Slow Motion Limit Research Articles

Revisit Birkhoff’s Theorem: The Post-Newtonian Metric of a Self-Gravitating and Collapsing Thin Spherical Shell

We calculate the metric of a self-gravitating and collapsing infinitely thin spherical shell in the weak-field and slow-motion limits, and we demonstrate that Birkhoff’s theorem is not consistent with the theory of the post-Newtonian approximation. More importantly, it is illustrated that performing a coordinate transformation in solving Einstein field equations may change the matter energy-momentum tensor, making the resultant solution not correspond to the original problem.

Time-dependent electron transfer and energy dissipation in condensed media.

We study a moving adsorbate interacting with a metal electrode immersed in a solvent using the time-dependent Newns-Anderson-Schmickler model Hamiltonian. We have adopted a semiclassical trajectory treatment of the adsorbate to discuss the electron and energy transfers that occur between the adsorbate and the electrode. Using Keldysh Green's function scheme, we found a non-adiabatically suppressed electron transfer caused by the motion of the adsorbate and coupling with bath phonons that model the solvent. The energy is thus dissipated into electron-hole pair excitations, which are hindered by interacting with the solvent modes and facilitated by the applied electrode potential. The average energy transfer rate is discussed in terms of the electron friction coefficient and given an analytical expression in the slow-motion limit.

Powering an autonomous clock with quantum electromechanics

We theoretically analyse an autonomous clock comprising a nanoelectromechanical system, which undergoes self-oscillations driven by electron tunnelling. The periodic mechanical motion behaves as the clockwork, similar to the swinging of a pendulum, while induced oscillations in the electrical current can be used to read out the ticks. We simulate the dynamics of the system in the quasi-adiabatic limit of slow mechanical motion, allowing us to infer statistical properties of the clock’s ticks from the current auto-correlation function. The distribution of individual ticks exhibits a tradeoff between accuracy, resolution, and dissipation, as expected from previous literature. Going beyond the distribution of individual ticks, we investigate how clock accuracy varies over different integration times by computing the Allan variance. We observe non-monotonic features in the Allan variance as a function of time and applied voltage, which can be explained by the presence of temporal correlations between ticks. These correlations are shown to yield a precision advantage for timekeeping over the timescales that the correlations persist. Our results illustrate the non-trivial features of the tick series produced by nanoscale clocks, and pave the way for experimental investigation of clock thermodynamics using nanoelectromechanical systems.

Pulsar Timing Array signature from oscillating metric perturbations due to ultra-light axion

Abstract A coherently oscillating ultra-light axion can behave as dark matter. In particular, its coherently oscillating pressure perturbations can source an oscillating scalar metric perturbation, with a characteristic oscillation frequency which is twice the axion Compton frequency. A candidate in the mass range 10(-24,-21) eV can provide a signal in the frequency range tested by current and future Pulsar Timing Array (PTA) programs. Involving the pressure perturbations in a highly nonlinear environment, such an analysis demands a relativistic and nonlinear treatment. Here, we provide a rigorous derivation of the effect assuming weak-gravity and slow-motion limit of Einstein's gravity in zero-shear gauge and show that dark matter's velocity potential determines the oscillation phase and frequency change. A monochromatic PTA signal correlated with the velocity field would confirm the prediction, for example, by cross-correlating the PTA results with the future local velocity flow measurements.

Khronometric Theories of Modified Newtonian Dynamics

In 2011 Blanchet and Marsat suggested a fully relativistic version of Milgrom's modified Newtonian dynamics in which the dynamical degrees of freedom consist of the spacetime metric and a foliation of spacetime, the khronon field. This theory is simpler than the alternative relativistic formulations. We show that the theory has a consistent nonrelativistic or slow-motion limit. Blanchet and Marsat showed that in the slow motion limit, the theory reproduces stationary solutions of modified Newtonian dynamics. We show that these solutions are stable to khronon perturbations in the low acceleration regime, for the cases of spherical, cylindrical, and planar symmetry. For nonstationary systems in the low acceleration regime, we show that the khronon field generally gives an order unity correction to the modified Newtonian dynamics. In particular, there will be an order unity correction to the MOND version of Kepler's third law, potentially relevant to ongoing efforts to test MOND using GAIA observations of wide binaries.

