Abstract:
A priori, there is nothing very special about shear-free or asymptotically shear-free null geodesic congruences. Surprisingly, however, they turn out to possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant effects. It is the purpose of this paper to try to fully develop these issues. This work starts with a detailed exposition of the theory of shear-free and asymptotically shear-free null geodesic congruences, i.e., congruences with shear that vanishes at future conformal null infinity. A major portion of the exposition lies in the analysis of the space of regular shear-free and asymptotically shear-free null geodesic congruences. This analysis leads to the space of complex analytic curves in an auxiliary four-complex dimensional space, H-space. They in turn play a dominant role in the applications. The applications center around the problem of extracting interior physical properties of an asymptotically-flat spacetime directly from the asymptotic gravitational (and Maxwell) field itself, in analogy with the determination of total charge by an integral over the Maxwell field at infinity or the identification of the interior mass (and its loss) by (Bondi's) integrals of the Weyl tensor, also at infinity. More specifically, we will see that the asymptotically shear-free congruences lead us to an asymptotic definition of the center-of-mass and its equations of motion. This includes a kinematic meaning, in terms of the center-of-mass motion, for the Bondi three-momentum. In addition, we obtain insights into intrinsic spin and, in general, angular momentum, including an angular-momentum--conservation law with well-defined flux terms. When a Maxwell field is present, the asymptotically shear-free congruences allow us to determine/define at infinity a center-of-charge world line and intrinsic magnetic dipole moment.

Abstract:
A priori, there is nothing very special about shear-free or asymptotically shear-free null geodesic congruences. Surprisingly, however, they turn out to possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant effects. It is the purpose of this paper to try to fully develop these issues. This work starts with a detailed exposition of the theory of shear-free and asymptotically shear-free null geodesic congruences, i.e., congruences with shear that vanishes at future conformal null infinity. A major portion of the exposition lies in the analysis of the space of regular shear-free and asymptotically shear-free null geodesic congruences. This analysis leads to the space of complex analytic curves in complex Minkowski space. They in turn play a dominant role in the applications. The applications center around the problem of extracting interior physical properties of an asymptotically-flat spacetime directly from the asymptotic gravitational (and Maxwell) field itself, in analogy with the determination of total charge by an integral over the Maxwell field at infinity or the identification of the interior mass (and its loss) by (Bondi’s) integrals of the Weyl tensor, also at infinity. More specifically, we will see that the asymptotically shear-free congruences lead us to an asymptotic definition of the center-of-mass and its equations of motion. This includes a kinematic meaning, in terms of the center-of-mass motion, for the Bondi three-momentum. In addition, we obtain insights into intrinsic spin and, in general, angular momentum, including an angular-momentum–conservation law with well-defined flux terms. When a Maxwell field is present, the asymptotically shear-free congruences allow us to determine/define at infinity a center-of-charge world line and intrinsic magnetic dipole moment.

