Curved Spacetime Research Articles

Space-time entropy, space of singularities and gravity origin: A case study

A new definition of entropy is introduced using a model that simulates an expanding space-time compatible with the fundamental principle of cosmology. The entropy is obtained by mean of a state function that measures the variation of the space-time normal curvature, from a highly compressed space to a lower compressed space. The defined entropy leads to work out a new understanding of the earliest conditions that last for a period estimated to 380,000 years after the Big Bang. It leads to understand via a short period of inflation the process that generates the uniform distribution of matter and energy at the surface of the last scattering. It involves gravitational singularities in a process of gradual decompression propitious to the incubation of matter recombination, and it allows to trace back gravity origin.

The Observer and the General Centrality of Particles

According to the latest cosmological theories, the universe that began expanding with the Big Bang is a homogeneous and isotropic universe whose expansion continues to this day. According to cosmologists, based on this principle, the universe does not have a specific spatial center, and the special geometry considered for this viewpoint necessitates that any point in space can be considered the center of the universe. On the other hand, cosmologists consider the Big Bang as a point on a timeline, not a spatial occurrence. They state that our current best understanding of the origin and evolution of the universe is in line with the Standard Model of Cosmology (SMC), also known as the Lambda-Cold Dark Matter (ΛCDM) model. This model is based on two main theories of physics: General Relativity (GR) and the Standard Model of Particle Physics (SMPP). General Relativity is used to understand how the expansion of the universe is affected by the matter and energy within it, with gravity being the curvature of spacetime caused by the presence of mass and energy. Also, in this theory, a dimensionless quantity can be used to compare the time experienced by different observers in a gravitational field. This dimensionless quantity is often shown as the time dilation factor (Lorentz factor with the symbol γ, gamma). Alongside this theory, the Standard Model of Particle Physics describes the fundamental particles constituting matter and the three forces: electromagnetic, strong nuclear, and weak nuclear, governing the interactions of these particles. In this model, physicists often express time inversely as energy with the unit of giga-electron volts (GeV) and use it as a framework to understand the behavior of various particles in the early universe. Parallel to these theories, T-Consciousness Cosmology introduces the Spherical Cosmos Model (SCM). This model challenges the spatial centrality found in the standard cosmological model, which is based on the principle of isotropy. By doing so, SCM assigns a special place to the Milky Way galaxy, and consequently, to Earth, which is supported by results obtained from measurements and observations made from the depths of the cosmos. In this viewpoint, time is divided into Longitudinal Time and Transverse Time. The type of time defined in the theory of relativity as a four-dimensional intertwined spacetime is introduced here as an Entropic Time Force, which is a subcategory of Transverse Time. This challenges the definitions provided in SMPP or GR, raising questions about whether time should be considered a dimension or a quantity. T-Consciousness Cosmology states that the cosmos and every one of its components are subject to the Taheri Uncertainty Principle, and further considers them as virtual, by virtue of how they are defined by this principle. This principle can be examined from various angles, and in this discussion, it is approached from the perspective of time. On the other hand, the Taheri Uncertainty Principle segues to theories such as the Temporal Centrality of Particles, Time Continuity, and Time Uncertainty based on the special vantage point of the human observer, highlighting the fundamental role of humans in giving meaning to these concepts.

