Principle Of Covariance Research Articles

Relativistic Guiding-Center Motion: Action Principle, Kinetic Theory and Hydrodynamics.

We treat the guiding-center dynamics in a varying external Maxwell field using a relativistically covariant action principle which reproduces the known Vandervoort expression for the drift velocity and extends it to curved spacetime. We derive the corresponding kinetic theory and ideal hydrodynamic theory. In contrast to conventional five-equation hydrodynamics, the guiding-center hydrodynamics needs only three equations due to a constraint on the motion across magnetic field. We argue that such a hydrodynamics is applicable to strongly coupled plasmas where kinetic theory fails.

On Augmented Dimensional Analysis

We present an innovative approach to dimensional analysis, referred to as augmented dimensional analysis and based on a representation theorem for complete quantity functions with a scaling-covariant scalar representation. This new theorem, grounded in a purely algebraic theory of quantity spaces, allows the classical π\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\pi $\\end{document} theorem to be restated in an explicit and precise form and its prerequisites to be clarified and relaxed. Augmented dimensional analysis, in contrast to classical dimensional analysis, is guaranteed to take into account all relations among the quantities involved. Several examples are given to show that the information thus gained, together with symmetry assumptions, can lead to new or stronger results. We also explore the connection between dimensional analysis and matroid theory, elucidating combinatorial aspects of dimensional analysis. It is emphasized that dimensional analysis rests on a principle of covariance.

Cosmological Consequence of Varying Speed of Light and Gravitational Constant

<p>The speed of light is taken to be a constant in a vacuum. This forms the basic tool for the principle of General Covariance, which asserts that all laws of Physics should take the same form in all frames of reference. Without putting inflation into consideration, the theory of varying speed of light (VSL) would solve basic problems of cosmology in the early universe. Furthermore, the Gravitational constant G that occurred in the Friedmann Equations may not have been real constants in the early universe but have some variation with the universe scale factor. Cosmological models with varying physical constants have been of interest in recent years with few works in the literature. A cosmological solution obtained by incorporating variable speed of light and gravitational constant gives a cosmic model which is free from the initial Big Bang singularity and horizon problem. It is also observed here that the early universe was dominated by dark matter, however, as the scale factor increases, the dark energy become dominant.</p>

Comparison of the imaginary parts of the atmospheric refractive index structure parameter and aerosol flux based on different measurement methods

Abstract. The complexity of aerosol particle properties and the diversity of characterizations make aerosol vertical transport flux measurements and analysis difficult. Although there are different methods, such as aerosol particle number concentration flux and aerosol mass flux based on the eddy covariance principle as well as aerosol mass flux measurements based on the light-propagated large-aperture scintillation principle, there is a lack of mutual validation among the different methods. In this paper, a comparison of aerosol mass flux measurements based on the eddy covariance principle and aerosol mass flux measurements based on the light-propagated large-aperture scintillation principle is carried out. The key idea of aerosol mass flux measurements based on the light-propagated large-aperture scintillation principle is the determination of the imaginary part of the atmospheric equivalent refractive index structure parameter (AERISP). In this paper, we first compare the AERISPs measured by two different methods and then compare the aerosol mass vertical transport fluxes obtained by different methods. The experiments were conducted on the campus of the University of Science and Technology of China (USTC). A light propagation experiment was carried out between two tall buildings to obtain the imaginary and real parts of the AERISPs for the whole path, which in turn can be used to obtain the aerosol vertical transport flux. An updated visibility meter is installed on the meteorological tower in the middle of the light path, which is utilized to obtain the single-point visibility, which is then converted to the imaginary part of the atmospheric equivalent refractive index (AERI). The imaginary parts of the AERISP were obtained via spectral analysis with visibility data. The results show that the imaginary parts of the AERISPs obtained by different methods are in good agreement. The imaginary part of the AERI measured by the visibility meter and the vertical wind speed obtained by the ultrasonic anemometer were used for covariance calculations to obtain the aerosol vertical transport flux. The trends in aerosol vertical transport fluxes obtained by the different methods are consistent, and there are differences in some details, which may be caused by the inhomogeneity in the vertical transport of aerosol fluxes. The experimental results also showed that urban green land is a sink area for aerosol particles.

Covariance and Uncertainty Principle for Dispersive Pulse Propagation

We develop the concept of covariance for waves and show that it plays a fundamental role in understanding the evolution of a propagating pulse. The concept clarifies several issues regarding the spread of a pulse and the motion of the mean. Exact results are obtained for the time dependence of the covariance between position and wavenumber and the covariance between position and group velocity. We also derive relevant uncertainty principles for waves.

