Conformal Sphere Research Articles

Asymptotic Antipodal Solutions as the Limit of Elliptic Relative Equilibria for the Two- and n-Body Problems in the Two-Dimensional Conformal Sphere

We consider the two- and n-body problems on the two-dimensional conformal sphere MR2, with a radius R>0. We employ an alternative potential free of singularities at antipodal points. We study the limit of relative equilibria under the SO(2) symmetry; we examine the specific conditions under which a pair of positive-mass particles, situated at antipodal points, can maintain a state of relative equilibrium as they traverse along a geodesic. It is identified that, under an appropriate radius–mass relationship, these particles experience an unrestricted and free movement in alignment with the geodesic of the canonical Killing vector field in MR2. An even number of bodies with pairwise conjugated positions, arranged in a regular n-gon, all with the same mass m, move freely on a geodesic with suitable velocities, where this geodesic motion behaves like a relative equilibrium. Also, a center of mass formula is included. A relation is found for the relative equilibrium in the two-body problem in the sphere similar to the Snell law.

A sphere theorem for Bach-flat manifolds with positive constant scalar curvature

A classic gap theorem for complete non-compact Ricci flat manifolds by M. Anderson suggests that one can get the rigidity of certain spaces simply by passing local curvature estimates on geodesic balls to global. Using similar ideas, Kim showed that a 4-dimensional complete non-compact Bach-flat manifold with vanishing scalar curvature and small L2-curvature tensor has to be flat. Unfortunately, this method does not generalize to compact manifolds without boundary. Applying a different approach, we show that a closed Bach-flat Riemannian manifold with positive constant scalar curvature has to be locally spherical if its Weyl and traceless Ricci tensors are small in the sense of either L∞ or Ln2-norm. These results generalize a rigidity theorem of positive Einstein manifolds due to M.-A. Singer. As an application, we can partially recover the well-known Chang–Gursky–Yang's 4-dimensional conformal sphere theorem.

Transformations and singularities of polarized curves

We study the limiting behaviour of Darboux and Calapso transforms of polarized curves in the conformal n-dimensional sphere when the polarization has a pole of first or second order at some point. We prove that for a pole of first order, as the singularity is approached, all Darboux transforms converge to the original curve and all Calapso transforms converge. For a pole of second order, a generic Darboux transform converges to the original curve while a Calapso transform has a limit point or a limit circle, depending on the value of the transformation parameter. In particular, our results apply to Darboux and Calapso transforms of isothermic surfaces when a singular umbilic with index frac{1}{2} or 1 is approached along a curvature line.

Flat space amplitudes and conformal symmetry of the celestial sphere

The four-dimensional (4D) Lorentz group $SL(2,\mathbb{C})$ acts as the two-dimensional (2D) global conformal group on the celestial sphere at infinity where asymptotic 4D scattering states are specified. Consequent similarities of 4D flat space amplitudes and 2D correlators on the conformal sphere are obscured by the fact that the former are usually expressed in terms of asymptotic wavefunctions which transform simply under spacetime translations rather than the Lorentz $SL(2,\mathbb{C})$. In this paper we construct on-shell massive scalar wavefunctions in 4D Minkowski space that transform as $SL(2,\mathbb{C})$ conformal primaries. Scattering amplitudes of these wavefunctions are $SL(2,\mathbb{C})$ covariant by construction. For certain mass relations, we show explicitly that their three-point amplitude reduces to the known unique form of a 2D CFT primary three-point function and compute the coefficient. The computation proceeds naturally via Witten-like diagrams on a hyperbolic slicing of Minkowski space and has a holographic flavor.

Large gauge symmetries and asymptotic states in QED

Large Gauge Transformations (LGT) are gauge transformations that do not vanish at infinity. Instead, they asymptotically approach arbitrary functions on the conformal sphere at infinity. Recently, it was argued that the LGT should be treated as an infinite set of global symmetries which are spontaneously broken by the vacuum. It was established that in QED, the Ward identities of their induced symmetries are equivalent to the Soft Photon Theorem. In this paper we study the implications of LGT on the S-matrix between physical asymptotic states in massive QED. In appose to the naively free scattering states, physical asymptotic states incorporate the long range electric field between asymptotic charged particles and were already constructed in 1970 by Kulish and Faddeev. We find that the LGT charge is independent of the particles' momenta and may be associated to the vacuum. The soft theorem's manifestation as a Ward identity turns out to be an outcome of not working with the physical asymptotic states.

