The aim of this paper is twofolded. (1) Showing that Newtonian mechanics of point particles in static potentials admits an alternative description in terms of effective riemannian

The last years have witnessed a great interest in the study of analog gravity [

Although the first analog models appeared in the early days of general relativity [

In this paper, I adopt a different perspective concerning “artificial” gravity. I analyze if effective spacetime techniques are suitable to describe some aspects of Newtonian mechanics of point particles and vice versa. As it is known, the relation between curved geometries and particle dynamics was studied at least since the time of Jacobi [

It will be shown that Newton’s mechanics of point particles in static potentials may provide a very simple analog model of gravitation. At first this statement seems to be suspicious once we will obtain an alternative description of Newtonian trajectories in terms of pseudo-Riemannian spacetimes without any reference to relativity theory. Nevertheless, using the well-known optical-mechanical analogy as a starting point we will see that, indeed, it is possible to give an effective spacetime description of the motion instead of using the traditional description in terms of forces. We will see that the resulting analog model works as a counterpart of Gordon’s metric in the context of optics, while the trajectory of the particle is mapped into null geodesics of an effective spacetime with metric

This new geometrized scenario of mechanics may give us some interesting hints to the study of future models because of three main reasons. (1) It introduces curved spacetimes from a very simple and well understood physics. (2) It may extend our laboratory perspectives to measure effective gravitation using, for instance, electronic optics. (3) It may provide an interesting and smooth transition to the issue of quantization since we are working with the mechanics of particles instead of fields.

The well-known optical-mechanical analogy (OMA) has been discussed often and from many different points of view [

Let us start by considering a particle with mass “

A similar situation appears in the context of geometrical optics. According to Fermat’s principle (see, for instance [

To complete the analogy we divide (

To a given optical trajectory inside an isotropic material with index of refraction “

It is important to note that the parallelism extends only to the geometrical form of the extremal curve. As was noted by Lanczos [

It is a well-known fact in optics that Fermat’s principle may be mapped in the problem of finding geodesics of a three-dimensional curved manifold [

Although this is a purely formal property of the action principle, one can convince himself/herself by explicitly substituting the identity

Less discussed in the literature of analogue gravity is the geometrical version of the mechanical counterpart of Fermat’s principle. Nevertheless, the similarity between (

Gordon was the first to develop effective metric techniques in the context of analog models. He was interested in trying to describe dielectric media by an effective metric while at the same time using the gravitational field in Einstein ansatz to mimic a dielectric medium. Gordon showed that the trajectories of light inside a dielectric medium was such that they could be mapped in null geodesics of a four-dimensional pseudo-Riemannian metric given by

Now, we are going to show that there exists an entirely equivalent situation in Newtonian mechanics of point paticles. We will see that Gordon’s metric provides the apparatus to describe Newtonian motions by means of the optical-mechanical analogy. Although it is possible to give a formal demonstration of our proposition we are going to adopt a simpler route by showing that Newton’s second law in the form (

We first note that the energy relation (

We are going to parametrize the trajectory of the particle in

Note that, for the effective metric to make sense it has to carry a hyperbolic (Lorentzian) signature. This is in complete agreement with the fact that the modulus of the three-dimensional momentum “

To conclude our four-dimensional geometrization we are going to show that, indeed, the null geodesics of Gordon’s metric coincide with Newton’s equation in the form (

The trajectories of particles with given energy

We thus define the effective four-dimensional element of distance as

We now turn to some explicit situations where our geometrization procedure may be applied. Our aim is to show that some mechanical properties of the systems may as well be described in terms of an effective geometrical language.

We start with a one-dimensional oscillator with potential energy

The null trajectories may be obtained by imposing the condition

A diagram for the trajectories is given in Figure

Harmonic oscillator space-time diagram.

At a given point of the diagram there exist only two admissible trajectories with fixed energy

The curves (

We now turn to the motion of point test particles with mass “

To be specific, let us consider the spherically symmetric potential

In this case we have a maximum admissible radii

This fact has an interesting geometrical interpretation in our scheme. First, let us note that the square root of the determinant is given by

In fact, the components of the Riemann tensor are

To develop some additional feeling on our geometrical description of dynamics, it is instructive to solve the well-known problem of finding the trajectories in the Newtonian gravitational potential in terms of the effective null geodesics (

The conservative potentials that appear in electromagnetism are, perhaps, the most interesting from the point of view of analog gravity in our context. This is not only because electromagnetic fields are simple to handle in laboratory sizes, but because many interesting configurations of static potentials occur in the advanced field of electronic optics. I will concentrate here on simple electrostatic configurations. The potential is obtained by the familiar formula

One interesting situation appears in the geometrization of the static electric dipole. If the dipole moment

As a final remark, I would like to point out that the geometrical framework may introduce techniques of general relativity and riemannian geometry in the realm of electron optics. We suspect that it is possible to envisage situations where electronic dispositives may be projected using these new tools. This last statement is strongly suggested by the recent achievements of transformation optics [

The relation between geometry and dynamics is an interesting problem that has been investigated from many different perspectives along the years. In this paper it is shown that Newtonian mechanics of point particles in static potentials admits a geometrical interpretation in terms of (

The author would like to thank M. Novello, F. T. Falciano, S. E. P. Bergliaffa, and M. Borba for usefull comments. This work is supported by Faperj, Brazil.