World Lines Of Particles Research Articles

A fourth-order indirect integration method for black hole perturbations: even modes

On the basis of a recently proposed strategy of finite element integration in time domain for partial differential equations with a singular source term, we present a fourth-order algorithm for non-rotating black hole perturbations in the Regge–Wheeler gauge. Herein, we address even perturbations induced by a particle plunging in. The forward time value at the upper node of the (r*, t) grid cell is obtained by an algebraic sum of (i) the preceding node values of the same cell, (ii) analytic expressions, related to the jump conditions on the wavefunction and its derivatives and (iii) the values of the wavefunction at adjacent cells. In this approach, the numerical integration does not deal with the source and potential terms directly, for cells crossed by the particle world line. This scheme has also been applied to circular and eccentric orbits and it will be the object of a forthcoming publication.

Kinematics of a relativistic particle with de Sitter momentum space

We discuss kinematical properties of a free relativistic particle with deformed phase space in which momentum space is given by (a submanifold of) de Sitter space. We provide a detailed derivation of the action, Hamiltonian structure and equations of motion for such a free particle. We study the action of deformed relativistic symmetries on the phase space and derive explicit formulae for the action of the deformed Poincaré group. Finally we provide a discussion on parametrization of the particle worldlines stressing analogies and differences with ordinary relativistic kinematics.

Self-force viam-mode regularization and2+1Devolution: Foundations and a scalar-field implementation on Schwarzschild spacetime

To model the radiative evolution of extreme mass-ratio binary inspirals (a key target of the LISA mission), the community needs efficient methods for computation of the gravitational self-force (SF) on the Kerr spacetime. Here we further develop a practical ``$m$-mode regularization'' scheme for SF calculations, and give the details of a first implementation. The key steps in the method are (i) removal of a singular part of the perturbation field with a suitable ``puncture'' to leave a sufficiently regular residual within a finite worldtube surrounding the particle's worldline, (ii) decomposition in azimuthal ($m$) modes, (iii) numerical evolution of the $m$ modes in $2+1\mathrm{D}$ with a finite-difference scheme, and (iv) reconstruction of the SF from the mode sum. The method relies on a judicious choice of puncture, based on the Detweiler-Whiting decomposition. We give a working definition for the ``order'' of the puncture, and show how it determines the convergence rate of the $m$-mode sum. The dissipative piece of the SF displays an exponentially convergent mode sum, while the $m$-mode sum for the conservative piece converges with a power law. In the latter case, the individual modal contributions fall off at large $m$ as ${m}^{\ensuremath{-}n}$ for even $n$ and as ${m}^{\ensuremath{-}n+1}$ for odd $n$, where $n$ is the puncture order. We describe an $m$-mode implementation with a 4th-order puncture to compute the scalar-field SF along circular geodesics on Schwarzschild. In a forthcoming companion paper we extend the calculation to the Kerr spacetime.

Interference in the radiation of two pointlike charges

Energy-momentum and angular momentum carried by electromagnetic field of two pointlike charged particles arbitrarily moving in flat space-time are presented. Apart from usual contributions to the Noether quantities produced separately by particles 1 and 2, the conservation laws contain also joint contribution due to the fields of both particles. The mixed part of Maxwell energy-momentum density is decomposed into bound and radiative components which are separately conserved off the world lines of particles. The former describes the deformation of electromagnetic clouds of “bare” charges due to mutual interaction while the latter defines the radiation which escapes to infinity. The bound terms contribute to particles’ individual 4-momenta while the radiative ones exert the radiation reaction. Analysis of energy-momentum and angular momentum balance equations results the Lorentz–Dirac equation as an equation of motion for a pointed charge under the influence of its own electromagnetic field as well as field produced by another charge.

