Newtonian Gravitation Research Articles

Open Star Clusters and Their Asymmetrical Tidal Tails

Stars that evaporate from their star cluster by the energy equipartition process end up in either a leading or a trailing tidal tail. In Newtonian gravitation and for open star clusters in the solar vicinity, the tidal threshold, or práh, for escape is symmetrical, such that the leading and trailing tails are equally populated. The data from six independent teams that applied the convergent point method to map out the tidal tails of four open clusters (the Hyades, Praesepe, Coma Berenices, and COIN-Gaia 13) using Gaia DR2 and DR3 are here applied to test for the expected symmetry. All tidal tails contain more stars in the leading tail. The combined confidence amounts to an 8σ falsification of the práh symmetry. The same test using Milgromian dynamics leads to consistency with the data. More effort needs to be exerted on this matter, but the data indicate with high confidence that the tidal práh of an open star cluster is asymmetrical, with the corresponding confidence that Newtonian gravitation is falsified. Open star clusters depopulate more rapidly in Milgromian than in Newtonian dynamics, and the COIN-Gaia 13 cluster is here found to be nearly completely dissolved. In view of these results, the wide-binary star test and the finding of a Keplerian Galactic rotation curve are briefly discussed.

Newtonian gravitational waves from a continuum

Gravitational waves are being shown to derive directly from Newtonian dynamics for a continuous mass distribution, e.g. compressible fluids or equivalent. It is shown that the equations governing a continuous mass distribution, i.e. the inviscid Navier–Stokes equations for a general variable gravitational field g ( t , x ) , are equivalent to a form identical to the Maxwell equations from electromagnetism, subject to a specified condition. The consequence of this equivalence is the creation of gravity waves that propagate at a finite speed. The latter implies that Newtonian gravitation as presented in this paper is not ‘spooky action at a distance’ but rather is similar to electromagnetic waves propagating at finite speed, despite the apparent form appearing in the integrated field formula. In addition, this proves that, in analogy to the Maxwell equations, the Newtonian gravitation equations are Lorentz invariant for waves propagating at the speed of light. Since gravitational waves were so far derived only from Einstein’s general relativity theory, it becomes appealing to check if there is a connection between the Newtonian waves presented in this paper and the general relativity type of waves at least in a certain limit of overlapping validity, i.e. as a flat-space approximation. The latter is left for follow-up research.

The ΛCDM-NG Cosmological Model: A Possible Resolution of the Hubble Tension

We offer a cosmological model based on conventional general relativity (no speculative physics) that may resolve the Hubble tension. A reanalysis of the foundation of the Lambda-CDM model shows that general relativity alone does not specify what fraction of the mass density acts as the source term in Friedmann’s equation and what fraction acts as the source of the gravitational potential of condensed objects. This observation opens the way to alternative cosmological models within conventional general relativity, and it proves that the ΛCDM model is not the unique solution of Einstein’s equations for the usual cosmological sources of gravitation. We emphasize that the source of the gravitation potential in the ΛCDM model is the deviation δ ρ m of the mass density away from its average value, and not the total density of condensed masses as in Newtonian theory. Though not often stated, this is a modification of Newtonian gravitation within the ΛCDM model. The ΛCDM-NG model uses the freedom to move matter between source terms to restore the source of gravitational potential to its Newtonian form. There is no Hubble tension in the ΛCDM-NG model if the gravitational potential of condensed objects (stars, galaxies, and dark matter clouds) falls in a certain range, a range that does not seem unreasonable for the actual Universe. The deceleration parameter in the ΛCDM-NG model differs from that in the ΛCDM model, suggesting a test to distinguish between the two models.

