Geodesic Approximation Research Articles

The dynamics of vortices on S2 near the Bradlow limit

The explicit solutions of the Bogomolny equations for N vortices on a sphere of radius R2>N are not known. In particular, this has prevented the use of the geodesic approximation to describe the low energy vortex dynamics. In this article we introduce an approximate general solution of the equations, valid for R2≳N, which has many properties of the true solutions, including the same moduli space CPN. Within the framework of the geodesic approximation, the metric on the moduli space is then computed to be proportional to the Fubini–Study metric, which leads to a complete description of the particle dynamics.

Fast and slow blowup in the S2 σ-model and the (4+1)-dimensional Yang-Mills model

We study singularity formation in spherically symmetric solutions ofthe charge-one and charge-two sector of the (2+1)-dimensional S2 σ-model and the (4+1)-dimensional Yang-Mills model, near the adiabatic limit. These equationsare nonintegrable, and so studies are performed numerically onrotationally symmetric solutions using an iterative finite differencingscheme that is numerically stable. We evaluate the accuracy of predictions made with thegeodesic approximation. We find that the geodesic approximation is extremelyaccurate for the charge-two σ-model and the Yang-Mills model, bothof which exhibit fast blowup to within experimental error. The charge-one σ-model exhibits slowblowup. There the geodesic approximation must be modified by applyingan infrared cutoff that depends on initial conditions.

Motion on moduli spaces with potentials

In the limit of small velocities, the dynamics of half-BPS Yang-Mills-Higgs solitons can be described by the geodesic approximation. Recently, it has been shown that quarter-BPS states require the addition of a potential term to this approximation. We explain the logic behind this modification for a larger class of models and then analyse in detail the dynamics of two five-dimensional dyonic instantons, using both analytical and numerical techniques. Nonzero-modes are shown to play a crucial role in the analysis of these systems, and we explain how these modes lead to qualitatively new types of dynamics.

Geodesic Incompleteness in the ℂP1 Model on a Compact Riemann Surface

It is proved that the moduli space of static solutions of the ℂP1 model on spacetime Σ ×∝, where Σ is any compact Riemann surface, is geodesically incomplete with respect to the metric induced by the kinetic energy functional. The geodesic approximation predicts, therefore, that lumps can collapse and form singularities in finite time in these models.

On the curvature of two-dimensional singular Regge lattices

We consider geodesic approximation of a two-dimensional Riemannian manifold, M, with a singular Regge lattice K and, in particular, relations between classical continuum curvature power actions and corresponding lattice actions. By the singular Regge lattice we mean a triangulated piecewise flat space having very thin triangles. It is shown that the continuum actions are well approximated by the lattice actions in the sense of measures, provided that the edge lengths of K are small, independently of whether very thin triangles are contained in K or not.

Symmetric monopoles

We discuss SU(2) Bogomolny monopoles of arbitrary charge k invariant under various symmetry groups. The analysis is largely in terms of the spectral curves, the rational maps, and the Nahm equations associated with monopoles. We consider monopoles invariant under inversion in a plane, monopoles with cyclic symmetry, and monopoles having the symmetry of a regular solid. We introduce the notion of a strongly centred monopole and show that the space of such monopoles is a geodesic submanifold of the monopole moduli space. By solving Nahm's equations we prove the existence of a tetrahedrally symmetric monopole of charge 3 and an octahedrally symmetric monopole of charge 4, and determine their spectral curves. Using the geodesic approximation to analyse the scattering of monopoles with cyclic symmetry, we discover a novel type of non-planar k-monopole scattering process.

Remarks on gauge vortex scattering

In the abelian Higgs model, among other situations, it has recently been realized that the head-on scattering of n solitons distributed symmetrically around the point of scattering is by an angle π n , independent of various details of the scattering. In this note, it is first observed that this result is in fact not entirely surprising: the above is one of only two possible outcomes. Then, a generalization of an argument given by Ruback for the case of two gauge theory vortices in the Bogomol'nyi limit is used to show that in the geodesic approximation the above result follows from purely geometric considerations.

