Explaining Leibniz-equivalence as difference of non-inertial appearances: dis-solution of the Hole Argument and physical individuation of point-events

"The last remnant of physical objectivity of space-time" is disclosed in the case of a continuous family of spatially non-compact models of general relativity (GR). The {\it physical individuation} of point-events is furnished by the intrinsic degrees of freedom of the gravitational field, (viz, the {\it Dirac observables}) that represent - as it were - the {\it ontic} part of the metric field. The physical role of the {\it epistemic} part (viz. the {\it gauge} variables) is likewise clarified as emboding the unavoidable non-inertial aspects of GR. At the end the philosophical import of the {\it Hole Argument} is substantially weakened and in fact the Argument itself dis-solved, while a specific four-dimensional {\it holistic and structuralist} view of space-time, (called {\it point-structuralism}), emerges, including elements common to the tradition of both {\it substantivalism} and {\it relationism}. The observables of our models undergo real {\it temporal change}: this gives new evidence to the fact that statements like the {\it frozen-time} character of evolution, as other ontological claims about GR, are {\it model dependent}. \medskip Forthcoming in Studies in History and Philosophy of Modern PhysicsComment: 37 pages, talk at Oxford Conference on Spacetime (2004), to appear in Studies in History and Philosophy of Modern Physics. Affiliations Correcte

The Physical Role of Gravitational and Gauge Degrees of Freedom in General Relativity - I: Dynamical Synchronization and Generalized Inertial Effects

This is the first of a couple of papers in which, by exploiting the capabilities of the Hamiltonian approach to general relativity, we get a number of technical achievements that are instrumental both for a disclosure of \emph{new} results concerning specific issues, and for new insights about \emph{old} foundational problems of the theory. The first paper includes: 1) a critical analysis of the various concepts of symmetry related to the Einstein-Hilbert Lagrangian viewpoint on the one hand, and to the Hamiltonian viewpoint, on the other. This analysis leads, in particular, to a re-interpretation of {\it active} diffeomorphisms as {\it passive and metric-dependent} dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose the (nearly unknown) connection of a subgroup of them to Hamiltonian gauge transformations {\it on-shell}; 2) a re-visitation of the canonical reduction of the ADM formulation of general relativity, with particular emphasis on the geometro-dynamical effects of the gauge-fixing procedure, which amounts to the definition of a \emph{global (non-inertial) space-time laboratory}. This analysis discloses the peculiar \emph{dynamical nature} that the traditional definition of distant simultaneity and clock-synchronization assume in general relativity, as well as the {\it gauge relatedness} of the "conventions" which generalize the classical Einstein's convention.Comment: 45 pages, Revtex4, some refinements adde

Marzke-Wheeler coordinates for accelerated observers in special relativity

In special relativity, the definition of coordinate systems adapted to generic accelerated observers is a long-standing problem, which has found unequivocal solutions only for the simplest motions. We show that the Marzke-Wheeler construction, an extension of the Einstein synchronization convention, produces accelerated systems of coordinates with desirable properties: (a) they reduce to Lorentz coordinates in a neighborhood of the observers' world-lines; (b) they index continuously and completely the causal envelope of the world-line (that is, the intersection of its causal past and its causal future: for well-behaved world-lines, the entire space-time). In particular, Marzke-Wheeler coordinates provide a smooth and consistent foliation of the causal envelope of any accelerated observer into space-like surfaces. We compare the Marzke-Wheeler procedure with other definitions of accelerated coordinates; we examine it in the special case of stationary motions, and we provide explicit coordinate transformations for uniformly accelerated and uniformly rotating observers. Finally, we employ the notion of Marzke-Wheeler simultaneity to clarify the relativistic paradox of the twins, by pinpointing the local origin of differential aging.Comment: AmsLaTeX, 22 pages, 8 eps figures; revised, references added. To appear in Foundations of Physics Letters, October 200

Ephemeral point-events: is there a last remnant of physical objectivity?

For the past two decades, Einstein's Hole Argument (which deals with the apparent indeterminateness of general relativity due to the general covariance of the field equations) and its resolution in terms of Leibniz equivalence (the statement that Riemannian geometries related by active diffeomorphisms represent the same physical solution) have been the starting point for a lively philosophical debate on the objectivity of the point-events of space-time. It seems that Leibniz equivalence makes it impossible to consider the points of the space-time manifold as physically individuated without recourse to dynamical individuating fields. Various authors have posited that the metric field itself can be used in this way, but nobody so far has considered the problem of explicitly distilling the metrical fingerprint of point-events from the gauge-dependent components of the metric field. Working in the Hamiltonian formulation of general relativity, and building on the results of Lusanna and Pauri (2002), we show how Bergmann and Komar's intrinsic pseudo-coordinates (based on the value of curvature invariants) can be used to provide a physical individuation of point-events in terms of the true degrees of freedom (the Dirac observables) of the gravitational field, and we suggest how this conceptual individuation could in principle be implemented with a well-defined empirical procedure. We argue from these results that point-events retain a significant kind of physical objectivity.Comment: LaTeX, natbib, 34 pages. Final journal versio

New Directions in Non-Relativistic and Relativistic Rotational and Multipole Kinematics for N-Body and Continuous Systems

