Mathisson-Papapetrou Equations Research Articles

General-relativistic wave–particle duality with torsion

We propose that the four-velocity of a Dirac particle is related to its relativistic wave function by ui=ψˉγiψ/ψˉψ . This relativistic wave–particle duality relation is demonstrated for a free particle related to a plane wave in a flat spacetime. For a curved spacetime with torsion, the momentum four-vector of a spinor is related to a generator of translation, given by a covariant derivative. The spin angular momentum four-tensor of a spinor is related to a generator of rotation in the Lorentz group. We use the covariant conservation laws for the spin and energy–momentum tensors for a spinor field in the presence of the Einstein–Cartan torsion to show that if the wave satisfies the curved Dirac equation, then the four-velocity, four-momentum, and spin satisfy the classical Mathisson–Papapetrou equations of motion. We show that these equations reduce to the geodesic equation. Consequently, the motion of a particle guided by the four-velocity in the pilot-wave quantum mechanics coincides with the geodesic motion determined by spacetime. We also show how the duality and the operator form of the Mathisson–Papapetrou equations arise from the covariant Heisenberg equation of motion in the presence of torsion.

Semiclassical analysis of Dirac fields on curved spacetime

We present a semiclassical analysis for Dirac fields on an arbitrary spacetime background and in the presence of a fixed electromagnetic field. Our approach is based on a Wentzel-Kramers-Brillouin approximation, and the results are analyzed at leading and next-to-leading order in the small expansion parameter $\ensuremath{\hbar}$. Taking into account the spin-orbit coupling between the internal and external degrees of freedom of wave packets, we derive effective ray equations with spin-dependent terms. These equations describe the gravitational spin Hall effect of localized Dirac wave packets. We treat both massive and massless Dirac fields and show how a covariantly defined Berry connection and the associated Berry curvature govern the semiclassical dynamics. The gravitational spin Hall equations are shown to be particular cases of the Mathisson-Papapetrou equations for spinning objects.

Spin Hall effects and the localization of massless spinning particles

The spin Hall effects of light represent a diverse class of polarization-dependent physical phenomena involving the dynamics of electromagnetic wave packets. In a medium with an inhomogeneous refractive index, wave packets can be effectively described by massless spinning particles following polarization-dependent trajectories. Similarly, in curved spacetime the gravitational spin Hall effect of light is represented by polarization-dependent deviations from null geodesics. In this paper, we analyze the equations of motion describing the gravitational spin Hall effect of light. We show that these equations are a special case of the Mathisson-Papapetrou equations for spinning objects in general relativity. This allows us to use several known results for the Mathisson-Papapetrou equations, and apply them to the study of electromagnetic wave packets. We derive conservation laws, we discuss the limits of validity of the spin Hall equations, and we study how the energy centroids of wave packets, effectively described as massless spinning particles, depend on the external choice of a timelike vector field, representing a family of observers. In flat spacetime, the relativistic Hall effect and the Wigner(-Souriau) translations are recovered, while our equations also provide a generalization of these effects in arbitrary spacetimes. We construct a large class of wave packets that can be described by the spin Hall equations, but also find its limits by giving examples of wave packets which are more general and are not described by the spin Hall equations. Lastly, we examine the assumption that electromagnetic wave packets are massless. While this is approximately true in many contexts, it is not exact. We show that failing to carefully account for the limitations of the massless approximation results in the appearance of unphysical ``centroids'' which are nowhere near the wave packet itself.

On the dynamics of spinning particle in nonlinear relativity

In order to study the dynamics of spinning particles in R-Minkowski space–time, first we have used the Bhabha–Corben model to describe how a spinning particle behave in a uniform electromagnetic field. Then, to extend the Mathisson–Papapetrou equations to R-Minkowski space–time, that correspond to de Sitter space–time given by a conformally flat metric, it was necessary to determine the Killing vectors, which allowed us to find the equations of motion that describe the dynamics of spinning particles.