The geodesic approximation for the Yang-Mills-Higgs equations

In this paper we consider the dynamics of the monopole solutions of Yang-Mills-Higgs theory on Minkowski space. The monopoles are solutions of the Yang-Mills-Higgs equations on three dimensional Euclidean space. It is of interest to understand how they evolve in time when considered as solutions of the Yang-Mills-Higgs equations on Minkowski space-i.e. the time dependent equations. It was suggested by Manton that in certain situations the monopole dynamics could be understood in terms of geodesics with respect to a certain, metric on the space of guage equivalence classes of monopoles-the moduli space. The metric is defined by taking theL 2 inner product of tangent vectors to this space. In this paper we will prove that Manton's approximation is indeed valid in the right circumstances, which correspond to the slow motion of monopoles. The metric on the moduli space of monopoles was analysed in a book by Atiyah and Hitchin, so together with the results of this paper a detailed and rigorous understanding of the low energy dynamics of monopoles in Yang-Mills-Higgs theory is obtained. The strategy of the proof is to develop asymptotic expansions using appropriate gauge conditions, and then to use energy estimates to prove their validity. For the case of monopoles to be considered here there is a technical obstacle to be overcome-when the equations are linearised about the monopole the continuous spectrum extends all the way to the origin. This is overcome by using a norm introduced by Taubes in a discussion of index, theory for the Yang-Mills-Higgs functional.

Low-energy dynamics of N = 2 supersymmetric monopoles

It is argued that the low-energy dynamics of k-monopoles in N = 2 supersymmetric Yang-Mills theory are determined by an N = 4 supersymmetric quantum mechanics based on the moduli space of k static monople solutions. This generalises Manton's “geodesic approximation” for studying the low-energy dynamics of (bosonic) BPS monopoles. We discuss some aspects of the quantisation and in particular argue that Dolbeault cohomology classes of the moduli space are related to bound states of the full quantum field theory.

Smooth Limits of Piecewise-Linear Approximations

We consider particular types of discrete approximations to tensor fields on manifolds suggested by triangulations. The approximations are objects of finite geometrical extent, parameterized by a finite set of numbers, so they are suitable for numerical computations. We study the limiting behaviour of sequences of approximations and construct the theory so that the limits are tensor fields on the manifold. We propose a Cauchy criterion for our approximations, which guarantees convergence to a limit. The specific examples include geodesic approximation to Riemannian and pseudo-Riemannian manifolds.

Differential cross sections and escape plots for low-energy solitonic SU(2) BPS magnetic monopole dynamics

We compute the low-energy classical differential scattering cross section for solitonic BPS SU(2) magnetic monopoles using the geodesic approximation to the actual dynamics and 32K virtual parallel processors on a CM2. We compare the classical solitonic differential cross section to the low-energy quantum BPS SU(2) magnetic monopole differential cross section obtained by Schroers. Our numerical experiments suggest that the classical solitonic BPS magnetic monopole differential cross section approximates well the quantum BPS magnetic monopole differential cross section. In particular, the expected quantum interference of identical bosons at scattering angle θ = π 2 (centre-of-mass frame) is not observed numerically, and indeed is contradicted. We argue that the lack of singularities in the two-body configuration space, related to the solitonic qualities of the BPS SU(2) magnetic monopole, is responsible for the agreement observed between the classical solitonic and the quantum mechanical cross sections. We also study the scattering and bounded classical motions of BPS SU(2) dyons and their global structure in phase space by constructing “escape plots”. The escape plots contain a surprising amount of structure. The escape plots suggest that the classical dynamics of two BPS SU(2) magnetic monopoles is non-integrable and that there are closed and bounded two-monopole motions with isolated energies.

Gamma-ray emission from accretion onto a rotating black hole

view Abstract Citations (15) References (16) Co-Reads Similar Papers Volume Content Graphics Metrics Export Citation NASA/ADS Gamma-Ray Emission from Accretion onto a Rotating Black Hole Colpi, M. ; Maraschi, L. ; Treves, A. Abstract Gamma-ray production from the decay of pi(0) mesons deriving from proton-proton interactions in a thermal plasma around a rotating black hole is considered. The temperature and density profile of the accreting plasma are calculated for an adiabatic flow in a Kerr metric with detached electron and proton populations and using the geodesic approximation. The computed gamma-ray luminosity turns out to be lower than that obtained by other authors. The relevance of the process for observable gamma-ray sources is discussed. Publication: The Astrophysical Journal Pub Date: December 1986 DOI: 10.1086/164760 Bibcode: 1986ApJ...311..150C Keywords: Black Holes (Astronomy); Galactic Rotation; Gamma Ray Astronomy; Stellar Mass Accretion; Adiabatic Conditions; Luminosity; Particle Production; X Ray Sources; Astrophysics; BLACK HOLES; GALAXIES: NUCLEI; GAMMA RAYS: GENERAL; RADIATION MECHANISMS; STARS: ACCRETION full text sources ADS | data products SIMBAD (1)