Mathematical Foundations of the Relativistic Theory of Quantum Gravity

Abstract

Starting from the action function, we have derived a theoretical background
that leads to the quantization of gravity and the deduction of a correlation
between the gravitational and the inertial masses, which depends on the kinetic
momentum of the particle. We show that the strong equivalence principle is
reaffirmed and, consequently, Einstein's equations are preserved. In fact, such
equations are deduced here directly from this new approach to Gravitation.
Moreover, we have obtained a generalized equation for the inertial forces,
which incorporates the Mach's principle into Gravitation. Also, we have deduced
the equation of Entropy; the Hamiltonian for a particle in an electromagnetic
field and the reciprocal fine structure constant directly from this new
approach. It was also possible to deduce the expression of the Casimir force
and to explain the Inflation Period and the Missing Matter, without assuming
existence of vacuum fluctuations. This new approach to Gravitation will allow
us to understand some crucial matters in Cosmology.

Full text

Page 1

Mathematical Foundations of the
Relativistic Theory of Quantum Gravity

Fran De Aquino
Maranhao State University, Physics Department, S.Luis/MA, Brazil.
Copyright © 2008-2011 by Fran De Aquino. All Rights Reserved

Abstract: Starting from the action function, we have derived a theoretical background that leads to
the quantization of gravity and the deduction of a correlation between the gravitational and the inertial
masses, which depends on the kinetic momentum of the particle. We show that the strong equivalence
principle is reaffirmed and, consequently, Einstein's equations are preserved. In fact, such equations
are deduced here directly from this new approach to Gravitation. Moreover, we have obtained a
generalized equation for the inertial forces, which incorporates the Mach's principle into Gravitation.
Also, we have deduced the equation of Entropy; the Hamiltonian for a particle in an electromagnetic
field and the reciprocal fine structure constant directly from this new approach. It was also possible to
deduce the expression of the Casimir force and to explain the Inflation Period and the Missing Matter,
without assuming existence of vacuum fluctuations. This new approach to Gravitation will allow us to
understand some crucial matters in Cosmology.

Key words: Quantum Gravity, Quantum Cosmology, Unified Field.
PACs: 04.60.-m; 98.80.Qc; 04.50. +h

Contents

1. Introduction 3

2. Theory 3

Generalization of Relativistic Time 4

Quantization of Space, Mass and Gravity 6

Quantization of Velocity 7

Quantization of Time 7

Correlation Between Gravitational and Inertial Masses 8

Generalization of Lorentz's Force 12

Gravity Control by means of the Angular Velocity 13

Gravitoelectromagnetic fields and gravitational shielding effect 14

Gravitational Effects produced by ELF radiation upon electric current 26

Magnetic Fields affect gravitational mass and the momentum 27

Gravitational Motor 28

Gravitational mass and Earthquakes 28

The Strong Equivalence Principle 30

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Incorporation of the Mach's Principle into Gravitation Theory 30

Deduction of the Equations of General Relativity 30

Gravitons: Gravitational Forces are also Gauge forces 31

Deduction of Entropy Equation starting from the Gravity Theory 31

Unification of the Electromagnetic and Gravitational Fields 32

Elementary Quantum of Matter and Continuous Universal Fluid 34

The Casimir Force is a gravitational effect related to the Uncertainty Principle 35