Resonant orbits for a spinning particle in Kerr spacetime

In the present article, we study the orbital resonance corresponds to an extended object approximated up to the dipole order term in Kerr spacetime. We start with the Mathisson-Papapetrou equations under the linear spin approximation and primarily concentrate on two particular events. First, when the orbits are nearly circular and executing a small oscillation about the equatorial plane and second, a generic trajectory confined on the equatorial plane. While in the first case, all the three fundamental frequencies, namely, radial $\Omega_r$, angular $\Omega_{\theta}$, azimuthal $\Omega_{\phi}$ can be commensurate with each others and give rise to the resonance phenomenon, the later is only accompanied with the resonance between $\Omega_r$ and $\Omega_{\phi}$ as we set $\theta=\pi/2$. We provide a detail derivation in locating the prograde resonant orbits in either of these cases and also study the role played by the spin of the black hole. The implications related to spin-spin interactions between the object and black hole are also demonstrated.

Relativistic center of mass in general relativity

The center of mass and spin for isolated sources of gravitational radiation that move at relativistic speeds are defined. As a first step, we also present these definitions in flat space. This contradicts some general wisdom given in textbooks claiming that such definitions are not covariant and thus, have no physical meaning. We then generalize the definitions to asymptotically flat spacetimes giving their equations of motion when gravitational radiation is emitted by the isolated sources. The resulting construction has some similarities with the Mathisson-Papapetrou equations which describes the motion of the particle in an external field. We analyze the relationship between the center of mass velocity and the Bondi linear momentum and show they are not proportional to each other. A similar situation happens between the total and intrinsic angular momentum when the Bondi momentum vanishes. We claim that extra terms should be added in other approaches to adequately describe the time evolution of isolated sources of gravitational radiation.

On Reaction of a Spinning Particle on the Spacetime Curvature

The reaction of a classical (nonquantum) spinning particle on the spacetime curvature according to the Mathisson–Papapetrou equations is analyzed. From the point of view of the observer comoving with the particle in Schwarzschild’s field, this reaction is a reaction on the gravitomagnetic components of the gravitational field. The values of these components significantly depend on the relativistic Lorentz factor calculated by the particle velocity relative to the Schwarzschild mass. As a result, the value of the spinning particle acceleration relative to the geodesic motion is proportional to the second power of the Lorentz factor. At the same time, the intensity of the electromagnetic radiation of a charged spinning particle is proportional to the fourth power of this factor. Some numerical estimates are presented.

Nonequatorial circular orbits of spinning particles in the Schwarzschild–de Sitter background

In this paper analytical solutions of the Mathisson-Papapetrou equations that describe nonequatorial circular orbits of a spinning particle in the Schwarzschild-de Sitter background are studied, and the role of the cosmological constant is emphasized. It is shown that generally speaking a highly relativistic velocity of the particle is a necessary condition of motion along this type of orbits, with an exception of orbits locating close to the position of the static equilibrium, where low velocities are possible as well. Depending on the correlation between the spin orientation of the particle and its orbital velocity some of the possible nonequatorial circular orbits exist due to the repulsive action on the particle caused by the spin-gravity coupling and the others are caused by the attractive action.Here values of the energy of the particle on the corresponding orbits are also analyzed.

