In this paper, we examine the division of forces into internal and external components and explore their connection to the conservation of energy across various branches of physics, including classical mechanics, electromagnetism, quantum mechanics, and the theories of special and general relativity. While the partitioning of forces is a well-established concept in classical mechanics, it is often overlooked or inadequately addressed in other areas of physics. This oversight has significant implications for our understanding of fundamental physical principles and may influence the development of future technologies.
he study of the minimal coupling of scalar field with gravity in a general comoving spherically symmetric system is reconsidered. Spherical symmetry implies the independence of the field on the angular variables and the equivalence of the consistency of Einstein equation condition to the vanishing of the divergence of the energy momentum tensor of the scalar field. The explicit system of equations is reduced to the solution of a Kepler like equation coupled to the scalar field equation. The equations are studied in the special case in which the scalar field is massless and depends only on the time coordinate. As a consequence the complexity of the system of equations strongly reduces. The equations can be finally integrated by variable separation. By using the asymptotic behaviors of the explicit solution for the physical radius one can see that the the induced cosmological model shows an inflation at short time and a constant expansion velocity at large time.
In this article, we explore implications of the Alternative Relativistic Mechanics (ARM) of a material particle, proposed by us [\textit{Advanced Studies in Theoretical Physics} \textbf{19, No.4}, 173 - 182 (2025)]. This ARM is a framework that introduces Lorenz invariance violation through the presence of a new fundamental assumption: variable rest mass. The rest mass of a particle $m_0$ is not a constant as in Einstein's special relativity, but a known function of the Lorenz factor $\gamma$, $m_0(\gamma)=M_0(1+\alpha \ln\gamma)$, where the parameter $\alpha=\text{const.} \geq 0$ quantifies the modification of Einstein's special relativity, $M_0=m_0(1)=\text{const.} >0$. To protect the ARM from being debunked by the ultra precise electron g-2 and muon g-2 data, we proposed [\textit{Advanced Studies in Theoretical Physics} \textbf{20, No.1}, 19 - 27 (2026)] that the parameter $\alpha$ is not a universal constant, but instead scales with the square of the particle's initial rest mass, $\alpha_i=kM_{0,i}^2$, where $k=2.5(0.511\,\text{MeV}/c^2)^{-2} \times10^{-15}$ is the new fundamental constant of the ``variable mass field'' and $M_{0,i}$ is the initial rest mass of the i lepton (an isolated particle). We applied $k$ only to leptons. In the present paper we focus on the formalization of the mass-scaling law for the parameter $\alpha$ for composite particles like protons and heavy ions, specifically for the lead ion ${}^{208}\text{Pb}^{82+}$. To save the ARM at the high energies of the Large Hadron Collider (LHC) at CERN without adding hidden variables or arbitrary dampers, the choice of the particle's initial rest mass (``dressed'' or ``bare'') in the mass-scaling law $\alpha=kM_0^2$ becomes a make-or-break decision for the ARM. By treating the proton and the lead ion as collections of ``bare'' constituents rather than a single composite ``dressed'' object, the Bare Mass Hypothesis proposed in the present paper, prevents the mass-scaling law $\alpha_{bare}=kM_{0,bare}^2$ from ``scaling'' out of control. By shifting to the Bare Mass Hypothesis in the ARM, we make a profound claim: relativistic mass scaling is a property of fundamental constituents, not composite systems.
This study presents a novel form of bistability in a mathematical model of snail RPa1 neurons. This model was previously reported as a system of nonlinear ordinary differential equations simulating the membrane potential oscillations of snail RPa1 neurons. Here, we perform a numerical simulation revealing that whether the model shows a chaotic bursting state or a depolarized steady state is dependent on the initial condition. Notably, a certain transient current pulse can change the dynamical state of the model from a chaotic bursting state to a depolarized steady state, or vice versa. Taken together, these results indicate that the snail RPa1 neuron model can show bistability between chaotic bursting and depolarized steady states.
