The Vacuum Einstein Equations via Holonomy around Closed Loops on Characteristic Surfaces

We reformulate the standard local equations of general relativity for asymptotically flat spacetimes in terms of two non-local quantities, the holonomy $H$ around certain closed null loops on characteristic surfaces and the light cone cut function $Z$, which describes the intersection of the future null cones from arbitrary spacetime points, with future null infinity. We obtain a set of differential equations for $H$ and $Z$ equivalent to the vacuum Einstein equations. By finding an algebraic relation between $H$ and $Z$ this set of equations is reduced to just two coupled equations: an integro-differential equation for $Z$ which yields the conformal structure of the underlying spacetime and a linear differential equation for the ``vacuum'' conformal factor. These equations, which apply to all vacuum asymptotically flat spacetimes, are however lengthy and complicated and we do not yet know of any solution generating technique. They nevertheless are amenable to an attractive perturbative scheme which has Minkowski space as a zeroth order solution.Comment: 28 pages, RevTeX, 3 PostScript figure

Ecuador's Economy Since 2007

Outside observers could be forgiven for expecting Ecuador to suffer terribly during and after the recent global crisis. Two of the country's largest sources of foreign earnings, petroleum exports and remittances from abroad, crashed during the global recession. Furthermore, lacking its own currency, the government's options for responding were limited. But instead of a deep, protracted recession, Ecuador lost only 1.3 percent of GDP during three quarters of contraction. After four additional quarters the economy returned to the pre-recession level of output, and two years after the recession started, it reached its 20-year growth trend

Asymptotically Shear-free and Twist-free Null Geodesic Congruences

We show that, though they are rare, there are asymptotically flat space-times that possess null geodesic congruences that are both asymptotically shear- free and twist-free (surface forming). In particular, we display the class of space-times that possess this property and demonstrate how these congruences can be found. A special case within this class are the Robinson- Trautman space-times. In addition, we show that in each case the congruence is isolated in the sense that there are no other neighboring congruences with this dual property.Comment: 10 page

Electromagnetic Dipole Radiation Fields, Shear-Free Congruences and Complex Center of Charge World Lines

We show that for asymptotically vanishing Maxwell fields in Minkowski space with non-vanishing total charge, one can find a unique geometric structure, a null direction field, at null infinity. From this structure a unique complex analytic world-line in complex Minkowski space that can be found and then identified as the complex center of charge. By ''sitting'' - in an imaginary sense, on this world-line both the (intrinsic) electric and magnetic dipole moments vanish. The (intrinsic) magnetic dipole moment is (in some sense) obtained from the `distance' the complex the world line is from the real space (times the charge). This point of view unifies the asymptotic treatment of the dipole moments For electromagnetic fields with vanishing magnetic dipole moments the world line is real and defines the real (ordinary center of charge). We illustrate these ideas with the Lienard-Wiechert Maxwell field. In the conclusion we discuss its generalization to general relativity where the complex center of charge world-line has its analogue in a complex center of mass allowing a definition of the spin and orbital angular momentum - the analogues of the magnetic and electric dipole moments.Comment: 17 page

The Real Meaning of Complex Minkowski-Space World-Lines

In connection with the study of shear-free null geodesics in Minkowski space, we investigate the real geometric effects in real Minkowski space that are induced by and associated with complex world-lines in complex Minkowski space. It was already known, in a formal manner, that complex analytic curves in complex Minkowski space induce shear-free null geodesic congruences. Here we look at the direct geometric connections of the complex line and the real structures. Among other items, we show, in particular, how a complex world-line projects into the real Minkowski space in the form of a real shear-free null geodesic congruence.Comment: 16 page

Asymptotically Stationary and Static Space-times and Shear-free Null Geodesic Congruences

In classical electromagnetic theory, one formally defines the complex dipole moment (the electric plus 'i' magnetic dipole) and then computes (and defines) the complex center of charge by transforming to a complex frame where the complex dipole moment vanishes. Analogously in asymptotically flat space-times it has been shown that one can determine the complex center of mass by transforming the complex gravitational dipole (mass dipole plus 'i' angular momentum) (via an asymptotic tetrad trasnformation) to a frame where the complex dipole vanishes. We apply this procedure to such space-times which are asymptotically stationary or static, and observe that the calculations can be performed exactly, without any use of the approximation schemes which must be employed in general. In particular, we are able to exactly calculate complex center of mass and charge world-lines for such space-times, and - as a special case - when these two complex world-lines coincide, we recover the Dirac value of the gyromagnetic ratio.Comment: 11 page

Center of Mass and spin for isolated sources of gravitational radiation

We define the center of mass and spin of an isolated system in General Relativity. The resulting relationships between these variables and the total linear and angular momentum of the gravitational system are remarkably similar to their Newtonian counterparts, though only variables at the null boundary of an asymptotically flat spacetime are used for their definition. We also derive equations of motion linking their time evolution to the emitted gravitational radiation. The results are then compared to other approaches. In particular one obtains unexpected similarities as well as some differences with results obtained in the Post Newtonian literature . These equations of motion should be useful when describing the radiation emitted by compact sources such as coalescing binaries capable of producing gravitational kicks, supernovas, or scattering of compact objects.Comment: 16 pages. Accepted for publication in Phys. Rev.

Astrophysical limits on quantum gravity motivated birefringence

We obtain observational upper bounds on a class of quantum gravity related birefringence effects, by analyzing the presence of linear polarization in the optical and ultraviolet spectrum of some distant sources. In the notation of Gambini and Pullin we find $\chi < 10^{-3}$.Comment: 4 pages, submitted to Phys. Rev. Let

Twisting Null Geodesic Congruences, Scri, H-Space and Spin-Angular Momentum

The purpose of this work is to return, with a new observation and rather unconventional point of view, to the study of asymptotically flat solutions of Einstein equations. The essential observation is that from a given asymptotically flat space-time with a given Bondi shear, one can find (by integrating a partial differential equation) a class of asymptotically shear-free (but, in general, twistiing) null geodesic congruences. The class is uniquely given up to the arbitrary choice of a complex analytic world-line in a four-parameter complex space. Surprisingly this parameter space turns out to be the H-space that is associated with the real physical space-time under consideration. The main development in this work is the demonstration of how this complex world-line can be made both unique and also given a physical meaning. More specifically by forcing or requiring a certain term in the asymptotic Weyl tensor to vanish, the world-line is uniquely determined and becomes (by several arguments) identified as the `complex center-of-mass'. Roughly, its imaginary part becomes identified with the intrinsic spin-angular momentum while the real part yields the orbital angular momentum.Comment: 26 pages, authors were relisted alphabeticall

Null Surfaces and Legendre Submanifolds

It is shown that the main variable Z of the Null Surface Formulation of GR is the generating function of a constrained Lagrange submanifold that lives on the energy surface H=0 and that its level surfaces Z=const. are Legendre submanifolds on that energy surface. The behaviour of the variable Z at the caustic points is analysed and a genralization of this variable is discussed.Comment: 28 pages, 7 figure