New symmetries for the gravitational S-matrix

In [15] we proposed a generalization of the BMS group G which is a semidirect product of supertranslations and smooth diffeomorphisms of the conformal sphere. Although an extension of BMS, G is a symmetry group of asymptotically flat space times. By taking G as a candidate symmetry group of the quantum gravity S-matrix, we argued that the Ward identities associated to the generators of Diff(S^2) were equivalent to the Cachazo-Strominger subleading soft graviton theorem. Our argument however was based on a proposed definition of the Diff(S^2) charges which we could not derive from first principles as G does not have a well defined action on the radiative phase space of gravity. Here we fill this gap and provide a first principles derivation of the Diff(S^2) charges. The result of this paper, in conjunction with the results of [4, 15] prove that the leading and subleading soft theorems are equivalent to the Ward identities associated to G.

New symmetries of massless QED

An infinite number of physically nontrivial symmetries are found for abelian gauge theories with massless charged particles. They are generated by large $U(1)$ gauge transformations that asymptotically approach an arbitrary function $\varepsilon(z,\bar{z})$ on the conformal sphere at future null infinity ($\mathscr I^+$) but are independent of the retarded time. The value of $\varepsilon$ at past null infinity ($\mathscr I^-$) is determined from that on $\mathscr I^+$ by the condition that it take the same value at either end of any light ray crossing Minkowski space. The $\varepsilon\neq$ constant symmetries are spontaneously broken in the usual vacuum. The associated Goldstone modes are zero-momentum photons and comprise a $U(1)$ boson living on the conformal sphere. The Ward identity associated with this asymptotic symmetry is shown to be the abelian soft photon theorem.

Low's subleading soft theorem as a symmetry of QED.

It was shown by Low in the 1950s that the subleading terms of soft-photon S-matrix elements obey a universal linear relation. In this Letter, we give a new interpretation to this old relation, for the case of massless QED, as an infinitesimal symmetry of the S matrix. The symmetry is shown to be locally generated by a vector field on the conformal sphere at null infinity. Explicit expressions are constructed for the associated charges as integrals over null infinity and shown to generate the symmetry. These charges are local generalizations of electric and magnetic dipole charges.

Semiclassical Virasoro symmetry of the quantum gravity S $$ \mathcal{S} $$ -matrix

It is shown that the tree-level S-matrix for quantum gravity in four-dimensional Minkowski space has a Virasoro symmetry which acts on the conformal sphere at null infinity.

Geometric Poisson brackets on Grassmannians and conformal spheres

We relate the geometric Poisson brackets on the 2-Grassmannian in ℝ4 and on the (2, 2) Möbius sphere. We show that, when written in terms of local moving frames, the geometric Poisson bracket on the Möbius sphere does not restrict to the space of differential invariants of Schwarzian type. But when the concept of conformal natural frame is transported from the conformal sphere into the Grassmannian, and the Poisson bracket is written in terms of the Grassmannian natural frame, it restricts and results in either a decoupled system or a complexly coupled system of Korteweg–de Vries (KdV) equations, depending on the character of the invariants. We also show that the bi-Hamiltonian Grassmannian geometric brackets are equivalent to the non-commutative KdV bi-Hamiltonian structure. Both integrable systems and Hamiltonian structure can be brought back to the conformal sphere.

On the geometry of curves and conformal geodesics in the Möbius space

This article deals with the study of some properties of immersed curves in the conformal sphere $${\mathbb{Q}_n}$$ , viewed as a homogeneous space under the action of the Möbius group. After an overview on general well-known facts, we briefly focus on the links between Euclidean and conformal curvatures, in the spirit of F. Klein’s Erlangen program. The core of this article is the study of conformal geodesics, defined as the critical points of the conformal arclength functional. After writing down their Euler–Lagrange equations for any n, we prove an interesting codimension reduction, namely that every conformal geodesic in $${\mathbb{Q}_n}$$ lies, in fact, in a totally umbilical 4-sphere $${\mathbb{Q}_4}$$ . We then extend and complete the work in Musso (Math Nachr 165:107–131, 1994) by solving the Euler–Lagrange equations for the curvatures and by providing an explicit expression even for those conformal geodesics not included in any conformal 3-sphere.