Quantum theory lives in abstract, infinite-dimensional, complex, linear Hilbert space, is unitary and non-dissipative, and has been proven not to be embeddable in spacetime for $N \geq 2$ quantum entities. It is per definition unobservable (in itself). Classical physics describes the causal, nonlinear dynamics of actual events, which lie in, and also define, four-dimensional spacetime. The ``Born Rule" maps abstract quantum theory into events, i.e. real outcomes, in classical spacetime. It must be postulated separately and cannot be deduced from quantum theory, as it is both non-unitary and irreversible, i.e. dissipative. Hence, the ``Born Rule" is ultimately and fundamentally responsible for dissipation, i.e. friction, and therefore all evolving complex systems, in the classical, observable world.
In this article we explore implications of the Alternative Relativistic Mechanics (ARM) of a material particle, proposed by us [\textit{Advanced Studies in Theoretical Physics} \textbf{19, No.4}, 173 - 182 (2025)]. This ARM is a framework that introduces Lorentz invariance violation through the presence of a new fundamental assumption: variable rest mass. The rest mass of a particle $m_0$ is not a constant as in Einstein's special relativity, but a known function of the Lorentz factor $\gamma$, $m_0(\gamma)=M_0(1+\alpha \ln{\gamma})$, where the free parameter $\alpha=\text{const.} \geq 0$ quantifies the modification of Einstein's special reality, $M_0=m_0(1)=\text{const.}>0$. Our research focuses on the impact of the ARM on the Cherenkov angle (the characteristic angle of Cherenkov radiation) taking into consideration the deviations from Einstein's special relativity, which we derived in the ARM. These ARM's deviations lead to different predictions for the Cherenkov angle compared to the framework of classical electrodynamics and the framework of Einstein's special relativity. We derive the expression for the Cherenkov angle within the ARM framework. This allows us to use Cherenkov detectors (like the LHCb experiment at CERN) to test the ARM and to set limits on its free parameter $\alpha$. Since in the ARM the parameter $\alpha$ governs the non-linear increase of mass-energy and mass-momentum at high $\gamma$, to protect the ARM from being immediately debunked by the ultra-precise electron g-2 and muon g-2 data, we abandon the idea of $\alpha$ as a global constant. We propose that the parameter $\alpha$ scales with the square of the particle's initial (at $v=0$) rest mass: $\alpha_i=kM_{0,i}^2$, where $k$ is a new fundamental constant of the "variable mass field" and $M_{0,i}$ is the initial rest mass of the i~particle (lepton).
This study employs birational transformations to derive explicit exact solutions of the complex Gerdjikov--Ivanov (GI) equation. Through travelling wave reduction and singularity analysis, we establish the system's complete integrability and its algebraic-geometric structure via elliptic curves. We construct a comprehensive set of solutions, including elliptic functions, bright and dark solitons, and rational solutions, with detailed analysis of their physical relevance to nonlinear optics, Bose–Einstein condensates, and wave focusing phenomena. Our approach highlights the power of algebraic geometry in solving nonlinear integrable PDEs and offers a general framework applicable to similar complex systems.
The derivation of Dirac equation previously studied in a curved space-time with torsion is reconsidered. As in the original paper the total action, sum of the Einstein Hilbert Cartan and of the Dirac action, is considered. The Dirac equation fallows, with improvements, by varying with respect to the Dirac spinor and to the torsion field, Variation with respect to the gravitational field is not considered. The equation is translated into the language of the 2 spinor formalism of Newman and Penrose with many improvement and clarifications. Contrarily to the original result, here there are at least two possible forms of the final equation. Both form have the same torsion dependent part. Instead the torsion free part in one case leads to Chandrasekhar like formulation (as in the original paper), while in the other case to the Penrose - Rindler's one. The torsion induced non linearity is represented by the interaction of the particle with its own current that remains conserved.