Space, Gravity-Time

Space and time, from Newton's point of view, were considered as two separate and absolute concepts, and gravity was also defined as a force that varies according to the mass of two objects and their distance from each other. However, Hermann Minkowski, using  Einstein's theory of relativity, offered a geometric interpretation of special relativity, integrating  time and the three dimensions of space into a unified four-dimensional model now known as Minkowski space. In the theory of general relativity, gravity is also defined as a geometric function and a consequence of the curvature of space-time, which itself  arises from the uneven distribution of mass and energy. One of the key outcomes of the theory of general relativity is that the motion of celestial bodies orbiting around other bodies in curved paths is not driven by a force known as gravity. Instead, bodies are actually following the closest objects to them through curved space-time, a bending that gravity is a result of. This movement happens along the shortest path, known as a geodesic. In contrast to Newton's view, the theory of relativity has shown that the speed at which gravity acts is limited to the speed of light. Modern physics proposes that this transmission of gravity might be carried out by hypothetical particles called gravitons. Another prediction of this theory is the concept of a gravitational potential well, which explains why the frequency of light increases or decreases as it falls into or escapes from this well, due to the curvature of space-time caused by a massive object. Time dilation near a massive object is also a phenomenon that occurs due to the strong gravitational field of that object. In this regard, Einstein's theory of special relativity explains how the laws of physics are the same for all observers moving at the same velocity and how the speed of light is constant in a vacuum. One of the consequences of this theory is that mass and energy are considered equivalent and interchangeable according to the famous E=mc2 equation. However, in the theories of T-Consciousness Cosmology, space is conceptualized independently and as a principle, in the form of a mesh, and time is described as an entropic force that acts opposite to gravity to break down objects that cause space to contract. In other words, the entropic force of time is introduced as a force that arises from mass to release space from stress, not as a fourth dimension perpendicular to the dimensions of space. Furthermore, this perspective considers varying viscosities for the space surrounding celestial objects. It introduces gravity as a force that is equivalent to the viscosity of space, rather than as a consequence of curved geometry, as typically presented in relativity. Gravity functions in accordance with the structure of space mesh. Consequently, phenomena such as gravitational redshift or blueshift are interpreted as outcomes of the viscosity of space. This perspective states that as light enters black holes, the mass-energy equivalence principle is violated because of the energetic resonance of the wave during its gravitational blackshift; implying that black holes are matter production factories. Furthermore, the mass-energy equivalence principle and the law of conservation of matter and energy do not hold true in the Cosmic Black Hole, which is the beginning of the cosmos, or after the final stage of space Rebound.

On Penrose’s Analogy between Curved Spacetime Regions and Optical Lenses

On Penrose’s Analogy between Curved Spacetime Regions and Optical Lenses

Radio Science Experiments during a Cruise Phase to Uranus

The exploration of Uranus, a key archetype for ice giant planets and a gateway to understanding distant exoplanets, is acquiring increasing interest in recent years, especially after the Uranus Orbiter and Probe (UOP) mission has been prioritized in the Planetary Science Decadal Survey 2023–2032. This paper presents the results of numerical simulations aimed at providing experimental constraints on the parameterized post-Newtonian (PPN) parameter γ, a measure of space–time curvature in general relativity (GR), during the cruise phase of a spacecraft travelling to Uranus. Leveraging advanced radio tracking systems akin to those aboard the JUICE and BepiColombo missions, we explore the potential of solar conjunction experiments (SCEs) to refine current measurements of γ by exploiting the spacecraft’s long journey in the outer Solar System. We discuss the anticipated enhancements over previous estimates, underscoring the prospect of detecting violations of GR. Our simulations predict that by using an advanced radio tracking system, it is possible to obtain an improvement in the estimation of γ up to more than an order of magnitude with respect to the latest measurement performed by the Cassini–Huygens mission in 2002, contingent on the calibration capabilities against solar plasma noise. The results reveal that a number of SCEs during the mission can substantially strengthen the validation of GR. In tandem with fundamental physics tests, the use of radio links during SCEs presents a valuable opportunity to dissect the solar corona’s plasma dynamics, contributing to solar physics and space weather forecasting. This paper also enumerates methodologies to analyze electron density, localize plasma features, and deduce solar wind velocity, enriching the scientific yield of the experiments beyond the primary objective of testing GR during the cruise phase of a mission to Uranus.

Polarized signatures of orbiting hot spots: Special relativity impact and probe of spacetime curvature

