Spinning Point Particle Research Articles

On the BTZ black hole and the spinning cosmic string

We review the Baňados–Teitelboim–Zanelli (BTZ) black hole solution in connection with the spinning string solution. We find a new exact solution, which can be related to the (2 + 1)-dimensional spinning point particle solution. There is no need for a cosmological constant, and so the solution can be up-lifted to (3 + 1) dimensions. The exact solution in a conformally invariant gravity model, where the space–time is written as $$g_{\mu \nu }=\omega ^2 {\tilde{g}}_{\mu \nu }$$ , is horizon free and has an ergo-circle, while $${\tilde{g}}_{\mu \nu }$$ is the BTZ solution. The dilaton $$\omega $$ determines the scale of the model. It is conjectured that the conformally invariant non-vacuum BTZ solution will solve the boundary and causality problems which one encounters in spinning cosmic string solutions.

Model independent tests of the Kerr bound with extreme mass ratio inspirals

An outstanding prediction of general relativity is the fact that the angular momentum S of an isolated black hole with mass μ is limited by the Kerr bound, S≤Gμ2/c. Testing this cornerstone is challenging due to the difficulty in modeling spinning compact objects that violate this bound. We argue that precise, model-independent tests can be achieved by measuring gravitational waves from an extreme mass ratio inspiral around a supermassive object, one of the main targets of the future LISA mission. In the extreme mass ratio limit, the dynamics of the small compact object depends only on its multipole moments, which are free parameters. At variance with the comparable-mass case, accurate waveforms are valid also when the spin of the small object greatly exceeds the Kerr bound. By computing the orbital dephasing and the gravitational-wave signal emitted by a spinning point particle in circular, nonprecessing, equatorial motion around a Kerr black hole, we estimate that LISA will be able to measure the spin of the small compact object at the level of 10%. Together with mass measurements, this will allow for theory-agnostic, unprecedented constraints on string-theory inspired objects such as “superspinars”, almost in their entire parameter space.

Geodesic structure of naked singularities in AdS3 spacetime

We present a complete study of the geodesics around naked singularities in AdS$_3$, the three-dimensional anti-de Sitter spacetime. These stationary spacetimes, characterized by two conserved charges --mass and angular momentum--, are obtained through identifications along spacelike Killing vectors with a fixed point. They are interpreted as massive spinning point particles, and can be viewed as three-dimensional analogues of cosmic strings in four spacetime dimensions. The geodesic equations are completely integrated and the solutions are expressed in terms of elementary functions. We classify different geodesics in terms of their radial bounds, which depend on the constants of motion. Null and spacelike geodesics approach the naked singularity from infinity and either fall into the singularity or wind around and go back to infinity, depending on the values of these constants, except for the extremal and massless cases for which a null geodesic could have a circular orbit. Timelike geodesics never escape to infinity and do not always fall into the singularity, namely, they can be permanently bounded between two radii. The spatial projections of the geodesics (orbits) exhibit self-intersections, whose number is particularly simple for null geodesics. As a particular application, we also compute the lengths of fixed-time spacelike geodesics of the static naked singularity using two different regularizations.

Quantum corrections for spinning particles in de Sitter

We compute the one-loop quantum corrections to the gravitational potentials of a spinning point particle in a de Sitter background, due to the vacuum polarisation induced by conformal fields in an effective field theory approach. We consider arbitrary conformal field theories, assuming only that the theory contains a large number N of fields in order to separate their contribution from the one induced by virtual gravitons. The corrections are described in a gauge-invariant way, classifying the induced metric perturbations around the de Sitter background according to their behaviour under transformations on equal-time hypersurfaces. There are six gauge-invariant modes: two scalar Bardeen potentials, one transverse vector and one transverse traceless tensor, of which one scalar and the vector couple to the spinning particle. The quantum corrections consist of three different parts: a generalisation of the flat-space correction, which is only significant at distances of the order of the Planck length; a constant correction depending on the undetermined parameters of the renormalised effective action; and a term which grows logarithmically with the distance from the particle. This last term is the most interesting, and when resummed gives a modified power law, enhancing the gravitational force at large distances. As a check on the accuracy of our calculation, we recover the linearised Kerr-de Sitter metric in the classical limit and the flat-space quantum correction in the limit of vanishing Hubble constant.

