- 13.-MACH'S PRINCIPLE. ANTIGRAVITY

The Mach principle impacts in the problem of the rotations observability, in the concept of inertia force and in the concept of centrifugal force. According to him the rotation movement and their effect, the centrifugal force, they are due to the relative movement regarding the background of immobile stars (universe rest). They are the cause of the inertia force and the centrifugal force in the accelerated objects. The model of Living Universe proposes a new vision of this problem.

Let us imagine an universe with only two objects, m_{a} and m_{b} separate a distance r, this situation is clearly unreal, however, we can imagine it. These two masses, in principle, would suffer a gravitational attraction that it could be observed when measuring the distance r, however, the system, if the two masses rotate around their center of masses with a such angular moment that compensates the gravitational force, it would seem equivalent and indistinguishable of a system in the one that doesn't exist the gravitational force and they didn't rotate or of a system with only a body with an internal rotation. The Mach principle says that a rotation in this system could not be observed if not a background of fixed stars existed. The model of Living Universe includes this principle. The expression (6.3) it postulated the existence of two speeds derived about the times t_{1} and t_{2}:

v^{2}= v_{2}^{2} + v_{1}^{2} (1)

I previously also defined systems in rest those with constant v. The speed v_{1} is defined about the time t_{1}. Let us consider the previous particles rotating with an angular moment L, this moment will be owed in its biggest part to the speed v_{2} (let us remind the postulate of the epigraph 1), let us consider to simplify a perfectly circular rotation (without eccentricity), the speed v_{2} we can decompose it in a radial component and another tangential component:

v_{2}^{2} = v_{2r}^{2} + v_{2t}^{2}

The angular moment L will depend exclusively on the tangential component, I have called it tangential for analogy with the speed in an adjournment, in fact this speed is the responsible for the angular moment, and this can be due to an adjournment movement or rotation on an internal axis, that is to say, in the following expressions, L is the only constant of the movement, being r or v dependent of the reference system that it is chosen, that is to say, in the following expression the values of v_{2t} and r is arbitrary as long as L is the correct value of the angular moment :

L^{2} = m^{2} v_{2t}^{2} r^{2} (2)

v_{2t}^{2} = L^{2} / m^{2} r^{2} (2)

m = m_{a} m_{b} / (m_{a} + m_{b})

Substituting in (1) the values of both speeds:

v^{2} = v_{2r}^{2} + L^{2}/ m^{2} r^{2} + c^{2}(r^{2} ρ(r) / R_{h}^{2} ρ(R_{h}) - c^{2} r^{2}/ R_{h}^{2}) (3)

The expression (3) is the one that this model contributes, of her some conclusions can be extracted: in the first place, in our imaginary universe r = R_{h} and ρ(r) = ρ(R_{h}), therefore v^{2} = v_{2}^{2}. That is to say, the centrifugal force nor the gravitational force would be observable neither and r would remain constant (does not exist radial speed v_{2}). However if r <> R_{h} and ρ(r) <> ρ(R_{h}), that is to say, exist other objects in the universe, will appear the centrifugal force as much as the gravitational one. If we suppose v_{2r} = 0:

v^{2} = L^{2}/ m^{2} r^{2} + 2 G M / r - c^{2} r^{2}/R_{h}^{2} (4)

If v = constant, that is to say, an inertial system:

v^{2} - L^{2} / m^{2}r^{2} = 2 G M / r - c^{2} r^{2}/ R_{h}^{2} (5)

If we apply the divergence we obtain:

- L^{2} / m^{2}r^{3} = G M / r^{2} + c^{2} r/ R_{h}^{2} (6)

Expression that it indicates that the angular moment on the time t_{2} has antigravitational character; the term - c^{2} r^{2}/R_{h}^{3} does not appear in (6) when in principle constant considering to R_{h}. The expression (6) is the classic expression that allows to maintain in orbit to our artificial satellites. Nevertheless, if we considered that L is a constant of the movement and that in the previous expressions meant total the angular moment including spin or intrinsic angular moment of each particle or object the result is clearly surprising: any object with an angular moment L could suffer a force equivalent antigravitatoria to the one expressed in (6). It is to say we could define the antigravitational force, despising the cosmological term, like:

F = L^{2} / m^{2} r^{3} (7)

Angular moment L is the total of the system including any internal angular moment that has the object. If we locate ourselves on the Earth, this model predicts that an object of mass m will float (depending on the orientation of L) with an angular moment L of value:

L = (G M m^{2} r)^{1/2} (8)

To practical effects, to take advantage of this property is quite complicated. If we apply (8) to an object in form of hoop of mass m and radio r _{0} that rotates around an axis with angular speed ω and inertia moment I = m r_{0}^{2} we obtain:

m r_{0}^{2} ω = (G M m^{2} r)^{1/2} and

ω = (G M r / r_{0}^{4})^{1/2}

To get the necessary angular speed, on the surface of the Earth, to get that the body floats with a tangential speed similar to 0.99 c, it would imply that r_{0} would have to have a minimum value of 169 meters; if the hoop had a mass of 1 Kg , the necessary energy to generate this turn would be of the order of 1.8 10^{44} ergs, the equivalent one to the disintegration of 2 10^{17} tons of matter, clearly disappointing. However, the photons have kinetic moment and intrinsically spin 1 (independent of their energy); would it be feasible to confine enough number of photons, appropiately organized, inside a vehicle in such a way that the group had the required angular moment? . In the case of the previous example, they would be necessary 2.39 10^{62} photons. If you could get photons with a frequency of 1.66 10^{-18} hz., the necessary energy would be 2.6 10^{18} ergs, equivalent at 2.9 10^{-3} grams of disintegrated matter.

The speed v_{1} can also generate an angular moment, at least, just as it was indicated in the postulate of the epigraph 1, ½ ħ . In general the angular moment due to v_{1} of an object m regarding an observer M you can express as:

L_{1}^{2} = (2GM/r) m^{2} r^{2} sen(α)^{2}

Where α is the angle that r and v_{1} form; if we consider that the minimum distance between m and M would be the event horizon of M, then sen(α) = 2GM / c^{2} r.

L_{1}^{2} = (2GM/r)^{3} m^{2} r^{2} / c^{2}

The speed v_{1} we can also consider it composed by a radial component and another tangential one :

v_{1}^{2} = v_{1r}^{2} + v_{1t}^{2}

The tangential component is the one that generates the angular moment and this has antigravitational character, we can think that in the Lorentz transformations it will be necessary to discount this speed . The value of v_{1t}, considering L_{1} as constant of the movement, it will be:

v_{1t}^{2}= L_{1}^{2} / m^{2} r^{2}

v_{1t}^{2} = (2GM/r)^{3} / c^{2}

The Lorentz trasformaciones could be generalized discounting this speed, calling r_{h} to the event horizon of M, in the following way :

γ = (1 - v_{2}^{2}/c^{2} - (2 G M / r c^{2})(1 - r_{h}^{2}/r^{2}) + r^{2}/ R_{h}^{2} )^{1/2} (9)

In this expression we can see that when r becomes similar to r_{h}, the speed about the time t_{1} is made 0. Likewise, in the event horizon of a black hole γ it is worth (1 - v_{2}^{2}/c^{2} + r^{2}/ R_{h}^{2} )^{1/2}, just as we had seen in the epigraph on the cosmological redshift. When it accumulates enough energy (a Planck mass), the black hole will absorb half Planck mass and it will expel the rest toward its universe mother.

The expression (9) we could have used it in the determination of the precession of the perihelion of Mercury in the epigraph 8. Making the same calculations would find a difference regarding the result of the invaluable General Theory of Relativity, however, this difference can be appreciable in intense gravitational fields or when r is small as, for example, in the case of the solar differential rotation .