Context. The Galactic Center supermassive black hole is well known to exhibit transient peaks of flux density on a daily basis across the spectrum. Recent infrared and millimeter observations have strengthened the case for the association between these flares and circular orbital motion in the vicinity of the event horizon. The strongly polarized synchrotron radiation associated with these events leads to specific observables called QU loops, that is, looping motion in the Stokes QU plane of linear polarization. These patterns have been observed by the Submillimeter Array, VLTI/GRAVITY, and ALMA. Aims. We want to deepen the understanding of the QU loops associated with orbiting hot spots. To this end, we computed such loops in Minkowski and Schwarzschild spacetimes in order to determine which aspects of the observed patterns are due to special- or general-relativistic phenomena. Methods. We considered a parcel of energized plasma in circular motion in Minkowski spacetime and in Keplerian orbit in the Schwarzschild spacetime. We computed, using the GYOTO ray-tracing code, the polarized radiative transfer associated with this orbiting hot spot and derived the evolution of the flux density, astrometry, and Stokes Q and U parameters. Results. We show that QU loops in Minkowski spacetime at low or moderate inclination i ≲ 45° (where i = 0° is a face-on view) share all the qualitative features of Schwarzschild QU loops. There exist QU loops for all setups considered (including for the face-on view and vertical magnetic field), there may be one or two QU loops per orbital period for a vertical magnetic field configuration, and there are always two QU loops in case of a toroidal magnetic field. The simplicity of Minkowski spacetime is a key asset for allowing us to provide analytical formulas that explain the details of this behavior. Moreover, we analyzed the flux variation of the hot spot and show that it is dictated either by the angular dependence of the radiative transfer coefficients or by relativistic beaming. In the former case, this can lead to extreme flux ratios, even at a moderate inclination. Finally, we highlight the increasing mirror asymmetry of the Schwarzschild QU track with increasing inclination and show that this behavior is a specific Schwarzschild feature caused by light bending. Conclusions. Although special-relativistic effects have not been extensively discussed in this context, they are a crucial part in generating the observed QU loops. However, general-relativistic light bending leads to a specific observable feature encoded in the asymmetry of the observed loops, and this feature might allow the spacetime curvature to be quantified.

BCS in the sky: signatures of inflationary fermion condensation

We consider a Bardeen-Cooper-Schrieffer (BCS)-like model in the inflationary background. We show that with an axial chemical potential, the attractive quartic fermion self-interaction can lead to a BCS-like condensation. In the rigid-de Sitter (dS) limit of inflation where backreaction from the inflaton and graviton is neglected, we perform the first computation of the non-perturbative effective potential that includes the full spacetime curvature effects in the presence of the chemical potential, subject to the mean-field approximation whose validity has been checked via the Ginzburg criterion. The corresponding BCS phase transition is always first-order, when the varying Hubble is interpreted as an effective Gibbons-Hawking temperature of dS spacetime. In the condensed phase, the theory can be understood from UV and IR sides as fermionic and bosonic, respectively. This leads to distinctive signatures in the primordial non-Gaussianity of curvature perturbations. Namely, the oscillatory cosmological collider signal is smoothly turned off at a finite momentum ratio, since different momentum ratios effectively probe different energy scales. In addition, such BCS phase transitions can also source stochastic gravitational waves, which are feasible for future experiments.

Riemann's Acceleration in Light of the Howusu Metric Tensor in Spherical Polar Coordinates and Its Effect on the Theory of Gravitation

This work explores the ramifications of introducing the Howusu Metric Tensor into the theoretical framework of Riemann's Acceleration within spherical polar coordinates, aiming to deepen our comprehension of gravitational interactions. Bridging the gap between traditional Cartesian coordinates and spherical polar coordinates, the research navigates the challenges posed by rotational symmetry to seamlessly integrate Riemann's Acceleration into the latter. The Howusu Metric Tensor emerges as a novel mathematical construct tailored to the intricacies of spherical polar coordinates, enabling a refined representation of spacetime curvature. The Results which are Riemann's acceleration in light of the Howusu metric tensor in spherical polar coordinates and consequently a generalization of the Newton's equations of motion showcase the modified Riemann's Acceleration equations in spherical polar coordinates, revealing subtle differences when compared to conventional Cartesian coordinates. Comparative analyses underscore the impact of the Howusu Metric Tensor, both quantitatively and qualitatively, offering insights into the altered dynamics of gravitational forces. This signifies a paradigm shift in our approach to gravitational theory, showcasing the potential of the Howusu Metric Tensor to unravel novel insights and broaden the horizons of theoretical physics. The findings not only advance our understanding of gravitational interactions but also set the stage for further exploration and refinement in the ever-evolving landscape of theoretical physics.

Tsallis Distribution as a Λ-Deformation of the Maxwell-Jüttner Distribution.

