6.-GRAVITATIONAL LENSES

In this and in the following sections, some consequences deduced from this model are described. The treatment is simple and it is only an attempt to show how this model can provide solutions to some current mysteries of physics. An in-depth mathematical treatment of these ideas could open new lines of research with surprising results. Below time and space are considered concepts after events, that is, time and space are not the scenario where the events take place, but rather they define the time and the space in which they are observed.

In the previous epigraph it was introduced in a natural way a new expression of the gravitation that coincided with that of Newton in the case peculiar of a punctual distribution or uniform spherical energy in an universe with critical density:

g= - 1/2 H² r ρ(r) / ρ(R_h) - r c²/ R_h² + r² c²/ R_h³     (6.1)

Where H is the Hubble Constant, R_h is the Hubble Radius, ρ (r) is the energy density of the revolution volume generated by the energy distribution origin of the gravitational field and the due deformation to its angular moment and ρ(R_h) the energy density of the universe.

If we consider that ρ(r) is extended to a spherical volume with uniform energy density, then ρ(r) = M / (4/3) π r³ and we can find that the potential of which (6.1) it can derive, put on as:

V = - 1/2 H² r² m ρ(r) / ρ(R_h) + 1/2 m c² r² / R_h²

or

V = 1/2 m c² [ r²/ R_h² - r² ρ(r) / R_h² ρ(R_h) ]     (6.2)

or

V = 1/2 m H² r² [1 - ρ(r) / ρ(R_h) ]

Since g derives of V, we can suppose that V is the general form of the gravitational potential and to generalize for any energy distribution, so that the r² ρ(r) value would be calculated making the integral extended to the volume of revolution of the energy distribution that is considered. The expression (6.2) can define it like a kinetic energy about the time t₁, so we can find an imaginary speed about the time t₁ with the following form:

v₁ = i c [ r² ρ(r) / R_h² ρ(R_h) - r²/ R_h²]^1/2    (6.3)

The times t₁ and t₂ are orthogonal coordinates, then we can postulate that the speed v of a particle can be expressed as the composition of two speeds, each one of them derived from the times t₁ and t₂:

v = v₂ + i c [ r² ρ(r) / R_h² ρ(R_h) - r²/ R_h²]    (6.4)

v² = v₂² + v₁²    (6.4)

v² = v₂² + 2 G M / r - c² r² / R_h²    (6.4.1)

Consequently, the deflection of light rays by a gravitational field is deduced directly from (6.4). Indeed, the speed of light is c, in any reference frame, and the phase velocity of light is ν λ = c. In the absence of gravitational fields, from equation (6.4) we have c² = v₂², where v₂² = v_o c, v_o is the apparent speed of light or of the group in the gravitational field and c is the phase velocity, speed v_o being equal to c in this case. In the presence of a gravitational field, (6.4) it would be c² = v_o c + v₁² and assuming that the gravitational field is generated by a homogeneous spherical distribution of energy with radius r and considering a universe with critical density, that is, R_h² ρ(R_h) = 3 c² /8 π G, we have:

c² = v_o c + 2 G M / r - c² r²/ R_h²    (6.5)

From (6.5) it can be deduced that the gravitational field will generate an r-dependent refraction index as follows:

n( r ) = c / v₂ = 1 / ( 1 - 2 G M / c² r + r²/ R_h²)    (6.6)

This equation coincides with the equivalent equation in General Relativity with the exception of the cosmological term r²/ R_h², which explains why the phenomenon of gravitational lenses is so localized. Absent this term, a sky full of reflections and false images of distant galaxies would be expected. Applying the Fermat principle to the variable index in (6.6), the deflection of the light ray would be: α = 4 G M /c² r

Also I want to make notice that in what it continues, as in this epigraph, I will use a system of reference anchored in the subspace of the times, that is to say, the speeds that appear will be recounted to the temporary coordinates t₁ and t₂, provided that always the modules will be used of these, the sense and direction in this system will come given implicitly. It is necessary equally have in bill that the speed that we observe coincides that the speed on the coordinate t₂, being the speed on the time t₁ unobservable directly; naturally the gravitation is observable, nevertheless, the movement of fall of a body is owed only of the speed on t₂ and answers to the conservation of the energy in the subspace of the times. It is for it that the bodies in free fall in the universe are inertial reference systems.

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