Spinning Body Research Articles

Spin precession in a black hole and naked singularity spacetimes

We propose here a specific criterion to address the existence or otherwise of Kerr naked singularities, in terms of the precession of the spin of a test gyroscope due to the frame dragging by the central spinning body. We show that there is indeed an important characteristic difference in the behavior of gyro spin precession frequency in the limit of approach to these compact objects, and this can be used, in principle, to differentiate the naked singularity from black hole. Specifically, if gyroscopes are fixed all along the polar axis upto the horizon of a Kerr black hole, the precession frequency becomes arbitrarily high, blowing up as the event horizon is approached. On the other hand, in the case of naked singularity, this frequency remains always finite and well-behaved. Interestingly, this behavior is intimately related to and is governed by the geometry of the ergoregion in each of these cases which we analyze here. One intriguing behavior that emerges is, in the Kerr naked singularity case, the Lense-Thirring precession frequency ($\Omega_{\rm LT}$) of the gyroscope due to frame-dragging effect decreases as ($\Omega_{\rm LT}\propto r$) after reaching a maximum, in the limit of $r= 0$, as opposed to $r^{-3}$ dependence in all other known astrophysical cases.

Electromagnetic analysis in magnetic suspension mechanism of a wind tunnel for spinning body

A wind tunnel system for a spinning body has been designed for the measurement of the hydrodynamic forces acting on the body. The body was successfully suspended and rotated by electromagnets. This existing arrangement is required to be enlarged to observe more accurate hydrodynamic forces acting o n the body. Therefore, for the further development and control, it is important to understand the exact magnetic flux distribution inside the system as well as the variation of the force with the increase of the current. In this paper, both 2D symmetric model and 3D model of the arrangement are taken in consideration for the analysis. The magnetic flux distributions as well as magnetic force acting on the suspended object are obtained. The magnetic flux densities are analyzed for the variation of current in electromagnets. Actual force acting on the body is also measured. It is observed that the experimental results support the results obtained by numerical analysis.

Spinning test body orbiting around a Schwarzschild black hole: Circular dynamics and gravitational-wave fluxes

We consider a spinning test-body in circular motion around a nonrotating black hole and analyze different prescriptions for the body's dynamics. We compare, for the first time, the Mathisson-Papapetrou formalism under the Tulczyjew spin-supplementary-condition (SSC), the Pirani SSC and the Ohashi-Kyrian-Semerak SSC, and the spinning particle limit of the effective-one-body Hamiltonian of [Phys.~Rev.~D.90,~044018(2014)]. We analyze the four different dynamics in terms of the ISCO shifts and in terms of the coordinate invariant binding energies, separating higher-order spin contributions from spin-orbit contributions. The asymptotic gravitational wave fluxes produced by the spinning body are computed by solving the inhomogeneous $(2+1)D$ Teukolsky equation and contrasted for the different cases. For small orbital frequencies $\Omega$, all the prescriptions reduce to the same dynamics and the same radiation fluxes. For large frequencies, ${x \equiv (M \Omega)^{2/3} >0.1 }$, where $M$ is the black hole mass, and especially for positive spins (aligned with orbital angular momentum) a significant disagreement between the different dynamics is observed. The ISCO shifts can differ up to a factor two for large positive spins; for the Ohashi-Kyrian-Semerak and the Pirani SSC the ISCO diverges around dimensionless spins $\sim0.52$ and $\sim0.94$ respectively. In the spin-orbit part of the energetics the deviation from the Hamiltonian dynamics is largest for the Ohashi-Kyrian-Semerak SSC; it exceeds $10\%$ for $x>0.17$. The Tulczyjew and the Pirani SSCs behave compatible across almost the whole spin and frequency range. Our results will have direct application in including spin effects to effective-one-body waveform models for circularized binaries in the extreme-mass-ratio limit.