Gravitational Magnus effect

It is well known that a spinning body moving in a fluid suffers a force orthogonal to its velocity and rotation axis --- it is called the Magnus effect. Recent simulations of spinning black holes and (indirect) theoretical predictions, suggest that a somewhat analogous effect may occur for purely gravitational phenomena. The magnitude and precise direction of this "gravitational Magnus effect" is still the subject of debate. Starting from the rigorous equations of motion for spinning bodies in General Relativity (Mathisson-Papapetrou equations), we show that indeed such an effect takes place and is a fundamental part of the spin-curvature force. The effect arises whenever there is a current of mass/energy, non-parallel to a body's spin. We compute the effect explicitly for some astrophysical systems of interest: a galactic dark matter halo, a black hole accretion disk, and the Friedmann-Lema\^itre-Robertson-Walker (FLRW) spacetime. It is seen to lead to secular orbital precessions potentially observable by future astrometric experiments and gravitational-wave detectors. Finally, we consider also the reciprocal problem: the "force" exerted by the body on the surrounding matter, and show that (from this perspective) the effect is due to the body's gravitomagnetic field. We compute it rigorously, showing the matching with its reciprocal, and clarifying common misconceptions in the literature regarding the action-reaction law in post-Newtonian gravity.

Periastron shift for a spinning test particle around naked singularities

In the present article, we investigate the Periastron precession for a spinning test particle moving in nearly circular orbits around naked singularities. We consider two well-known solutions that can produce a spacetime with naked singularity---(a) first, the Reissner-Nordstr\om metric, which is a static charged solution with spherical symmetry, and (b) second, the stationary, axisymmetric Kerr metric. For simplicity, we only consider the motion confined on the equatorial plane in both these cases and solve exactly the Mathisson-Papapetrou equations. In addition, we analytically compute the Periastron precession within the framework of linear spin approximation. The inclusion of the spin parameter modifies the results with nonspinning particles and also reflects some interesting properties of the naked geometries. Furthermore, we carried out a numerical approach without any assumptions to probe the large order spin values. The implication of the spin-curvature coupling in connection with the naked geometries is also discussed.

Stirring a black hole

We present novel asymptotically global AdS4 solutions, constructed by turning on a dipolar differential rotation at the conformal boundary. At fixed energy and boundary profile, we find two different geometries: a horizonless spacetime, and a deformed, hourglass shaped black hole with zero net angular momentum. Both solutions exist up to some maximum amplitudes of the boundary profile, and develop an ergoregion attached to the boundary before the maximum amplitude is reached. We show that both spacetimes develop hair as soon as the ergoregion develops. Furthermore, we discuss the full phase diagram, including the possibility of phases with disconnected horizons, by considering the Mathisson-Papapetrou equations for a spinning test particle. Finally, we provide a first principle derivation of the superradiant bound purely from CFT data, and outline possible scenarios for the late time evolution of the system.

Highly relativistic spin-gravity- Λ coupling

The effects of highly relativistic spin-gravity coupling in the Schwarzschild-de Sitter background which follow from the Mathisson-Papapetrou equations are investigated. The dependence of gravitoelectric and gravitomagnetic components of gravitational field on the velocity of an observer which is moving in Schwarzschild-de Sitter's background is estimated. The action of gravitomagnetic components on a fast moving spinning particle is considered. Different cases of the highly relativistic circular orbits of a spinning particle which essentially differ from the corresponding geodesic orbits are described.

Antigravity: Spin-gravity coupling in action

The typical motions of a spinning test particle in Schwarzschild's background which show the strong repulsive action of the highly relativistic spin-gravity coupling are considered using the exact Mathisson-Papapetrou equations. An approximated approach to choice solutions of these equations which describe motions of the particle's proper center of mass is developed.

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.

Asymptotic gravitational wave fluxes from a spinning particle in circular equatorial orbits around a rotating black hole

