2.-A NEW ROAD TO THE QUANTUM GRAVITY

PACS: 04.60.-m, 98.80.-k

Is our universe an isolated system? All the cosmological models to date are based on considering our universe as an isolated system, using the General Theory of Relativity as valid instrument for their study and analysis. There is no an objective reason that it guarantees this axiom as valid, in fact, in the well-known universe does not exist no example of strictly isolated system. This article explores the possibility that our universe is not an isolated system.

If the universe is not an isolated system it has sense to make certain questions: Which is the nature of its borders? How is the interaction that allows to the interchange of energy and information between our universe and the outside? The propose answer to the first question will be the initial hypothesis of this work. The border of the universe is the time in the quantum sense of the term, all the events that constitute it are developed in their present like temporary parameter of the wave function of the universe, as much the future as the past they are outside him. A common present (temporary parameter of the wave function of the universe) to all the observers exists, although each one of them only can observe the past of the rest. The answer to the second question we can obtain it when investigating on what it is possible to be understood by unified interaction. In a unified theory it is tried that all the interactions can be deduced from an only hamiltonian with the highest group of symmetries possible. This would happen to unimaginably high energies, for example in the Big Bang or the event horizon of a black hole. The hypothesis of this work and the definition of unified interaction will allow finding a new way towards the quantum gravity. This way will allow discovering a quantum expression equivalent to the Newton gravitation.

2. Hypothesis

Time and space are symmetrical three-dimensional subspaces and together they form the space-time of events. The symmetry plane would be on the event horizon of a black hole. That is, the event horizon of a black hole would separate two symmetrical universes. The spatial dimensions of the mother universe would constitute the temporal dimensions of the son universe. Our universe would be a black hole inside another external universe. The spatial dimensions are generated at the beginning of the new universe from three microscopic dimensions of the mother universe (coiled dimensions). The three temporal dimensions of the mother universe could give place to three microscopic dimensions in the field of elementary particles (coiled dimensions) of the son universe. The group of a mother universe and another son could have nine dimensions. These dimensions are grouped by three, alternating their functionality in each generation.

The concept of time used in this hypothesis coincides with the time concept of the quantum mechanics; that is to say, each one of the three temporary dimensions are temporary parameters of other as many wave functions. Of analogous way to as the observable position is defined, this hypothesis would allow defining the concept of time observable that would coincide with the ordinary concept of the time. We can denominate electroweak time to the parameter t₂, gravitational time to parameter t₁ and cosmological time to parameter t₃.

The wave function of a particle would be of the following form:

Ψ(r,t₁,t₂,t₃) = C + iΨ₁(r, t₁) + jΨ₂(r, t₂) + kΨ₃(r, t₃)

This hypothesis proposes a new point of view of our universe, the outer point of view to the universe. Also it allows defining the concept of "volume of the universe" as the volume of the sphere with center in the observer and radius the distance to the more distant observable particle. That is to say, V = (4/3) π R_h³, where R_h is the radius of Hubble.

Our universe could be considered as a three-dimensional hologram inscribed in a bidimensional flat spherical surface of temporary coordinates. The cosmological common present would be the radius of this sphere; the surface would constitute the temporary plane constituted by the gravitational and electroweak coordinates.

3. Unified Interaction

The answer to the question of how is the interaction that allows the exchange of information between our universe and the outside can give the answer us to this other: How a black hole grows? Which will be the minimum amount of energy and in what minimum time would be absorbed? Applying strictly the uncertainty principle this question it only admits an answer. In effect, a black hole, when growing, cannot lose its identity of black hole and the absorption of energy must take place so in a time such that ΔE Δt >= h / 4 π. In this time, the event horizon will expand a length Δr. If this horizon also is subject to the limit of c as permissible terminal velocity then Δr / Δt = c . On the other hand, the characteristic equation of a Schwarzschild black hole with energy M and radius of the event horizon R is c² = 2 G M /R or R = 2 G M / c². Let us apply this equation to a black hole before and after growing:

R₀ = 2 G M / c²

R₀ + Δr =2 G (M + Δm) / c²

ΔE = Δm c²

Of these three expressions is deduced that Δr = 2 G ΔE / c⁴ for the minimum amount of absorbed energy. If the absorption is due to produce in the minimum time, then ΔE Δt = h / 4 π. Dividing this and the previous equality we obtain:

Δr / ΔE Δt = 8 π G ΔE / h c²

ΔE² = c³ h / 8 π G

If we called ћ to h / 2 π and considering that the Planck mass (m_p) is defined as m_p = (ћ c / G)^1/2, we reached the conclusion that a black hole grows absorbing packages of energy by value of half Planck mass every Planck time (t_p = (G ћ / c⁵)^1/2), expanding its event horizon a Planck length (l_p = (G ћ / c³)^1/2). The mass of a black hole would come given by expression M = N m_p / 2 and the radius by N l_p, where N is a whole number that indicates the order of the last process of absorption. The density of a black hole can calculate:

ρ = M / (4/3) π R³

ρ = (N m_p / 2) / (4/3) π N³ l_p³, and replacing the values of l_p and m_p we obtain:

ρ = 3 c² / 8 π G R², that is to say, a black hole has critical energy density.

The previous result allows postulating that the interaction responsible for the exchange of information between a universe and its surroundings consists of the interchange or absorption of half Planck mass every Planck time. The universe (black hole) would accumulate energy in the outside of its event horizon, when arriving at a Planck mass the universe would absorb half Planck mass and the rest would return to the outer universe.

4. Quantum Black Hole

Let us consider in the first place a quantum system with a number of N² equal elementary oscillating, each one with a angular momentum with respect to the system of ћ/2 and equal wavelength of 2 R_h, where R_h = N l_p. It is easy to verify that this system can describe to a black hole as the one that we saw in previous epigraph. In effect, the energy of each oscillator is worth E = m₀ c², where m₀=ћ / 2 c R_h. If we multiplied m₀ by N² we obtain a mass (energy) total of M = N² ћ / 2 c N l_p, is to say to M = N m_p/2. And consequently M = c² R_h / 2 G, definition of the black hole of Schwarzschild.

If the previous quantum system coincides indeed with a black hole, we could describe of qualitative way the interaction between universes postulated in previous epigraph. As we saw, the black hole would be accumulating energy in its event horizon, when reaching the amount of a Planck mass it would absorb half Planck mass and the rest would return to the universe. It is easy to verify that so that the accumulated number of elementary oscillating with angular moment ћ/2 is N², in each absorption would have to add an angular moment equal to (2N-1) ћ. Since the accumulated Planck mass in the event horizon has an angular moment of m_p c N l_p = 2 N ћ. The particle half Planck mass absorbed will inject an angular moment of (2N-1) ћ and the rest of energy would be expelled to the universe in form of gamma ray (spin 1). These extraordinarily power gamma rays could be the causes of the gamma ray bursts (GRBs Gamma Ray Bursts) observed in distant galaxies, triggering a chain of creation / annihilation of particles and antiparticles. As this phenomenon is known (Gamma Ray Bursts) lacked until the moment a mechanism well-known origin. The cosmic rays that we see in our atmosphere also may be caused by this phenomenon. It is necessary to insist on that the particles absorbed by the black hole fundamentally consist of energy and angular moment, does not become no hypothesis of as the "interior" of the black hole is reorganized. Only interests to know that N² elementary oscillating exist with angular moment of ћ/2 each one.

5. Quantum Gravity

In Marcelo A. Crotti article [1] a model of particles is exposed that allows to deduce the Lorentz transformations directly from the particles characteristics. It is only necessary to postulate a primordial medium where interactions occur. The basic constituents of this primordial medium are postulated as one-dimension oscillators (like superstring theory does). The basic difference with superstring theory is that the oscillators are not only considered as the basic constituents of particles. Linear oscillators are postulated as filling (and defining) all the Universe, while particles are only the physical manifestation of their coordinate interaction. As an analogy, waves in the sea and whirlpool in air are only manifestation of the coordinate interaction of water or air molecules. Whirlpools are not constituted by a new kind of element although, from our point of view, they could behave like not "only" air molecules.

As it will be shown, this new approach is compatible with special relativity formulations but avoids its typical paradoxes. The following points are remarkable:

The proposed model is compatible with the electromagnetic theory as well as the special theory of relativity since it reaches the same equations.

This model proposes that the dimensional and time changes for systems in relative motion are real in the same fashion as the transformation of mass to energy is real.

In this model, "moving" and "stationary" observers would state that clocks in a moving system run more slowly than those in a stationary system. This does not imply the existence of an "absolute" reference system and mobile systems would only exist with respect to the basic frame that forms the space. This concept is analogous to the movement of waves and currents with respect to the extended mass of water of which they form part. There are not water molecules that can claim to be at absolute rest. Still the velocity of transportation phenomena becomes meaningful only when compared to the "stationary" local water extension.