Quadratic-in-spin effects in the orbital dynamics and gravitational-wave energy flux of compact binaries at the 3PN order

We investigate the dynamics of spinning binaries of compact objects at the next-to-leading order in the quadratic-in-spin effects, which corresponds to the third post-Newtonian order (3PN). Using a Dixon-type multipolar formalism for spinning point particles endowed with spin-induced quadrupoles and computing iteratively in harmonic coordinates the relevant pieces of the PN metric within the near zone, we derive the post-Newtonian equations of motion as well as the equations of spin precession. We find full equivalence with available results. We then focus on the far-zone field produced by those systems and obtain the previously unknown 3PN spin contributions to the gravitational-wave energy flux by means of the multipolar post-Minkowskian wave generation formalism. Our results are presented in the center-of-mass frame for generic orbits, before being further specialized to the case of spin-aligned, circular orbits. Based on the energy balance equation, we derive the orbital phase of the binary at the order 3PN including all conservative spin effects and briefly discuss the relevance of the new terms.

On the Gyromagnetic and Gyrogravito-Magnetic Ratios of the Electron

The magnetic dipole moment of the Kerr–Newman metric, defined by mass \(M\), electrical charge \(Q\) and angular momentum \(J\), is \(\mu =QJ/M\), corresponding, for all values of \(J/M\), to a gyromagnetic ratio \(g_1=2\), which is also the value of the intrinsic gyromagnetic ratio of the electron, as first noted by Carter. Here, we argue that this result can be understood in terms of the particle-wave complementarity principle. For \(\mu \) can only be defined at asymptotic spatial infinity, where the metric appears to describe a spinning point particle, and therefore setting \(M=m\), \(Q=e\), we necessarily have a model of the electron. From the Dirac equation we can construct a covariantly conserved four-current \(J_i\) that is the source of the electromagnetic field generated by the charge \(e\). The result \(g_1=2\) then follows from the minimal gauge principle \(\partial _j\rightarrow \partial _j -\mathrm {i}e A_j\) which is implicit in the formulation of the spinorial wave equation, and which can also be justified from the line action for a spin-1/2 point particle interacting with an external electromagnetic field, due to Berezin and Marinov. By contrast, analysis of the gyrogravito-magnetic effect, investigated classically by Wald and quantum mechanically by Adler et al., yields the result \(\tilde{g}=1\) in all non-relativistic cases, which can be explained from the principle of equivalence. The results are in accord with the correspondence principle.

First law of mechanics for black hole binaries with spins

We use the canonical Hamiltonian formalism to generalize to spinning point particles the first law of mechanics established for binary systems of nonspinning point masses moving on circular orbits [A. Le Tiec, L. Blanchet, and B. F. Whiting, Phys. Rev. D 85, 064039 (2012)]. We find that the redshift observable of each particle is related in a very simple manner to the canonical Hamiltonian and, more generally, to a class of Fokker-type Hamiltonians. Our results are valid through linear order in the spin of each particle, but hold also for quadratic couplings between the spins of different particles. The knowledge of spin effects in the Hamiltonian allows us to compute spin-orbit terms in the redshift variable through 2.5PN order, for circular orbits and spins aligned or anti-aligned with the orbital angular momentum. To describe extended bodies such as black holes, we supplement the first law for spinning point-particle binaries with some ``constitutive relations'' that can be used for diagnosis of spin measurements in quasi-equilibrium initial data.

Supersymmetry in the spacetime of higher-dimensional rotating black holes

General higher-dimensional rotating black hole spacetimes of any dimensions admit the Killing and Killing-Yano tensors, which generate the hidden symmetries just as in four-dimensional Kerr spacetime. We study these properties of the black holes using the formalism of supersymmetric mechanics of pseudoclassical spinning point particles. We present two nontrivial supercharges, corresponding to the Killing-Yano and conformal Killing-Yano tensors of the second rank. We demonstrate that an unusual extended Poisson-Dirac algebra of these supercharges results in two independent Killing tensors in spacetime dimensions $D\ensuremath{\ge}6$, giving explicit examples for the Myers-Perry black holes in $D=6$ dimensions.

