Essay written for the Gravity Research Foundation 2020 Awards for Essays on Gravitation

rjbeery@gmail.com

**Abstract**

In this speculative paper, we show that electromagnetic (EM) mass and general relativistic time dilation are sufficient to predict gravitational attraction.

**Time Dilation as Refraction**

First, we consider light moving slowly through a local medium with a large refractive index; we then observe a remote light ray moving slowly in a large gravitational field due to relativistic time dilation, such that their respective apparent velocities are equal, and recognize the opportunity for a potential equivalence. Exploring this, we create a spherical refractive medium whose index varies with the distance from its center by the following:

where r is the distance from the center of the object and rs is the Schwarzschild radius of some gravitational object O with mass m which we are attempting to emulate.

What we discover is that light passing through such an object at a given radius r will behave identically as it would while passing by O at the same radius. This phenomenon, known as the optical-mechanical analogy (or more recently as F=ma optics), has been well-established and extensively studied over the last century. [ref 1-4]

As Sir Arthur Eddington wrote [ref 5] in his famous 1920 summary of General Relativity, “Space, Time and Gravitation”:

We can thus imitate the gravitational effect on light precisely, if we imagine the space round the sun filled with a refracting medium which gives the appropriate velocity of light. To give the velocity 1 − 2m/r, the refractive index must be 1/(1 − 2m/r), or, very approximately, 1 + 2m/r. At the surface of the sun, r = 697, 000 km., m = 1.47 km., hence the necessary refractive index is 1.00000424. At a height above the sun equal to the radius it is 1.00000212.

Any problem on the paths of rays near the sun can now be solved by the methods of geometrical optics applied to the equivalent refracting medium. It is not difficult to show that the total deflection of a ray of light passing at a distance r from the centre of the sun is (in circular measure)

whereas the deflection of the same ray calculated on the Newtonian theory would be

For a ray grazing the surface of the sun the numerical value of this deflection is

1”.75 (Einstein’s theory),

00”.87 (Newton’s theory).

The efficacy of the “F=ma optics” is without doubt, however, respective authors on the subject are careful to stress the purely analogous nature of the relationship. We would like to suggest that it isn’t an analogy at all, but rather a literal equivalence.

**EM Mass**

Let us envision an electromagnetic wave, with a wavelength of 2.43 * 10-12 m, moving in a periodic cycle which takes it back upon itself such that it becomes a self-reinforcing soliton. The complete orbital path length of this EM wave is equal to its wavelength but is such that it makes a double-loop. (see Fig 1)

Such a quasi-symmetrical object, if stable, would resemble an electron. It would have a physical radius of on the order of 2.43 * 10-12 m / 4, an electric field, a magnetic dipole, and a half-integral spin [ref 6]. It would also offer a physical manifestation of Einstein’s mass/energy equivalence (e.g. “releasing” the photon from its self-contained path would result in a burst equal to its “rest energy”).

**Cosmic Speed Limit**

Philosophically, many of us have been mystified by the limiting nature of c. EM mass might provide a straight-forward explanation -- a photon turning back upon itself does not follow the traditional geodesic between two points. As an EM mass particle is accelerated, a larger portion of its photon’s circuit is thus spent moving in the direction of its velocity; this percentage can be arbitrarily close to, but not quite, 1. (see Fig 2)

**Transverse Waves**

If we refer back to our sphere of graded refractive index, we would expect that the path of light moving radially to it would remain unaffected; only a light’s path with a transverse component would be altered. The photon of an EM mass particle moving in a closed circuit within the sphere would possess a transverse portion of its path relative to the center of the medium in a range between .5 and 1, depending upon the relative velocities of the sphere and the particle. This could manifest as relativistic mass.

**Conclusion**

In this paper we have shown the connection of optics to the gravitational bending of light in a graded time dilation field. Additionally we have shown that if mass were to possess an electromagnetic nature moving in a cyclic fashion (i.e. “EM mass”) then we are able to precisely predict the gravitational behavior of that mass in the presence of such a time dilation field without invoking any other mechanism related to General Relativity. Lastly, we are able to show that this model may plausibly explain other aspects of Relativity, such as the limiting speed of light and relativistic mass. We feel that these aggregate theories provide ample potential to warrant further investigation.

References

[1]The optical-mechanical analogy for stationary metrics in general relativity; Paul M. Alsing; American Journal of Physics 66, 779 (1998); https://doi.org/10.1119/1.18957

[2] The optical-mechanical analogy in general relativity: Exact Newtonian forms for the equations of motion of particles and photons; James Evans, Kamal K. Nandi & Anwarul Islam; General Relativity and Gravitation volume 28, pages 413–439(1996)

[3] ‘‘F=ma’’ optics; James Evans and Mark Rosenquist; American Journal of Physics 54, 876 (1986); https://doi.org/10.1119/1.14861

[4] Sumana Bhadra, Electromagnetic Mass Models in General Theory of Relativity: https://arxiv.org/abs/0710.5619

[5] Space, Time and Gravitation; Sir Arthur Eddington; http://www.gutenberg.org/files/29782/29782-pdf.pdf

[6] Is the electron a photon with toroidal topology? J.G. Williamson and M.B. van der Mark, 1997; Annales de la Fondation Louis de Broglie, Volume 22, no.2, 133. http://home.claranet.nl/users/benschop/homepg2/electron.pdf