With support in the previous article and from the initial hypothesis we can say that the particles of our universe (the waves in the ocean) are described by the function of classic wave of the well-known quantum mechanics, its temporary parameter will be electroweak time t₂. The elementary oscillating substratum (the ocean) can be considered as a state mixes with density of probability ρ_p(r, t₁, t₃) = Ψ*(r, t₁, t₃) Ψ(r, t₁, t₃), where t₁ is the gravitational time and t₃ is the cosmological time, to see epigraph 2. The subscript p indicates that it is density of probability to differentiate it from the energy density.

The oscillating cannot be located at any moment, nevertheless, the density function of probability |Ψ₁(r, t₁)|² can be used for the study of this system. The observable particles in our world would be accumulations of probability density in form of "soliton". They would be stable fluctuations of this density function of probability. The complete wave function of a particle would be Ψ(r, t₁, t₂, t₃) and would have quaternion character, that is to say, it would be of the following form:

Ψ(r,t₁,t₂,t₃) = C + kΨ₃(r, t₃) + iΨ₁(r, t₁) + jΨ₂(r, t₂)       (2.1)

In previous expression as much i as j and k they are imaginary units.

The matrix function of probability density based on t₁, t₂ and t₃ could calculate:

ρ_p(r,t₁,t₂,t₃) = |Ψ(r,t₁,t₂,t₃)|² = |Ψ₃(r, t₃)|² + |Ψ₁(r, t₁)|² + |Ψ₂(r, t₂)|²       (2.2)

We could identify the gravitation with the displacement of the elementary oscillating towards places of greater density of probability, just as the current of the river drags the waves produced by the fall of a stone, the particles would be dragged by this current of probability.

The hypothesis of the three-dimensional time allows writing the relativistic expression of the energy of the following form:

E² = m₀²c⁴ + c²p₁² + c²p₂²

The energy in rest can be identified with the energy on the cosmological axis of the time. The kinetic moment observed would be a composition of the kinetic moments responsible for the kinetic energies on the cosmological, gravitational and electroweak axes.

The following expositions are limited a non relativistic scope, that is to say, for small speeds and weak gravitational fields. It will be considered that the expression of the mechanical energy is E = 1/2 m v² - V(r) and applicable therefore the Schrödinger wave equation. This equation for the gravitational time would be of the following form:

i ћ δ Ψ₁(r, t₁) / Δt₁ = - (ћ²/2m) Δ²Ψ₁(r, t₁) / Δ²t₁ - V(r)Ψ₁(r, t₁)

In this function the V(r) potential does not depend on time t₁, would be identifiable, for example, with the dependent electromagnetic potential of t₂. For a particle in free fall, that is to say, free of potentials it would be 0.

With the object of calculating the value of |Ψ₁(r, t₁)|² we can consider that all the energy manifestations are ultimately conformed by the elementary oscillating indicated previously. Therefore, a good approach to the previous value, defined in the volume generated by a closed surface of equal superficial density of energy, would be the quotient between the total energy inside this volume and the total energy of the universe, that is to say m/M. From the average of energy density in the interior of the volume it would be of the form: ρ(r) v / ρ(R_h) V. For systems with spherical symmetry, which they are those that contemplates the systems to which the law of Newton of the gravitation is applied, we can approximate so much v as V to r³ and R_h³ respectively. We would obtain the probability inside the closed surface indicated previously:

|Ψ₁(r, t₁)|² = ρ(r) r³ / ρ(R_h) R_h³      (2.3)

The expression |Ψ₃(r, t₃)|² can calculate considering that on time t₃ are no space references. We can say therefore that the probability of finding a particle in volume v is v/V, where V is the volume of the universe.

|Ψ₂(r, t₂)|² + |Ψ₃(r, t₃)|² + |Ψ₁(r, t₁)|² = |Ψ₂(r, t₂)|² + r³ / R_h³ + ρ(r) r³ / ρ(R_h) R_h³      (2.4)

This expression would be valid in absence of events. Since in effect a particle m is detected, that is to say, an observable event takes place to distance r, is necessary to modify this probability distribution applying the definition of conditional probability. In order to calculate the conditional probability to the fact that the particle to distance r is detected, it is necessary, applying the definition of conditional probability, to dividing by the probability that it is to a maximum distance r, that is to say, of dividing by r/R_h. The expression (2.4) would be of the following form (the expression |Ψ₂(r, t₂)|² includes the factor r/R_h):

|Ψ₂(r, t₂)|² + |Ψ₁(r, t₁)|² + |Ψ₃(r, t₃)|² = |Ψ₂(r, t₂)|² + r² / R_h² + ρ(r) r² / ρ(R_h) R_h²      (2.5)