In this paper, cross-coupled metric perturbations of the form $h_{xz}=\epsilon\,t^{s}\sin(kz)$ are analysed in a Kasner-type anisotropic background, $ds^{2}=-dt^{2}+t^{2p}dx^{2}+t^{2q}dy^{2}+t^{2r}dz^{2}$. Working at first order in the small parameter $\epsilon$ and using both traceless and harmonic gauges, It is shown that the vacuum Einstein equations admit free long-wavelength solutions provided the single resonance condition $s=1+p+r$ is satisfied. Under this relation the perturbation amplitude scales as $h_{xz}\!\propto t^{\,2-q}$, while its contribution to the Weyl invariant obeys $\Delta C^{2}\!\propto t^{\,2p-2}$. Expansion along the $y$-axis therefore governs the wave amplitude, whereas the $x$-axis fixes the scalar-curvature imprint, producing a distinctive cross-directional signature. Potential consequences for the primordial stochastic gravitational-wave background and for constraining early-universe anisotropies are briefly discussed.
We introduce new relativistic mechanics of a material particle, where the {rest mass of the particle} is not a constant, but a known function of the Lorentz factor $\gamma$, alternative to the relativistic mechanics of Einstein's special relativity, where the {rest mass of the particle} is a constant. We first introduce a new relativistic linear momentum alternative to the relativistic linear momentum in special relativity, and then we find a new relativistic force law alternative to the relativistic force law in special relativity. After that, we introduce a new relativistic electromagnetic force law alternative to the Lorentz force law in special relativity, and then we use the muon g-2 experiments, done by J. Bailey \textit{et al}. [\textit{Nature} \textbf{268}, 301-305 (1977)] and G.W. Bennett \textit{et al}. [\textit{Phys. Rev. D} \textbf{73}, 072003 (2006)], to see whether those experiments are supportive evidence for the new law alternative to the Lorentz force law in special relativity.
Considering the formula of Plank for the spectrum of blackbody radiation as a product of two terms derived from a particle and a wave picture representing the blue and red ends, Einstein studied Wein’s equation and concluded in 1905 that light must be composed of independent packets (quanta/wave- particles/soft corpuscles) each of constant energy h. That is; the discreteness postulate of Plank was not limited to radiation/light interactions with matter but extends to light itself. This was not intuitive and was not accepted before Compton did his electron-radiation scattering calculations in 1923 using only particles with momentum and energy. In this paper, we adopt Einstein’s wave-particle concept and show that it’s singly (with the known characteristics of light) and with no added assumptions, can describe all the interactions of physics! We start with the concepts of space, time, mass, charge, force, inertia, discreteness, symmetry, elementary particles, and end with the all-important conservation laws. Radiation spreads from a point symmetrically in all directions, it conserves momentum, has infinite lifetime and moves in empty space at a constant speed c. It can also condense to create matter via E=mc^2, confirmed by the e-p pair experiments, with matter attributes emerging from this condensation [1]. Further interactions with radiation, causes matter to accelerate according to the negative density gradient of energy (radiation) [4]. A non-stop energy-matter interaction becomes the basis for building larger matter elements, growing up from elementary particles to elements and compounds to cover all space. The present work suggests that the interactions of physics can all be explained using the Einstein energy quanta together with the known radiation characteristics and we find that this remarkable success of the quanta model (and also that of the weight-function of QM) in representing nature, is basically due to the use of the ‘sine’ function of mathematics. It is a solution to both the classical and QM wave equations- carrying the characteristics of spacetime symmetry as well as its discreteness and continuity!
Based on the existing literature, this paper presents a diagonal four-manifold for the Universe to account for the wave-particle duality, where the wave spacetime in itself possesses energies and by nature a quotient topology so that the entire particle spacetime can be identified with effectively one point (the Planck length) in the wave spacetime as from the Big Bang and therefore actions-at-a-distance and quantum entanglement.