We present a new computation of the asymptotic gravitational wave energy fluxes emitted by a {\it spinning} particle in circular equatorial orbits about a Kerr black hole. The particle dynamics is computed in the pole-dipole approximation, solving the Mathisson-Papapetrou equations with the Tulczyjew spin-supplementary-condition. The fluxes are computed, for the first time, by solving the 2+1 Teukolsky equation in the time-domain using hyperboloidal and horizon-penetrating coordinates. Denoting by $M$ the black hole mass and by $\mu$ the particle mass, we cover dimensionless background spins $a/M=(0,\pm0.9)$ and dimensionless particle spins $-0.9\leq S/\mu^2 \leq +0.9$. Our results span orbits of Boyer-Lindquist coordinate radii $4\leq r/M \leq 30$; notably, we investigate the strong-field regime, in some cases even beyond the last-stable-orbit. We confirm, numerically, the Tanaka {\it et al.} [Phys.\ Rev.\ D 54, 3762] 2.5th order accurate Post-Newtonian (PN) predictions for the gravitational wave fluxes of a spinning particle in circular equatorial orbit: we find an unambiguos trend of the PN-prediction towards the numerical results when $r$ is large. At $r/M=30$ the fractional agreement between the full numerical flux, approximated as the sum over the modes $m=1,2,3$, and the PN prediction is $\lesssim0.5\%$ in all cases tested. This is close to our fractional numerical accuracy ($\sim 0.2\%$). For smaller radii, the agreement between the 2.5PN prediction and the numerical result progressively deteriorates. Our numerical data will be essential to develop suitably resummed expressions of the 2.5PN analytical fluxes in order to improve their accuracy in the strong-field regime.

Comment on “Highly relativistic spin-gravity coupling for fermions”

We exhibit difficulties of different sorts which appear when using the Mathisson-Papapetrou equations, in particular in the description of highly relativistic particles presented in R. Plyatsko and M. Fenyk [Phys. Rev. D 91, 064033 (2015)]. We compare some results of this theory and of the aforementioned work with the ones obtained using a Lagrangian formulation for massive spinning particles and show that the issues mentioned in the preceding sentence do not appear in the Lagrangian treatment.

Reply to “Comment on ‘Highly relativistic spin-gravity coupling for fermions’ ”

In the context of results presented in [S. A. Hojman and F. A. Asenjo, Classical Quantum Gravity 30, 025008 (2013)], we discuss different formulations of equations which are used to describe motions of a spinning test body (particle) in general relativity. The specific features of the spin-gravity interaction for highly relativistic spinning particles in Schwarzschild's background according to the Mathisson-Papapetrou equations are stressed.

Highly relativistic spin-gravity coupling for fermions

Descriptions of highly relativistic fermions in a gravitational field in the classical (nonquantum) and quantum approaches are discussed. The results following from the Mathisson-Papapetrou equations for a fast spinning particle in Schwarzschild's and Kerr's background are considered. Numerical estimates for electron, proton and neutrino in the gravitational field of black holes are presented.The general relativistic Dirac equation is analyzed from the point of view it is using for the adequate description of highly relativistic fermions in a gravitational field, in the linear and nonlinear spin approximation. It is necessary to have some corrected Dirac equation for a highly relativistic fermion with strong spin-gravity coupling.

Investigating spinning test particles: Spin supplementary conditions and the Hamiltonian formalism

In this paper we report the results of a thorough numerical study of the motion of spinning particles in Kerr spacetime with different prescriptions. We first evaluate the Mathisson-Papapetrou equations with two different spin supplementary conditions, namely, the Tulczyjew and the Newton-Wigner, and make a comparison of these two cases. We then use the Hamiltonian formalism given by Barausse, Racine, and Buonanno in [Phys. Rev. D, 80, 104025 (2009)] to evolve the orbits and compare them with the corresponding orbits provided by the Mathisson-Papapetrou equations. We include a full description of how to treat the issues arising in the numerical implementation.

Mathisson–Papapetrou Equations as Conditions for the Compatibility of General Relativity and Continuum Physics

In continuum physics is presupposed that general-relativistic balance equations are valid which are created from the Lorentz-covariant ones by application of the equivalence principle. Consequently, the question arises, how to make these general-covariant balances compatible with Einstein’s field equations, if no Lagrangian is available. The compatibility conditions are derived by performing a modified Belinfante–Rosenfeld symmetrization for the non-symmetric and not divergence-free general-relativistic energy-momentum tensor of continuum physics. The procedure results in the Mathisson–Papapetrou equations.