Stationary spinning strings and symmetries of classical spacetimes

We explore the symmetries of classical stationary spacetimes in terms of the dynamics of a spinning string described by a worldsheet supersymmetric action. We show that for stationary configurations of the string, the action reduces to that for a pseudo-classical spinning point particle in an effective space, which is a conformally scaled quotient space of the original spacetime. As an example, we consider the stationary spinning string in the Kerr–Newman spacetime, whose motion is equivalent to that of the spinning point particle in the three-dimensional effective space. We present the Killing tensor as well as the spin-valued Killing vector of this space. However, the nongeneric supersymmetry corresponding to the Killing–Yano tensor of the Kerr–Newman spacetime is lost in the effective space.

Group field theory formulation of 3D quantum gravity coupled to matter fields

We present a new group field theory describing 3D Riemannian quantum gravity coupled to matter fields for any choice of spin and mass. The perturbative expansion of the partition function produces fat graphs coloured with SU(2) algebraic data, from which one can reconstruct at once a three-dimensional simplicial complex representing spacetime and its geometry, like in the Ponzano–Regge formulation of pure 3D quantum gravity, and the Feynman graphs for the matter fields. The model then assigns quantum amplitudes to these fat graphs given by spin foam models for gravity coupled to interacting massive spinning point particles, whose properties we discuss.

Dynamics of a classical spinning particle in Vaidya space-time

Based on the Mathisson-Papapetrou-Dixon (MPD) equations and the Vaidya metric, the motion of a spinning point particle orbiting a non-rotating star while undergoing radiation-induced gravitational collapse is studied in detail. A comprehensive analysis of the orbital dynamics is performed assuming distinct central mass functions which satisfy the weak energy condition, in order to determine a correspondence between the choice of mass function and the spinning particle's orbital response, as reflected in the gravitational waves emitted by the particle. The analysis presented here is likely most beneficial for the observation of rotating solar mass black holes or neutron stars in orbit around intermediate-sized Schwarzschild black holes undergoing radiation collapse. The possibility of detecting the effects of realistic mass accretion based on this approach is considered. While it seems unlikely to observe such effects based on present technology, they may perhaps become observable with the advent of future detectors.

Three-dimensional loop quantum gravity: coupling to point particles

We consider the coupling between three-dimensional gravity with zero cosmological constant and massive spinning point particles. First, we study the classical canonical analysis of the coupled system. Then, we go to the Hamiltonian quantization generalizing loop quantum gravity techniques. We give a complete description of the kinematical Hilbert space of the coupled system. Finally, we define the physical Hilbert space of the system of self-gravitating massive spinning point particles using Rovelli's generalized projection operator which can be represented as a sum over spin-foam amplitudes. In addition we provide an explicit expression of the classical distance operator between two particles which is defined as an observable.

DUAL METRICS AND NONGENERIC SUPERSYMMETRIES FOR A CLASS OF SIKLOS SPACE–TIMES

The presence of Killing–Yano tensors implies the existence of nongeneric supercharges in spinning point particle theories on curved backgrounds. Dual metrics are defined through their associated nondegenerate Killing tensors of valence two. Siklos space–times, which are the only nontrivial Einstein spaces conformal to nonflat pp-waves are investigated with regard to the existence of their corresponding Killing and Killing–Yano tensors. It is found that a class of Siklos space–times admit dual metrics and nongeneric supercharges.

Structure of a spinning point particle at rest

The momentum, angular momentum, and energy of a spinning particle each consists of two terms, as if the particle were composed of two other particles that move coherently in different orbits about their common center of mass. The conditions that the momentum terms cancel each other and that the orbital angular momenta should be quantized lead to a series of quantized mass-energy levels and mass ratios that describe the relevant experimental data to better than 0.25%. The correlated particles move as if they have a potential energy that is constant for very small distances from the fixed center of mass and that increases linearly for large distances, but no quarks or gluons are postulated. Baryons and mesons are linked to each other, as are baryons and antibaryons.