Investigating Nonlinear Dynamics in Atmospheric Aerosols during the Transition from Laminar to Turbulent Flow

This paper investigates the nonlinear dynamics of atmospheric aerosols during the transition from laminar to turbulent flows using the framework of Scale Relativity Theory. It is proposed that the transition from multifractal to non-multifractal scales (in the dynamics of the atmospheric aerosols) can be assimilated to the transition between laminar and turbulent states. These transitions are determined by the multifractal diffusion and deposition processes. The methodology used involves the application of the principle of scale covariance, which states that the laws of atmospheric physics remain invariant with respect to spatial and temporal transformations as well as scale transformations. Based on this principle, several conservation laws are constructed. In such context, the conservation law of the density of states associated with the multifractal-non-multifractal scale transition in a one-dimensional case is then considered. The model describes the non-linear behaviour of atmospheric aerosols undergoing diffusion and deposition processes. The theoretical approach was correlated using experimental data from a ceilometer and radar reflectivity factor data.

Quantum theory of three-dimensional de Sitter space

We sketch the construction of a quantum model of three-dimensional de Sitter space, based on the covariant entropy principle and the observation that semiclassical physics suggests the possibility of a consistent theory of a finite number of unstable massive particles with purely gravitational interactions. Our model is holographic, finite, unitary, causal, plausibly exhibits fast scrambling, and qualitatively reproduces features of semiclassical de Sitter physics. In an appendix we outline some calculations that might lead to further tests of the model. Published by the American Physical Society 2024

Manifestly covariant variational principle for gauge theories of gravity

A variational principle for gauge theories of gravity is presented, which maintains manifest covariance under the symmetries to which the action is invariant, throughout the calculation of the equations of motion and conservation laws. This is performed by deriving explicit manifestly covariant expressions for the Euler-Lagrange variational derivatives and Noether’s theorems for a generic action of the form typically assumed in gauge theories of gravity. The approach is illustrated by application to two scale-invariant gravitational gauge theories, namely Weyl gauge theory (WGT) and the recently proposed ‘extended’ Weyl gauge theory (eWGT), where the latter may be considered as a novel gauging of the conformal group; the method can also be straightforwardly applied to other theories with smaller or larger symmetry groups. In addition, the approach enables one easily to establish the relationship between manifestly covariant forms of variational derivatives obtained when one or more of the gauge field strengths is set to zero either before or after the variation is performed. This is illustrated explicitly for both WGT and eWGT in the case where the translational gauge field strength (or torsion) is set to zero before and after performing the variation, respectively. Published by the American Physical Society 2024

A Fock space structure for the diffeomorphism invariant Hilbert space of loop quantum gravity and its applications

Loop quantum gravity (LQG) is a quantization program for gravity based on the principles of QFT and general covariance of general relativity. Quantum states of LQG describe gravitational excitations based on graphs embedded in a spatial slice of spacetime. We show that, under certain assumptions on the class of diffeomorphisms, the space of diffeomorphism invariant states carries a Fock space structure. The role of one-particle excitations for this structure is played by the diffeomorphism invariant states based on graphs with a single (linked) component. This means, however, that a lot of the structure of the diffeomorphism invariant Hilbert space remains unresolved by this structure. We show how the Fock structure allows to write at least some condensate states of group field theory as diffeomorphism invariant coherent states of LQG in a precise sense. We also show how to construct other interesting states using this Fock structure. We finally explore the quantum geometry of single- and multi-particle states and tentatively observe some resemblance to geometries with a single or multiple components, respectively.

Gravity theories with local energy-momentum exchange: a closer look at Rastall-like gravity

Einstein’s famous equivalence principle is certainly one of the most striking features of the gravitational interaction. In a strict reading, it states that the effects of gravity can be made to disappear locally by a convenient choice of reference frame. As a consequence, no covariantly-defined net gravitational (point-wise) force should exist and energy-momentum of all matter and interaction fields combined, with gravity excluded, should be locally conserved. Although elegant, this represents a strong constraint on the dynamics of a gravitating system and it is only logical to question its naturality and observational basis. This is the purpose of the present work. For concreteness sake, we analyze, in the context of metric (torsion-free) theories of gravity, the simplest phenomenological model which allows for local energy-momentum exchange between the spacetime and matter/interaction fields while preserving the seemingly more natural principle of general covariance. This concrete model turns out to be a generalization of the socalled Rastall’s theory, with one important advantage: criticisms made to the latter, which are often used to dismiss it as a viable or interesting model, do not apply to the former in a Universe containing ‘dark’ ingredients, such as ours—a connection which seems to have been overlooked thus far. We conclude by exploring the consequences of our Rastall-like model to standard (astrophysical and cosmological) gravitational scenarios.