The Descartes Code (Spin Orbital Rotation of Photons)–III. The Cosmological Redshift and the Hubble Constant

A new alternative model to the Friedmann-Lemaître-Robertson-Walker metric is presented based on Descartes’ code—the old color theory published in 1637. In this model, the photon spin-orbital speed is slowly decreasing in the free space of the Universe. The formulae for the wavelength, frequency, local time, momentum, energy, temperature, and photon energy dilation are presented for cosmological photons outside of a gravitational field. The Hubble constant H⊕ = 2.3195 ∗ 10−18 s−1 (= 71.572 kms−1Mpc−1) is calculated as (G M⊕)/(R⊕ c3) ∗ dcrotational/dt where G is the Newtonian gravitation constant, M⊕ is the Earth’s mass, R⊕ is the Earth’s radius, and crotational is the rotational speed of Descartes’ light “globules” without any fitting. The cosmological photons expand their wavelength in the Earth’s gravitational field based on the age of those photons determined via their redshift given as (1 + z). This newly defined Hubble constant can be falsified in the gravitational field of the Moon, and Mars. The simple formulae for the angular diameter distance dA, the luminosity distance dL, the distance modulus m—M, the concentration of galaxies at the redshift z can be tested on the recent data. E.g., the distance modulus was several times tested on the cosmological objects till their redshift z = 5 (references to those studies are given). The formula for the angular diameter distance dA was recently tested for the data taken using the James Webb Space Telescope (JWST) by Lovyagin et al. and the value of dA followed the data trend better than the standard model. This Descartes’ hypothesis can start a discussion of Descartes’ cosmological photons can newly explain experimental data without fitting in Euclidean space.

A METHOD FOR EMPIRICAL ESTIMATION OF PLANCK’S LENGTH, MASS, AND TIME THROUGH THE CHARACTERISTICS OF AN ELECTRON. IMPROVING THE ACCURACY OF SOME PHYSICAL CONSTANTS

A method for empirical estimation of Planck's length, mass, and time is proposed, which is based on the characteristics of the electron, Avogadro, and Euler numbers, and the fine structure constant. Basic physical constants can be expressed in terms of Planck's length, mass, and time. The disadvantage of this method is that Planck's elementary particle does not exist in nature. Planck's particle is presented in a hypothetical, virtual form, its characteristics are theoretically calculated through the reduced Planck's constant, Newton's gravitation constant, and the constant speed of light in a vacuum, and the accuracy of these characteristics is low. This is due to the low accuracy of the Newtonian constant of gravity. This shortcoming can be eliminated if the characteristics of the hypothetical Planck particle are associated with a real elementary particle, for example, with an electron, and through its characteristics with the characteristics of leptons and baryons. Since the characteristics of leptons and baryons are determined experimentally and are among the most accurate, for example, for a proton now is 11 decimal places, establishing a connection with them of a hypothetical Planck particle will increase the accuracy of the values: of Planck's length, mass and time, of Planck constant, of elementary electric charge, of the Newtonian gravitational constant, of electron mass, of Planck temperature up to the accuracy level of a proton. As is known, the 26th General Conference on Weights and Measures decided to define one of the basic SI units of measurement through physical constants. In particular, with this solution, the kilogram is now defined in terms of Planck's constant, and the ampere is now determined in terms of the value of the elementary electric charge. Accordingly, with increasing accuracy of the values of Planck's constant and elementary electric charge, the accuracy of the values of a kilogram and Ampere will increase.

How Newton Was Misled by Kepler: A Classical Derivation of Newtonian Gravitation from Precessing Orbits

How Newton Was Misled by Kepler: A Classical Derivation of Newtonian Gravitation from Precessing Orbits

Direct correspondence between Newtonian gravitation and general relativity

Direct correspondence between Newtonian gravitation and general relativity

Dielectric constant and magnetic permeability in the Newtonian gravitational field derived by the elastic energy

The pressure distribution in the Earth’s interior is expressed using the Adams‐ Williamson equation. The elastic energy is stored inside the Earth by the pressure. The Newtonian gravitation is derived by this energy. The Newtonian gravitation refracts light at the surface of the Sun. When light is an electromagnetic wave, this refraction is expressed by the dielectric constant and magnetic permeability in the Newtonian gravitational field. The propagation of electromagnetic waves in a black hole is discussed using the dielectric constant and magnetic permeability.

