Haramein, Rauscher - A consideration of torgue and Coriolis forces in Einstein's field equations.pdf

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THE ORIGIN OF SPIN: A CONSIDERATION OF TORQUE AND CORIOLIS FORCES
IN EINSTEIN’S FIELD EQUATIONS AND GRAND UNIFICATION THEORY
N. Haramein and E.A. Rauscher §
The Resonance Project Foundation, haramein@theresonanceproject.org
§
Tecnic Research Laboratory, 3500 S. Tomahawk Rd., Bldg. 188, Apache Junction, AZ 85219 USA
Received January 1, 2004
Abstract. We address the nature of torque and the Coriolis forces as dynamic properties of the spacetime metric and
the stress-energy tensor. The inclusion of torque and Coriolis effects in Einstein’s field equations may lead to
significant advancements in describing novae and supernovae structures, galactic formations, their center super-
massive black holes, polar jets, accretion disks, spiral arms, galactic halo formations and advancements in
unification theory as demonstrated in section five. We formulate these additional torque and Coriolis forces terms to
amend Einstein’s field equations and solve for a modified Kerr-Newman metric. Lorentz invariance conditions are
reconciled by utilizing a modified metrical space, which is not the usual Minkowski space, but the U 4 space. This
space is a consequence of the Coriolis force acting as a secondary effect generated from the torque terms. The
equivalence principle is preserved using an unsymmetric affine connection. Further, the U 1 Weyl gauge is associated
with the electromagnetic field, where the U 4 space is four copies of U 1 . Thus, the form of metric generates the dual
torus as two copies of U 1 x U 1 , which we demonstrate through the S 3 spherical space, is related to the SU 2 group and
other Lie groups. Hence, the S 4 octahedral group and the cuboctahedron group of the GUT (Grand Unification
Theory) may be related to our U 4 space in which we formulate solutions to Einstein’s field equations with the
inclusion of torque and Coriolis forces.
1. INTRODUCTION
Current standard theory assumes spin/rotation to be the result of an initial impulse generated in the Big Bang
conserved over billions of years of evolution in a frictionless environment. Although this first theoretical
approximation may have been adequate to bring us to our current advanced theoretical models, the necessity to
better describe the origin and evolution of spin/rotation, in an environment now observed to have various plasma
viscosity densities and high field interaction dynamics which is inconsistent with a frictionless ideal environment,
may be paramount to a complete theoretical model. We do so by formulating torque and Coriolis forces into
Einstein’s field equations and developing a modified Kerr-Newman solution where the spacetime torque, Coriolis
effect and torsion of the manifold becomes the source of spin/rotation. Thus, incorporating torque in Einstein’s
stress energy term may lead to a more comprehensive description of the dynamic rotational structures of organized
matter in the universe such as galactic formations, polar jets, accretion disks, spiral arms, and galactic halos without
the need to resort to dark matter/dark energy constructs. These additions to Einsteinian spacetime may as well help
describe atomic and subatomic particle interactions and produce a unification of fundamental forces as preliminarily
described in section five of this paper.
Modification of the field equations with the inclusion of torque requires an unsymmetric affine connection to
preserve the Principle of Equivalence and inhomogeneous Lorentz invariance, which includes translational
invariance as well as rotational invariance and, hence, spin. The antisymmetric torsion term in the stress-energy
tensor accommodates gauge invariance and maintains field transformations. Although the affine connection is not
always a tensor, its antisymmetric components relate to torsion as a tensor. This is the case because when only the
unsymmetric part is taken, the affine connections no longer disallows the existence of the tensor terms. We
demonstrate that such new terms lead to an intrinsic spin density of matter which results from torque and gyroscopic
effects in spacetime. The conditions on the Riemannian geometry in Einstein’s field equations and solutions are also
modified for torque and Coriolis forces and spacetime torsion condition. The torque and torsion terms are coupled
algebraically to stress-energy tensor. The effect of the torque term leads to secondary effects of the Coriolis forces
that are expressed in the metric. Torsion is a state of stress set up in a system by twisting from applying torque.
Hence, torque acts as a force and torsion as a geometric deformation. The gauge conditions for a rotational gauge
potential,
are used.
153
R. L. Amoroso, B. Lehnert & J-P Vigier ( eds. ) Beyond The Standard Model : Searching For Unity In Physics, 153-168,
© 2005 The Noetic Press, Printed in the United States of America.
154
N. Haramein & E. A. Rauscher
The affine connection relates to transformations as translations and rotations in a uniform manner and
represents the plasticity of the metric tensor in general relativity. Connections can carry straight lines into straight
lines and not into parallel lines, but they may alter the distance between points and angles between lines. The affine
connection
has 64 components or 4 3 components of A 4 . Each index can take on one of four values yielding 64
components. The symmetric part of
has 40 independent components where the two symmetric indices give ten
components including the times four for the third index. The torsion tensor  has 24 independent components
and it is antisymmetric in the first two indices, which gives us six independent components and four independent
components for the third index (indices run 1 to 4). These independent components relate to dimensions in analogy
to the sixteen components of the metric tensor g . If this tensor is symmetric then it has ten independent
components. Note for a trace zero, tr 0 symmetric tensor, we have six independent components. The components of
a tensor are, hence, related to dimensionality.
It appears that the only method to formulate the modified Einstein’s equations, to include torque and Coriolis
terms, is to utilize the U 4 spacetime and not the usual four-dimensional Minkowski space, M 4 . This is the case
because the vectors of the space in spherical topology have directionality generating a discontinuity or part in the
hairs of a sphere whereas a torus topology can have its vectors curl around its short axis having no parts in the
hairs so that no discontinuity of the vector space exists. Thus all the vectors of the space obey invariance
conditions. Also, absolute parallelism is maintained. The U 4 space appears to be the only representation in which we
can express torsion, resulting from torque, in terms of the Christoffel covariant derivative, which is used in place of
the full affine connections where represents the covariant derivative in U 4 spacetime using the full unsymmetric
connections. Thus we are able to construct a complete, self-consistent theory of gravitation with dynamic torque
terms and which results in modified curvature conditions from metrical effects from torsion. In the vacuum case, we
assume
4 x
R where R is denoted as the scalar curvature density in U 4 spacetime. This new approach to
the affine connection may allow the preservations of the equivalence principle. The usual nonsymmetric stress-
energy tensor is combined with its antisymmetric torque tensor. The U 4 is key to the structure of matter affected by
the structure of spacetime. We present in detail the manner in which the U 4 group space relates to the unification of
the four force fields. The structure of U 4 is four copies of U 1 , the Weyl group, as
d
 0
 where
U 1 x U 1 represents the torus. Hence U 4 represents the dual torus structure. In this case we believe the U 4 spacetime,
which allows a domain of action of torque and Coriolis effects, is a model of the manner in which dynamical
properties of matter-energy arise.
Further, in section five we show that the 24 elements of the torsion tensor can be related to the 24 element
octahedral gauge group S 4 which are inscribed in S 2 , and that the 24 element octahedral gauge is related to the cube
through its being inscribed in S 2 . The 24 element group through S 2 yields the cuboctahedral group which we can
relate to the U 4 space; thus, we can demonstrate a direct relationship between GUT theory to Einstein’s field
equations in which a torque tensor and a Coriolis effect is developed and incorporated.
U
U
U
U
U
4
1
1
1
1
2. ANALYSIS OF TORQUE AND CORIOLIS FORCES
In this section we present some of the fundamental descriptions of the properties of the torque and Coriolis forces.
We examine the forces, which appear to yield a picture of galactic, nebula, and supernova formation. We apply
these concepts to Einstein’s field equations and their solutions. The angular momentum is
L
L
r
p
and
where r is a radial variable and p is a linear momentum. The torque
d
L
(1)
r
F
dt
where F is force and the conservation theorem for the angular momentum of a particle states that if the total torque
is zero then
d
L
L
0
(2)
dt
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The Origin of Spin
155
0
then L is not conserved. Torque is a
and thus the angular momentum is conserved. In the case where
twisting or turning action. Whereby
d
p
d
 