E. C. G. Sudarshan and symmetry in classical dynamics, optics and quantum mechanics

We review work initiated and inspired by Sudarshan in relativistic dynamics, beam optics, partial coherence theory, Wigner distribution methods, multimode quantum optical squeezing, and geometric phases. The 1963 No Interaction Theorem using Dirac's instant form and particle World Line Conditions is recalled. Later attempts to overcome this result exploiting constrained Hamiltonian theory, reformulation of the World Line Conditions and extending Dirac's formalism, are reviewed.Dirac's front form leads to a formulation of Fourier Optics for the Maxwell field, determining the actions of First Order Systems (corresponding to matrices of Sp(2,R) and Sp(4,R)) on polarization in a consistent manner. These groups also help characterize properties and propagation of partially coherent Gaussian Schell Model beams, leading to invariant quality parameters and the new Twist phase.The higher dimensional groups Sp(2n,R) appear in the theory of Wigner distributions and in quantum optics. Elegant criteria for a Gaussian phase space function to be a Wigner distribution, expressions for multimode uncertainty principles and squeezing are described.In geometric phase theory we highlight the use of invariance properties that lead to a kinematical formulation and the important role of Bargmann invariants. Special features of these phases arising from unitary Lie group representations, and a new formulation based on the idea of Null Phase Curves, are presented.

Bohmian mechanics at space–time singularities: II. Spacelike singularities

We develop an extension of Bohmian mechanics by defining Bohm-like trajectories for quantum particles in a curved background space-time containing a spacelike singularity. As an example of such a metric we use the Schwarzschild metric, which contains two spacelike singularities, one in the past and one in the future. Since the particle world lines are everywhere timelike or lightlike, particles can be annihilated but not created at a future spacelike singularity, and created but not annihilated at a past spacelike singularity. It is argued that in the presence of future (past) spacelike singularities, there is a unique natural Bohm-like evolution law directed to the future (past). This law differs from the one in non-singular space-times mainly in two ways: it involves Fock space since the particle number is not conserved, and the wave function is replaced by a density matrix. In particular, we determine the evolution equation for the density matrix, a pure-to-mixed evolution equation of a quasi-Lindblad form. We have to leave open whether a curvature cut-off needs to be introduced for this equation to be well defined.

Frequency-domain calculation of the self-force: The high-frequency problem and its resolution

The mode-sum method provides a practical means for calculating the self-force acting on a small particle orbiting a larger black hole. In this method, one first computes the spherical-harmonic $l$-mode contributions ${F}_{l}^{\ensuremath{\mu}}$ of the ``full-force'' field ${F}^{\ensuremath{\mu}}$, evaluated at the particle's location, and then sums over $l$ subject to a certain regularization scheme. In the frequency-domain variant of this procedure the quantities ${F}_{l}^{\ensuremath{\mu}}$ are obtained by fully decomposing the particle's self-field into Fourier-harmonic modes $lm\ensuremath{\omega}$, calculating the contribution of each such mode to ${F}_{l}^{\ensuremath{\mu}}$, and then summing over $\ensuremath{\omega}$ and $m$ for given $l$. This procedure has the advantage that one only encounters ordinary differential equations. However, for eccentric orbits, the sum over $\ensuremath{\omega}$ is found to converge badly at the particle's location. This problem (reminiscent of the familiar Gibbs phenomenon of Fourier analysis) results from the discontinuity of the time-domain ${F}_{l}^{\ensuremath{\mu}}$ field at the particle's worldline. Here we propose a simple and practical method to resolve this problem. The method utilizes the homogeneous modes $lm\ensuremath{\omega}$ of the self-field to construct ${F}_{l}^{\ensuremath{\mu}}$ (rather than the inhomogeneous modes, as in the standard method), which guarantees an exponentially fast convergence to the correct value of ${F}_{l}^{\ensuremath{\mu}}$, even at the particle's location. We illustrate the application of the method with the example of the monopole scalar-field perturbation from a scalar charge in an eccentric orbit around a Schwarzschild black hole. Our method, however, should be applicable to a wider range of problems, including the calculation of the gravitational self-force using either Teukolsky's formalism, or a direct integration of the metric perturbation equations.