Radiation has momentum and on reflection from confining surfaces exerts a pressure due to momentum reversal. Radiation is also a quanta of energy given by Plank’s formula E=hf. We show that it is possible using these results to calculate a Carnot efficiency for a process in which the hot and cold surfaces are both in the form of radiation. We then connect this with the general formula for entropy and give a new definition for entropy as a measure of uniformity rather than chaos. This allows a statistical variance to be used as a measure for entropy. We show that this goes in line with the basic definition of thermodynamic entropy, which is found to be in essence a ‘non-dimensional’ quantity describing the ‘energy distribution’ or the configuration state of energy in a system. The dimensions normally given to entropy are not an essential part of it- it is an imposed thing. A connection between entropy and gravity is found next, showing that gravity is a source of negative entropy, with big stars and blackholes acting as stores of low entropy radiation and matter.
Einstein field equations in their most basic form give the bending of spacetime- the ‘R’ tensor, in response to an energy momentum density tensor input ‘T’ that expresses the distribution of various energy types in space. The idea of spacetime bending in response to energy in space is not intuitive to many, leading to a question if this is the true physical interpretation of these equations. We show here that it is possible to change this and make the equations intuitive and simply understood, if we take the gradient of the two sides of the equations. That is so because; the gradient of energy density on the RHS gives an ‘acceleration’/ force that is equalled by the gradient of the LHS curvature of the path of a moving mass. This can be immediately visible by checking the units involved- in which energy density divided by a length (to get a gradient) has the units of acceleration. And the gradient of the curvature of the LHS is the second derivative along the path and gives units of acceleration too. This amounts to saying that gravity acceleration/force at a point is simply the result of the negative gradient of the energy density at that point. This is the same rule that drives fluids due to pressure and electrons in wires due to electric potential, and also heat due to the (random) kinetic energy density gradient. The explanation is clear; particles move from high to low energy densities and pocket the difference. A gravity derived from an energy gradient idea has many perks. Gravity become both local and nonlocal- since a gradient is local but the energy on which it depends obeys an integral equation that requires summation over all space. It further makes voids in space as important as matter filled regions as both affect the gradient along a specified path. It also allows the calculation of the universal gravity constant G as a gradient of the energy density of the distant masses as Mach suggested before. This work compliments an earlier work that showed a similar property of Einstein geodesic equation [2].
This paper presents a unified semi-classical framework that bridges quantum mechanics and relativity to investigate nucleon structure through quark dynamics, introducing the Quantum Turning Point—a fundamental threshold defined by the product of an elementary particle’s mass and classical radius that distinguishes quantum-scale from classical-scale. By anchoring this threshold to Planck-scale parameters, we demonstrate that nucleon masses emerge not from quark rest masses alone but from relativistic quark dynamics, characterized by large Lorentz factors (γu ≈ 968, γd ≈ 3871) reflecting near-light-speed motion. A key achievement is the derivation of the proton-to-electron mass ratio (∼1837) from first principles, aligning with empirical observations and suggesting a deeper connection between fundamental constants. Remarkably, our unified nucleon mass formula incorporates the fine-structure constant (α) as a scaling factor, revealing an unexpected interplay between electromagnetic and strong interactions in nucleon mass generation. The empirical factor 0.476 in this formula further reflects a geometric symmetry in quark binding, explaining the near-equality of proton and neutron masses despite their differing quark compositions (Nu = 2, Nd = 1 vs. Nu = 1, Nd = 2). These results challenge the conventional separation of forces in the Standard Model, proposing instead that constants like α may emerge from internal particle dynamics rather than being externally imposed. While heuristic, our framework offers numerically consistent predictions and opens new pathways to unify quantum mechanics, relativity, and subatomic structure. This study advances our understanding of nucleon mass origins and hints at a deeper geometric or dynamical symmetry underlying fundamental physics.