Currently, there is no widely accepted consensus regarding a consistent thermodynamic framework within the special relativity paradigm. However, by postulating that the inverse temperature 4-vector, denoted as β, is future-directed and time-like, intriguing insights emerge. Specifically, it is demonstrated that the q-dependent Tsallis distribution can be conceptualized as a de Sitterian deformation of the relativistic Maxwell-Jüttner distribution. In this context, the curvature of the de Sitter space-time is characterized by Λ/3, where Λ represents the cosmological constant within the ΛCDM standard model for cosmology. For a simple gas composed of particles with proper mass m, and within the framework of quantum statistical de Sitterian considerations, the Tsallis parameter q exhibits a dependence on the cosmological constant given by q=1+ℓcΛ/n, where ℓc=ℏ/mc is the Compton length of the particle and n is a positive numerical factor, the determination of which awaits observational confirmation. This formulation establishes a novel connection between the Tsallis distribution, quantum statistics, and the cosmological constant, shedding light on the intricate interplay between relativistic thermodynamics and fundamental cosmological parameters.

Particle production from non-minimal coupling in a symmetry breaking potential transporting vacuum energy

We propose an inflationary scenario where the inflaton field is non-minimally coupled to spacetime curvature and inflation is driven by a vacuum energy symmetry breaking potential without specifying a priori whether the inflaton field is small or large. As we incorporate vacuum energy into our analysis, we further explore the implications of a non-zero potential offset within inflationary dynamics. We propose that vacuum energy can transform into particles as a result of the transition triggered by spontaneous symmetry breaking. This entails a vacuum energy cancellation that yields an effective cosmological constant during inflation by virtue of a quasi-de Sitter evolution and shows that vacuum energy contribution can manifest as geometric particles produced by inflaton fluctuations, with particular emphasis on super-Hubble modes. We conjecture these particles as quasi-particles arising from interaction between the inflaton and spacetime geometry, enhanced by non-minimal coupling. Specifically, we propose that dark matter arises from a pure geometric quasi-particle contribution, quantifying the corresponding dark matter candidate ranges of mass. In this scenario, we further find that a zero potential offset leads to a bare cosmological constant at the end of inflation, while a negative offset would require an additional kinetic (or potential) contribution in order to be fully-canceled. In this regard, we conclude that the scenario of large-field inflation is preferred since it necessitates a more appropriate selection of the offset. Our conclusion is reinforced as small-field inflation would lead to a significant screening of the Newtonian gravitational constant as inflation ends.

Curvature Quantization based on the Ehrenfest Paradox in the Bohr Atomic Model

Ehrenfest Paradox has been studied in the Bohr Atomic Model as a theoretical procedure for expressing the atomic coordinate curvature in the term of electromagnetic fine structure/coupling constant α = 1/137. The strength of the curvature corresponding to the gravitational aspect depends on the principal quantum number via a new constant 16α2a02=3,0348×1017m−2, which shows that the value of the curvature is quantised. For instance n = 1, the value is 3, 0349 × 1017 m−2. The curvature value in the Bohr atomic model can be used as a standard to compare how strong the curvatures of all systems are. This procedure can also be generalized to the strong and weak interactions in such a way that the value of the curvature can be represented by their own coupling constants. In the real situation, we can take the case of the atom near supermassive objects such as: blackhole, neutron star, white dwarf, etc. In this case, the atom is in the curved space-time, while the space-time curvature is quantized by the atom. Principally, the idea of the curvature and quantization are correlated. The excitation (the change of the state) of the atom or the atomic nuclei will generate the change of the space-time curvature ΔGμν(α)n that manifests a quantized curvature propagation (curvaton) through the space-time coordinate in the form of quantum stress tensor ΔTμν(q2B)=ℏΔσnΔGμν(α)n. It can be viewed as a unit of moving curvature reduced to a gravitational wave. This theory can be considered to expand the unification between quantum mechanics and gravitation.

Measuring gravity with milligram levitated masses.

Gravity differs from all other known fundamental forces because it is best described as a curvature of space-time. For that reason, it remains resistant to unifications with quantum theory. Gravitational interaction is fundamentally weak and becomes prominent only at macroscopic scales. This means, we do not know what happens to gravity in the microscopic regime where quantum effects dominate and whether quantum coherent effects of gravity become apparent. Levitated mechanical systems of mesoscopic size offer a probe of gravity, while still allowing quantum control over their motional state. This regime opens the possibility of table-top testing of quantum superposition and entanglement in gravitating systems. Here, we show gravitational coupling between a levitated submillimeter-scale magnetic particle inside a type I superconducting trap and kilogram source masses, placed approximately half a meter away. Our results extend gravity measurements to low gravitational forces of attonewton and underline the importance of levitated mechanical sensors.