Dynamic Universe Model Based on Newtonian Gravitation Giving Results Well from Universe Level to Galaxy Level to Solar System Level to Earth Level to Electron Level to Energy Level and to Nanobio Particle Level Now

Dynamic universe model’s SITA simulations can be used to solve many problems in physics, chemistry, Biology etc. The Dynamic Universe Model is based on Newtonian gravitation and finds the total attractive force of all masses in a combined way on any single mass. This SITA approach provides accurate results from the Universe level down to the Galaxy, Solar System, Earth, Electron, Energy, and even Nanobio Particle levels. Our thinking and exploration should not be limited to just 2-body problem and Universe model limited to cosmology in Physics. A small software program (SITA) takes your thinking to: - What are the inherent properties of inter-molecular attraction forces? - How does the Galaxy balance its rotations of stars? - Why did the Pioneer satellite have an irregular movement? - Why there are so many blue shifted Galaxies in the universe? - What is the structure of Galaxy center which will be stable and can support full Galaxy? - How insolvable contaminations in water or in any other liquids diffuse? - What happens to all the energy emitted by sun and other stars and Galaxies? - How local system of Galaxies balance each other? - How are electrons or positrons generated in the universe? - Exploring galaxy rotation curves - Can these be explained…the Origin, Propagation and Uniformity of CMB?? - How hydrogen atoms and other atoms are formed? - How are three states of water (H2O) formed? - How astronomical Jets are formed? - How are the mathematical equations proved? - What about SINGULARITIES in the software and in the Model?

Education

The Education section in this issue presents two contributions. In `"Nesterov's Method for Convex Optimization," Noel J. Walkington proposes a teaching guide for a first course in optimization of this well-known algorithm for computing the minimum of a convex function. This algorithm, first proposed in 1983 by Yuri Nesterov, though well recognized in computational optimization in the presence of large data as a more efficient tool than the steepest descent method, is still absent in most modern textbooks on optimization. The author of the present article develops an elementary analysis of Nesterov's first order algorithm that parallels that of steepest descent but with an additional requirement proposed by Nesterov. Two cases are discussed. The first concerns an unconstrained minimization problem, while the second includes closed convex constraints represented using infinite penalization of the cost. More generally, the cost function becomes the sum of a smooth convex function and a lower semicontinuous convex function. Several student-level exercises are included in this paper. Results are nicely illustrated by an example of a signal recovery problem and a discussion of the Uzawa algorithm for optimization problems with constraints defined by inequalities involving convex functions. The second paper, "A Comprehensive Proof of Bertrand's Theorem," is presented by Patrick De Leenheer, John Musgrove, and Tyler Schimleck. It concerns the behavior of the solutions of the classical two-body problem and states that, among all possible gravitational laws, there are only two exhibiting the property that all bounded orbits are closed: Newtonian gravitation and Hookean gravitation. Historically, even if Newton was aware that there are to specific gravitational laws having the above property, it was only two centuries later, in 1873, that Bertrand realized that these are the only ones. Bertrand's theorem, due to its important consequences, has been integrated into the undergraduate curriculum in theoretical mechanics, but its proof, accessible to undergraduate mathematics or physics students, seems to be absent from modern textbooks. Although Bertrand's original paper did not contain a precise proof, V. Arnold proposed a sketch of it based on six subproblems. Among other contributions, this article provides a complete proof of the sixth subproblem under a specific assumption imposed on the magnitude of the force in the motion model. Under this assumption, a complete proof of Bertrand's theorem is then given, incorporating also earlier contributions by other authors. Still, comprehensive does not mean simple here, and this paper may be used to conceive several research projects for advanced-level undergraduate students in mathematics or physics.

A Comprehensive Proof of Bertrand's Theorem

A cornerstone result in Newtonian mechanics is Bertrand's Theorem concerning the behavior of the solutions of the classical two-body problem. It states that among all possible gravitational laws there are only two exhibiting the property that all bounded orbits are closed. One of these is Newtonian gravitation, the other being Hookean gravitation. We present a comprehensive proof of Bertrand's Theorem that is accessible to undergraduate students who are familiar with basic notions from advanced calculus and differential equations.

