Physical Degrees Of Freedom Research Articles

Canonical analysis of linearized [formula omitted] gravity plus a Chern–Simons term

The Hamiltonian analysis for the linearized λR gravity plus a Chern–Simons term is performed. The first-class and second-class constraints for arbitrary values of λ are presented, and one physical degree of freedom is reported. The second-class constraints are removed, and the corresponding generalized Dirac brackets are constructed; then, the difference between theories with different values of λ is remarked.

Gravitational waves on Kerr black holes: I. Reconstruction of linearized metric perturbations

Abstract The gravitational perturbations of a rotating Kerr black hole are notoriously complicated, even at the linear level. In 1973, Teukolsky showed that their physical degrees of freedom are encoded in two gauge-invariant Weyl curvature scalars that obey a separable wave equation. Determining these scalars is sufficient for many purposes, such as the computation of energy fluxes. However, some applications -- such as second-order perturbation theory -- require the reconstruction of metric perturbations. In principle, this problem was solved long ago, but in practice, the solution has never been worked out explicitly. Here, we do so by writing down the metric perturbation (in either ingoing or outgoing radiation gauge) that corresponds to a given mode of either Weyl scalar. Our formulas make no reference to the Hertz potential (an intermediate quantity that plays no fundamental role) and involve only the radial and angular Kerr modes, but not their derivatives, which can be altogether eliminated using the Teukolsky--Starobinsky identities. We expect these analytic results to prove useful in numerical studies and for extending black hole perturbation theory beyond the linear regime.

Faddeev–Jackiw Hamiltonian formulation for general exotic bi-gravity

General exotic bi-gravity, obtained in Ozkan et al. (Phys Rev Lett 123(3):031303, 2019), is a unitary parity-preserving model which describes two interacting spin-two fields in three-dimensional spacetime. Adopting a symplectic viewpoint, we investigate the dynamical structure of general exotic bi-gravity theory. In particular, by exploiting the properties of the corresponding pre-symplectic matrix and its associated zero-modes, we explicitly derive all constraints of the theory, including the integrability conditions and scalar relationships between all the parameters and fields defining the model. Then, as an application, these scalar relationships are used for studying the anti-de Sitter background. After that, we derive the gauge transformations for the dynamical variables from the structure of the remaining zero-modes, meaning that such zero-modes are indeed the generators of the gauge symmetry of the theory. Finally, by switching off one of the four coupling constants βn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\beta _{n}$$\\end{document} and assuming the invertibility of some of the dreibeins, we find that the general exotic bi-gravity theory has two physical degrees of freedom.

Unifying ordinary and null memory

Based on a recently proposed reinterpretation of gravitational wave memory that builds up on the definition of gravitational waves pioneered by Isaacson, we provide a unifying framework to derive both ordinary and null memory from a single well-defined equation at leading order in the asymptotic expansion. This allows us to formulate a memory equation that is valid for any unbound asymptotic energy-flux that preserves local Lorentz invariance. Using Horndeski gravity as a concrete example metric theory with an additional potentially massive scalar degree of freedom in the gravitational sector, the general memory formula is put into practice by presenting the first account of the memory correction sourced by the emission of massive field waves. Throughout the work, physical degrees of freedom are identified by constructing manifestly gauge invariant perturbation variables within an SVT decomposition on top of the asymptotic Minkowski background, which will in particular prove useful in future studies of gravitational wave memory within vector tensor theories.

Hamiltonian analysis for perturbative [formula omitted] gravity

The Hamiltonian analysis for the linearized λR gravity around the Minkowski background is performed. The first-class and second-class constraints for arbitrary values of λ are presented, and two physical degrees of freedom are reported. In addition, we remove the second-class constraints, and the generalized Dirac brackets are constructed; then, the equivalence between General Relativity and the λR theory is shown.

In-sensor human gait analysis with machine learning in a wearable microfabricated accelerometer

In-sensor computing could become a fundamentally new approach to the deployment of machine learning in small devices that must operate securely with limited energy resources, such as wearable medical devices and devices for the Internet of Things. Progress in this field has been slowed by the difficulty to find appropriate computing devices that operate using physical degrees of freedom that can be coupled directly to degrees of freedom that perform sensing. Here we leverage reservoir computing as a natural framework to do machine learning with the degrees of freedom of a physical system, to show that a micro-electromechanical system can implement computing and the sensing of accelerations by coupling the displacement of suspended microstructures. We present a complete wearable system that can be attached to the foot to identify the gait patterns of human subjects in real-time. The computing efficiency and the power consumption of this in-sensor computing system is then compared to a conventional system with a separate sensor and digital computer. For similar computing capabilities, a much better power efficiency can be expected for the highly-integrated in-sensor computing devices, thus providing a path for the ubiquitous deployment of machine learning in edge computing devices.