The Shape of the Universe and Maximum speed of Tachyons 36

The expanding Universe is accelerating and not slowing down 38

Gravitational and Inertial Masses of the Photon 39

What causes the fundamental particles to have masses? 40

Electron’s Imaginary Masses 41

Transitions to the Imaginary space-time 43

Explanation for red-shift anomalies 48

Superparticles (hypermassive Higgs bosons) and Big-Bang 49

Deduction of Reciprocal Fine Structure Constant and the Uncertainty Principle 51

Dark Matter, Dark Energy and Inflation Period 51

The Origin of the Universe 57

Solution for the Black Hole Information Paradox 59

A Creator’s need 61

The Origin of Gravity and Genesis of the Gravitational Energy 62

Explanation for the anomalous acceleration of Pioneer 10 64

New type of interaction 66

Appendix A 69

Allais effect explained 69

Appendix B 72


References 73

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1. INTRODUCTION

Quantum Gravity was originally
studied, by Dirac and others, as the
problem of quantizing
Relativity. This approach presents
many difficulties, detailed by Isham
[1]. In the 1970's, physicists tried an
even more conventional approach:
simplifying Einstein's equations by
assuming that they are almost linear,
and then applying the standard
methods of quantum field theory to
the thus oversimplified equations. But
this method, too, failed. In the 1980's
a very different approach, known as
string theory, became popular. Thus
far, there are many enthusiasts of
string theory. But the mathematical
difficulties in string theory are
formidable, and it is far from clear that
they will be resolved any time soon.
At the end of 1997, Isham [2] pointed
out several "Structural Problems
Facing Quantum Gravity Theory". At
the beginning of this new century,
the problem of quantizing the
gravitational field was still open.
In this work, we propose a new
approach to Quantum
Starting from the generalization of the
action function we have derived a
theoretical background that leads to
the quantization of gravity. Einstein's
General Relativity equations are
deduced directly from this theory of
Quantum Gravity. Also, this theory
leads to a complete description of the
Electromagnetic Field, providing a
consistent unification of gravity with
electromagnetism.

2. THEORY

We start with the action for a
free-particle that, as we know, is
given by


General
Gravity.

∫
quantity
−=
a
b
adsS
α
where
characterizes the particle.
In Relativistic Mechanics, the
action can be written in the following
form [3]:
tt
∫∫
11
where
cL
−= α
is the Lagrange's function.
In Classical Mechanics, the
Lagrange's function for a free-particle
is, as we know, given by:
where V is the speed of the particle
and is a quantity hypothetically [
a
given by:
ma =
where is the mass of the particle.
However, there is no distinction about
the kind of mass (if gravitational
mass,, or inertial mass
about its sign
The correlation between and
α can be established based on the
fact that, on the limit
relativistic expression for
reduced to the classic expression
.The result [
aVL =
Therefore, if
ac =
2
=α
. Now, we must decide if
aVL =
mm =
or
m =
. We will see in this
work that the definition of
. Thus, the right option is
i
m
ma
=
Consequently,
=
α
generalized expression for the action
of a free-particle will have the
following form:
−=
g
cmS
or
α is which
dtcVcLdtS
tt
−−==
22
22
1
α
22
1
cV
−

2
aVL =
4]
2
) neither
m
g
m
i
m
( ) ± .
a
, the
∞→
L must be
c
2
5] is:
cVL
, we obtain
2
2
α=
.
mc
2
g
i m
includes
, i.e.,
g
m
g
m
.
c
g2
mg
and the
( ) 1
∫
b
a
ds

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4
( ) 21
222
2
1
dtcVcmS
t
t
g
−−=∫
where the Lagrange's function is
mL
g
−=
=∫
preceded by the plus sign, cannot
have a minimum. Thus, the integrand
of Eq.(2) must be always positive.
Therefore, if
; if , then
possibility of is based on the
well-known equation
of Einstein's Theory.
Thus if the gravitational mass
of a particle is positive, then t is also
positive and, therefore, given by
1
cVtt
−+=
. This leads to the
well-known relativistic prediction that
the particle goes to the future, if
. However, if the gravitational
mass of the particle is negative, then
is negative
t
1
cVtt
−−=
. In this case, the
prediction is that the particle goes to
the past, if
is the necessary condition for
the particle to go to the past. Further
on, a correlation
gravitational and the inertial masses
will be derived, which contains the
possibility of
0
<
g
m
The Lorentz's transforms follow
the same rule for
m
( ) 31
222
.cVc
−
The integral
dtcVcmS
t
t
g
2221
2
1
−
,
, then necessarily
<
t
. The
0
>
g
m
0
>
t
0
<
t
g
m
0
0
<
22
0
1
cVtt
−±=