An Inverse Conformal Projection of the Spherical and Ellipsoidal Geodetic Elements

Many studies, involving Earth's science, made by geodesists and geometers revealed the great importance of the two remarkably influential geodetic curves, which are known as geodesic and loxodrome. The most recent observations assembled from artificial satellites paid a great attention to the ellipticity of the Earth's equator. The biaxial ellipsoidal model was always manipulated as a simulate representation in favour of the triaxial model of the earth. Because the triaxial ellipsoidal surface has a changing curvature along the curve of constant latitude, the mathematical manipulation becomes complex and the computations are labour intensive. Also because the conformal map is flat or spherical in any sense, the formulae of spherical or plane trigonometry are applied. Unfortunately, the mathematics in the development of a geodesic on the ellipsoid and its image on the map are fairly complex, requiring advanced mathematics. Constructive geometry of the earth's surface has for a long time been devoted to the spherical model. The present paper offers the geometric model by which the constructive transformation of some ellipsoidal curves can be facilitated. The conformal transformation from/onto the ellipsoid and a spherical surface is utilized to set up the equations of the loxodrome on the ellipsoidal surface and the inverse of the geodesic curve from the ellipsoid to the conformal sphere. The procedure is illustrated and its merits are introduced by means of some practical geodetic problems.

Isothermic submanifolds of Euclidean space

We study the problem posed by F. Burstall of developing a theory of isothermic Euclidean submanifolds of dimension greater than or equal to three. As a natural extension of the definition in the surface case, we call a Euclidean submanifold isothermic if it is locally the image of a conformal immersion of a Riemannian product of Riemannian manifolds whose second fundamental form is adapted to the product net of the manifold. Our main result is a complete classification of all such conformal immersions of Riemannian products of dimension greater than or equal to three. We derive several consequences of this result. We also study whether the classical characterizations of isothermic surfaces as solutions of Christoffel’s problem and as envelopes of nontrivial conformal sphere congruences extend to higher dimensions. MSC 2000: 53 C15, 53 C12, 53 C50, 53 B25.

Lorentzian Kleinian groups

In this article we introduce some basic tools for the study of Lorentzian Kleinian groups. These groups are discrete subgroups of the Lorentzian Möbius group O(2,n) , acting properly discontinuously on some nonempty open subset of Einstein's universe, the Lorentzian analogue of the conformal sphere.

Homogeneous regions on the conformal sphere

All regions on a sphere which are similar with respect to a group of conformal transformations of this sphere are determined. Such regions are: the whole sphere, a hemisphere, the complement of a sphere of lower dimension, and the complement of a point.

The behavior of finite particles in conformal-covariant weak field theory

A finite classical particle is shown to be associable with the new concept of particle implied by the use of the conformal sphere geometry of space-time. The introduction of the finite particle is not only Lorentz but also conformal-covariant. The mass-energy relation for these free finite particles has an extra term generalizing the Einstein special relativistic mass-energy relation for point particles. In the conjugate (position) picture a similar extra term occurs in the Einstein proper time-time relation for point particles which extends the definition of proper time to finite particles. In §1 conformal form-invariant field and motion equations for the weak field case are given. The Maxwell Theory in vacuo is a special case. The finite effects are shown to have an electromagnetic origin; e.g., the size of the finite particle is the « classical radius » defined by its charge-mass ratio. In §2, the « extended » gravitational solutions are investigated. They contain theEinstein weak gravitation theory as a special case. These forces are shown to be charge-independent. In §3 the connection with radiation damping of the ultra-Maxwell terms in the « extended » electromagnetic solutions is investigated. Several new possibilities for avoiding the difficulties of point particles « self-force » are discussed. In §4 some aspects of the proposed exact equations are remarked. It is shown that general relativity is properly contained in conformal relativity in the sense that every exact solution of the former gives a « pure gravitational » exact solution of the latter.