Interface reduction technique for Enhanced Craig-Bampton method

Substructure coupling and model order reduction using Component Mode Synthesis (CMS) have, over recent years, gained considerable attention in the vibroacoustic analysis of complex structures. In the CMS methodology, the interior dynamics of each subcomponent in a substructured system are represented by a truncated set of normal modes within the lower frequency range, while all physical degrees of freedom (DOFs) at the interface are retained. In cases where there are many interconnected subcomponents within a system, in particular when these components are finely discretised in the Finite Element (FE) domain, the reduced system matrices may still involve a significant number of equations. This, in turn, leads to a considerable computational workload. To address this issue, further reduction of the system matrices concerning the interface DOFs by using a set of truncated interface modes can be considered. However, the accuracy of the reduced matrices depends on the representation of the truncated dynamics in the reduction process. In this work, two interface reduction techniques are presented to truncate the interface dynamics of the Enhanced Craig-Bampton (ECB) equations of motion. The first technique is a classical interface reduction approach that assumes decoupled internal and interface dynamics. The second approach is an extension of the first one by incorporating an additional coupling term that accounts for interactions between the truncated internal and interface dynamics. The performance of each interface reduction technique is evaluated by applying them to three practical engineering examples. In these instances, resonance frequencies, associated errors, transfer functions, and normal modes are compared to those obtained using both the classical CB method and the full model.

Propagation of anisotropic gravitational and electromagnetic waves at very high energies

We analyze the dispersion relation for an anisotropic gravity-electromagnetic theory at very high energies. In particular for photons of very high energy. We start by introducing the anisotropic gravity-gauge vector field model. It is invariant under spacelike diffeomorphisms, time parametrization, and U(1) gauge transformations. It includes high-order spacelike derivatives as well as polynomial expressions of the Riemann and field strength tensor fields. It is based on the Hořava-Lifshitz anisotropic proposal. We show its consistency, and the stability of the Minkowski ground state. Finally, we determine the exact zone at which the physical degrees of freedom, i.e. the transverse-traceless tensorial degrees of freedom and the transverse vectorial degrees of freedom propagate according to a linear wave equation. This is so, in spite of the fact that there exists in the zone a non-trivial Newtonian background of the same order. The wave equation contains spatial derivatives up to the sixth order, in the lowest order it exactly matches the relativistic wave equation. We then analyze the dispersion relation at very high energies in the context of recent experimental data. The qualitative predictions of the proposed model, concerning the propagation of highly energetic photons, are different from the ones obtained from the modified dispersion relation of the LIV models.

The Origin of the Difference between Space and Time

All differences between the role of space and time in nature are explained by proposing principles in which none of the spacetime coordinates has an a priori special role. Spacetime is treated as a non-dynamical manifold, with a fixed global RD topology. The dynamical theory of gravity determines only the metric tensor on a fixed manifold. All dynamics is treated as a Cauchy problem, so it follows that one coordinate takes a special role. It is proposed that any boundary condition that is finite everywhere leads to a solution which is also finite everywhere. This explains the (1,D−1) signature of the metric, the boundedness of energy from below, the absence of tachyons, and other related properties of nature. The time arrow is explained by proposing that the boundary condition should be ordered. The quantization is considered as a boundary condition for field operators. Only the physical degrees of freedom are quantized.

Theoretical research on the transverse spin of structured optical fields inside a waveguide

Structured optical fields inside a waveguide possess the transverse spin, i.e., the spin angular momentum perpendicular to the direction of the waveguide. The physical origin of the transverse spin can be attributed to the presence of an effective rest mass of photons in guided waves, or equivalently, to the existence of a longitudinal field component, such that the transverse and longitudinal fields together form an elliptical polarization plane. In contrary to the traditional viewpoint, the transverse spin of photons in guided waves is also quantized, and its quantization form is related to the ellipticity of the polarization ellipse. The direction of the transverse spin depends on the propagation direction of electromagnetic waves along the waveguide, such a spin-momentum locking may have important applications in spin-dependent unidirectional optical interfaces. By means of a coupling between the transverse spin of guided waves and some physical degrees of freedom, one can develop an optical analogy of spintronics, i.e., spinoptics.