Background Independence and Gauge Invariance in General Relativity Part 2—Covariant Quantum Gravity

Background independence is often being claimed as the characteristic property of several current and past models of Quantum Gravity. In actual fact, such a notion has a wider connotation and must be rooted into the validity of the general covariance principle, demanding its logical connection with the notions of manifest covariance and (quantum) gauge invariance. In fact, as we intend to show here, it involves (a) the existence of a well-defined, albeit arbitrary, classical background space-time; and (b) the suitable realization of a dynamical equation for the related background metric field tensor, referred to as quantum-modified Einstein tensor field equation, which actually determines it in a suitable functional setting. Remarkably, it is proved that in the context of the theory of Covariant Quantum Gravity (CQG-theory), recently developed by Cremaschini and Tessarotto (2015–2022), background independence implies that such an equation “emerges” rigorously from the same CQG-theory. This follows in terms of a stochastic quantum expectation value evaluated with respect to the corresponding characteristic quantum PDE. It is shown that an analogous emergence property applies also to the background metric field tensor in terms of stochastic fluctuations of the corresponding underlying quantum tensor of gravitational field. These results warrant the consistent validity of background independence for the prescription of the space-time metric tensor in CQG-theory.

The uncertainty principle for the octonion Fourier transform

The octonion Fourier transform (OFT) is a hypercomplex Fourier transform that generalizes the quaternion Fourier transform. However, in octonion algebra, there are two major obstacles that are presented in the loss of associativity and commutativity. Researchers have been trying to extend the results of the Euclidean Fourier transform to quaternion‐valued signals using special techniques to overcome these two problems. In this context, we intend to generalize the Heisenberg uncertainty principles associated with covariance and Hardy's uncertainty principle for octonion multivector valued signals over using the polar form of an octonion, the quaternion decomposition, and the relationship between the OFT and the three‐dimensional (3D)‐Clifford‐Fourier transform.

Covariant Space-Time Line Elements in the Friedmann–Lemaitre–Robertson–Walker Geometry

Most quantum gravity theories quantize space-time on the order of Planck length (ℓp ). Some of these theories, such as loop quantum gravity (LQG), predict that this discreetness could be manifested through Lorentz invariance violations (LIV) over travelling particles at astronomical length distances. However, reports on LIV are controversial, and space discreetness could still be compatible with Lorentz invariance. Here, it is tested whether space quantization on the order of Planck length could still be compatible with Lorentz invariance through the application of a covariant geometric uncertainty principle (GeUP) as a constraint over geodesics in FRW geometries. Space-time line elements compatible with the uncertainty principle are calculated for a homogeneous, isotropic expanding Universe represented by the Friedmann–Lemaitre–Robertson–Walker solution to General Relativity (FLRW or FRW metric). A generic expression for the quadratic proper space-time line element is derived, proportional to Planck length-squared, and dependent on two contributions. The first is associated to the energy–time uncertainty, and the second depends on the Hubble function. The results are in agreement with space-time quantization on the expected length orders, according to quantum gravity theories, and within experimental constraints on putative LIV.

Unruh Effect for Mixed Neutrinos and the KMS Condition

The quantization of mixed (neutrino) fields in an accelerated background reveals a non-thermal nature for Unruh radiation, which can be fitted by a Tsallis-like distribution function. However, for relativistic flavor neutrinos, which are represented by the standard Pontecorvo states, such a correction turns out to be negligible and thermality is restored. We show that the usage of Pontecorvo states for the calculation of the decay rate of an accelerated proton in the laboratory and comoving frames leads to consistent results and correctly implements the KMS thermal condition. Thus, the employment of these states in the above framework is not at odds with the principle of general covariance, in contrast to recent claims in the literature.

A multisymplectic Lie algebra variational integrator for flexible multibody dynamics on the special Euclidean group SE (3)

A multisymplectic Lie algebra variational integrator (MLAVI) for simulating the dynamics of a flexible multibody system is proposed. The flexible bodies are described by the geometrically exact beam elements established on the special Euclidean Lie group SE(3). A lemma is also provided to demonstrate that the used discrete compatibility equation can make temporal and spatial interpolation independent. Then, the reduced Lagrangian density on the Lie algebra is derived, and the discrete dynamics of motions are established by the discretization of the covariant variational principle. Furthermore, another novel closed-form equality relation lemma on the right tangent maps of the associated exponential map and Cayley map is originally proved, which greatly helps to simplify the Jacobi matrix of the proposed MLAVI in the Newton iteration process. The accuracy and efficiency of the proposed MLAVI are validated by six static and dynamic numerical examples. Numerical results show that the proposed MLAVI can preserve the system's symplectic structure, momentum, and energy for long-time simulation.