Weak field and slow motion limits in energy–momentum powered gravity

We explore the weak field and slow motion limits, Newtonian and Post-Newtonian limits, of the energy–momentum powered gravity (EMPG), viz., the energy–momentum squared gravity (EMSG) of the form f(TμνTμν)=α(TμνTμν)η with α and η being constants. We have shown that EMPG with η≥0 and general relativity (GR) are not distinguishable by local tests, say, the Solar System tests; as they lead to the same gravitational potential form, PPN parameters, and geodesics for the test particles. However, within the EMPG framework, Mast, the mass of an astrophysical object inferred from astronomical observations such as planetary orbits and deflection of light, corresponds to the effective mass Meff(α,η,M)=M+Mempg(α,η,M), M being the actual physical mass and Mempg being the modification due to EMPG. Accordingly, while in GR we simply have the relation Mast=M, in EMPG we have Mast=M+Mempg. Within the framework of EMPG, if there is information about the values of {α,η} pair or M from other independent phenomena (from cosmological observations, structure of the astrophysical object, etc.), then in principle it is possible to infer not only Mast alone from astronomical observations, but M and Mempg separately. For a proper analysis within EMPG framework, it is necessary to describe the slow motion condition (also related to the Newtonian limit approximation) by |peff/ρeff|≪1 (where peff=p+pempg and ρeff=ρ+ρempg), whereas this condition leads to |p/ρ|≪1 in GR.

Parameterized post-Newtonian cosmology

Einstein’s theory of gravity has been extensively tested on solar system scales, and for isolated astrophysical systems, using the perturbative framework known as the parameterized post-Newtonian (PPN) formalism. This framework is designed for use in the weak-field and slow-motion limit of gravity, and can be used to constrain a large class of metric theories of gravity with data collected from the aforementioned systems. Given the potential of future surveys to probe cosmological scales to high precision, it is a topic of much contemporary interest to construct a similar framework to link Einstein’s theory of gravity and its alternatives to observations on cosmological scales. Our approach to this problem is to adapt and extend the existing PPN formalism for use in cosmology. We derive a set of equations that use the same parameters to consistently model both weak fields and cosmology. This allows us to parameterize a large class of modified theories of gravity and dark energy models on cosmological scales, using just four functions of time. These four functions can be directly linked to the background expansion of the universe, first-order cosmological perturbations, and the weak-field limit of the theory. They also reduce to the standard PPN parameters on solar system scales. We illustrate how dark energy models and scalar-tensor and vector-tensor theories of gravity fit into this framework, which we refer to as ‘parameterized post-Newtonian cosmology’ (PPNC).

Ngravs: Distinct gravitational interactions in gadget-2

We discuss an extension of the massively parallel cosmological simulation code gadget-2, which now enables investigation of multiple and distinct gravitational force laws, provided they are dominated by a constant scaling of the Newtonian force. In addition to simplifying investigations of a universally modified force law, the ngravs extension provides a foundation for state-of-the-art collisionless cosmological simulations of exotic gravitational scenarios. We briefly review the algorithms used by gadget-2, and present our extension to multiple gravities, highlighting additional features that facilitate consideration of exotic force laws. We discuss the accuracy and performance of the ngravs extension, both internally and with an unaltered gadget-2, in the relevant operational modes. The ngravs extension is publicly released to the research community. Program summaryProgram title: Gadget-2.0.7-ngravsCatalogue identifier: AFAV_v1_0Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AFAV_v1_0.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: GNU GPL v2No. of lines in distributed program, including test data, etc.: 36504No. of bytes in distributed program, including test data, etc.: 3919508Distribution format: tar.gzProgramming language: C.Computer: Commodity.Operating system: Unix.RAM: 256MB+Classification: 1.9, 4.12.External routines: GSL, FFTW3, HDF5 (optional)Nature of problem:N-body cosmological codes are traditionally designed to investigate a single gravitating species interacting via the Newtonian force law. There exist viable extensions to General Relativity [1], however, which predict weakfield, slow-motion limits featuring distinct gravitational force laws between distinct particle species. To enable investigation and constraint of these theories with available astrophysical data, a necessary first step is to extend an N-body simulator to handle distinct gravitating species.Solution method:The massively parallel Barnes–Hut tree, Fast Fourier Transform, and sorting routines of the versatile and well-vetted N-body simulator [2] gadget-2 were extended to support D distinct gravitationally interacting species. The tree implementation now vectorizes over each species’ monopole masses and positions, the Fourier routines now handle active and passive gravitational masses separately, and the sorting routines now group all particle data by type. An additional file was introduced allowing the user to specify all D2 gravitational interactions: real space, Fourier space, and lattice summation corrections. To improve monopole approximations in scenarios where the scale of the gravitational interaction depends on the mass itself, an optional tracking of the number of bodies contributing to any particular monopole approximation has been written.Restrictions:Force laws with mass-dependent scale lengths are not amenable to the implemented Fourier methods (or even the traditional [3] Fourier approach). Nodes containing highly heterogeneous collections of particles with different mass-dependent scale lengths may not be well-approximated, even with the additional tracking introduced. The collisional “gas” species can only interact via a single gravitational force law.Unusual features:The extension allows consideration of quite general and exotic interactions between bodies, and will serve as a common platform for model-dependent adjustments to the cosmological background evolution.Additional comments:Data file format is identical to that of gadget-2. Configuration file format is unchanged, save for the addition of required bindings between particle species and gravitational type.Running time:Typical running times are ≤2Dx those of gadget-2, where D is an integer between 1 and 6.