Negative energy densities in extended sources generating closed timelike curves in general relativity with and without torsion.

Near a spinning point particle in (2+1)-dimensional gravity (or near an infinitely thin, straight, spinning string in 3+1 dimensions) there is a region of space-time with closed timelike curves. Exact solutions for extended sources with apparently physically acceptable energy-momentum tensors, have produced the same exterior space-time structure. Here it is pointed out that in the case with torsion, closed timelike curves appear only for spin densities so high that the spin energy density is higher than the net effective energy density. In models without torsion, the presence of closed time-like curves is related to a heat flow of unphysical magnitude. This corroborates earlier arguments against the possibility of closed timelike curves in space-time geometries generated by physical sources.

Radiation reaction for the classical relativistic spinning particle in scalar, tensor and linearized gravitational fields

We use the method of analytic continuation of the equation of motion including the self-fields to evaluate the radiation reaction for a classical relativistic spinning point particle in interaction with scalar, tensor and linearized gravitational fields in flat space-time. In the limit these equations reduce to those of spinless particles. We also show the renormalizability of these theories.

Spinning particle and gauge theories as integrability conditions

Starting from a new four dimensional spinning point particle we obtain new representations of the standard four dimensional gauge field equations in terms of a generalized space (Minkowski + light cone). In terms of this new formulation we define linear systems whose integrability conditions imply the massive Dirac-Maxwell and the Yang-Mills equations.

A spinning string

The Einstein-Cartan field equations are solved for a string source with spin-polarisation along the axis of symmetry. The interior solution is matched to an exterior vacuum space-time using Arkuszewski-Kopczynski-Ponomariev junction conditions. The exterior solution is a four dimensional extension of the space-time outside a spinning point particle in three dimensional Einstein theory. It reduces to the geometry outside a conventional straight cosmic string in the case of vanishing spin.

Gravitational anyons, chern-simons-witten gravity and the gravitational aharonov-bohm effect

The anyonic behaviour of massive point particles in the (2 + 1)-dimensional theory of topologically massive gravity, as first suggested by Deser, is further investigated. At the linearised level, we recall that the exotic statistics of the particles may be understood in terms of the gravitational Aharonov-Bohm effect discussed by several authors in the 1960's. This approach suggests that massive spinning point particles in ordinary Einstein gravity may also exhibit exotic statistics, and that it is the Einstein term which takes on the role carried out by the Chern-Simons term in the standard U(1) anyon model. We investigate this further in the Chern-Simons-Witten formulation of 2 + 1 gravity, where the Einstein term is indeed a Chern-Simons term. In this context the exotic statistics may be understood in terms of a non-abelian Aharonov-Bohm effect, giving a description in terms of the fully non-linear theory of gravity, and making explicit the relationship between gravitational anyons and Chern-Simons anyons of (2 + 1)-dimensional U(1) gauge theory. The analogue of magnetic flux in this context is seen to be both torsion and Riemann curvature. We finally show the correspondence between the statistics phase obtained in this approach and that obtained by Deser and McCarthy in the linearised approximation of topologically massive gravity.

On the electrodynamics of spinning particles

We consider the electrodynamics of spinning point particles. We introduce a modification of the Lorentz force law which can be interpreted as a classical limit of the Dirac-Klein-Gordon equation. An improved version of the inhomogeneous Maxwell equations is constructed for the description of the classical fields of spinning particles. Both classical and quantum electrodynamics are shown to predict relativistic time-dilation effects for spinning particles in an electromagnetic field, even in the limit of zero velocity. The life-time of unstable charged particles moving in a Coulomb field is computed for both spin-0 and spin- 1 2 particles. Comparison shows (relatively small) spin effects to be present. The magnitude of further spin-dependent corrections from hyperfine interactions is computed. A measurement of the life-time of muons in atomic bound states separated by such spin-dependent hyperfine interactions would provide a clean test for the effect we predict. Similar effects are shown to arise in non-abelian gauge theories such as QCD.