Spinning bodies in general relativity

A covariant Hamiltonian formalism for the dynamics of compact spinning bodies in curved space-time in the test-particle limit is described. The construction allows a large class of Hamiltonians accounting for specific properties and interactions of spinning bodies. The dynamics for a minimal and a specific non-minimal Hamiltonian is discussed. An independent derivation of the equations of motion from an appropriate energy–momentum tensor is provided. It is shown how to derive constants of motion, both background-independent and background-dependent ones.

Post-Newtonian dynamics basing on Mathisson-Papapetrou equations with Corinaldesi-Papapetrou condition in Kerr spacetime

We derive the post-Newtonian dynamics for a spinning body with Corinaldesi-Papapetrou spin supplementary condition in Kerr spacetime. Both the equations of motion for the center of mass of the body and the spin evolution are obtained. For the non-relativistic case, our calculations show that the magnitude of spin measured in the rest frame of the body’s center of mass does not change with time, though the center of mass does not move along the geodesic. Moreover, we find that the effects of the spin-orbit and spin-spin couplings will be suppressed by the Lorentz factor when the body has a relativistic velocity.

Spinning bodies in curved spacetime

We study the motion of neutral and charged spinning bodies in curved space-time in the test-particle limit. We construct equations of motion using a closed covariant Poisson-Dirac bracket formulation which allows for different choices of the hamiltonian. We derive conditions for the existence of constants of motion and apply the formalism to the case of spherically symmetric space-times. We show that the periastron of a spinning body in a stable orbit in a Schwarzschild or Reissner-Nordstr{\o}m background not only precesses, but also varies radially. By analysing the stability conditions for circular motion we find the innermost stable circular orbit (ISCO) as a function of spin. It turns out that there is an absolute lower limit on the ISCOs for increasing prograde spin. Finally we establish that the equations of motion can also be derived from the Einstein equations using an appropriate energy-momentum tensor for spinning particles.

Spinning particles in vacuum spacetimes of different curvature types: Natural reference tetrads and massless particles

In a previous paper, we considered the motion of massive spinning test particles in the ``pole-dipole'' approximation, as described by the Mathisson-Papapetrou-Dixon (MPD) equations and examined its properties in dependence on the spin supplementary condition. We decomposed the equations in the orthonormal tetrad based on the timelike vector fixing the spin condition and on the corresponding spin, while representing the curvature in terms of the Weyl scalars obtained in the Newman-Penrose null tetrad naturally associated with the orthonormal one; the projections thus obtained did not contain the Weyl scalars ${\mathrm{\ensuremath{\Psi}}}_{0}$ and ${\mathrm{\ensuremath{\Psi}}}_{4}$. In the present paper, we choose the interpretation tetrad in a different way, attaching it to the tangent ${u}^{\ensuremath{\mu}}$ of the worldline representing the history of the spinning body. Actually two tetrads are suggested, both given ``intrinsically'' by the problem and each of them incompatible with one specific spin condition. The decomposition of the MPD equation, again supplemented by writing its right-hand side in terms of the Weyl scalars, is slightly less efficient than in the massive case, because ${u}^{\ensuremath{\mu}}$ cannot be freely chosen (in contrast to ${V}^{\ensuremath{\mu}}$) and so the ${u}^{\ensuremath{\mu}}$-based tetrad is less flexible. In the second part of this paper, a similar analysis is performed for massless spinning particles; in particular, a certain intrinsic interpretation tetrad is again found. The respective decomposition of the MPD equation of motion is considerably simpler than in the massive case, containing only ${\mathrm{\ensuremath{\Psi}}}_{1}$ and ${\mathrm{\ensuremath{\Psi}}}_{2}$ scalars and not the cosmological constant. An option to span the spin-bivector eigenplane, besides the worldline null tangent, by a main principal null direction of the Weyl tensor can lead to an even simpler result.