Scalar-field perturbations from a particle orbiting a black hole using numerical evolution in2+1dimensions

We present a new technique for time-domain numerical evolution of the scalar- field generated by a pointlike scalar charge orbiting a black hole. Time-domain evolution offers an efficient way for calculating black hole perturbations, especially as input for computations of the local self force acting on orbiting particles. In Kerr geometry, the field equations are not fully separable in the time domain, and one has to tackle them in 2+1 dimensions (two spatial dimensions and time; the azimuthal dependence is still separable). A technical difficulty arises when the source of the field is a pointlike particle, as the 2+1-dimensional perturbation is then singular: Each of the azimuthal modes diverges logarithmically at the particle. To deal with this problem we split the numerical domain into two regions: Inside a thin worldtube surrounding the particle's worldline we solve for a regularized variable, obtained from the full field by subtracting out a suitable puncture function, given analytically. Outside this worldtube we solve for the full, original field. The value of the evolution variable is adjusted across the boundary of the worldtube. In this work we demonstrate the applicability of this method in the example of circular orbits around a Schwarzschild black hole (refrainingmore » from exploiting the spherical symmetry of the background, and working in 2+1 dimensions)« less

Dipolar particles in general relativity

The dynamics of ‘dipolar particles’, i.e. particles endowed with a 4-vector mass dipole moment, is investigated using an action principle in general relativity. The action is a specific functional of the particle's worldline, and of the dipole moment vector, considered as a dynamical variable. The first part of the action is inspired by that of a particle with spin moving on an arbitrary gravitational background. The second part is intended to describe, at some effective level, the internal non-gravitational force linking together the ‘microscopic’ constituents of the dipole. We find that some solutions of the equations of motion and evolution of the dipolar particles correspond to an equilibrium state for the dipole moment in a gravitational field. Under some hypothesis we show that a fluid of dipolar particles, supposed to constitute the dark matter, reproduces the modified Newtonian dynamics (MOND) in the non-relativistic limit. We recover the main characteristics of a recently proposed quasi-Newtonian model of ‘gravitational polarization’.

Scalar self-force on eccentric geodesics in Schwarzschild spacetime: A time-domain computation

We calculate the self-force acting on a particle with scalar charge moving on a generic geodesic around a Schwarzschild black hole. This calculation requires an accurate computation of the retarded scalar field produced by the moving charge; this is done numerically with the help of a fourth-order convergent finite-difference scheme formulated in the time domain. The calculation also requires a regularization procedure, because the retarded field is singular on the particle's world line; this is handled mode-by-mode via the mode-sum regularization scheme first introduced by Barack and Ori. This paper presents the numerical method, various numerical tests, and a sample of results for mildly eccentric orbits as well as ''zoom-whirl'' orbits.

Determinate Values for Quantum Observables

This is a comment on J. A. Barrett's article ``The Preferred-Basis Problem and the Quantum Mechanics of Everything'' in Brit. J. Phil. Sci. 56 (2005), which concerns theories postulating that certain quantum observables have determinate values, corresponding to additional (often called ``hidden'') variables. I point out that it is far from clear, for most observables, what such a postulate is supposed to mean, unless the postulated additional variable is related to a clear ontology in space-time, such as particle world lines, string world sheets, or fields.