Research on the Innovation and Development of Gravitational Wave Detection Technology

According to Einstein’s general theory of relativity, gravity is described as the curvature of space-time caused by gravitational sources, and to prove the existence of gravitational waves, many scientific research institutes around the world have begun to build equipment to try to detect gravitational waves in the vast background signals of the universe. At present, gravitational waves have been detected by many scientific research institutions and scientists. Now, the detection of gravitational waves has shifted to the direction of high precision and accuracy. This paper starts with the laser interferometer, the most basic instrument for gravitational wave detection. It expounds on the latest development progress and key technologies of gravitational wave detection in the current physical world, including noise suppression and gravitational wave detection spacecraft projects. In terms of laser interferometers, this paper describes their principle and key technologies and puts forward the difficult problems that still need to be tackled and studied. In terms of noise suppression, this paper describes the noise and interference that may be generated and how to suppress the noise to avoid interfering with the accuracy of the experiment. It also points out the shortcomings of current noise suppression techniques. For gravitational wave detection spacecraft projects, this paper first focuses on several key projects in the world, then describes the technologies and shortcomings of spacecraft, and puts forward the direction of improvement and in-depth research on these technologies. This paper aims to summarize the key technologies of gravitational wave detection in the world today and point out the direction of its future development.

Analytic States in Quantum Field Theory on Curved Spacetimes

AbstractWe discuss high energy properties of states for (possibly interacting) quantum fields in curved spacetimes. In particular, if the spacetime is real analytic, we show that an analogue of the timelike tube theorem and the Reeh–Schlieder property hold with respect to states satisfying a weak form of microlocal analyticity condition. The former means the von Neumann algebra of observables of a spacelike tube equals the von Neumann algebra of observables of a significantly bigger region that is obtained by deforming the boundary of the tube in a timelike manner. This generalizes theorems by Araki (Helv Phys Acta 36:132–139, 1963) and Borchers (Nuovo Cim (10) 19:787–793, 1961) to curved spacetimes.

Change of polarization degree of light beams on propagation in curved space

Even in free space, which is commonly considered of as a flat space–time in most settings, the degree of polarization of a partially spatially coherent light beam changes as it travels. Similarly, the polarization degree would change when a partially spatially coherent light beam propagates in a curved space–time. The difference of the polarization degree between the curved space and flat space can reveal the essential structure of the curved space. In this work, we consider a simplest case of curved space known as Schwarzschild spacetime. We can simulate the Schwarzschild space–time as an optical material with an effective refractive index. The difference of the polarization degree of a light beam propagating in curved space and flat space can be achieved up to 5%, which is detectable in practical measurement. In addition, we have found that the partially spatially coherent light source is necessary for obtaining significant changes in polarization degree. Our results provide an alternative method to estimate the Schwarzschild radius of a massive object with the optical polarization degree measurement.

Do Magnetic Monopoles Exist?

In this paper, we continue our previous work by discussing the consequences of curvature in spacetime caused by the presence of mass and the implications of this. We address the question raised by Dirac regarding the impact of a magnetic monopole on the quantum world. We discuss quantization concerning the solid angles observed, which may have a connection with a magnetic monopole.

The Ricci Rotation Coefficients in the Description of Trajectories of Spinning Test Particles Off-equatorial Plane in the Gravitational Field of a Rotating Source

We describe the trajectories of circular orbits of spinless and spinning test particles around a rotating body in equatorial and non-equatorial planes via the Mathisson-Papapetrou-Dixon equations. In this paper, these equations include the Ricci rotation coefficients with the purpose of describing not only the curvature of space time, but also the rotation of the spinning test particles that orbit around a rotating massive body. We found a numerical difference between the trajectories of spinless test particles and spinning test particles in the order of 10-7\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\varvec{10}}^{\\varvec{-7}}$$\\end{document}. We take as parameters: radius, energy, Carter’s constant and angular momentum.

