There is a special relationship between **permittivity and permeability **in the Sciama model that I am not aware that Sciama himself was aware of.

There does exist in literature, a direct analogy of the gravitational permeability and permittivity. It is stated that [math]\epsilon_G = \frac{1}{4 \pi G}[/math] is the gravitational analogue of the vacuum permittivity [math]\epsilon[/math]. Gravitational waves have been experimentally varified to move at the speed of light - this is related in analogy (Sivaram, Sabatta) with electromagnetism [math]c = \frac{1}{\sqrt{\epsilon_0 \mu_0}}[/math] and that a corresponding gravitational interpretation is [math]c = \frac{1}{\sqrt{\epsilon_G \mu_G}}[/math]. This implies the gravitational permeability is [math]\mu_G = \frac{4 \pi G}{c^2}[/math]. Sciama defined the gravielectric field as using cgs units for the gravitational constant:

[math](\frac{\Phi + \phi}{c^2}) = -\frac{1}{G}[/math]

*the units chosen for G ensures that the ratio of the scalar and the speed of light squared is not dimensionless, as often encountered in physics.

This equation may be understood when reformulated to represent an inverse gravitational permeability:

[math]\Phi = -\frac{c^2}{4 \pi G}[/math]

That mean's Sciama's gravielectric scalar potential is itself the definition of the inverse permeability. A unique property of these special units is that the ratio and inverse ratio have the relationships:

[math]\frac{c^2}{\Phi} = \mu_G c^2 = -\frac{4 \pi G c^2}{c^2} = -4 \pi G = \frac{1}{\epsilon_G}[/math]

[math]\frac{\Phi}{c^2} = \frac{1}{\mu_G c^2} = -\frac{c^2}{4 \pi G c^2} = -\frac{1}{4 \pi G} = \epsilon_G[/math]

Both these quantities appear within the framework of a single gravitomagnetic coupling equation to angular momentum

[math]\mathbf{B} = \frac{1}{mc^2 r}(\frac{\Phi}{c^2})\frac{\partial v}{\partial t} \mathbf{J} \approx \frac{1}{mc^2 r}(\frac{1}{c^2} \frac{c^2}{4 \pi G})\frac{\partial v}{\partial t} \mathbf{J} = -\frac{1}{mc^2 r}\frac{1}{4 \pi G}\frac{\partial v}{\partial t} \mathbf{J}[/math]

Sciama never attributed the inverse G to the permeability but doing so makes Sciama's equation an approximate.

**Edited by Dubbelosix, 28 March 2018 - 10:36 AM.**