Residual gauge symmetry in light-cone electromagnetism

We analyze the residual gauge freedom in light-cone electromagnetism in four dimensions. The standard boundary conditions involved in the so-called lc2formalism, which contains only the two physical degrees of freedom, allow for a subset of residual gauge transformations. We relax the boundary conditions imposed on the fields in order to obtain all the residual gauge transformations. We compute the canonical generators for Poincaré and gauge transformations with these relaxed boundary conditions. This enables us to distinguish between the trivial (proper) and large (improper) gauge transformations in light-cone electromagnetism. We then employ the Newman-Penrose formalism to identify the incoming and outgoing radiation fields. We comment on the quadratic form structure of light-cone Hamiltonians, often encountered in lc2 gauge theories.

ADM formulation and Hamiltonian analysis of f(Q) gravity

$f(Q)$ gravity is an extension of the symmetric teleparallel equivalent to general relativity. We demonstrate the Hamiltonian analysis of $f(Q)$ gravity with fixing the coincident gauge condition. Using the standard Dirac-Bergmann algorithm, we show that $f(Q)$ gravity has 8 physical degrees of freedom. This result reflects that the diffeomorphism symmetry of $f(Q)$ gravity is completely broken due to the gauge fixing. Moreover, in terms of the perturbations, we discuss the possible mode decomposition of these degrees of freedom.

Implications of NICER for Neutron Star Matter: The QHC21 Equation of State

The recent NICER measurement of the radius of the neutron star PSR J0740+6620, and the inferred small variation of radii from 1.4 to 2.1 M ⊙, reveal key features of the equation of state of neutron star matter. The pressure rises rapidly in the regime of baryon density n ∼ 2–4 times nuclear saturation density, n 0—the region where we expect hadronic matter to be undergoing transformation into quark matter—and the pressure in the nuclear regime is greater than predicted by microscopic many-body variational calculations of nuclear matter. To incorporate these insights into the microscopic physics from the nuclear to the quark matter regimes, we construct an equation of state, QHC21, within the framework of quark–hadron crossover. We include nuclear matter results primarily based on the state-of-the-art chiral effective field theory, but also note results of using nuclear matter variational calculations based on empirical nuclear forces. We employ explicit nuclear degrees of freedom only up to n ∼ 1.5 n 0, in order to explore the possibility of further physical degrees of freedom than nucleonic here. The resulting QHC21, which has a peak in sound velocity in ∼2–4 n 0, is stiffer than the earlier QHC19 below 2 n 0, predicting larger radii in substantial agreement with the NICER data.

Questioning the question: Exploring how physical degrees of freedom are retrieved with neural networks

When studying a physical system, it is crucial to identify the degrees of freedom that characterize that system. Recently, specific neural networks have been designed to retrieve these underlying degrees of freedom automatically. Indeed, fed with data from a physical system, a variational autoencoder can learn a latent representation of that system that directly corresponds to its underlying degrees of freedom. However, the understanding of these neural networks is limited on two fronts. First, very little is known about the impact of the question vector, a key parameter in designing performant autoencoders. Second, there is the mystery of why the correct degrees of freedom are found in the latent representation, not an arbitrary function of these parameters. Both gaps in our understanding are addressed in this paper. To study the first question on the optimal design of the question vector, we investigate physical systems characterized by analytical expressions with a limited set of degrees of freedom. We empirically show how the type of question influences the learned latent representation. We find that the stochasticity of a random question is fundamental in learning physically meaningful representations. Furthermore, the dimensionality of the question vector should not be too large. To address the second question, we make use of a symmetry argument. We show that the learning of the degrees of freedom in the latent space is related to the symmetry group of the input data. This result holds for linear and nonlinear transformations of the degrees of freedom. In this way, in this paper, we contribute to the research on automated systems for discovery and knowledge creation.

Covariant BRST quantization of unimodular gravity. II. Formulation with a vector antighost

In our previous paper, we have presented a covariant BRST quantization of unimodular gravity which may account for the smallness of the cosmological constant, and have shown that the physical degrees of freedom in the theory are the same as general relativity. The formulation has been given by using rank-2 antisymmetric tensor fields for both ghosts and antighosts. Here we give an alternative formulation using a vector field for the antighost but keeping the same structure for the ghosts. This gives a significantly simpler covariant quantization with less ghosts and no tripole modes in the ghost sector. We show that this also gives only two physical transverse modes as in general relativity.