A string-theoretical analog of non-maximal chaos in some Sachdev-Ye-Kitaev-like models

Very recently two of the present authors have studied the chaos exponent of some Sachdev-Ye-Kitaev (SYK)-like models for arbitrary interaction strength [1]. These models carry supersymmetric (SUSY) or SUSY-like structures. Namely, bosons and Majorana fermions are both present and each of them interacts with (q − 1) particles, but the model is not necessarily supersymmetric. It was found that the chaos exponents in different models, no matter whether they carry SUSY(-like) structures or not, all follow a universal single-parameter scaling law for large q, and by tuning that parameter continuously a flow from maximally chaotic to completely regular motion results. Here we report a string-theoretical analog of this chaotic phenomenon. Specifically, we consider closed string scattering near the two-sided AdS black hole, whose amplitude grows exponentially in the Schwarzschild time, with a rate determined by the Regge spin of the Pomeron exchanged during string scattering. We calculate the Pomeron Regge spin for strings of different types, including the bosonic string, the type II superstring and the heterotic superstring. We find that the Pomeron Regge spin also displays a single-parameter scaling behavior independent of string types, with the parameter depending on the string length and the length scale characterizing the spacetime curvature; moreover, the scaling function has the same limiting behaviors as that for the chaos exponent of SYK-like models. Remarkably, the flow from maximally chaotic to completely regular motion in SYK-like models corresponds to the flow of the Pomeron Regge spin from 2 to 1.

Chaos and integrability of relativistic homogeneous potentials in curved space

Relativistic Hamiltonian systems of n degrees of freedom in static curved spaces are considered. The source of space-time curvature is a scalar potential V(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\varvec{q})$$\\end{document}. In the limit of weak potential 2V(q)/mc2≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2V(\\varvec{q})/mc^2\\ll 1$$\\end{document}, and small momentum |p|/mc≪1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$|\\varvec{p} |/ mc\\ll 1$$\\end{document}, these systems transform into the corresponding non-relativistic flat Hamiltonian’s with the standard sum of kinetic energy plus potential V(q)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V(\\varvec{q})$$\\end{document}. We compare the dynamics of the classical and the corresponding relativistic curved counterparts on examples of important physical models: the Hénon–Heiles system, the Armbruster–Guckenheimer–Kim galactic system and swinging Atwood’s machine. Our main results are formulated for relativistic Hamiltonian systems with homogeneous potentials of non-zero integer degree k of homogeneity. First, we show that the integrability of a non-relativistic flat Hamiltonian with a homogeneous potential is a necessary condition for the integrability of its relativistic counterpart in curved space-time with the same homogeneous potential or with a non-homogeneous potential that the lowest homogenous part coincides with this homogeneous potential. Moreover, we formulate necessary integrability conditions for relativistic Hamiltonian systems with homogeneous potentials in curved space-time. These conditions were obtained from analysis of the differential Galois group of variational equations along a particular straight-line solution defined by a non-zero vector d\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{d}$$\\end{document} satisfying V′(d)=γd\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V'(\\varvec{d})=\\gamma \\varvec{d}$$\\end{document} for a certain γ≠0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma \ e 0$$\\end{document}. They are very strong: if the relativistic system is integrable in the Liouville sense, then either k=±2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=\\pm 2$$\\end{document}, or all non-trivial eigenvalues of the re-scaled Hessian γ-1V′′(d)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\gamma ^{-1}V''(\\varvec{d})$$\\end{document} are either 0, or 1. Certain integrable relativistic systems are presented.

On CCGG, the De Donder‐Weyl Hamiltonian formulation of canonical gauge gravity

AbstractThis short paper gives a brief overview of the manifestly covariant canonical gauge gravity (CCGG) that is rooted in the De Donder‐Weyl Hamiltonian formulation of relativistic field theories, and the proven methodology of the canonical transformation theory. That framework derives, from a few basic physical and mathematical assumptions, equations describing generic matter and gravity dynamics with the spin connection emerging as a Yang Mills‐type gauge field. While the interaction of any matter field with spacetime is fixed just by the transformation property of that field, a concrete gravity ansatz is introduced by the choice of the free (kinetic) gravity Hamiltonian. The key elements of this approach are discussed and its implications for particle dynamics and cosmology are presented. New insights: Anomalous Pauli coupling of spinors to curvature and torsion of spacetime, spacetime with (A)dS ground state, inertia, torsion and geometrical vacuum energy, Zero‐energy balance of the Universe leading to a vanishing cosmological constant and torsional dark energy.