7.-PRECESSION OF MERCURY PERIHELION

In the last epigraph exposed as this model is equal to the models based on the General Theory of Relativity (GTR) when explaining the one phenomenon of the gravitational lenses. In this epigraph, I will show, as it is capable also of explaining the precession of the perihelion of Mercury. It will also serve as example of the use of the new Lorentz transformations of the previous epigraph.

The planets' orbits around the sun are ellipses with the sun in one of their focuses. The ellipse equation is the following:

r = r₀ (1 + e) / (1 + e cos(φ))

Where r is the distance to the focus where the Sun is at, r_o is the minimum distance between the planet and the star (perihelion) and e is the eccentricity. For a circular orbit, e is zero. The perihelion precession may be observed for all the planets with an eccentric orbit, its observation being more difficult in less eccentric orbits. This movement, for the most part, is due to the gravitational influence of the rest of the planets, however, there is an excess, more evident in Mercury, that cannot be explained by Newtonian gravitation. This excess is what satisfactorily explains the GTR. This does not mean that a circular orbit does not undergo such movement, however it is nearly indistinguishable. The angle of precession predicted by the GTR is given by the following equation:

r = r₀ (1 + e) / (1 + e cos(φ - Δφ))

Where

Δφ = 6 π G M / c² r₀ (1 + e)

To simplify the calculation, let us consider a circular orbit (e = 0) with a radius of 58 10⁶ Km., which is the average radius of the orbit of Mercury. The period of orbit is 88 days and it completes 415 orbits every 100 years. M is the sun mass (2 10³³ g.). With these data, according to the GTR, the angle of precession is Δφ = 4.8235 10^-7 rad. That multiplied by the 415.01391912 orbits per century, gives 2.0018354 10^-4 rad. In seconds of arc, it would be 41.29 seconds of arc (in fact, considering the eccentricity, it is 43 seconds of arc).

The Living Universe model, starts out with the following equation, in which there am considering a homogeneous spherical distribution of energy:

v²= v₂² + 2 G M / r - c² r²/R_h²    (7.1)

In this expression, v is the speed of the planet mercury and v₂ is the speed that is observed.

If we apply the expression of Lorentz's transformations indicated in (3.1), we obtain:

γ = [ 1 - v₂² /c² + (r² H² / c²) (1 - ρ(r) / ρ(R_h) ) ]^1/2

If we consider, since it is possible in this case, a spherical distribution and uniform of energy, the previous expression remains like:

γ = (1 - v₂² /c² - 2 G M / r c² + r²/ R_h²)^1/2    (7.2)

In this particular case it is possible to calculate v₂, In fact, since the orbit is circular, it holds that v₂² / r = G M / r²; equation (7.2) is then:

γ = (1 - 3 G M / r c² + r²/ R_h²)^1/2    (7.3)

If we ignore the cosmological term and apply this equation to the calculation of the temporal dilation and space contraction undergone by Mercury during its orbit, we have, denoting as t_m the time measured on Mercury and as t_t the time measured by the observer, the following equation:

t_t = t_m / γ    (7.4)

The contraction in the orbit of Mercury is:

S_t = S_m γ

The average speed measure in Mercury will be :

v_m = S_m / t_m

And the measure in the Earth will be: :

v_t = S_t / t_t
v_t = S_m γ / (t_m / γ)
v_t = v_m γ²

Using the mass values of the Sun and radio of the orbit of Mercury finds that the difference of speeds among the real of Mercury and the one observed in the Earth is of 0,36817 cm/sg. This difference accumulated along one century generates a desfase in the orbit of Mercury 1.161.064.532 cm. that is equal to an arch of 2,0018 10^-4 rad.

This is the same value predicted by the GTR.

In the paragraphs above, the problem was simplified for circular orbits. Below, it is attempted to generalize the problem of particle movement in the heart of a gravitational field.

The phase velocity of a particle with energy E can be defined as w = ν λ, where ν = E / h and λ = h / p. Frequency ν can also be defined as the reverse of the period υ = 1 / τ.

To solve the problem of finding the particle's trajectory in the heart of a gravitational field, we can think of it as similar to the case of light ray deflection by the gravitational lenses, in that the gravitational field generates a variable refraction index n(r) defined by the quotient w_o/w, where w_o is the phase velocity in the vacuum and w the phase velocity in the heart of the gravitational field.

Keeping in mind equation (3.1), considering a homogeneous spherical distribution and ignoring the cosmological term, we have γ = (1 - v₂²/c² - 2 G M /c² r)^1/2 . The particle wavelength contracts λ = λ_oγ and time dilates τ = τ_o/ γ. We therefore arrive at the following equation of the refraction index:

n(r) = w_o / w     
n(r) = ν_o λ_o / ((λ_o γ)(γ / τ_o ))    
n(r) = 1 / γ²    (7.4)

The real trajectory of the particle is the one that makes the optic path extremal:

n(r,φ,Θ) = 1 / (1 - v₂(r,φ,Θ)²/c² - c² r² ρ(r,φ,Θ)/ R_h²ρ(R_h) + r²/R_h²)     (7.5)

δS = δ ∫ n(r,φ,Θ) ds = 0     (7.6)

Could it be applied to these expressions on the way the path integral methodology of Richard Feynman ?

In the case of the Mercury perihelion, from (7.3) we have the following variable index:

n( r ) = 1 / (1 - 3 G M / r c² + r²/ R_h²)

The theoretical speed of Mercury in its orbit according to Newtonian mechanics is v = 4,795,831 cm/sg, refraction index which makes the real speed of Mercury less: v_r = v / n(r), This speed is thus 4,795,830 cm/sg, The difference from the first is 0.36817 cm/sg; In 100 years, this will accumulate a 1,161,064,532 cm. lag equal to an arc of 2.0018 10^-4 rad, which is the same value obtained above.

_{© Jorge Ales, 2002. http://www.livinguniverseweb.com}