Spatial covariance analysis reveals the residue-by-residue thermodynamic contribution of variation to the CFTR fold

Although the impact of genome variation on the thermodynamic properties of function on the protein fold has been studied in vitro, it remains a challenge to assign these relationships across the entire polypeptide sequence in vivo. Using the Gaussian process regression based principle of Spatial CoVariance, we globally assign on a residue-by-residue basis the biological thermodynamic properties that contribute to the functional fold of CFTR in the cell. We demonstrate the existence of a thermodynamically sensitive region of the CFTR fold involving the interface between NBD1 and ICL4 that contributes to its export from endoplasmic reticulum. At the cell surface a new set of residues contribute uniquely to the management of channel function. These results support a general ‘quality assurance’ view of global protein fold management as an SCV principle describing the differential pre- and post-ER residue interactions contributing to compartmentalization of the energetics of the protein fold for function. Our results set the stage for future analyses of the quality systems managing protein sequence-to-function-to-structure broadly encompassing genome design leading to protein function in complex cellular relationships responsible for diversity and fitness in biology in response to the environment.

Quantum superposition of spacetimes obeys Einstein's equivalence principle

We challenge the view that there is a basic conflict between the fundamental principles of Quantum Theory and General Relativity and, in particular, the fact that a superposition of massive bodies would lead to a violation of the Equivalence Principle. It has been argued that this violation implies that such a superposition must inevitably spontaneously collapse (like in the Diósi–Penrose model). We identify the origin of such an assertion in the impossibility of finding a local and classical reference frame in which Einstein's Equivalence Principle would hold. In contrast, we argue that the formulation of the Equivalence Principle can be generalized so that it holds for reference frames that are associated with quantum systems in a superposition of spacetimes. The core of this new formulation is the introduction of a quantum diffeomorphism to such Quantum Reference Frames. This procedure reconciles the principle of linear superposition in Quantum Theory with the principle of general covariance and the Equivalence Principle of General Relativity. Hence, it is not necessary to invoke a gravity-induced spontaneous state reduction when a massive body is prepared in a spatial superposition.

Observer-invariant time derivatives on moving surfaces

Observer-invariance is regarded as a minimum requirement for an appropriate definition of time derivatives. We derive various time derivatives systematically from a spacetime setting, where observer-invariance is a special case of a covariance principle and covered by Ricci-calculus. The analysis is considered for tangential n-tensor fields on moving surfaces and provides formulations which are applicable for numerical computations. For various special cases, e.g., vector fields (n=1) and symmetric and trace-less tensor fields (n=2) we compare material and convected derivatives and demonstrate the different underlying physics.

Wave Functional of the Universe and Time

A version of the quantum theory of gravity based on the concept of the wave functional of the universe is proposed. To determine the physical wave functional, the quantum principle of least action is formulated as a secular equation for the corresponding action operator. Its solution, the wave functional, is an invariant of general covariant transformations of spacetime. In the new formulation, the history of the evolution of the universe is described in terms of coordinate time together with arbitrary lapse and shift functions, which makes this description close to the formulation of the principle of general covariance in the classical theory of Einstein’s gravity. In the new formulation of quantum theory, an invariant parameter of the evolutionary time of the universe is defined, which is a generalization of the classical geodesic time measured by a standard clock along time-like geodesics.

Constraints on General Relativity Geodesics by a Covariant Geometric Uncertainty Principle

The classical uncertainty principle inequalities are imposed over the general relativity geodesic equation as a mathematical constraint. In this way, the uncertainty principle is reformulated in terms of proper space–time length element, Planck length and a geodesic-derived scalar, leading to a geometric expression for the uncertainty principle (GeUP). This re-formulation confirms the need for a minimum length of space–time line element in the geodesic, which depends on a Lorentz-covariant geodesic-derived scalar. In agreement with quantum gravity theories, GeUP imposes a perturbation over the background Minkowski metric unrelated to classical gravity. When applied to the Schwarzschild metric, a geodesic exclusion zone is found around the singularity where uncertainty in space-time diverged to infinity.