r
F
r
m
r
r
p
(3)
dt
dt
for r is a constant. The force F is orthogonal to , and r is not parallel to F . The centrifugal term is then given as
(4)
2 r
c
where is the rotation of a spherical body, such as the earth’s angular velocity or rotation and r its radius and
is the angle of latitude. The Coriolis term is proportional to
 cos
0
2 and is responsible for the rotation of the plane
of oscillation of a Foucault pendulum. This is a method whereby the Coriolis force can be detected and measured.
The key to the gyroscopic effect is that the rate of change in its angular momentum is always equal to the applied
torque. The direction of change of a gyroscope, therefore, occurs only when a torque is ap plied. The torque is
v
r
F
(5)
2
due to F which is perpendicular to r and L is the vector angular momentum
L
r
p
m
r
v
where the
vector r is taken along the axis of the gyroscope, and is a phase angle in the more general case.
A spinning system along an axis r with an angular momentum L has a torque in equation ( 1 ) when the force F
is directed towards the center of gravity. If the total force,
F
0
p
0
then
and linear momentum is conserved.
Angular frequency,
2
1
d
V
2
(6)
2
m
dr
E where E is the total energy, V is the potential energy, T is kinetic energy
and m is the mass of the system. A revolving of a particle has angular velocity
V
T
in the generalized case where
d
L
(7)
.
dt
2
mr
The rate of revolution decreases as r increases. If r = constant, then the areas swept out by the radius from the origin
to the particle when it moves for a small angle
d
, then
1
dA
r
2
d
(8)
2
L
mr
2
then
and has an area A. Then
dA
1
1
d
L
r
2
r
2
(9)
dt
2
2
dt
2
m
the radius vector r moves through
d
and for a central force, if the motion is periodic, for integration over a
Lt
t of motion, we have the area of the orbit
A
complete period
0
.
2
m
For a rigid uniform bar on a frictionless fulcrum, the moment of a force, or torque, in the simplest of mechanical
terms, is the mass times the length of the arm. The product of the force and the perpendicular distance from the axis
line of the action of the force is called the force arm or movement arm. The product of the force and its force arm is
called the moment of the force or the torque . In more detail, we can describe torque in terms of a force couple
exerted on the end of a rod for a solid or highly viscous material producing a twist displacement and hence shear
stress and shear stain
Shear
stress
F
/
A
M
(10)
Shear
strain
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156
N. Haramein & E. A. Rauscher
where F is the force, A is the area, is the angle of distortion and M is the shear modulus. Torsion is a state of
stress set up in a system by twisting from an applied torque. Torque creates action or work. The external twisting
effect is opposed by the shear stresses included in a solid or highly viscous material. That is, torsion is the angular
strain produced by applying torque, which is a twisting force, to a body or system, which occurs when, for example,
a rod or wire is fixed at one end (i.e., has an equal and opposite torque exerted on it) and rotated at the other.
Therefore, torque is a force and torsion is a geometric deformation in the medium given by the torsion
Mr
2
4