Abstract:
Nell?ambito di un progetto di ricerca sull?ecologia dello scoiattolo comune in boschi di conifere delle Alpi, abbiamo avviato uno studio con la radiotelemetria per indagare i fattori che influiscono sull?uso dello spazio da parte degli animali. I risultati riportati nel presente lavoro si riferiscono a due anni successivi caratterizzati da una diversa disponibilità alimentare. L?area di studio si trovava nel Parco Nazionale del Gran Paradiso, in Val di Rhemes, all?interno di una pecceta subalpina (Picea abies 85%, Larix decidua 11%, alberi morti 4%). La produzione energetica del bosco (semi delle conifere) è stata valutata moltiplicando il n. di piante per ettaro x il n. medio di coni prodotti (contati su 60 alberi campione) x il n. medio di semi per cono x il peso medio dei semi, trasformando poi la biomassa in Mj. Le catture sono state effettuate tre volte l?anno nel 2001 e 2002 con 30 trappole incruente Tomahawk tipo 201. Diciotto scoiattoli nel 2001 e 13 nel 2002 sono stati dotati di radiocollare (PD-2C Holohil Systems Ltd.) e seguiti in estate e autunno. Sono stati calcolati i seguenti parametri: home range MCP 100%, MCP 95% (animali con singole escursioni), 100% Cluster-based (animali che usavano differenti aree di attività); stime delle core-area mono e multinucleari effettuate con la tecnica della Cluster Analysis 85%; sovrapposizione delle core-area. Nel 2001, all?inizio dell?estate, 4 maschi su 8 e 7 femmine su 8 sono emigrati nella valle adiacente o a quote più basse. Nel 2002, tutti gli individui sono rimasti residenti. La dimensione media degli home range stagionali nel 2001 è stata di 83,30 ± 48,72 ha (n = 30) contro 31,04 ± 16,65 ha (n = 19) nel 2002, la media delle core-area è stata 18,18 ± 17,74 ha nel 2001 e 9,36 ± 5,40 ha in 2002 (Kruskal-Wallis ANOVA: home range H = 22,6; g.l. = 1, P < 0,0001; core-area H = 4,55, g.l. = 1, P = 0,033). La sovrapposizione delle core-area maschio/femmina e femmina/maschio è stata maggiore nel 2001 rispetto al 2002. Nel 2001 e nell?estate del 2002 le core-area femmina/femmina sono risultate fortemente sovrapposte, mentre nell?autunno 2002 le femmine presentavano core-area esclusive. L?elevata dimensione degli home range e la grande sovrapposizione delle core-area riscontrata nel 2001, suggeriscono che si sia verificata un?alterazione dell?organizzazione sociale tipica della specie e dei consueti pattern d?uso dello spazio. Normalmente nello scoiattolo comune le femmine difendono core-area esclusivi da altre femmine adulte; i maschi adulti hanno home-range più estesi, con core-area che si sovrappongono in

Abstract:
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.

Abstract:
We investigate the geometry of a particular class of null surfaces in space-time called vacuum Non-Expanding Horizons (NEHs). Using the spin-coefficient equation, we provide a complete description of the horizon geometry, as well as fixing a canonical choice of null tetrad and coordinates on a NEH. By looking for particular classes of null geodesic congruences which live exterior to NEHs but have the special property that their shear vanishes at the intersection with the horizon, a good cut formalism for NEHs is developed which closely mirrors asymptotic theory. In particular, we show that such null geodesic congruences are generated by arbitrary choice of a complex world-line in a complex four dimensional space, each such choice induces a CR structure on the horizon, and a particular world-line (and hence CR structure) may be chosen by transforming to a privileged tetrad frame.

Abstract:
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.

Abstract:
We study the physical consequences of two diffferent but closely related perturbation schemes applied to the Einstein-Maxwell equations. In one case the starting space-time is flat while in the other case it is Schwarzschild. In both cases the perturbation is due to a combined electric and magnetic dipole field. We can see, within the Einstein-Maxwell equations a variety of physical consequences. They range from induced gravitational energy-momentum loss, to a well defined spin angular momentum with its loss and a center-of-mass with its equations of motion.

Abstract:
The properties of null geodesic congruences (NGCs) in Lorentzian manifolds are a topic of considerable importance. More specifically NGCs with the special property of being shear-free or asymptotically shear-free (as either infinity or a horizon is approached) have received a great deal of recent attention for a variety of reasons. Such congruences are most easily studied via solutions to what has been referred to as the 'good cut equation' or the 'generalization good cut equation'. It is the purpose of this note to study these equations and show their relationship to each other. In particular we show how they all have a four complex dimensional manifold (known as H-space, or in a special case as complex Minkowski space) as a solution space.

Abstract:
We study geometric structures associated with shear-free null geodesic congruences in Minkowski space-time and asymptotically shear-free null geodesic congruences in asymptotically flat space-times. We show how in both the flat and asymptotically flat settings, complexified future null infinity acts as a "holographic screen," interpolating between two dual descriptions of the null geodesic congruence. One description constructs a complex null geodesic congruence in a complex space-time whose source is a complex world-line; a virtual source as viewed from the holographic screen. This complex null geodesic congruence intersects the real asymptotic boundary when its source lies on a particular open-string type structure in the complex space-time. The other description constructs a real, twisting, shear-free or asymptotically shear-free null geodesic congruence in the real space-time, whose source (at least in Minkowski space) is in general a closed-string structure: the caustic set of the congruence. Finally we show that virtually all of the interior space-time physical quantities that are identified at null infinity (center of mass, spin, angular momentum, linear momentum, force) are given kinematic meaning and dynamical descriptions in terms of the complex world-line.