Complete solution of the Einstein field equations for a spherical distribution of polytropic matter

We determine the complete solution of the Einstein field equations for the case of a spherically symmetric distribution of gaseous matter, characterized by a polytropic equation of state. We show that the field equations automatically generate two sharp boundaries for the gas, an inner one and an outer one, given by radial positions $$r_{1}$$ and $$r_{2}$$ , and thus define a shell of gaseous matter outside of which the energy density is exactly zero. Hence this shell is surrounded by an outer vacuum region, and surrounds an inner vacuum region. Therefore, the solution is given in three regions, one being the well-known analytical Schwarzschild exterior solution in the outer vacuum region, one being determined analytically in the inner vacuum region, and one being determined partially analytically and partially numerically, within the matter region, between the two boundary values $$r_{1}$$ and $$r_{2}$$ of the Schwarzschild radial coordinate r. This solution is therefore somewhat similar to the one previously found for a spherically symmetric shell of liquid fluid, and is in fact exactly the same in the cases of the inner and outer vacuum regions. The main difference is that here the boundary values $$r_{1}$$ and $$r_{2}$$ are not chosen arbitrarily, but are instead determined by the dynamics of the system. As was shown in the case of the liquid shell, also in this solution there is a singularity at the origin, that just as in that case does not correspond to an infinite concentration of matter, but in fact to zero matter energy density at the center. Also as in the case of the liquid shell, the spacetime within the spherical cavity is not flat, so that there is a non-trivial gravitational field there, in contrast with Newtonian gravitation. This inner gravitational field has the effect of repelling matter and energy away from the origin, thus avoiding a concentration of matter at that point.

Exact solution of the Einstein field equations for a spherical shell of fluid matter

We determine the exact solution of the Einstein field equations for the case of a spherically symmetric shell of liquid matter, characterized by an energy density which is constant with the Schwarzschild radial coordinate r between two values $$r_{1}$$ and $$r_{2}$$ . The solution is given in three regions, one being the well-known analytical Schwarzschild solution in the outer vacuum region, one being determined analytically in the inner vacuum region, and one being determined mostly analytically but partially numerically, within the matter region. The solutions for the temporal coefficient of the metric and for the pressure within this region are given in terms of a non-elementary but fairly straightforward real integral. For some values of the parameters this integral can be written in terms of elementary functions. We show that in this solution there is a singularity at the origin, and give the parameters of that singularity in terms of the geometrical and physical parameters of the shell. This does not correspond to an infinite concentration of matter, but in fact to zero energy density at the center. It does, however, imply that the spacetime within the spherical cavity is not flat, so that there is a non-trivial gravitational field there, in contrast with Newtonian gravitation. This gravitational field is repulsive with respect to the origin, and thus has the effect of stabilizing the geometrical configuration of the matter, since any particle of the matter that wanders out into either one of the vacuum regions tends to be brought back to the bulk of the matter by the gravitational field.

Gravitational potential in spherical topologies

We study the properties of the Newtonian gravitational potential in a spherical Universe for different topologies. For this, we use the non-Euclidean Newtonian theory developed in Vigneron [On non-Euclidean Newtonian theories and their cosmological backreaction, Classical Quantum Gravity 39, 155006 (2022)]. describing Newtonian gravitation in a spherical or hyperbolic Universe. The potential is calculated for a point mass in all the globally homogeneous regular spherical topologies, i.e., whose fundamental domain is unique and is a platonic solid. We provide the exact solution and the Taylor expansion series of the potential at a test position near the point mass. We show that the odd terms of the expansion can be interpreted as coming from the presence of a nonzero spatial scalar curvature, while the even terms relate to the closed nature of the topological space. A consequence is that, compared to the point mass solution in a 3-torus, widely used in Newtonian cosmological simulations, the spherical cases all feature an additional attractive first order term dependent solely on the spatial curvature. The choice of topology only affects the potential at second order and higher. For typical estimates of cosmological scales (curvature and topology), the strongest topological effect occurs in the case of the Poincar\'e dodecahedral space, but in general the effect of curvature dominates over topology. We also provide the set of equations that can be used to perform $N$-body simulations of structure formation in spherical topologies.