(11)
d
where r is the radius and d is the length or distance in flat space. The torque for such a system is defined by
 or
4
Mr
2
(12)
d
where is in units of dyne-cm, M is the shear modulus and relates to the distortion of the shaft in dyne / cm 2 and
is the angle in radians through which one end of the shaft is twisted relative to the other. The moment of inertia
is denoted as I and we substitute
from equation ( 6 ).
1
1
1
 
2
2
2
2
E k
mv
mr
I
(13)
.
2
2
2
In our case, the term W for a generalized modulus in a medium that relates to the shear tensor of a fluid
torsion (Ellis, 1971) is utilized. We employ a torque tensor as the
m ,
E
which is a term in Einstein’s

stress-energy tensor T  where torque is given as
4
Wr
2

(14)

R
where R is the scalar curvature path in U 4 space over which torque acts and r is the radius of twist produced by the
torquing force acting over R. In order to define the scalar sustained for maximum curvature, hence maximum torque
in spacetime, we express the spatial gradient of R along the vector length
R  . This is the tensor form
that can be utilized in Einstein’s field equations. The distance or length is now denoted as R in a generalized curved
space. We can denote R as R . The quantity is a tensor in which rotation is included, and hence requires
inhomogeneous Lorentz transformations and requires a modification of the topology of space from M 4 into U 4
space, which has intrinsic rotational components. In order to convert from Minkowski space to U 4 space we must
define the relationship of the metric tensor and the coordinates for each space. We have the usual Minkowski metric
R as
R
2
2
ds
g
dx
dx
ds
dx
dx
and the metric of U 4 space is given as
. We relate the metrics of the M 4


x
x
g
T than
T v
T
space and U 4 space as
. For any tensor
(all indices run 1 to 4). Then


v

x
x
4

as
x
d
0
under the gauge transformation for an arbitrary
, we have
in



in Minkowski space.
Note that the spin field is the source of torsion and is the key to the manner in which spin exists in particle
physics and astrophysics. The formulation of torque is not included in Einstein’s field equations in any manner and
is not incorporated in
4

g
d
xT
U 4 space in analogy to
T terms without modifications. Currently it appears that torque and Coriolis
forces are eliminated by attaching the observer to a rotating reference frame and by assuming an absolute symmetry
of the stress-energy tensor
R
,
v g
and
v
v