Geometric gyrokinetic theory for edge plasmas

It turns out that gyrokinetic theory can be geometrically formulated as a special case of a geometrically generalized Vlasov-Maxwell system. It is proposed that the phase space of the space-time is a seven-dimensional fiber bundle P over the four-dimensional space-time M, and that a Poincaré-Cartan-Einstein 1-form γ on the seven-dimensional phase space determines a particle's worldline in the phase space. Through Liouville 6-form Ω and fiber integral, the 1-form γ also uniquely defines a geometrically generalized Vlasov-Maxwell system as a field theory for the collective electromagnetic field. The geometric gyrokinetic theory is then developed as a special case of the geometrically generalized Vlasov-Maxwell system. In its most general form, gyrokinetic theory is about a symmetry, called gyrosymmetry, for magnetized plasmas, and the 1-form γ again uniquely defines the gyrosymmetry. The objective is to decouple the gyrophase dynamics from the rest of the particle dynamics by finding the gyrosymmetry in γ. Compared to other methods of deriving the gyrokinetic equations, the advantage of the geometric approach is that it allows any approximation based on mathematical simplification or physical intuition to be made at the 1-form level, and yet the field theories still have the desirable exact conservation properties, such as phase space volume conservation and energy-momentum conservation if the 1-form does not depend on the space-time coordinate explicitly. A set of generalized gyrokinetic equations valid for the edge plasmas is then derived using this geometric method. This formalism allows large-amplitude, time-dependent background electromagnetic fields to be developed fully nonlinearly in addition to small-amplitude, short-wavelength electromagnetic perturbations. The fact that we adopted the geometric method in the present study does not necessarily imply that the major results reported here cannot be achieved using classical methods. What the geometric method offers is a systematic treatment and simplified calculations.

Classical string in curved backgrounds

The Mathisson-Papapetrou method is originally used for derivation of the particle world line equation from the covariant conservation of its stress-energy tensor. We generalize this method to extended objects, such as a string. Without specifying the type of matter the string is made of, we obtain both the equations of motion and boundary conditions of the string. The world sheet equations turn out to be more general than the familiar minimal surface equations. In particular, they depend on the internal structure of the string. The relevant cases are classified by examining canonical forms of the effective 2-dimensional stress-energy tensor. The case of homogeneously distributed matter with the tension that equals its mass density is shown to define the familiar Nambu-Goto dynamics. The other three cases include physically relevant massive and massless strings, and unphysical tachyonic strings.

Magnetized Tolman-Bondi collapse

We investigate the gravitational implosion of magnetized matter by studying the inhomogeneous collapse of a weakly magnetized Tolman-Bondi spacetime. The role of the field is analyzed by looking at the convergence of neighboring particle world lines. In particular, we identify the magnetically related stresses in the Raychaudhuri equation and use the Tolman-Bondi metric to evaluate their impact on the collapsing dust. We find that, despite the low energy level of the field, the Lorentz force dominates the advanced stages of the collapse, leading to a strongly anisotropic contraction. In addition, of all the magnetic stresses, those that resist the collapse are found to grow faster.

MOTION OF A VECTOR PARTICLE IN A CURVED SPACETIME. II: FIRST-ORDER CORRECTION TO A GEODESIC IN A SCHWARZSCHILD BACKGROUND

The influence of spin on a photon's motion in a Schwarzschild and FRW spacetimes is studied. The first-order correction to the geodesic motion is found. It is shown that unlike the worldlines of spinless particles, the photons worldlines do not lie in a plane.

Coulomb effects and hopping transport in granular metals

We investigate effects of Coulomb interaction and hopping transport in the insulator phase of granular metals and quantum dot arrays. We consider a spatially periodic as well as an irregular array, including disorder in a form of a random on-site electrostatic potential. We study the Mott transition between the insulating and metallic states in the regular system and find the dependence of the Mott gap upon the intergranular coupling. The conductivity of a strictly periodic array has an activation form with the Mott gap as an activation energy. Considering irregular systems we concentrate on the transport properties in the dielectric, low coupling limit and derive the Efros-Shklovskii law for hopping conductivity. In the irregular arrays electrostatic disorder results in the finite density of states on the Fermi level giving rise to the variable range hopping mechanism. We develop a theory of tunneling through a chain of grains and discuss in detail both elastic and inelastic cotunneling mechanisms; the former dominates at very low temperatures and/or very low applied electric fields, while the inelastic mechanism controls tunneling at high temperature/fields. Our results are obtained within the framework of the new technique based on the mapping of quantum electronic problem onto the classical gas of Coulomb charges. The processes of quantum tunnelling of real electrons are represented in this technique as trajectories (world lines) of charged classical particles in $d+1$ dimensions. The Mott gap is related to the dielectric susceptibility of the Coulomb gas in the direction of the imaginary time axis.