It is necessary to standardize the previous equation. If we make the calculation of probabiliad extended to all the universe it seems that we would obtain a value of 3. Nevertheless if we considered that on the cosmological time references do not exist (all the observers they share the same temporary coordinate) we can adjudge a negative sign to him to the cosmological term obtaining with it the normalization of the previous density function.

|Ψ₂(r, t₂)|² + |Ψ₁(r, t₁)|² + |Ψ₃(r, t₃)|² = |Ψ₂(r, t₂)|² - r² / R_h² + ρ(r) r² / ρ(R_h) R_h²      (2.5.1)

The expression (2.5.1) would allow calculating the different average values of the quantum observable, in this case of the position, the kinetic energy or the moment. Nevertheless in this exhibition we are limiting ourselves the non relativistic scope and therefore we will not consider the term on time t₃, since this one is bound to the energy in rest. The density of probability on gravitational time t₁ would allow to find the kinetic moment on this time; the density of probability on time t₂ the kinetic moment on this other. The observable kinetic moment would come by the following expression:

p² = p₁² + p₂²

We can use the expression (2.5.1) to calculate the average of kinetic energy that corresponding to t₁ and t₃ of a particle located in some point on the closed surface previous. When calculating the energies in t₁ or t₃, we can simplify the calculation considering the dependent term of time of the wave function is of the form e^{-i E t / ћ}, where E = 1/2 m₀ c² in both temporary coordinates. Applying to the hamiltonian operator it is obtained:

E_c = E_c2 + (1/2) m c² ρ(r) r² / ρ(R_h) R_h² - (1/2) m c² r² / R_h²      (2.6)

It is to say:
(1/2) m v² = (1/2) m v₂² + (1/2) m v₁² - (1/2) m v₃²       (2.6.1)

In (2.6), m is the mass of the observed particle, in this case, m = n*m0 (n is a whole number). Expression (1/2) m c² r² / R_h² can be considered as the kinetic energy due to the expansion of the observed universe; we will not consider it in which it follows. We can replace the expression of the kinetic energy on the gravitational time in the expression p² = p₁² + p₂². We will obtain, calling K to the kinetic energy observed and K₂ the kinetic energy on the electroweak time, the following thing:

K = K₂ + (1/2) m c² ρ(r) r² / ρ(R_h) R_h²

And the expression of K₂ will be:

K₂ = K – (1/2) m c² ρ(r) r² / ρ(R_h) R_h²

This expression can be compared with the one of non relativistic mechanical energy:

E_m = (1/2) m v² – (1/2) m c² ρ(r) r² / ρ(R_h) R_h²

Considering that c / R_h is the constant of Hubble H, we can write the previous expression as:

E_m = (1/2) m v² – (1/2) m H² r² ρ(r) / ρ(R_h)

For a spherical distribution, we can replace ρ(r) by M / (4/3) π r³ (M is the total energy inside the surface) and ρ(R_h) by 3 c² / 8 π G R_h², obtaining:

E_m = 1/2 m v² - G M m / r

This expression is the already well-known one of the classic mechanical energy and it allows us to identify this mechanical energy with the kinetic energy on the electroweak time. Also we can denominate newtonian quantum gravity to the following expression.

Replacing the constant of Hubble by its value:

The latter makes it easy to verify that expression is Lorentz covariant. Both r and Rh are affected by the Lorentz transformations in the same direction.

6. Conclusion

The hypothesis of the three-dimensional time, exposed in epigraph 2, can allow opening a way to the quantum treatment of the gravitation of a natural way, giving a quantum origin typically to the gravitation and solving some mysteries of modern cosmology. Additionally, the expression of probability density (2.5) eliminates the singularities problem of the General Theory of Relativity; the energy density in the considered volume could not grow indefinitely since the probability of detecting a particle cannot be greater than 1. The greater possible density is the density of the particle half Planck mass, in her the expression (2.5) is worth 1 and therefore it has sense to consider it as a universe embryo. The absorption of later packages of value energy half Planck mass maintains the value of the expression (2.5) with value 1.

Naturally the development until now is elementary, but it shows the strategy of this new point of view in the attempt to introduce the gravitation in the formalism of the quantum mechanics. A rigorous formal treatment could give place to a complete quantum formulation of the gravity.

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