Spherically symmetric solution off(R,G)gravity at low energy

The weak-field and slow-motion limit of $f(R,\mathcal{G})$ gravity is developed up to $(v/c)^{4}$ order in a spherically symmetric background. Considering the Taylor expansion of a general function $f$ around vanishing values of $R$ and $\mathcal{G}$, we present general vacuum solutions up to $(v/c)^{4}$ order for the gravitational field generated by a ball-like source. The spatial behaviors at $(v/c)^{2}$ order are the same for $f(R,\mathcal{G})$ gravity and $f(R)$ gravity, and their corresponding real valued static behaviors are presented and compared with the one in general relativity. The static Yukawa-like behavior is proved to be compatible with the previous result of the most general fourth-order theory. At $(v/c)^{4}$ order, the static corrections to the Yukawa-like behavior for $f(R,\mathcal{G})$ gravity, $f(R)$ gravity, and the Starobinsky gravity are presented and compared with the one in general relativity.

Testing the gravitational interaction in the field of the Earth via satellite laser ranging and the Laser Ranged Satellites Experiment (LARASE)

In this work, the Laser Ranged Satellites Experiment (LARASE) is presented. This is a research program that aims to perform new refined tests and measurements of gravitation in the field of the Earth in the weak field and slow motion (WFSM) limit of general relativity (GR). For this objective we use the free available data relative to geodetic passive satellite lasers tracked from a network of ground stations by means of the satellite laser ranging (SLR) technique. After a brief introduction to GR and its WFSM limit, which aims to contextualize the physical background of the tests and measurements that LARASE will carry out, we focus on the current limits of validation of GR and on current constraints on the alternative theories of gravity that have been obtained with the precise SLR measurements of the two LAGEOS satellites performed so far. Afterward, we present the scientific goals of LARASE in terms of upcoming measurements and tests of relativistic physics. Finally, we introduce our activities and we give a number of new results regarding the improvements to the modelling of both gravitational and non-gravitational perturbations to the orbit of the satellites. These activities are a needed prerequisite to improve the forthcoming new measurements of gravitation. An innovation with respect to the past is the specialization of the models to the LARES satellite, especially for what concerns the modelling of its spin evolution, the neutral drag perturbation and the impact of Earth's solid tides on the satellite orbit.

Constraints on Covariant Horava-Lifshitz Gravity from frame-dragging experiment

The effects of Horava-Lifshitz corrections to the gravito-magnetic field are analyzed. Solutions in the weak field, slow motion limit, referring to the motion of a satellite around the Earth are considered. The post-newtonian paradigm is used to evaluate constraints on the Horava-Lifshitz parameter space from current satellite and terrestrial experiments data. In particular, we focus on GRAVITY PROBE B, LAGEOS and the more recent LARES mission, as well as a forthcoming terrestrial project, GINGER.

Deviation of quadrupolar bodies from geodesic motion in a Kerr spacetime

The deviation from geodesic motion of the world line of an extended body endowed with multipolar structure up to the mass quadrupole moment is studied in the Kerr background according to the Mathisson-Papapetrou-Dixon model. The properties of the quadrupole tensor are clarified by identifying the relevant components which enter the equations of motion, leading to the definition of an effective quadrupole tensor sharing its own algebraic symmetries, but also obeying those implied by the Mathisson-Papapetrou-Dixon model itself. The equations of motion are then solved analytically in the limit of small values of the characteristic length scales associated with the spin and quadrupole variables in comparison with the one associated with the background curvature and under special assumptions on the body's structure and motion. The resulting quasicircular orbit is parametrized in a Keplerian-like form, so that temporal, radial, and azimuthal eccentricities as well as semimajor axis, period, and periastron advance are explicitly computed and expressed in terms of gauge-invariant variables in the weak field and slow motion limit. A companion numerical study of the equations of motion is performed too.