T so to make the torque vanish [ 1] . We believe that inclusion of torque is
essential to understanding the mechanics of spacetime, which may better explain cosmological structures and
potentially the origin of rotation.
T
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The Origin of Spin
157
3. INCLUSION OF TORQUE AND CORIOLIS FORCE TERMS IN EINSTEIN’S FIELD EQUATIONS
In order to include torque, we must modify the original form of Einstein’s field equations. The homogeneous and
inhomogeneous Lorentz transformations involve linear translations and rotation, and hence angular momentum is
accommodated. The time derivative of angular momentum, or torque, is not included in its field equations.
Researchers have attempted to include torsion by different methods since Elie Cartan’s letter to Einstein in the early
1930’s [2 ]. However, we feel that an inclusion of torsion in Einstein’s Field Equations demands a torque term to be
present in the stress-energy tensor in order to have physical effects.
Two currently held key issues are addressed in which torque and Coriolis forces are eliminated. First, in
reference [1 ] the complications of fractional differences are avoided by formulating them in terms of the size of
spatial lower limit Planck length dimension, and the earth’s gravitational acceleration g ~ 10 3 cm/sec 2 . The choice
of
g  is made so that the accelerated frames undergo small accelerations which yields an approximately
inertial frame. Black hole dynamical processes requires a relaxation
1
g  . If one considers a vacuum structure
having a lattice form, then the conditions to include torque and Coriolis forces require a relaxation of the
1
g 
1
g . Second, the
torque and Coriolis forces are eliminated in a nonrelativistic manner by carefully choosing the observer’s state of
coordinates by preventing the latticework from rotating, i.e. by tying the frame of reference to a gyroscope that
accelerates in such a manner that its centers of mass are chosen to eliminate these forces [ 1 ]. Hence, we have a
major clue for including torque so as to fix our frame of reference to the fundamental lattice states, which includes
rotation terms, and does not eliminate them. Then, for
g
1
~1
condition to be consistent with black hole physics and torque terms in relativity, then
or
d
e
m
a
a
u
e
(15)
dt
so that
a is eliminated, noting that u is the four vector velocity and e is a basis vector in analogy to x, y, z.
The incorrect transport equation is the Fermi-Walker transport equation because it is formulated in a rotating frame
that eliminates torque. This equation acts at the center of mass so that I, the moment of inertia, is zero; hence this
cannot be our reference frame.
It appears that we must utilize a different kind of rotational frame of reference. We have utilized this frame using
the Kerr-Newman or Reissman-Nordstrom solutions with spin, as well as atomic spin and the spin of the whole
universe as in our scaling law [ 3 - 8 ]. We thus generate a torus from our new basis vector set e [ 9 ].
Given these two conditions, we proceed to account for a torque term in Einstein’s Field Equations. The angular
momentum vector L for a system must change in order to have torque. Hence L is not orthogonal to u , the four
velocity; thus, a torque can be utilized in Einstein’s field equations. Then
e
d
L
(
u
a
)
L
(16)
dt
whereas in the Fermi-Walker transport case
d
L
(
u
a
)
L
(17)
dt
where a is the four acceleration. The fact that a non-zero solution exists allows us to choose frames of reference
that do not move with the system and include torque, which requires a variable acceleration. No longer is
2
3
2
L
(18)
4
constant because torque,
d
L
L
0
(19)
dt
where L is the angular momentum.
Key to the inclusion of torque terms and its torsion effects is the modification of Einstein’s field equations
formulated in the generalized U 4 spacetime. This approach can be reconciled with conditions for affine connections
and extended Lorentz invariance. Torsion resulting from torque is introduced as the antisymmetric part of the affine
connection. The U 4 space appears to be the only spacetime metric that yields an unsymmetric affine connection and
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