22
0
and given by
cV →
22
0
. Consequently,
between the
.
cV →
0
<
g
m
and
0
>
g
0
<
g
m
,
i.e., the sign before
when
( )
+
The momentum, as we know,
is the vector
p
=
Eq.(3) we obtain
m
p
−±
1
The sign in the equation above
will be used when
m
22
1
( )
−
c
m
V
−
will be
0
<
. and if
0
>
g
m
g
VL
r
r
∂∂
.Thus, from
VM
cV
V
g
g
r
r
r
==
22
and the
( )
+
0
>
g
( )
−
sign if
will be always positive. Consequently,
we will express the momentum
the following form
Vm
p
−
The derivative
inertial force which acts on the
particle. If the force is perpendicular
to the speed, we have
m
F
i
−
However, if the force and the speed
have the same direction, we find that
0
<
g
m
. This means that
g
M
pr in
( ) 4
1
22
VM
cV
g
g
r
r
r
==
dtpdr
is the
iF
( ) 5
1
22
dt
Vd
cV
g
r
r
=
(
1
)
( ) 6
2
3
22
dt
Vd
cV
m
F
g
i
r
r
−
=
From Mechanics [6], we know that
LVp
−⋅
denotes the energy of the
particle. Thus, we can write
cm
LVpE
g
−
Note that is not null for
g
E
that it has the finite value
mE
g
Equation (7) can be rewritten in
the following form:
2
2
g
gg
cV
−
r
r
( ) 7
1
2
22
2
cM
cV
g
g
==−⋅=
r
r
0
=
V
, but
( ) 8
2
00
c
g =
()( ) 9
1
4
1
0
2
2
4
2
2
2
2
22
i
i
g
Kii
i
g
E
i
i
i
i
g
g
E
m
m
EE
m
m
cm
cV
cm
−
4
cm
m
m
cm
cm
cmE
Ki
=+=
=
⎥
⎦
⎥
⎥
⎥
⎥
⎤
⎢
⎣
⎢
⎢
⎢
⎢
⎡
⎟
⎠
⎟
⎞
⎜
⎝
1
⎜
⎛
−+=
=−−=
44342
By analogy to Eq. (8),
the equation above, is the
inertial energy at
EEE
+=
0
is the total inertial
into
rest. Thus,
2
00
cmE
ii=
Kiii

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5
energy, where
inertial energy. From Eqs. (7) and (9)
we thus obtain
2
0
cV
−
For small velocities
obtain
2
0
cmE
ii
≈
where we recognize the classical
expression for the inertial kinetic
energy of the particle.
The expression
gravitational kinetic energy,
easily deduced by comparing Eq.(7)
with Eq.(9). The result is
m
E
i
In the presented picture, we
can say that the gravity,
gravitational field produced by a
particle of gravitational mass
depends on the particle's gravitational
energy,(given by Eq.(7)), because
we can write
M
G
c
r
Due to
rg
∂Φ∂=
, the expression of
the relativistic
potential, , is given by
Φ
is the kinetic
Ki
E
( )
10
1
2
22
.
cM
cm
E
i
i
i
==
, we
(
cV<< )
( )
11
2
2
1
V
i
m
+
for the
, is
Kg
E
( )
12
.E
m
Ki
g
Kg=
, in a
,
gr
g
M
g
E
( )
13
222
2
22
r
M
G
c
r
cE
Gg
ggg
===
gravitational
22
1
cV
m
r
G
r
GM
gg
−
−=−=Φ
Then, it follows that

2222
11
cVcVr
Gm
r
GM
gg
−
=
−
−=−=Φ
φ

where
Then we get
r Gmg
−=φ
.
22222
11
cVr
Gm
cVr
r
g
−
=
−∂
∂
=
∂
Φ∂
φ
whence we conclude that
2221
the
cVr
Gm
r
g
−
=
∂
Φ∂
By
potential
gravitational mass of a particle inside
a gravitational field is equal to the
gravitational potential
Thus, we can write that
U
=Φ
definition, gravitational
per unit energy of
of the field.
Φ
( )
m
′
g
r

Then, it follows that

U
F
g
∂
( )
r
2221
cVr
mm
G
r
m
r
g
′
g
g
′
−
−=
∂
Φ∂
−=
∂
−=

If
m
force will never be null due to the
existence of a minimum value for
(see Eq. (24)). However, if
and
0
<
′g
m
, or
m
the force will be attractive. Just
for
ig
mm =
and
the Newton's attraction law.
On the other hand, as we
know, the gravitational force is
conservative. Thus,
energy, in agreement with the energy
conservation law, can be expressed
by the decrease of the inertial energy,
i.e.,
E
Δ
=
This equation expresses the fact that
a decrease of gravitational energy
corresponds to an increase of the
inertial energy.
Therefore, a variation
yields a variation
Thus
iii
EEE
Δ
+=
0
;
g
E
and
EE
=+
Comparison between (7) and (10)
shows that
00
ig
EE
=
Consequently, we have
and
the force will be repulsive; the
0
>
g
m
0
<
′g
m
, or and
0
<
g
m
0
>
′g



g
m
<
>
0
0
g
′g
m
m
and
0
>
g
we obtain
i
′
g
′
mm
=
gravitational
( )
14
ig
E
Δ
−
i E
Δ
in
E
in
E
E
Δ
−
i E
ig
+
E
=
E
Δ
−
E
Δ
Δ
E
=
.
g
iggg
=
00
( )
15
00
igig
EE
+
, i.e.,
00
ig
mm
=
.