Mathematical Properties of the Moments of Truncated Dynamic Magnetic Resonance Spectra and Their Potential Application

AbstractThe moments of truncated dynamic magnetic resonance spectra, Mn(L), are expanded in terms of power series of the integration range −L to +L. The expansion consists of three contributions: 1) an L‐independent term that, for the first three moments (n=1 to 3), is independent of the motion and equals the corresponding moment, Mn, of the rigid powder spectrum; 2) a limited number of (positive) terms, diverging as Lk (k<n, odd), reflecting the broadening effect due to motion; these terms vanish for the first three moments and become nonzero, with motion‐dependent coefficients, only from the fourth moment on; and 3) an infinite series of converging (negative) terms, in powers of 1/Lk (k is odd), reflecting the reduction of the moments due to the truncation of the spectra; these terms are motion dependent for all moments. The convergence properties of this series are discussed and expressions for the lower (truncated) moments in the slow and fast motion limits of a secular Hamiltonian are derived. For the slow motion limit, it is shown how the L‐dependence of the moments can be used to estimate the magnetic and dynamic parameters. The procedure is demonstrated using computer‐simulated spectra. In the fast motion regime, closed expressions are obtained in a similar form to those of the relaxation equations. The effect of natural line width and strong‐collision dynamics on the various moments as well as that of nonsecular terms in the Hamiltonian are also briefly discussed.

WIDE BINARY EFFECTS ON ASYMMETRIES IN ASYMPTOTIC GIANT BRANCH CIRCUMSTELLAR ENVELOPES

Observations of increasingly higher spatial resolution reveal the existence of asymmetries in the circumstellar envelopes of a small fraction of asymptotic giant branch (AGB) stars. Although there is no general consensus for their origin, a binary companion star may be responsible. Within this framework, we investigate the gravitational effects associated with a sufficiently wide binary system, where Roche lobe overflow is unimportant, on the outflowing envelopes of AGB stars using three dimensional hydrodynamic simulations. The effects due to individual binary components are separately studied, enabling investigation of the stellar and circumstellar characteristics in detail. The reflex motion of the AGB star alters the wind velocity distribution, thereby, determining the overall shape of the outflowing envelope. On the other hand, the interaction of the companion with the envelope produces a gravitational wake, which exhibits a vertically thinner shape. The two patterns overlap and form clumpy structures. To illustrate the diversity of shapes, we present the numerical results as a function of inclination angle. Not only is spiral structure produced by the binary interaction, but arc patterns are also found that represent the former structure when viewed at different inclinations. The arcs reveal a systematic shift of their centers of curvature for cases when the orbital speed of the AGB star is comparable to its wind speed. They take on the shape of a peanut for inclinations nearly edge-on. In the limit of slow orbital motion of the AGB star relative to the wind speed, the arc pattern becomes nearly spherically symmetric. We find that the aspect ratio of the overall oblate shape of the pattern is an important diagnostic probe of the binary as it can be used to constrain the orbital velocity of the AGB star, and moreover the binary mass ratio.

Quantum noise in the mirror–field system: A field theoretic approach

We revisit the quantum noise problem in the mirror–field system by a field-theoretic approach. Here a perfectly reflecting mirror is illuminated by a single-mode coherent state of the massless scalar field. The associated radiation pressure is described by a surface integral of the stress-tensor of the field. The read-out field is measured by a monopole detector, from which the effective distance between the detector and mirror can be obtained. In the slow-motion limit of the mirror, this field-theoretic approach allows to identify various sources of quantum noise that all in all leads to uncertainty of the read-out measurement. In addition to well-known sources from shot noise and radiation pressure fluctuations, a new source of noise is found from field fluctuations modified by the mirror’s displacement. Correlation between different sources of noise can be established in the read-out measurement as the consequence of interference between the incident field and the field reflected off the mirror. In the case of negative correlation, we found that the uncertainty can be lowered than the value predicted by the standard quantum limit. Since the particle-number approach is often used in quantum optics, we compared results obtained by both approaches and examine its validity. We also derive a Langevin equation that describes the stochastic dynamics of the mirror. The underlying fluctuation–dissipation relation is briefly mentioned. Finally we discuss the backreaction induced by the radiation pressure. It will alter the mean displacement of the mirror, but we argue this backreaction can be ignored for a slowly moving mirror.