Canonical description of exotic general massive gravity

Exotic General Massive Gravity is the next-to-simplest gravitational theory fulfilling the so-called third-way consistency, the simplest being Minimal Massive Gravity. We investigate the canonical structure of the first-order formulation of Exotic General Massive Gravity. By using the Dirac Hamiltonian formalism, we systematically discover the complete set of physical constraints, including primary, secondary, and tertiary ones, and explicitly compute the Poisson bracket algebra between them. In particular, we demonstrate that the consistency condition for the tertiary constraints provides explicit expressions which can be solved algebraically for the auxiliary fields f and h in terms of the dreibein e. In this configuration, to confirm that the theory is ghost-free, the whole set of constraints is classified into first and second-class ones showing the existence of only two physical degrees of freedom corresponding to one massive graviton. Furthermore, we identify the transformation laws for all of the dynamical variables corresponding essentially to gauge symmetries, generated by the first-class constraints. Finally, by taking into account all the second-class constraints, we explicitly compute the Dirac matrix together with the Dirac’s brackets.

Very special linear gravity: A gauge-invariant graviton mass

Linearized gravity in the Very Special Relativity (VSR) framework is considered. We prove that this theory allows for a non-zero graviton mass mg without breaking gauge invariance nor modifying the relativistic dispersion relation. We find the analytic solution for the new equations of motion in our gauge choice, verifying as expected the existence of only two physical degrees of freedom. Finally, through the geodesic deviation equation, we confront some results for classic gravitational waves (GW) with the VSR ones: we see that the ratios between VSR effects and classical ones are proportional to (mg/E)2, E being the energy of a graviton in the GW. For GW detectable by the interferometers LIGO and VIRGO this ratio is at most 10−20. However, for GW in the lower frequency range of future detectors, like LISA, the ratio increases significantly to 10−10, that combined with the anisotropic nature of VSR phenomena may lead to observable effects.

Local spin base invariance from a global differential–geometrical point of view

This article gives a geometric interpretation of the spin base formulation with local spin base invariance of spinors on a curved space-time and, in particular, of a central element, the global Dirac structure, in terms of principal and vector bundles and their endomorphisms. It is shown that this is intimately related to Spin and SpinC structures in the sense that the existence of one of those implies the existence of a Dirac structure and allows for an extension to local spin base invariance. Vice versa, as a central result, the existence of a Dirac structure implies the existence of a SpinC structure. Nevertheless, the spin base invariant setting may be considered more general, allowing for more physical degrees of freedom. Furthermore, arguments are given that the Dirac structure is a more natural choice as a variable for (quantum) gravity than tetrads/vielbeins.

Insights through Modal Minimal Models for Analysis of Linear and Nonlinear Dynamic Problems

Dynamic analysis of three-dimensional structures is common practice in industry to optimize products and in research to gain insights on the influence of parameters. The complexity differs based on the linearity or nonlinearity of the underlying problem, the type of excitation, e.g., forced or self-excitation, and the number of degrees of freedom that need to be examined. Reduced order models and optimized numerical methods are used to optimize the time and computational power needed to gain a certain insight. This article focusses on a specific class of problems where the modes of the structure do not or do not significantly change through the (damping) device or force that is added to the structure. Herein, lumped mass models are commonly used for analysis of the dynamic response of the system. In the article, it is highlighted that lumped mass models can give quantitative insight but modally reduced models allow a direct optimization of the problems with respect to the physical degree of freedom that, for example, is subject to self-excitation or dampened. The benefit of modal minimal models and its limitations are shown and discussed for different linear and nonlinear dynamic problems.

A consistent two-skyrmion configuration space from instantons

To study a nuclear system in the Skyrme model one must first construct a space of low energy Skyrme configurations. However, there is no mathematical definition of this configuration space and there is not even consensus on its fundamental properties, such as its dimension. Here, we propose that the full instanton moduli space can be used to construct a consistent skyrmion configuration space, provided that the Skyrme model is coupled to a vector meson which we identify with the ρ-meson. Each instanton generates a unique skyrmion and we reinterpret the 8N instanton moduli as physical degrees of freedom in the Skyrme model. In this picture a single skyrmion has six zero modes and two non-zero modes: one controls the overall scale of the solution and one the energy of the ρ-meson field. We study the N = 1 and N = 2 systems in detail. Two interacting skyrmions can excite the ρ through scattering, suggesting that the ρ and Skyrme fields are intrinsically linked. Our proposal is the first consistent manifold description of the two-skyrmion configuration space. The method can also be generalised to higher N and thus provides a general framework to study any skyrmion configuration space.