Isochronous Spacetimes

Abstract

The possibility has been recently demonstrated to manufacture (nonrelativistic, Hamiltonian) many-body problems which feature an isochronous time evolution with an arbitrarily assigned period T yet mimic with good approximation, or even exactly, any given many-body problem (within a quite large class, encompassing most of nonrelativistic physics) over timesT which may also be arbitrarily large (but of course such thatT < T ). In this paper we review and further explore the possibility to extend this finding to a general relativity context, so that it becomes relevant for cosmology.

Isochronous solutions of Einstein’s equations and their Newtonian limit

“…As in the case of the treatments of non-relativistic dynamical systems, including Nbody problems with Newtonian equations of motion ("accelerations equal forces") [1,2], the findings we recently reported [3,4] have an unpalatable connotation: they indicate that the hope to ascertain which is the correct description of our cosmos is doomed by the possibility to identify quite different cosmologies-for instance displaying or not displaying the peculiar feature to be isochronous, or featuring the effects of a Big Bang without actually experiencing it-which are however indistinguishable by any, reasonably conceivable, human experiment [3,4]. The unpleasant feeling caused by this notion might perhaps resemble that experienced by physicists educated before the discovery of quantum mechanics, when they were confronted by the notion that it is actually impossible to measure simultaneously with unlimited accuracy the position and the velocity of a moving particle.…”

“…In [4] it has been pointed out that the realization of the isochronous spacetimes is not restricted to homogeneous, isotropic and spatially flat metrics, but it can be easily extended to any synchronous metric. Therefore our result is quite general, since any metric can be written in synchronous form by a diffeomorphic change of coordinates.…”

“…Therefore our result is quite general, since any metric can be written in synchronous form by a diffeomorphic change of coordinates. Furthermore, it has been shown that isochronous spacetimes can be realized in two different ways: the first by means of isochronous metrics that are degenerate at a discrete set of instants t n when the time reversals occur; the second via non-degenerate metrics featuring a jump in their first derivatives at the inversion times t n , which then implies a distributional contribution in the stressenergy tensor at t n , see the discussion in [4].…”

“…In particular we show that in this limit every isochronous metric reduces to a solution of an isochronous dynamical systemD that is obtained from a Newtonian N -body system through the techniques [1,2] which have been introduced in the context of autonomous dynamical systems. In fact, our purpose and scope is to provide a bridge among the results valid respectively in the general relativity and in the classical mechanics contexts, by revisiting in the new context [3,4,5] the standard derivation of Newtonian gravitational physics from general relativity. This paper is organized as follows: in section 2 we review some mathematical aspects of isochronous metrics, which shall be useful for the study of the Newtonian limit; this treatment includes an extended review of previous findings [3,4,5], which we considered appropriate given the somewhat "unpleasant"-yet in our opinion cogent-implications of these findings, which do imply a substantial limitation of all cosmological theories based on general relativity.…”

“…In fact, our purpose and scope is to provide a bridge among the results valid respectively in the general relativity and in the classical mechanics contexts, by revisiting in the new context [3,4,5] the standard derivation of Newtonian gravitational physics from general relativity. This paper is organized as follows: in section 2 we review some mathematical aspects of isochronous metrics, which shall be useful for the study of the Newtonian limit; this treatment includes an extended review of previous findings [3,4,5], which we considered appropriate given the somewhat "unpleasant"-yet in our opinion cogent-implications of these findings, which do imply a substantial limitation of all cosmological theories based on general relativity. In section 3 we investigate the weak field limit of our treatment of general relativity and thereby derive the corresponding Newtonian framework.…”