arXiv subjects
Publications and source records attributed to E. Minguzzi.
The full causal ladder of spacetimes is constructed, and their updated main properties are developed. Old concepts and alternative definitions of each level of the ladder are revisited, with emphasis in minimum hypotheses. The implications of the recently solved ``folk questions on smoothability'', and alternative proposals (as recent isocausality), are also summarized.
Causal continuity is usually defined by imposing the conditions (i) distinction and (ii) reflectivity. It is proved here that a new causality property which stays between weak distinction and causality, called feeble distinction, can actually replace distinction in the definition of causal continuity. An intermediate proof shows that feeble distinction and future (past) reflectivity implies past (resp. future) distinction. Some new characterizations of weak distinction and reflectivity are given.
A unifying framework for the study of causal relations is presented. The causal relations are regarded as subsets of M x M and the role of the corresponding antisymmetry conditions in the construction of the causal ladder is stressed. The causal hierarchy of spacetime is built from chronology up to K-causality and new characterizations of the distinction and strong causality properties are obtained. The closure of the causal future is not transitive, as a consequence its repeated composition leads to an infinite causal subladder between strong causality and K-causality - the A-causality subladder. A spacetime example is given which proves that K-causality differs from infinite A-causality.
Hawking's stable causality implies Sorkin and Woolgar's K-causality. The work investigates the possible equivalence between the two causality requirements, an issue which was first considered by H. Seifert and then raised again by R. Low after the introduction of K-causality. First, a new proof is given that a spacetime is stably causal iff the Seifert causal relation is a partial order. It is then shown that given a K-causal spacetime and chosen an event, the light cones can be widened in a neighborhood of the event without spoiling K-causality. The idea is that this widening of the light cones can be continued leading to a global one. Unfortunately, due to some difficulties in the inductive process the author was not able to complete the program for a proof as originally conceived by H. Seifert. Nevertheless, it is proved that if K-causality coincides with stable causality then in any K-causal spacetime the K future coincides with the Seifert future. Explicit examples are provided which show that the K^+ future may differ from the Seifert relation in causal spacetimes.
It is shown that the warped product spacetime P=M *_f H, where H is a complete Riemannian manifold, and the original spacetime M share necessarily the same causality properties, the only exceptions being the properties of causal continuity and causal simplicity which present some subtleties. For instance, it is shown that if diamH=+\infty, the direct product spacetime P=M*H is causally simple if and only if (M,g) is causally simple, the Lorentzian distance on M is continuous and any two causally related events at finite distance are connected by a maximizing geodesic. Similar conditions are found for the causal continuity property. Some new results concerning the behavior of the Lorentzian distance on distinguishing, causally continuous, and causally simple spacetimes are obtained. Finally, a formula which gives the Lorentzian distance on the direct product in terms of the distances on the two factors (M,g) and (H,h) is obtained.
This paper has been withdrawn by the author, since it is now a part of "Classical aspects of lightlike dimensional reduction" Class.Quant.Grav.23:7085-7110, 2006, gr-qc/0610011
We give a causal version of Eisenhart's geodesic characterization of classical mechanics. We emphasize the geometric, coordinate independent properties needed to express Eisenhart's theorem in light of modern studies on the Bargmann structures (lightlike dimensional reduction, pp-waves). The construction of the space metric, Coriolis 1-form and scalar potential through which the theorem is formulated is shown in detail, and in particular it is proved a one-to-one correspondence between Newtonian frames and Abelian connections on suitable lightlike principal bundles. The relation of Eisenhart's theorem in the lightlike case with a Fermat type principle is pointed out. The operation of lightlike lift is introduced and the existence of minimizers for the classical action is related to the causal simplicity of Eisenhart's spacetime.
It is well known that the Lorentzian length of a timelike curve in Minkowski spacetime is smaller than the Lorentzian length of the geodesic connecting its initial and final endpoints. The difference is known as the 'differential aging' and its calculation in terms of the proper acceleration history of the timelike curve would provide an important tool for the autonomous spacetime navigation of non-inertial observers. I give a solution in 3+1 dimensions which holds whenever the acceleration is decomposed with respect to a lightlike transported frame (lightlike transport will be defined), the analogous and more natural problem for a Fermi-Walker decomposition being still open.
Some aspects of lightlike dimensional reduction in flat spacetime are studied with emphasis to classical applications. Among them the Galilean transformation of shadows induced by inertial frame changes is studied in detail by proving that, (i) the shadow of an object has the same shape in every orthogonal-to-light screen, (ii) if two shadows are simultaneous in an orthogonal-to-light screen then they are simultaneous in any such screen. In particular, the Galilean group in 2+1 dimensions is recognized as an exact symmetry of Nature which acts on the shadows of the events instead that on the events themselves. The group theoretical approach to lightlike dimensional reduction is used to solve the reconstruction problem of a trajectory starting from its acceleration history or from its projected (shadow) trajectory. The possibility of obtaining a Galilean projected physics starting from a Poincaré invariant physics is stressed through the example of relativistic collisions. In particular, it is shown that the projection of a relativistic collision between massless particles gives a non-relativistic collision in which the kinetic energy is conserved.