It is shown that it is the (extended) relativity principle alone from which all of General and Special Relativity, namely Einstein’s field equation, is derived. This is done by operations in which the principles of conservation of mass and momentum (whose observation is required by the relativity principle for any observer at rest), and the covariant divergence of tensors play crucial roles. Kaluza’s attempt of a unification of gravitation and electromagnetism by the introduction of a fourth spatial dimension (so that the accessible universe constitutes an ultra-thin brane) must then be re-considered. Instead of introducing five new tensor elements g40 , g41 , g42, g43, g44 (as Kaluza did), five new tensor elements T40 , T41 , T42, T43, T44 are introduced that are an expression of conservation of charge. In the face of the new method of deriving Einstein’s field equation, the adding of the principle of conservation of charge and hence of a fourth spatial dimension is a necessity. The new elements of T have two physical meanings each: Mass-/momentum-flux in the 4th spatial direction on the one hand, and charge-density/charge-flux in all four spatial directions on the other hand. But it turns out that their dimensions are identical in basic units. Thus no ambiguity exists. The result of the tensor-expansion is stunning (even though electric force cannot be “transformed away”): Maxwell’s equations can be extracted, and the introduction of evenly distributed electric charge in the interior of a non-spinning spherical mass affects the metric tensor gµ nu not only because of the energy of the electric field, but in an additional manner. The necessity of adding a fourth spatial dimension, that is, the switching from symmetrical 4 x 4 to 5 x 5 tensors, comes with testable consequences. These consequences are solutions both to the Trouton-Noble and the Ehrenfest paradox.
The Dirac equation, previously formulated in a general space time with torsion by the Newman Penrose formalism, is considered in the context of the Schwarzschild space time with torsion. Based on a suitable null tetrad frame, the equation is separated by a variable separation method. The separated angular equations are integrated. The separated radial dependence is reduced, on the base of elementary properties of the solutions, to the solution of a single non linear one dimensional differential equation in the wave function and in its complex conjugate.
This study presents numerical simulations of a mathematical model of a type-A medial vestibular nucleus neuron (mVNn). This model is described by a system of nonlinear ordinary differential equations based on the Hodgkin–Huxley concept. The focus is on three system parameters: the injected current (Iapp), the maximal transient potassium conductance (gA), and the inactivation time constant of this conductance (τb). Simulations showed that as Iapp decreases, the model’s dynamical state transitions as follows: repetitive spiking state → mixed-mode oscillation state → quiescent state. Additionally, the sensitivities of these three dynamical states to variations in gA and τb are revealed. These findings deepen our understanding of the dynamics of the type-A mVNn model.
The curvature of space-time is replaced by the variable energy density of the time-invariant superfluid space, where the variable energy density of space carries gravity. The gravity vector points in the direction from a higher energy density to a lower energy density. When light moves in the direction of the gravity vector, it causes a blue shift. When light moves in the opposite direction, it causes a redshift. The Doppler effect in an expanding space has not been experimentally confirmed. The cosmological redshift originates from the gravitational redshift. Universal space does not expand.
Current literature presents about a dozen probability (P) interpretations: frequentist, Bayesian etc. Each one illustrates a partial aspect of indeterminism and turns out to be incompatible with the others on the logical plane. Basically, every model of P deals with a specific relation of P with the world and this fragmentary theorization inevitably has negative impact on quantum physics which is intrinsically indeterministic. Popper believed that the base issues of quantum mechanics cannot be untangled unless the unified probability theory is set up. So, we have conducted an attempt to integrate all the probability models and used the theoretical results to interpret quantum duality, wave collapse and measurement. The entire work, published in a recent book, exceeds the limits of this paper. Here we put forward a summary that concisely recalls the main definitions and five theorems out fifteen that have been proved. Finally, these theoretical tools will be used to discuss a thought experiment of Einstein and the Wheeler experiment. This proposal has the following features: (1) It addresses classical and quantum issues using the same theorems; (2) It develops a quantum interpretation that agrees with intuition and refutes disputable and bizarre views.