Dimensional regularization of the third post-Newtonian gravitational wave generation from two point masses

Dimensional regularization is applied to the computation of the gravitational wave field generated by compact binaries at the third post-Newtonian (3PN) approximation. We generalize the wave generation formalism from isolated post-Newtonian matter systems to $d$ spatial dimensions, and apply it to point masses (without spins), modeled by delta-function singularities. We find that the quadrupole moment of point-particle binaries in harmonic coordinates contains a pole when $\ensuremath{\epsilon}\ensuremath{\equiv}d\ensuremath{-}3\ensuremath{\rightarrow}0$ at the 3PN order. It is proved that the pole can be renormalized away by means of the same shifts of the particle world lines as in our recent derivation of the 3PN equations of motion. The resulting renormalized (finite when $\ensuremath{\epsilon}\ensuremath{\rightarrow}0$) quadrupole moment leads to unique values for the ambiguity parameters $\ensuremath{\xi}$, $\ensuremath{\kappa}$, and $\ensuremath{\zeta}$, which were introduced in previous computations using Hadamard's regularization. Several checks of these values are presented. These results complete the derivation of the gravitational waves emitted by inspiralling compact binaries up to the 3.5PN level of accuracy which is needed for detection and analysis of the signals in the gravitational wave antennas LIGO/VIRGO and LISA.

Are all particles real?

In Bohmian mechanics elementary particles exist objectively, as point particles moving according to a law determined by a wavefunction. In this context, questions as to whether the particles of a certain species are real—questions such as, Do photons exist? Electrons? Or just the quarks?—have a clear meaning. We explain that, whatever the answer, there is a corresponding Bohm-type theory, and no experiment can ever decide between these theories. Another question that has a clear meaning is whether particles are intrinsically distinguishable, i.e., whether particle world lines have labels indicating the species. We discuss the intriguing possibility that the answer is no, and particles are points— just points.

Uniformly Accelerated Charge in a Quantum Field: From Radiation Reaction to Unruh Effect

We present a stochastic theory for the nonequilibrium dynamics of charges moving in a quantum scalar field based on the worldline influence functional and the close-time-path (CTP or in-in) coarse-grained effective action method. We summarize (1) the steps leading to a derivation of a modified Abraham-Lorentz-Dirac equation whose solutions describe a causal semiclassical theory free of runaway solutions and without pre-acceleration patholigies, and (2) the transformation to a stochastic effective action which generates Abraham-Lorentz-Dirac-Langevin equations depicting the fluctuations of a particle's worldline around its semiclassical trajectory. We point out the misconceptions in trying to directly relate radiation reaction to vacuum fluctuations, and discuss how, in the framework that we have developed, an array of phenomena, from classical radiation and radiation reaction to the Unruh effect, are interrelated to each other as manifestations at the classical, stochastic and quantum levels. Using this method we give a derivation of the Unruh effect for the spacetime worldline coordinates of an accelerating charge. Our stochastic particle-field model, which was inspired by earlier work in cosmological backreaction, can be used as an analog to the black hole backreaction problem describing the stochastic dynamics of a black hole event horizon.

Fat Euclidean gravity with small cosmological constant

The cosmological constant problem is usually considered an inevitable feature of any effective theory capturing well-tested gravitational and matter physics, without regard to the details of short-distance gravitational couplings. In this paper, a subtle effective description avoiding the problem is presented in a first quantized language, consistent with experiments and the Equivalence Principle. First quantization allows a minimal domain of validity to be carved out by cutting on the proper length of particle worldlines. This is facilitated by working in (locally) Euclidean spacetime, although considerations of unitarity are still addressed by analytic continuation from Lorentzian spacetime. The new effective description demonstrates that the cosmological constant problem is sensitive to short-distance details of gravity, which can be probed experimentally. “Fat Gravity” toy models are presented, illustrating how gravity might shut off at short but testable distances, in a generally covariant manner that suppresses the cosmological constant. This paper improves on previous work by allowing generalizations to massless matter, non-trivial spins, non-perturbative phenomena, and multiple (metastable) vacua.