Poincaré on the Foundation of Geometry in the Understanding

This paper is about Poincaré’s view of the foundations of geometry. According to the established view, which has been inherited from the logical positivists, Poincaré, like Hilbert, held that axioms in geometry are schemata that provide implicit definitions of geometric terms, a view he expresses by stating that the axioms of geometry are “definitions in disguise.” I argue that this view does not accord well with Poincaré’s core commitment in the philosophy of geometry: the view that geometry is the study of groups of operations. In place of the established view I offer a revised view, according to which Poincaré held that axioms in geometry are in fact assertions about invariants of groups. Groups, as forms of the understanding, are prior in conception to the objects of geometry and afford the proper definition of those objects, according to Poincaré. Poincaré’s view therefore contrasts sharply with Kant’s foundation of geometry in a unique form of sensibility. According to my interpretation, axioms are not definitions in disguise because they themselves implicitly define their terms, but rather because they disguise the definitions which imply them

Restricted three-body problem in effective-field-theory models of gravity

One of the outstanding problems of classical celestial mechanics was the restricted 3-body prob- lem, in which a planetoid of small mass is subject to the Newtonian attraction of two celestial bodies of large mass, as it occurs, for example, in the sun-earth-moon system. On the other hand, over the last decades, a systematic investigation of quantum corrections to the Newtonian potential has been carried out in the literature on quantum gravity. The present paper studies the effect of these tiny quantum corrections on the evaluation of equilibrium points. It is shown that, despite the extreme smallness of the corrections, there exists no choice of sign of these corrections for which all qualitative features of the restricted 3-body problem in Newtonian theory remain unaffected. Moreover, first-order stability of equilibrium points is characterized by solving a pair of algebraic equations of fifth degree, where some coefficients depend on the Planck length. The coordinates of stable equilibrium points are slightly changed with respect to Newtonian theory, because the planetoid is no longer at equal distance from the two bodies of large mass. The effect is conceptually interesting but too small to be observed, at least for the restricted 3-body problems available in the solar system.Comment: 20 pages, latex, 8 figure

Maximal mass of uniformly rotating homogeneous stars in Einsteinian gravity

Using a multi domain spectral method, we investigate systematically the general-relativistic model for axisymmetric uniformly rotating, homogeneous fluid bodies generalizing the analytically known Maclaurin and Schwarzschild solutions. Apart from the curves associated with these solutions and a further curve of configurations that rotate at the mass shedding limit, two more curves are found to border the corresponding two parameter set of solutions. One of them is a Newtonian lens shaped sequence bifurcating from the Maclaurin spheroid sequence, while the other one corresponds to highly relativistic bodies with an infinite central pressure. The properties of the configuration for which both the gravitational and the baryonic masses, moreover angular velocity, angular momentum as well as polar red shift obtain their maximal values are discussed in detail. In particular, by comparison with the static Schwarzschild solution, we obtain an increase of 34.25% in the gravitational mass. Moreover, we provide exemplarily a discussion of angular velocity and gravitational mass on the entire solution class.Comment: 4 pages, 4 figures, 1 table, submitted to A&A, corrected eq. for W, W' in 3.

Libration driven elliptical instability

The elliptical instability is a generic instability which takes place in any rotating flow whose streamlines are elliptically deformed. Up to now, it has been widely studied in the case of a constant, non-zero differential rotation between the fluid and the elliptical distortion with applications in turbulence, aeronautics, planetology and astrophysics. In this letter, we extend previous analytical studies and report the first numerical and experimental evidence that elliptical instability can also be driven by libration, i.e. periodic oscillations of the differential rotation between the fluid and the elliptical distortion, with a zero mean value. Our results suggest that intermittent, space-filling turbulence due to this instability can exist in the liquid cores and sub-surface oceans of so-called synchronized planets and moons

Electric charge in the field of a magnetic event in three-dimensional spacetime

We analyze the motion of an electric charge in the field of a magnetically charged event in three-dimensional spacetime. We start by exhibiting a first integral of the equations of motion in terms of the three conserved components of the spacetime angular momentum, and then proceed numerically. After crossing the light cone of the event, an electric charge initially at rest starts rotating and slowing down. There are two lengths appearing in the problem: (i) the characteristic length $\frac{q g}{2 \pi m}$, where $q$ and $m$ are the electric charge and mass of the particle, and $g$ is the magnetic charge of the event; and (ii) the spacetime impact parameter $r_0$. For $r_0 \gg \frac{q g}{2 \pi m}$, after a time of order $r_0$, the particle makes sharply a quarter of a turn and comes to rest at the same spatial position at which the event happened in the past. This jump is the main signature of the presence of the magnetic event as felt by an electric charge. A derivation of the expression for the angular momentum that uses Noether's theorem in the magnetic representation is given in the Appendix.Comment: Version to appear in Phys. Rev.

