General Relativity In Newtonian Mechanics

by Salim Georges Khoury

We will show in this paper that there exists in "Classical Newtonian Mechanics" a relationship between the curvature of space and the density of energy similar to that of "General Relativity". The Theory of General Relativity seems to replace, in this relationship of Newtonian Mechanics the Scalar quantities by their equivalent Tensors.

DESCRIPTION

Consider a point mass which is not subject to any external force. According to classical mechanics, its motion is uniform rectilinear and its velocity is constant equal to

If this point mass passes near a mass with , it starts to turn around under the effect of gravitational force. Its trajectory is quasi-circular with radius .

First Remark:
The mass seems to curve the linear space of the mass .

The force exerted by the mass on is:

therefore

which is equal at the same time to the centripetal acceleration of the uniform circular motion

hence

         

       = curvature of the circular trajectory of radius

  = areal energy density of . It is distributed over the surface of the sphere where the mass moves.

Second Remark:

Equation (1) shows a relationship between curvature and energy density which looks like the equation of General Relativity (2)

 

where
= Einstein tensor, it represents the curvature of space-time
= Energy-impulse tensor, it represents the total energy density (mass and energy)

Third Remark:

The areal density of energy looks like the expression of the power radiated by an isotropic antenna. It's as if the gravitational energy of propagates like the electrmagnetic wave energy and the mass behaves like a receiving antenna.

CONCLUSION

We have shown with a simple classic example the relationship between curvature of space and energy density without going through the non-intuitive tensorial equation of General Relativity.