Flat deformation of a spacetime admitting two Killing fields

It is shown that given an analytic Lorentzian metric on a 4-manifold, $g$, which admits two Killing vector fields, then it exists a local deformation law $\eta = a g + b H$, where $H$ is a 2-dimensional projector, such that $\eta$ is flat and admits the same Killing vectors. We also characterize the particular case when the projector $H$ coincides with the quotient metric. We apply some of our results to general stationary axisymmetric spacetime

Rigid motion revisited: rigid quasilocal frames

We introduce the notion of a rigid quasilocal frame (RQF) as a geometrically natural way to define a "system" in general relativity. An RQF is defined as a two-parameter family of timelike worldlines comprising the worldtube boundary of the history of a finite spatial volume, with the rigidity conditions that the congruence of worldlines is expansion-free (constant size) and shear-free (constant shape). This definition of a system is anticipated to yield simple, exact geometrical insights into the problem of motion in general relativity. It begins by answering the questions what is in motion (a rigid two-dimensional system boundary), and what motions of this rigid boundary are possible. Nearly a century ago Herglotz and Noether showed that a three-parameter family of timelike worldlines in Minkowski space satisfying Born's 1909 rigidity conditions has only three degrees of freedom instead of the six we are familiar with from Newtonian mechanics. We argue that in fact we can implement Born's notion of rigid motion in both flat spacetime (this paper) and arbitrary curved spacetimes containing sources (subsequent papers) - with precisely the expected three translational and three rotational degrees of freedom - provided the system is defined quasilocally as the two-dimensional set of points comprising the boundary of a finite spatial volume, rather than the three-dimensional set of points within the volume.Comment: 10 pages (two column), 24 pages (preprint), 1 figur

Flat deformation theorem and symmetries in spacetime

The \emph{flat deformation theorem} states that given a semi-Riemannian analytic metric $g$ on a manifold, locally there always exists a two-form $F$, a scalar function $c$, and an arbitrarily prescribed scalar constraint depending on the point $x$ of the manifold and on $F$ and $c$, say $\Psi (c, F, x)=0$, such that the \emph{deformed metric} $\eta = cg -\epsilon F^2$ is semi-Riemannian and flat. In this paper we first show that the above result implies that every (Lorentzian analytic) metric $g$ may be written in the \emph{extended Kerr-Schild form}, namely $\eta_{ab} := a g_{ab} - 2 b k_{(a} l_{b)}$ where $\eta$ is flat and $k_a, l_a$ are two null covectors such that $k_a l^a= -1$; next we show how the symmetries of $g$ are connected to those of $\eta$, more precisely; we show that if the original metric $g$ admits a Conformal Killing vector (including Killing vectors and homotheties), then the deformation may be carried out in a way such that the flat deformed metric $\eta$ `inherits' that symmetry.Comment: 30 pages, 0 figure

The Ostrogradskian Instability of Lagrangians with Nonlocality of Finite Extent

I reply to the objections recently raised by J. Llosa to my constructive proof that Lagrangians with nonlocality of finite extent inherit the full Ostrogradskian instability.Comment: 9 pages, LaTeX 2 epsilon, no figure

Hamiltonian Formalism for Space-time Non-commutative Theories

Space-time non-commutative theories are non-local in time. We develop the Hamiltonian formalism for non-local field theories in d space-time dimensions by considering auxiliary d+1 dimensional field theories which are local with respect to the evolution time. The Hamiltonian path integral quantization is considered and the Feynman rules in the Lagrangian formalism are derived. The case of non-commutative \phi^3 theory is considered as an example.Comment: 6 pages, A new section is added with other comments and references. To appear in PR

An exactly solvable model in P.R.M.: quantization. II

The quantization of a priori hamiltonian predictive systems is studied. It is applied to quantize the family of N-particle relativistic models presented in ref. 1. The relativistic two-particle oscillator is studied in detail and finally our results are compared with others that already appeared in the literature.On étudié la quantification de systèmes hamiltoniens pre-prédictifs a priori. Comme application, on quantifie la famille de modèles relativistes à N particules présentes dans la référence 1. On étudié en detail l’oscillateur relativiste a deux particules. Finalement on compare les résultats obtenus avec d’autres parus précédemment.Peer ReviewedPostprint (published version

Osteosíntesis mínimamente invasiva con placa en fracturas de radio distal tipo C

Introducción. La tendencia hacia el uso de técnicas mínimamente invasivas (MIPO) en el tratamiento de fracturas se justifica, por la reducción de la exposición de la fractura y la disección de los planos musculares, lo que facilita la recuperación precoz. Material y Método. Se realizó un estudio en dos fases, la primera descriptiva retrospectiva de una serie de 48 casos, y la segunda, del estudio longitudinal prospectivo de una serie de 23 casos, que se sometieron a MIPO tras una fractura intraarticular de radio distal tipo C de AO. Además de las variables: edad, sexo, lateralidad y mecanismo de lesión, se evaluaron los resultados funcionales, dolor, rango de movimiento y fuerza, a través de Mayo Wrist Score escala y el cuestionario DASH. Las fracturas se clasificaron utilizando las directrices AO, se evaluaron los resultados radiológicos y complicaciones. Resultados. Se realizó la descripción del abordaje quirúrgico de los71 pacientes: 43 mujeres y 28 hombres, con una media de edad de 57 años. El lado más afectado fue el izquierdo con 51 casos. El resultado funcional fue 68,33 en el primer estudio (retrospectivo) y 85,86 en el estudio prospectivo. Observamos una pérdida de fuerza en comparación con el lado contralateral, medida con un dinamómetro, de 5,19 Kg. (retrospectivo) y 2,98 Kg. (prospectivo). Radiográ- ficamente no hubo colapso intraarticular o pérdida de la reducción en el estudio prospectivo, y la consolidación se obtuvo en el 100% de los casos. Se observó, en el segundo estudio, un caso de síndrome de dolor regional complejo y sólo un caso requirió la retirada de material debido a la protrusión dorsal de un tornillo. Conclusiones. Se trata de una técnica segura que permite la reducción y estabilización de las fracturas de la extremidad distal del radio. No solo la estética, sino también, los resultados funcionales son mejores con la técnica MIPO.Introduction. The current trend towards the use of minimally invasive techniques (MIPO) in the treatment of fractures is justified by the reduced exposure of the fracture and dissection of the muscle planes, which facilitates early recovery. Materials and Methods. We performed a double study, a retrospective descriptive study of a series of 48 cases and a prospective longitudinal study of a series of 23 patients that underwent MIPO after intra-articular distal radius fracture. In addition to the variables: age, sex, side and injury mechanism, etc., functional outcomes, pain, range of motion and strength, through Mayo Wrist Score scale and DASH were assessed. Fractures were classified using the AO guidelines, radiological results and complications were assessed. Results. We conducted the description of surgical approach with surgical images. We analyzed a total of 71 patients, 43 women and 28 men with a mean age of 57 years. The most affected side was the left with 51 cases. According to the Mayo Wrist Score, functional outcome was 68,33 in the first study (retrospective) and 85.86 in the prospective study. A mean loss in strength, measured with a dynamometer of 5,19 Kg. (retrospective) and 2,98 Kg. (prospective), was observed when compared to the contralateral side. Radiographically no intraarticular collapse or loss reduction were observed in prospective study, and consolidation was obtained in 100% of the cases. One case of complex regional pain syndrome was observed and only one case required the removal of material because of dorsal protrusion of a screw in this study. Conclusions. It is a safe technique that allows the reduction and stabilization of fractures of the distal radius. Not only the aesthetic but also functional results are better with MIPO technique