Gravitational Energy Loss and Binary Pulsars in the Scalar Ether-Theory of Gravitation

Motivation is given for trying a theory of gravity with a preferred reference frame (``ether'' for short). One such theory is summarized, that is a scalar bimetric theory. Dynamics is governed by an extension of Newton's second law. In the static case, geodesic motion is recovered together with Newton's attraction field. In the static spherical case, Schwarzschild's metric is got. An asymptotic scheme of post-Minkowskian (PM) approximation is built by associating a conceptual family of systems with the given weakly-gravitating system. It is more general than the post-Newtonian scheme in that the velocity may be comparable with $c$. This allows to justify why the 0PM approximation of the energy rate may be equated to the rate of the Newtonian energy, as is usually done. At the 0PM approximation of this theory, an isolated system loses energy by quadrupole radiation, without any monopole or dipole term. It seems plausible that the observations on binary pulsars (the pulse data) could be nicely fitted with a timing model based on this theory.Comment: Text of a talk given at the 4th Conf. on Physics Beyond the Standard Model, Tegernsee, June 2003, submitted to the Proceedings (H. V. Klapdor-Kleingrothaus, ed.

Revealing the state space of turbulent pipe flow by symmetry reduction

Symmetry reduction by the method of slices is applied to pipe flow in order to quotient the stream-wise translation and azimuthal rotation symmetries of turbulent flow states. Within the symmetry-reduced state space, all travelling wave solutions reduce to equilibria, and all relative periodic orbits reduce to periodic orbits. Projections of these solutions and their unstable manifolds from their $\infty$-dimensional symmetry-reduced state space onto suitably chosen 2- or 3-dimensional subspaces reveal their interrelations and the role they play in organising turbulence in wall-bounded shear flows. Visualisations of the flow within the slice and its linearisation at equilibria enable us to trace out the unstable manifolds, determine close recurrences, identify connections between different travelling wave solutions, and find, for the first time for pipe flows, relative periodic orbits that are embedded within the chaotic attractor, which capture turbulent dynamics at transitional Reynolds numbers.Comment: 24 pages, 12 figure

Lorentz Transformation from Symmetry of Reference Principle

The Lorentz Transformation is traditionally derived requiring the Principle of Relativity and light-speed universality. While the latter can be relaxed, the Principle of Relativity is seen as core to the transformation. The present letter relaxes both statements to the weaker, Symmetry of Reference Principle. Thus the resulting Lorentz transformation and its consequences (time dilatation, length contraction) are, in turn, effects of how we manage space and time.Comment: 2 page

Remarks on endomorphisms and rational points

Let X be a variety over a number field and let f: X --> X be an "interesting" rational self-map with a fixed point q. We make some general remarks concerning the possibility of using the behaviour of f near q to produce many rational points on X. As an application, we give a simplified proof of the potential density of rational points on the variety of lines of a cubic fourfold (originally obtained by Claire Voisin and the first author in 2007).Comment: LaTeX, 22 pages. v2: some minor observations added, misprints corrected, appendix modified

Gravity as Archimedes' thrust and a bifurcation in that theory

Euler's interpretation of Newton's gravity (NG) as Archimedes' thrust in a fluid ether is presented in some detail. Then a semi-heuristic mechanism for gravity, close to Euler's, is recalled and compared with the latter. None of these two "gravitational ethers" can obey classical mechanics. This is logical since the ether defines the very reference frame, in which mechanics is defined. This concept is used to build a scalar theory of gravity: NG corresponds to an incompressible ether, a compressible ether leads to gravitational waves. In the Lorentz-Poincar\'e version, special relativity is compatible with the ether, but, with the heterogeneous ether of gravity, it applies only locally. A correspondence between metrical effects of uniform motion and gravitation is assumed, yet in two possible versions (one is new). Dynamics is based on a (non-trivial) extension of Newton's second law. The observational status for the theory with the older version of the correspondence is summarized.Comment: 24 pages, invited contribution to the Franco Selleri Festschrift, to appear in Found. Physics. v3: Endnote 45 on absolute simultaneity improved (formerly footnote 6: class file changed to revtex4), a few references updated (and now with titles). v2: minor correction in subsect. 3.2, some wording improvements, and a few references adde