Isotropic reorientations of fullerene C70 triplet molecules in solid glassy matrices revealed by light-induced electron paramagnetic resonance

Continuous-wave X-band electron paramagnetic resonance (EPR) of fullerene C(70) molecules excited to a triplet state by continuous light illumination was studied in molecular glasses of o-terphenyl and cis/trans-decaline and in the glassy polymers polymethylmethacrylate (PMMA) and polystyrene (PS). Above ∼100 K, a distinct narrowing of EPR lineshape of the triplet was observed, which was very similar for all systems studied. EPR lineshape was simulated reasonably well within a framework of a simple model of random jumps, which implies that the C(70) molecule performs isotropic orientational motion by sudden jumps of arbitrary angles. In simulations, a single correlation time τ(c) was used, varying in the range of 10(-7)-10(-8) s. Near and below 100 K electron spin echo (ESE) signals were also obtained which were found to decay exponentially. Correlation times τ(c) obtained from simulation of the EPR spectra in the slow-motion limit (τ(c) close to 10(-7) s) turned out to be in good agreement with the phase memory times T(M) of the ESE decay, which additionally supports the employed simple model. The observed motional effects provide evidence that the nanostructure of the solid glassy media of different origins is soft enough to allow a large asymmetric C(70) molecule to reorient rapidly. Except for the EPR spectra of the triplet, in the center of the spectra, a small admixture of a narrow line was also observed; its possible nature is briefly discussed.

Experimental Verification of the Theory of Nuclear Quadrupole Relaxation in Liquids over the Entire Range of Molecular Tumbling Motion

Nuclearmagneticresonance(NMR)spectraofquadrupolarnuclei(I>1/2)in liquids often consist of broad resonances, making it difficult to obtain useful chemical information. The poor NMR spectral resolution commonly observed for quadrupolar nuclei isadirectconsequenceofnuclearquadrupolerelaxationprocesses.Althoughallkeyaspectsof nuclear quadrupole relaxation processes have been known for decades within the framework of the Redfield relaxation theory, direct experimental NMR relaxation data that cover a wide range of molecular motion in liquids for quadrupolar nuclei are generally lacking. Here we reportacompletesetofexperimentalnuclearquadrupolerelaxationdatathatareobtainedfor 17 O, a half-integer quadrupolar nucleus, over the entire range of molecular motion within the limit of the Redfield theory. Ageneral approachutilizing the quadrupole relaxation properties in the slow motion limit will be particularly beneficial for studies of quadrupolar nuclei in biomolecules of medium and large sizes.

Exact scaling relations in relativistic hydrodynamic turbulence

We consider the steady state statistics of turbulence in general classes of dissipative hydrodynamic equations, where the fluctuations are sustained by a random source concentrated at large scales. It is well known that in some particular cases, such as non-relativistic incompressible turbulence, a Kolmogorov-type exact scaling relation for a correlation function holds. We show that all such scaling relations follow from a general relation on the current-density correlation function. The derivation does not require an energy cascade picture and suggests that this traditional interpretation of the Kolmogorov relation for incompressible turbulence may be misleading. Using this we derive exact scaling results for compressible turbulence in relativistic hydrodynamics, which reduce in the slow motion limit to the Kolmogorov relation. We discuss the experimental implications of the results.

2 mm waveband saturation transfer electron paramagnetic resonance of conducting polymers

The 2 mm waveband (140 GHz) saturation transfer electron paramagnetic resonance (ST-EPR) spectroscopy has been employed to characterize the very slow microsecond to millisecond librational macromolecular dynamics of a wide range of conducting polymers. It is possible at this waveband to determine separately spin relaxation and dynamics affecting ST-EPR spectra. Higher microwave frequency provides substantial increases sensitivity of the method to the anisotropic macromolecular motion in conducting polymers and broadens the interval of correlation times up to 1-80 ms, thereby extending the slow-motion limit for ST-EPR by two orders of magnitude compared with convenient wavebands EPR.

The 140-GHz (D-Band) Saturation Transfer Electron Paramagnetic Resonance Studies of Macromolecular Dynamics in Conducting Polymers

The high-field/high-frequency (5 T, 140 GHz) saturation transfer electron paramagnetic resonance (ST-EPR) method was used for the study of superslow librational macromolecular dynamics in various conducting polymers. It was shown that the increase of the electron precession frequency allows separate determination of spin relaxation and dynamics differently affecting effective ST-EPR spectra. Higher microwave frequency increases significantly the spectral resolution of the method and its sensitivity to the anisotropic macromolecular motion in conducting polymers. This broadens the interval of correlation time measured, thereby extending the slow-motion limit for ST-EPR by at least 2 orders of magnitude compared with convenient waveband EPR.