We consider a generic minimal modification of the Newtonian potential, that is a modification that introduces only one additional dimensional parameter. The modified potential depends on a function whose behavior for large and small distances can be fixed in order to obtain respectively (i) galactic flat rotational curves and (ii) a universal constant acceleration independent of the masses of the interacting bodies (Pioneer anomaly). Then using a dimensional argument we show that the Tully-Fisher relation for the maximal rotational velocity of spiral galaxies follows without any further assumptions. This result suggests that the Pioneer anomalous acceleration and the flat rotational curves of galaxies could have a common origin in a modified gravitational theory. The relation of these results with the Modified Newtonian Dynamics (MOND) is discussed.
Starting from a weak gauge principle we give a new and critical revision of the argument leading to charge quantization on arbitrary spacetimes. The main differences of our approach with respect to previous works appear on spacetimes with non trivial torsion elements on its second integral cohomology group. We show that in these spacetimes there can be topologically non-trivial configurations of charged fields which do not imply charge quantization. However, the existence of a non-exact electromagnetic field always implies the quantization of charges. Another consequence of the theory for spacetimes with torsion is the fact that it gives rise to two natural quantization units that could be identified with the electric quantization unit (realized inside the quarks) and with the electron charge. In this framework the color charge can have a topological origin, with the number of colors being related to the order of the torsion subgroup. Finally, we discuss the possibility that the quantization of charge may be due to a weak non-exact component of the electromagnetic field extended over cosmological scales.
We study the behavior of clocks in 1+1 spacetime assuming the relativity principle, the principle of constancy of the speed of light and the clock hypothesis. These requirements are satisfied by a class of Finslerian theories parametrized by a real coefficient $β$, special relativity being recovered for $β=0$. The effect of differential aging is studied for the different values of $β$. Below the critical values $|β| =1/c$ the differential aging has the usual direction - after a round trip the accelerated observer returns younger than the twin at rest in the inertial frame - while above the critical values the differential aging changes sign. The non-relativistic case is treated by introducing a formal analogy with thermodynamics.
We give a closed expression for the Minkowski (1+1)-dimensional metric in the radar coordinates of an arbitrary non-inertial observer O in terms of O's proper acceleration. Knowledge of the metric allows the non-inertial observer to perform experiments in spacetime without making reference to inertial frames. To clarify the relation between inertial and non-inertial observers the coordinate transformation between radar and inertial coordinates, also is given. We show that every conformally flat coordinate system can be regarded as the radar coordinate system of a suitable observer for a suitable parametrization of the observer worldline. Therefore, the coordinate transformation between arbitrarily moving observers is a conformal transformation and conformally invariant (1+1)-dimensional theories lead to the same physics for all observers, independently of their relative motion.
We present some basic facts concerning simultaneity in both special and general relativity. We discuss Weyl's proof of the consistence of Einstein's synchronization convention and consider the general relativistic problem of assigning a time function to a congruence of timelike curves.
Recent results on the maximization of the charged-particle action I in a globally hyperbolic spacetime are discussed and generalized. We focus on the maximization of I over a given causal homotopy class C of curves connecting two causally related events x_0 <= x_1. Action I is proved to admit a maximum on C, and also one in the adherence of each timelike homotopy class. Moreover, the maximum on C is timelike if C contains a timelike curve (and the degree of differentiability of all the elements is at least C^2). In particular, this last result yields a complete Avez-Seifert type solution to the problem of connectedness through trajectories of charged particles in a globally hyperbolic spacetime endowed with an exact electromagnetic field: fixed any charge-to-mass ratio q/m, any two chronologically related events x_0 << x_1 can be connected by means of a timelike solution of the Lorentz force equation (LFE) corresponding to q/m. The accuracy of the approach is stressed by many examples, including an explicit counterexample (valid for all q/m) in the non-exact case. As a relevant previous step, new properties of the causal path space, causal homotopy classes and cut points on lightlike geodesics are studied.
There exist several approaches that investigate the connectedness of spacetime events through solutions of the Lorentz force equation. These approaches separate into three categories, that consider different equations. We clarify the physical meaning of each equation showing that only one method is based on the Lorentz force equation. The other two approaches lead respectively to a less restrictive equation that defines an electromagnetic flow on the cotangent fiber bundle, or to an unphysical constraint between charge-to-mass ratio and proper length of the solution. We outline the physical meaning of each approach studying the variational formulations and clarifying the results obtained in the explored directions.
As shown by the development of Special Relativity the simultaneity concept should be related to that of reference frame. Poincare' proposed to define the simultaneity of two events by means of light signals following what is nowadays known as the Einstein simultaneity convention. The need of a simultaneity definition is present also in general relativity and in curved spacetimes in order to provide the observers with a coordinate time. It is recognized that the old Einstein simultaneity convention is nothing but a connection on a suitable trivial bundle that defines the reference frame. Unfortunately, it has a non vanishing holonomy in curved and even in flat spacetimes a fact that makes it almost useless. We point out the advantage of local simultaneity conventions showing that they are represented by local simultaneity connections. Among them there is one, uniquely determined by the reference frame, which is particularly well-behaved globally.
In a globally hyperbolic spacetime any pair of chronologically related events admits a connecting geodesic. We present two theorems which prove that, more generally, under weak assumptions, given a charge-to-mass ratio there is always a connecting solution of the Lorentz force equation having that ratio. A geometrical interpretation of the charged-particle action is given which shows that the constructed solutions are maximizing.