In non-relativistic mechanics the center of mass of an isolated system is easily separated out from the relative variables. For a N-body system these latter are usually described by a set of Jacobi normal coordinates, based on the clustering of the centers of mass of sub-clusters. The Jacobi variables are then the starting point for separating {\it orientational} variables, connected with the angular momentum constants of motion, from {\it shape} (or {\it vibrational}) variables. Jacobi variables, however, cannot be extended to special relativity. We show by group-theoretical methods that two new sets of relative variables can be defined in terms of a {\it clustering of the angular momenta of sub-clusters} and directly related to the so-called {\it dynamical body frames} and {\it canonical spin bases}. The underlying group-theoretical structure allows a direct extension of such notions from a non-relativistic to a special- relativistic context if one exploits the {\it rest-frame instant form of dynamics}. The various known definitions of relativistic center of mass are recovered. The separation of suitable relative variables from the so-called {\it canonical internal} center of mass leads to the correct kinematical framework for the relativistic theory of the orbits for a N-body system with action -at-a-distance interactions. The rest-frame instant form is also shown to be the correct kinematical framework for introducing the Dixon multi-poles for closed and open N-body systems, as well as for continuous systems, exemplified here by the configurations of the Klein-Gordon field that are compatible with the previous notions of center of mass.Comment: Latex, p.75, Invited contribution for the book {\it Atomic and Molecular Clusters: New Research} (Nova Science

The Physical Role of Gravitational and Gauge Degrees of Freedom in General Relativity - II: Dirac versus Bergmann observables and the Objectivity of Space-Time

(abridged)The achievements of the present work include: a) A clarification of the multiple definition given by Bergmann of the concept of {\it (Bergmann) observable. This clarification leads to the proposal of a {\it main conjecture} asserting the existence of i) special Dirac's observables which are also Bergmann's observables, ii) gauge variables that are coordinate independent (namely they behave like the tetradic scalar fields of the Newman-Penrose formalism). b) The analysis of the so-called {\it Hole} phenomenology in strict connection with the Hamiltonian treatment of the initial value problem in metric gravity for the class of Christoudoulou -Klainermann space-times, in which the temporal evolution is ruled by the {\it weak} ADM energy. It is crucial the re-interpretation of {\it active} diffeomorphisms as {\it passive and metric-dependent} dynamical symmetries of Einstein's equations, a re-interpretation which enables to disclose their (nearly unknown) connection to gauge transformations on-shell; this is expounded in the first paper (gr-qc/0403081). The use of the Bergmann-Komar {\it intrinsic pseudo-coordinates} allows to construct a {\it physical atlas} of 4-coordinate systems for the 4-dimensional {\it mathematical} manifold, in terms of the highly non-local degrees of freedom of the gravitational field (its four independent {\it Dirac observables}), and to realize the {\it physical individuation} of the points of space-time as {\it point-events} as a gauge-fixing problem, also associating a non-commutative structure to each 4-coordinate system.Comment: 41 pages, Revtex

Dynamical Body Frames, Orientation-Shape Variables and Canonical Spin Bases for the Non-Relativistic N-Body Problem

After the separation of the center-of-mass motion, a new privileged class of canonical Darboux bases is proposed for the non-relativistic N-body problem by exploiting a geometrical and group theoretical approach to the definition of {\it body frame} for deformable bodies. This basis is adapted to the rotation group SO(3), whose canonical realization is associated with a symmetry Hamiltonian {\it left action}. The analysis of the SO(3) coadjoint orbits contained in the N-body phase space implies the existence of a {\it spin frame} for the N-body system. Then, the existence of appropriate non-symmetry Hamiltonian {\it right actions} for non-rigid systems leads to the construction of a N-dependent discrete number of {\it dynamical body frames} for the N-body system, hence to the associated notions of {\it dynamical} and {\it measurable} orientation and shape variables, angular velocity, rotational and vibrational configurations. For N=3 the dynamical body frame turns out to be unique and our approach reproduces the {\it xxzz gauge} of the gauge theory associated with the {\it orientation-shape} SO(3) principal bundle approach of Littlejohn and Reinsch. For $N \geq 4$ our description is different, since the dynamical body frames turn out to be {\it momentum dependent}. The resulting Darboux bases for $N\geq 4$ are connected to the coupling of the {\it spins} of particle clusters rather than the coupling of the {\it centers of mass} (based on Jacobi relative normal coordinates). One of the advantages of the spin coupling is that, unlike the center-of-mass coupling, it admits a relativistic generalization

Bound and Radiation Fields in the Rindler Frame

The energy-momentum tensor of the Li\'enard-Wiechert field is split into bound and emitted parts in the Rindler frame, by generalizing the reasoning of Teitelboim applied in the inertial frame. Our analysis proceeds by invoking the concept of ``energy'' defined with respect to the Killing vector field attached to the frame. We obtain the radiation formula in the Rindler frame (the Rindler version of the Larmor formula), and it is found that the radiation power is proportional to the square of acceleration $\alpha^\mu$ of the charge relative to the Rindler frame. This result leads us to split the Li\'enard-Wiechert field into a part II', which is linear in $\alpha^\mu$, and a part I', which is independent of $\alpha^\mu$. By using these, we split the energy-momentum tensor into two parts. We find that these are properly interpreted as the emitted and bound parts of the tensor in the Rindler frame. In our identification of radiation, a charge radiates neither in the case that the charge is fixed in the Rindler frame, nor in the case that the charge satisfies the equation $\alpha^\mu=0$. We then investigate this equation. We consider four gedanken experiments related to the observer dependence of the concept of radiation.Comment: 30 pages 2 figure