Homotopy Simulation of Nonlinear Unsteady Rotating Nanofluid Flow from a Spinning Body

The development of new applications of nanofluids in chemical engineering and other technologies has stimulated significant interest in computational simulations. Motivated by coating applications of nanomaterials, we investigate the transient nanofluid flow from a time-dependent spinning sphere using laminar boundary layer theory. The free stream velocity varies continuously with time. The unsteady conservations equations are normalized with appropriate similarity transformations and rendered into a ninth-order system of nonlinear coupled, multidegree ordinary differential equations. The transformed nonlinear boundary value problem is solved using the homotopy analysis method (HAM), a semicomputational procedure achieving fast convergence. Computations are verified with an Adomian decomposition method (ADM). The influence of acceleration parameter, rotational body force parameter, Brownian motion number, thermophoresis number, Lewis number, and Prandtl number on surface shear stress, heat, and mass (nanoparticle volume fraction) transfer rates is evaluated. The influence on boundary layer behavior is also investigated. HAM demonstrates excellent stability and leads to highly accurate solutions.

Tidal invariants along the worldline of an extended body in Kerr spacetime

An extended body orbiting a compact object undergoes tidal deformations by the background gravitational field. Tidal invariants built up with the Riemann tensor and their derivatives evaluated along the worldline of the body are essential tools to investigate both geometrical and physical properties of the tidal interaction. For example, one can determine the tidal potential in the neighborhood of the body by constructing a body-fixed frame, which requires Fermi-type coordinates attached to the body itself, the latter being in turn related to the spacetime metric and curvature along the considered worldline. Similarly, in an effective field theory description of extended bodies, finite size effects are taken into account by adding to the point mass action certain nonminimal couplings which involve integrals of tidal invariants along the orbit of the body. In both cases such a computation of tidal tensors is required. Here we consider the case of a spinning body also endowed with a nonvanishing quadrupole moment in a Kerr spacetime. The structure of the body is modeled by a multipolar expansion around the ``center-of-mass line'' according to the Mathisson-Papapetrou-Dixon model truncated at the quadrupolar order. The quadrupole tensor is assumed to be quadratic in spin, accounting for rotational deformations. The behavior of tidal invariants of both electric and magnetic type is discussed in terms of gauge-invariant quantities when the body is moving along a circular orbit as well as in the case of an arbitrary (equatorial) motion. The analysis is completed by examining the associated eigenvalues and eigenvectors of the tidal tensors. The limiting situation of the Schwarzschild solution is also explored both in the strong field regime and in the weak field limit.

An Angular Momentum and Energy Conserving Lie-Group Integration Scheme for Rigid Body Rotational Dynamics Originating From Störmer–Verlet Algorithm

The paper presents two novel second order conservative Lie-group geometric methods for integration of rigid body rotational dynamics. First proposed algorithm is a fully explicit scheme that exactly conserves spatial angular momentum of a free spinning body. The method is inspired by the Störmer–Verlet integration algorithm for solving ordinary differential equations (ODEs), which is also momentum conservative when dealing with ODEs in linear spaces but loses its conservative properties in a nonlinear regime, such as nonlinear SO(3) rotational group. Then, we proposed an algorithm that is an implicit integration scheme with a direct update in SO(3). The method is algorithmically designed to conserve exactly both of the two “main” motion integrals of a rotational rigid body, i.e., spatial angular momentum of a torque-free body as well as its kinetic energy. As it is shown in the paper, both methods also preserve Lagrangian top integrals of motion in a very good manner, and generally better than some of the most successful conservative schemes to which the proposed methods were compared within the presented numerical examples. The proposed schemes can be easily applied within the integration algorithms of the dynamics of general rigid body systems.

Magnetic fields of the spinning bodies

In this paper we show that the Thomas precession of the spinning bodies, which is in general case constrained in all rigid bodies, induces magnetic field of the spinning bodies. This is one of the main reasons for the magnetic field of the spinning bodies. The general formula for this magnetic field is deduced and if it is applied to the Earth, its magnetic field changes between 0.295 G at the equator and 0.59 G at the poles, assuming that the density inside the Earth is uniform.