Disjointed Equivalence of Gravitational and Inertial Mass

Problem—Contemporary physics offers no underlying reason for the equivalence of inertial and gravitational mass. Approach—The equivalence is examined from the new physics provided by the cordus theory, being a non-local hidden-variable (NLHV) theory. Mathematical formalisms are derived for masses and observers in different fabric densities. Findings—A disjointed equivalence is predicted, whereby inertial and gravitational masses are equivalent in any one situation, but a different equivalence holds when the fabric densities change. Consequently this theory predicts that the gravitational constant G varies with fabric density, and hence would be different across the universe and across time. Not only is the gravitational constant non-constant, but the formulation of gravitation changes with fabric density. Specifically, the theory predicts gravity is stronger at genesis (and the end of the universe) such that orbit velocity vB ∝ (where rB is orbit radius), compared to weaker gravitation at middle life epochs with rB ∝ . The current Earth location and epoch correspond to the latter case, i.e. Newtonian gravitation is recovered. The findings disfavour the existence of both dark energy and dark matter, and instead attribute these effects to differences in the fabric density. Originality—The work makes the contribution of deriving a mass equivalence relationship that includes fabric density, identifying a disjointed mass equivalence, and showing that the gravitation formulation itself changes with relative fabric densities.

The kinematics of Keplerian velocity imposes another interpretation of Newtonian gravitation

The velocity of any Keplerian orbiter is well known, but its time derivative is a centripetal acceleration, not an attractive one. Furthermore the rectilinear accelerated trajectory of Newton’s attraction is not part of the Keplerian conics. Newton’s postulate of attraction is therefore not consistent with Kepler’s laws. We demonstrate this geometric reality by the factual kinematics and expose its consequences from the bodies falling, to the rotation speed of the galaxies, passing through Einstein’s equivalence principle or the stability of the solar system.

Spherically Symmetric Problem of General Relativity for an Elastic Solid Sphere

The paper is devoted to a spherically symmetric problem of General Relativity (GR) for an elastic solid sphere. Originally developed to describe gravitation in continuum (vacuum, gas, fluid and solid) GR does not provide the complete set of equations for solids and, in contrast to the Newton gravitation theory, does not allow us to study the stresses induced by gravitation in solids, because the compatibility equations which are attracted in the Euclidean space for this purpose do not exist in the Riemannian space. To solve the problem within the framework of GR, a special geometry of the Riemannian space induced by gravitation is proposed. According to this geometry, the four-dimensional Riemannian space is assumed to be Euclidean with respect to the space coordinates and Riemannian with respect to the time coordinate. Such interpretation of the Riemannian space in GR allows us to supplement the conservation equations for the energy-momentum tensor with compatibility equations of the theory of elasticity and to arrive to the complete set of equations for stresses. The analytical solution of the Einstein equations for the empty space surrounding the sphere and the numerical solution for the internal space inside the sphere with the proposed geometry are presented and discussed.

Propagation of Newtonian gravitation derived by the elastic energy

The Earth is compressed by its own Newtonian gravitation. The pressure is generated inside the Earth for this compression. The pressure distribution is expressed using the Adams‐Williamson equation (AWE). The elastic energy is stored inside the Earth by the pressure. The Newtonian gravitation is derived by this energy. The propagation of Newtonian gravitation at the speed of light is expressed by using the AWE as a wave.

The Nature of a Constant of Nature: The Case of G

AbstractPhysics presents us with a symphony of natural constants: $G$, ℏ, $c$, and so forth. Up to this point, constants have received comparatively little philosophical attention. In this article, I provide an account of dimensionful constants, in particular the gravitational constant. I propose that they represent interquantity structure in the form of relations between quantities with different dimensions. I use this account of $G$ to settle a debate over whether mass scalings are symmetries of Newtonian gravitation. I argue that they are not, but only if we interpret mass anti-quidditistically. This is analogous to anti-haecceitism in the presence of spacetime symmetries.

Newtonian gravitation derived by the elastic energy in the Earth’s interior

The pressure distribution in the Earth’s interior is expressed using the Adams‐Williamson equation. The elastic energy is stored inside the Earth by the pressure. The Newtonian gravitation is expressed by this energy.