User:Christillin/Newton's law

WP:COPYARTICLE, old copy of Newton's law of universal gravitation with the following additions:

Post Newtonian corrections
The non linearity of gravitation can be simply accounted for by incorporating in Newton's law the mass energy equivalence. Since the energy is the source of gravitation, the mass (energy) m is not the mass the second object possesses at infinity. The interaction renormalizes it in a space dependent way
 * $$ m => m' = m_{0} (1 - G M  /c^2 r  )$$

where m0  stands for the bare mass (without the influence of M), so that, relying again on Einstein's mass-energy equivalence ,
 * $$\phi' = - (G M  m_{0}/ r)(1 -  G M  /c^2 r  )    $$

This is, mutatis mutandis, the gravitational analogue of mass renormalization in the e.m. case, where virtual processes have a different effect for a free and a bound electron, with an ensuing measurable differences (of two infinities).

Here, apart from (non) quantization, the gravitational interaction has decreased the total energy of the system and hence of its mass which we approximately attribute to the lighter one (but remember that also in the Schwarzschild solution space is supposed to be curved only by the heavier mass), modifying in turn the interaction. It is then clear that the round bracket in the previous equations incorporating energy conservation, should be modified and go into $$ 1 -GM/c^2 r (1-GM/c^2 r) = 1 -x (1-x) $$. The modification due to the self energy term can be interpreted in two alternative ways.

First, one realizes that the former equation can be recast into a modified "effective" potential with an additional small exponent (e.g. for the Sun at Mercury  $$  GM  /c^2 r  \sim 10^{-8} $$ )
 * $$\phi' = - G M /r^{(1+\alpha)}   $$

where the repulsive extra term modifies  in a parameter free way the 1/r2  Newton's power law, which then reads
 * $$\mathbf{F} = - G M m \mathbf{r}  /{r^3}  (1- 2 r'_{S}/r)  $$

The role of the extra term becomes more relevant at short distances, which makes it plausible why only in the case of Mercury it has played some role. In turn this implies a violation of Gauss's theorem, if expressed as usual in terms of a given constant mass. But, probably more interesting, one can also define
 * $$\phi' = - G M /r'     $$

where
 * $$r' = r /  (1 -  G M  /c^2 r  )  \simeq r/ \sqrt{   (1 -  r_{S}/ r  )  }   $$

where the last passage holds true up to second order terms in G M  /c2 r and, because of energy conservation,
 * $$t' = t   (1 -  G M  /c^2 r  )   $$

Notice however that no singularity appears  since the renormalized mass can be zero at worst in the defining equations.

The correction factor x = 1 - G M  /c2 r   intervenes in a two-fold way.

In a first instance one keeps the usual Newtonian potential for the standard gravitational attraction. Just because of energy conservation one predicts the first order gravitational red shift. Then self energy effects are taken into account. They can be disposed of in the traditional language just by rescaling dt and 1/ dr , by the factor  (1- Gm/c2 r) , hence causing a " first order   curvature of space- time ". The angular momentum L must be obviously conserved. The quantities entering L at the Sun  are changed as seen from the earth because of the previous relations. Hence the photon which was locally assumed to be perpendicular, is necessarily perceived to deviate by an angle $$ \Delta\theta'$$


 * $$c L / \hbar = \omega_{S}   R_{S}  \rightarrow  (1   + GM/ (c^2 R_{S}))  \omega_{T} R_{T}  / (1   - GM/ (c^2 R_{S}) )    \Delta  \theta'

\simeq ( (1  + 2 GM/ c^2 R_{S}) \omega_{T} R_{T}  \Delta  \theta' $$ where the small angle approximation has been made.

Thus the final  deflection is given by  by  $$ \Delta  \phi'  = 2 \Delta  \theta'$$  or


 * $$\Delta \phi' =  - 4 G M_{S} /(c^2 R_{S}) $$

the minus sign meaning that the photon must be of course attracted by the Sun. Similarly the standard expression for the precession
 * $$\Delta \omega' /\omega '  =  \Delta  \phi' /\phi'   = 6 \; G M_{S} /(c^2 a (1-e^2) ) $$

follows along standard lines. In terms of the factor GM/(c2 r) the coefficients 2 and 3 entering respectively light bending and the perihelion precession have  a transparent meaning. It is not superfluous to underline why the despised proposal of modifying Newton's law in an ad hoc manner was also successful for Mercury :  the present parameter free treatment  justifies the equivalence of the two approaches !