Analytic determination of high-order post-Newtonian self-force contributions to gravitational spin precession

Continuing our analytic computation of the first-order self-force contribution to the ``geodetic'' spin precession frequency of a small spinning body orbiting a large (nonspinning) body, we provide the exact expressions of the 10 and 10.5 post-Newtonian terms. We also introduce a new approach to the analytic computation of self-force regularization parameters based on a WKB analysis of the radial and angular equations satisfied by the metric perturbations.

How small can an over-spinning body be in general relativity?

The angular momentum of the Kerr singularity should not be larger than a threshold value so that it is enclosed by an event horizon: The Kerr singularity with the angular momentum exceeding the threshold value is naked. This fact suggests that if the cosmic censorship exists in our Universe, an over-spinning body without releasing its angular momentum cannot collapse to spacetime singularities. A simple kinematical estimate of two particles approaching each other supports this expectation and suggests the existence of a minimum size of an over-spinning body. But this does not imply that the geometry near the naked singularity cannot appear. By analyzing initial data, i.e., a snapshot of a spinning body, we see that an over-spinning body may produce a geometry close to the Kerr naked singularity around itself at least as a transient configuration.

Two-body gravitational spin-orbit interaction at linear order in the mass ratio

We analytically compute, to linear order in the mass-ratio, the "geodetic" spin precession frequency of a small spinning body orbiting a large (non-spinning) body to the eight-and-a-half post-Newtonian order, thereby extending previous analytical knowledge which was limited to the third post-Newtonian level. These results are obtained applying analytical gravitational self-force theory to the first-derivative level generalization of Detweiler's gauge-invariant redshift variable. We compare our analytic results with strong-field numerical data recently obtained by S.~R.~Dolan et al. [Phys.\ Rev.\ D {\bf 89}, 064011 (2014)]. Our new, high-post-Newtonian-order results capture the strong-field features exhibited by the numerical data. We argue that the spin-precession will diverge as $\approx -0.14/(1-3y)$ as the light-ring is approached. We transcribe our kinematical spin-precession results into a corresponding improved analytic knowledge of one of the two (gauge-invariant) effective gyro-gravitomagnetic ratios characterizing spin-orbit couplings within the effective-one-body formalism. We provide simple, accurate analytic fits both for spin-precession and the effective gyro-gravitomagnetic ratio. The latter fit predicts that the linear-in-mass-ratio correction to the gyro-gravitomagnetic ratio changes sign before reaching the light-ring. This strong-field prediction might be important for improving the analytic modeling of coalescing spinning binaries.

Frenkel electron and a spinning body in a curved background

We develop a variational formulation of a particle with spin in a curved space-time background. The model is based on a singular Lagrangian which provides equations of motion, a fixed value of spin and Frenkel condition on spin-tensor. Comparing our equations with those of Papapetrou we conclude that the Frenkel electron in a gravitational field has the same behavior as a rotating body in the pole-dipole and leading-spin approximation. Due to constraints presented in the formulation, position space is endowed with a noncommutative structure induced by the spin of the particle. Therefore, the model provides a physically interesting example of a noncommutative particle in a curved background.

Gender- and hydration- associated differences in the physiological response to spinning.

