical interaction, a hyperbolized transition process is simulated under artificially close moving masses.

Self-Supervised Deep Learning for 3D Gravity Inversion

The gravity method is one of the non-destructive geophysical methods, which aims to estimate the 3D subsurface density distribution of geological bodies from the observed 2D surface gravity anomalies. Recently, deep learning has achieved great success in solving ill-posed problems including gravity inversion. The limitation of the current deep learning methods for gravity inversion is the difference between synthetic and field data. Thus, we introduce a self-supervised estimation method for 3D gravity inversion (SSGI). SSGI learns the field data directly by closed-loop of the inversion model and forward model. The proposed inversion model contains an encoder, an expander, a decoder, and a 3D refiner. Since the forward model is built according to the law of universal gravitation, SSGI can optimize the inversion model by minimizing the mean absolute error of the original and reconstructed gravity anomalies. Besides, SSGI constrains the inversion model by a guide-line in the auxiliary loop. Since the guide-line corresponds to the sampling or average of the density matrix, minimizing the mean absolute error between the original guide-line and the generated guide-line can reduce the uncertainty of inversion. The experimental results demonstrate that the proposed SSGI achieves state-of-the-art performance in 3D gravity inversion.

Noncommutative Geometry and MOND

Modified Newtonian dynamics (MOND) is a hypothesized modification of Newton’s law of universal gravitation to account for the flat rotation curves in the outer regions of galaxies, thereby eliminating the need for dark matter. Although a highly successful model, it is not a self-contained physical theory since it is based entirely on observations. It is proposed in this paper that noncommutative geometry, an offshoot of string theory, can account for the flat rotation curves and thereby provide an explanation for MOND. This paper extends an earlier heuristic argument by the author.

God-Pantocrator and God-Coordinator: The Status of a Miracle in the Scientific Revolution

Scientists and philosophers of the 17th century, with all the novelty of their ideas, at the same time were in no hurry to reject the concept of a miracle, although many of them, such as I. Newton, rejected the understanding of a miracle as a violation of the laws of nature, its “ordinary course”. On the whole, with regard to the Christian concept of the miracle in the natural philosophy of the early modern period, a very uncertain situation developed. On the one hand, in the era of the Scientific Revolution, there was a clear tendency to explain extraordinary phenomena by the action of natural causes, which in theology found its meaningful expression in the Protestant concept of the cessation of miracles (cessatio miraculorum) in post-apostolic times, and in philosophy (more precisely, in the philosophico-theological literature), especially in the teachings of B. Pascal, R. Descartes, B. Spinoza, D. Hume and other authors, in an effort to build a rational theology in which the status of a miracle turned out to be very uncertain. On the other hand, the difficulties that arose in science after I. Newton's discovery of the law of universal gravitation and associated with the problem of actio in distans, forced researchers to resort to theological concepts and images in natural-philosophical reasoning, in particular, to refer to the concept of a miracle. The latter circumstance required the development of a new understanding of miracles, namely the concept of “coincidence miracles”, which made it possible to preserve the apologetic functions of miracles and at the same time to neutralize the philosophical and theological criticism of the concept of miracle by B. Spinoza and D. Hume. My aim in this article is to demonstrate that the relationship between theological and scientific (more precisely, natural-philosophical) problems is by no means reduced to the use of theological concepts in the process of the formation of classical science in the mode of general reasoning by analogy or as general ideological statements. Theological concepts turned out to be included in the natural-philosophical discourse on a par with purely physical arguments, and, on the contrary, theological thought had to somehow react to natural-philosophical discoveries, which ultimately led to a mutual adjustment of both natural-philosophical and theological concepts.

MATERIALITY OF THE GRAVITATIONAL FIELD AND THE PROCESS OF DEVELOPMENT OF MACROSCALE GRAVITATIONAL COLLAPSE

The materiality of the gravitational field is taken into account on the basis of the law of universal gravitation, accepted as an exact law describing the pairwise interactions of massive bodies. Unlike Brillouin and Lucas, who were the first to carry out such an account and obtain a negative value of the field mass, the field mass in our work has the same sign as the mass itself. Replacing the "mass-gravitational field" representation with "mass-field mass" distinguishes gravity from other interactions, leads to an increase in mass in such interactions, indicates the existence of a double effect of gravity and allows its physical modeling. In particular, it has been shown that, despite the small value of the relative mass gain in pair interactions, during the formation of clusters of stars with a large number of bodies, the relative mass gain increases nonlinearly. Under certain conditions, this increase becomes infinite, symbolizing the onset of a macroscale gravitational collapse, resulting in the formation of supermassive black holes. Attention is focused on the fact that the final mass of a supermassive black hole (invisible mass) can be tens and hundreds of times greater than the initial mass of the cluster (visible mass). Moreover, half of the black hole's mass is outside the gravitational radius of the black hole, forming a massive invisible halo. According to the authors, a macroscale collapse based on taking into account the materiality of the gravitational field can be considered as one of the effective mechanisms for the formation of invisible (dark) matter in the Universe.