Simonetta Frittelli

Spacetime perspective of Schwarzschild lensing

(November 29, 1999)We propose a definition of an exact lens equation without reference to a background spacetime,and construct the exact lens equation explicitly in the case of Schwarzschild spacetime. For theSchwarzschild case, we give exact expressions for the angular-diameter distance to the sources aswell as for the magnification factor and time of arrival of the images. We compare the exactlens equation with the standard lens equation, derived under the thin-lens-weak-field assumption(where the light rays are geodesics of the background with sharp bending in the lens plane, andthe gravitational field is weak), and verify the fact that the standard weak-field thin-lens equationis inadequate at small impact parameter. We show that the second-order correction to the weak-field thin-lens equation is inaccurate as well. Finally, we compare the exact lens equation with therecently proposed strong-field thin-lens equation, obtained under the assumption of straight pathsbut without the small angle approximation, i.e., with allowed large bending angles. We show thatthe strong-field thin-lens equation is remarkably accurate, even for lightrays that take several turnsaround the lens before reaching the observer.I. INTRODUCTION

The complete spectrum of the area from recoupling theory in loop quantum gravity

We compute the complete spectrum of the area operator in the loop representation of quantum gravity, using recoupling theory. This result extends previous derivations, which did not include the `degenerate sector, and agrees with the recently computed spectrum of the connection-representation area operator.

GR via characteristic surfaces

We reformulate the Einstein equations as equations for families of surfaces on a four‐manifold. These surfaces eventually become characteristic surfaces for an Einstein metric (with or without sources). In particular they are formulated in terms of two functions on R4×S2, i.e., the sphere bundle over space–time, one of the functions playing the role of a conformal factor for a family of associated conformal metrics, the other function describing an S2’s worth of surfaces at each space–time point. It is from these families of surfaces themselves that the conformal metric, conformal to an Einstein metric, is constructed; the conformal factor turns them into Einstein metrics. The surfaces are null surfaces with respect to this metric.

First order symmetric hyperbolic Einstein equations with arbitrary fixed gauge

We find a one-parameter family of variables which recast the 3+1 Einstein equations into first-order symmetric-hyperbolic form for any fixed choice of gauge. Hyperbolicity considerations lead us to a redefinition of the lapse in terms of an arbitrary factor times a power of the determinant of the 3-metric; under certain assumptions, the exponent can be chosen arbitrarily, but positive, with no implication of gauge-fixing.

Lorentzian metrics from characteristic surfaces

The following issue is raised and discussed; when do families of foliations by hypersurfaces on a given four‐dimensional manifold without further structure become the null surfaces of some unknown, but to be determined, metric gab(x). Explicit conditions for these surfaces are found, so that they do define a unique conformal metric with the surfaces themselves being characteristics of that metric. By giving an additional function (to be the conformal factor), full knowledge of the metric is determined. It is clear from these results that one can use these surfaces (and the conformal factor) as fundamental variables for describing any Lorentzian geometry and in particular for its use in general relativity.

An Exact universal gravitational lensing equation

We first define what we mean by gravitational lensing equations in a general space-time. A set of exact relations are then derived that can be used as the gravitational lens equations in all physical situations. The caveat is that into these equations there must be inserted a function, a two-parameter family of solutions to the eikonal equation, not easily obtained, that codes all the relevant (conformal) space-time information for this lens equation construction. Knowledge of this two-parameter family of solutions replaces knowledge of the solutions to the null geodesic equations. The formalism is then applied to the Schwarzschild lensing problem.

Linearized Einstein theory via null surfaces

Recently there has been developed a reformulation of general relativity (GR)—referred to as the null surface version of GR—where instead of the metric field as the basic variable of the theory, families of three‐surfaces in a four‐manifold become basic. From these surfaces themselves, a conformal metric, conformal to an Einstein metric, can be constructed. A choice of conformal factor turns it into an Einstein metric. The surfaces are then automatically characteristic surfaces of this metric. In the present paper we explore the linearization of this null surface theory and compare it with the standard linear GR. This allows a better understanding of many of the subtle mathematical issues and sheds light on some of the obscure points of the null surface theory. It furthermore permits a very simple solution generating scheme for the linear theory and the beginning of a perturbation scheme for the full theory.

On the dynamics of characteristic surfaces

We formulate the vacuum Einstein equations as differential equations for two functions, one complex and one real on a six‐dimensional manifold, M×S2, with M eventually becoming the space–time and the S2 becoming the sphere of null directions over M. At the start there is no other further structure available: the structure arising from the two functions. The complex function, referred to as Λ[M×S2], encodes information about a sphere’s worth of surfaces through each point of M. From knowledge of Λ one can define a second rank tensor on M which can be interpreted as a conformal metric, so that the ‘‘surfaces’’ are automatically null or characteristics of this conformal metric. The real function, Ω, plays the role of a conformal factor: it converts the conformal metric into a vacuum Einstein metric. Locally, all Einstein metrics can be obtained in this manner. In this work, we fully develop this ‘‘null surface version of general relativity (GR):’’ we display, discuss and analyze the equations, we show that m...

Image distortion in nonperturbative gravitational lensing

We introduce the idea of {it shape parameters} to describe the shape of the pencil of rays connecting an observer with a source lying on his past lightcone. On the basis of these shape parameters, we discuss a setting of image distortion in a generic (exact) spacetime, in the form of three {it distortion parameters}. The fundamental tool in our discussion is the use of geodesic deviation fields along a null geodesic to study how source shapes are propagated and distorted on the path to an observer. We illustrate this non-perturbative treatment of image distortion in the case of lensing by a Schwarzschild black hole. We conclude by showing that there is a non-perturbative generalization of the use of Fermats principle in lensing in the thin-lens approximation.

Image distortion from optical scalars in nonperturbative gravitational lensing

In a previous article concerning image distortion in non-perturbative gravitational lensing theory we described how to introduce shape and distortion parameters for small sources. We also showed how they could be expressed in terms of the scalar products of the geodesic deviation vectors of the sources pencil of rays in the past lightcone of an observer. In the present work we give an alternative approach to the description of the shape and distortion parameters and their evolution along the null geodesic from the source to the observer, but now in terms of the optical scalars (the convergence and shear of null vector field of the observers lightcone) and the associated optical equations, which relate the optical scalars to the curvature of the spacetime.