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
wellknown 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 threedimensional
subspaces and together they form the spacetime 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 t_{2}, gravitational time to parameter t_{1} and cosmological time to parameter t_{3}.
The
wave function of a particle would be of the following form:
Ψ(r,t_{1},t_{2},t_{3}) =
C + iΨ_{1}(r, t_{1}) + jΨ_{2}(r, t_{2}) + kΨ_{3}(r, t_{3})
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_{h}^{3}, where R_{h}
is the radius of Hubble.
Our universe could be considered as a threedimensional
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 c^{2} = 2 G M /R or R = 2 G M / c^{2}.
Let us apply this equation to a black hole before and after growing:
R_{0} = 2 G M / c^{2}
R_{0} + Δr =2 G (M + Δm) / c^{2}
ΔE = Δm c^{2}
Of these three expressions is deduced that Δr = 2 G ΔE / c^{4}
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 c^{2}
ΔE^{2} = c^{3} 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 / c^{5})^{1/2}),
expanding its event horizon a Planck length (l_{p} = (G ћ / c^{3})^{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 R^{3}
ρ = (N m_{p} / 2) / (4/3) π N^{3} l_{p}^{3}, and
replacing the values of l_{p} and m_{p} we obtain:
ρ = 3 c^{2} / 8 π
G R^{2}, 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 N^{2} 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 = m_{0} c^{2}, where m_{0}=ћ
/ 2 c R_{h}. If we multiplied m_{0} by N^{2} we
obtain a mass (energy) total of M = N^{2} ћ / 2 c N l_{p},
is to say to M = N m_{p}/2. And consequently M = c^{2} 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 N^{2}, in each absorption would have to add an
angular moment equal to (2N1) 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 (2N1) 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 wellknown 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 N^{2}
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 onedimension 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 wellknown quantum mechanics, its temporary parameter will be
electroweak time t_{2}. The elementary oscillating substratum
(the ocean) can be considered as a state mixes with density of
probability ρ_{p}(r, t_{1},
t_{3}) = Ψ*(r, t_{1}, t_{3})
Ψ(r, t_{1}, t_{3}),
where t_{1} is the gravitational time and t_{3} 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 Ψ_{1}(r,
t_{1})^{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, t_{1},
t_{2}, t_{3}) and would have quaternion character, that
is to say, it would be of the following form:
Ψ(r,t_{1},t_{2},t_{3}) =
C + kΨ_{3}(r, t_{3}) + iΨ_{1}(r, t_{1}) + jΨ_{2}(r, t_{2}) (2.1)
In previous expression as much i as j and k they are
imaginary units.
The matrix function of probability density based on t_{1},
t_{2} and t_{3} could calculate:
ρ_{p}(r,t_{1},t_{2},t_{3})
= Ψ(r,t_{1},t_{2},t_{3})^{2}
= Ψ_{3}(r, t_{3})^{2}
+ Ψ_{1}(r, t_{1})^{2}
+ Ψ_{2}(r, t_{2})^{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 threedimensional time allows
writing the relativistic expression of the energy of the following
form:
E^{2}
= m_{0}^{2}c^{4} + c^{2}p_{1}^{2}
+ c^{2}p_{2}^{2}
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 v^{2}  V(r) and applicable
therefore the Schrödinger wave equation. This equation for the
gravitational time would be of the following form:
i
ћ δ Ψ_{1}(r, t_{1}) / Δt_{1} =  (ћ^{2}/2m) Δ^{2}Ψ_{1}(r,
t_{1}) / Δ^{2}t_{1}
 V(r)Ψ_{1}(r, t_{1})
In this function the V(r) potential does not depend on
time t_{1}, would be identifiable, for example, with the
dependent electromagnetic potential of t_{2}. 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 Ψ_{1}(r, t_{1})^{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 r^{3} and R_{h}^{3}
respectively. We would obtain the probability inside the closed surface
indicated previously:
Ψ_{1}(r, t_{1})^{2} = ρ(r) r^{3} / ρ(R_{h})
R_{h}^{3} (2.3)
The expression Ψ_{3}(r,
t_{3})^{2} can calculate considering that on time t_{3}
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.
Ψ_{2}(r, t_{2})^{2} + Ψ_{3}(r, t_{3})^{2} + Ψ_{1}(r, t_{1})^{2} = Ψ_{2}(r, t_{2})^{2} + r^{3}
/ R_{h}^{3} + ρ(r) r^{3}
/ ρ(R_{h}) R_{h}^{3}
(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 Ψ_{2}(r, t_{2})^{2} includes the factor r/R_{h}):
Ψ_{2}(r, t_{2})^{2} + Ψ_{1}(r, t_{1})^{2} + Ψ_{3}(r, t_{3})^{2} = Ψ_{2}(r, t_{2})^{2} + r^{2}
/ R_{h}^{2} + ρ(r) r^{2}
/ ρ(R_{h}) R_{h}^{2}
(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.
Ψ_{2}(r, t_{2})^{2} + Ψ_{1}(r, t_{1})^{2} + Ψ_{3}(r, t_{3})^{2} = Ψ_{2}(r, t_{2})^{2}  r^{2}
/ R_{h}^{2} + ρ(r) r^{2}
/ ρ(R_{h}) R_{h}^{2}
(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 t_{3}, since
this one is bound to the energy in rest. The density of probability on
gravitational time t_{1} would allow to find the kinetic moment
on this time; the density of probability on time t_{2} the
kinetic moment on this other. The observable kinetic moment would come
by the following expression:
p^{2} = p_{1}^{2} + p_{2}^{2}
We can use the expression (2.5.1) to calculate the
average of kinetic energy that corresponding to t_{1} and t_{3}
of a particle located in some point on the closed surface previous.
When calculating the energies in t_{1} or t_{3}, 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 m_{0}
c^{2} in both temporary coordinates. Applying to the
hamiltonian operator it is obtained:
E_{c}
= E_{c2} + (1/2) m c^{2} ρ(r) r^{2}
/ ρ(R_{h}) R_{h}^{2}
 (1/2) m c^{2} r^{2} / R_{h}^{2} (2.6)
It is to say:
(1/2) m v^{2}
= (1/2) m v_{2}^{2} + (1/2) m v_{1}^{2}  (1/2) m v_{3}^{2}
(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 c^{2}
r^{2} / R_{h}^{2 }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 p^{2}
= p_{1}^{2} + p_{2}^{2}. We will
obtain, calling K to the kinetic energy observed and K_{2} the
kinetic energy on the electroweak time, the following thing:
K = K_{2} + (1/2) m c^{2} ρ(r) r^{2} / ρ(R_{h}) R_{h}^{2}
And the expression of K_{2} will be:
K_{2}
= K – (1/2) m c^{2} ρ(r) r^{2}
/ ρ(R_{h}) R_{h}^{2}
This expression can be compared with the one of non
relativistic mechanical energy:
E_{m}
= (1/2) m v^{2} – (1/2) m c^{2} ρ(r) r^{2} / ρ(R_{h})
R_{h}^{2}
Considering
that c / R_{h} is the constant of Hubble H, we can write the
previous expression as:
E_{m}
= (1/2) m v^{2} – (1/2) m H^{2} r^{2} ρ(r) / ρ(R_{h})
For a spherical distribution, we can replace ρ(r) by M / (4/3)
π r^{3} (M is the total energy inside the surface)
and ρ(R_{h}) by 3 c^{2}
/ 8 π G R_{h}^{2},
obtaining:
E_{m}
= 1/2 m v^{2}  G M m / r
This expression is the already wellknown 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 threedimensional 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}
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