There is scarce and inconsistent information about gender-related differences in the hydration of sports persons, as well as about the effects of hydration on performance, especially during indoor sports. To determine the physiological differences between genders during in indoor physical exercise, with and without hydration. 21 spinning sportspeople (12 men and 9 women) participated in three controlled, randomly assigned and non-sequential hydration protocols, including no fluid intake and hydration with plain water or a sports drink (volume adjusted to each individual every 15 min), during 90 min of spinning exercise. The response variables included body mass, body temperature, heart rate and blood pressure. During exercise without hydration, men and women lost ~2% of body mass, and showed higher body temperature (~0.2°C), blood pressure (~4 mmHg) and heart rate (~7 beats/min) compared to exercises with hydration. Body temperature and blood pressure were higher for men than for women during exercise without hydration, differences not observed during exercise with hydration. Between 42-99% of variance in body temperature, blood pressure and heart rate could be explained by the physical characteristics of subjects and the work done. During exercise with hydration (either with water or sport drink), the physiological response was similar for both genders. Exercise without hydration produced physical stress, which could be prevented with either of the fluids (plain water was sufficient). Gender differences in the physiological response to spinning (body temperature, mean blood pressure and heart rate) can be explained in part by the distinct physical characteristics of each individual.

Dynamics of spin effects of compact binaries

For a 10-dimensional post-Newtonian canonical conservative Hamiltonian system of spinning compact binaries, where the orbital part is accurate to the third-order post-Newtonian expansion and the spin-orbit contributions are up to the next-to-next-to-leading order, its dynamics is integrable and not at all chaotic due to the presence of five independent isolating integrals, including the total energy, three components of the total angular momentum vector and the length of the orbital angular momentum. As the spin-spin effects of the two spinning bodies are further included, only the length of the orbital angular momentum is no longer conserved, so that the dynamics becomes typically non-integrable. Numerical simulations support the onset of chaos. Above all, many chaotic orbits whose initial radii are larger than 10M and whose Lyapunov times are less than the corresponding inspiral decay times are found. In addition, a threshold value of the maximum ratio of the spin-spin Hamiltonian to the whole Hamiltonian for distinguishing between the ordered and chaotic cases is also given.

On the geometry of the space-time and motion of the spinning bodies

Abstract In this paper an alternative theory about space-time is given. First some preliminaries about 3-dimensional time and the reasons for its introduction are presented. Alongside the 3-dimensional space (S) the 3-dimensional space of spatial rotations (SR) is considered independently from the 3-dimensional space. Then it is given a model of the universe, based on the Lie groups of real and complex orthogonal 3 × 3 matrices in this 3+3+3-dimensional space. Special attention is dedicated for introduction and study of the space S × SR, which appears to be isomorphic to SO(3,ℝ) × SO(3,ℝ) or S 3 × S 3. The influence of the gravitational acceleration to the spinning bodies is considered. Some important applications of these results about spinning bodies are given, which naturally lead to violation of Newton’s third law in its classical formulation. The precession of the spinning axis is also considered.

Mathisson’s helical motions for a spinning particle: Are they unphysical?

It has been asserted in the literature that Mathisson's helical motions are unphysical, with the argument that their radius can be arbitrarily large. We revisit Mathisson's helical motions of a free spinning particle, and observe that such statement is unfounded. Their radius is finite and confined to the disk of centroids. We argue that the helical motions are perfectly valid and physically equivalent descriptions of the motion of a spinning body, the difference between them being the choice of the representative point of the particle, thus a gauge choice. We discuss the kinematical explanation of these motions, and we dynamically interpret them through the concept of hidden momentum. We also show that, contrary to previous claims, the frequency of the helical motions coincides, even in the relativistic limit, with the zitterbewegung frequency of the Dirac equation for the electron.

Spin-geodesic deviations in the Kerr spacetime

The dynamics of extended spinning bodies in the Kerr spacetime is investigated in the pole-dipole particle approximation and under the assumption that the spin-curvature force only slightly deviates the particle from a geodesic path. The spin parameter is thus assumed to be very small and the back reaction on the spacetime geometry neglected. This approach naturally leads to solve the Mathisson-Papapetrou-Dixon equations linearized in the spin variables as well as in the deviation vector, with the same initial conditions as for geodesic motion. General deviations from generic geodesic motion are studied, generalizing previous results limited to the very special case of an equatorial circular geodesic as the reference path.