-
On the Matter of Space:An Already-Unified Gauge Field Theory
Brian Lee Scipioni∗
16638 E IvanhoeMontgomery, TX 77316(Dated: November 9, 2019)
This already-unified gauge field theory uses solutions to
classical force field equations as back-ground dependent metric
gauge transformations. Based upon their analytic properties they
areidentified as the different local force fields with a built-in
hierarchy; scalar, vector, tensor. It entailsquantization of
charge. The Einstein field equation becomes an identity and a
background indepen-dent wrapper for the flat space fields with
singularities removed from the theory. The stress-energytensor
becomes a covariant consequence of the local fields. The Laue
scalar, total mass, stress-energytensor are calculated for massive
and massless gauge fields. Exact solutions for neutral spin
zeromasses, electromagnetic plane waves, gravitational waves, the
Aharonov-Bohm effect and galacticrotation curves are analyzed. The
static, spherically symmetric gravitational case is asymptoti-cally
the Schwarzchild solution, but without singularity. The Laue scalar
integrated over all space isshown to correctly give the
gravitational mass. Black holes become ”red holes”, almost
identical, butdistinguishable, and without an event horizon. The
gravitational wave stress-energy tensor is shownto be identical to
both the Landau-Lifshitz pseudotensor and the TT-mode Isaacson
pseudotensorfrom the linearized theory, only exact and covariant.
Electromagnetic plane wave solutions give aflat space. Charged
particles are found to rely on gravity for their existence. The
Aharonov-Bohmeffect is shown to be purely geometrical and the
magnetic energy is found to be stored outside themagnet as
gravitational energy. Dark Matter is a natural consequence and is
shown to fit galaxyrotation curves without the need for a modeled
DM density profile. Neutral spin zero masses areautomatically
prevented from singularity by a Yukawa term having identical
properties to the Higgsboson, providing inertial mass, but being
classical in nature. If gravitational masses are allowed tobe
negative, they have negative energy and are found to be repelled by
positive energy masses. Asmall asymmetry is found between positive
and negative energy fields, although they are identicalin their
masses. Their symmetry is found to be restored under a Mostly-Minus
to Mostly-Plusmetric transformation. The baryon asymmetry is
explained and the cosmological implications arediscussed.
Experiments to validate and refute are proposed.a
PACS numbers: 04.20.Cv, 98.80.-k, 95.35.+d, 12.10.-g, 04.20.Dw,
04.30.-wKeywords: General relativity formalism, Unified field
theories, Cosmology, Dark matter, Singularities, Grav-itational
waves
I. INTRODUCTION
Since the discovery of the nuclear forces and the suc-cess of
Quantum Electrodynamics (QED) the focus ofunification has primarily
been on quantum field theory[1]. Since 3 of the 4 forces are
represented in the Stan-dard Model (SM) it seems reasonable to try
to quantizethe gravitational field to complete the task.
Howeverthis could be a category error. It is possible that quan-tum
principles do not apply to General Relativity(GR)[2] since it is a
background independent theory. Althoughattempts at classical
unification have been unsuccessful,there is a good argument to
proceed in that direction.Electromagnetism (EM) has one foot in
each world. It isa long range force with a successful well
developed clas-sical theory like gravitation. It is an integral
part of theSM and closely tied to the weak force. Unification of
EMwith gravity is therefore certain to define the
relationshipbetween gravity and quantum theory.
∗ [email protected]; https://thematterofspace.coma
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
The approaches to unification, classical and quantum,all appear
to have one aspect in common. They gener-alize, extend or add
degrees of freedom to the mathe-matical framework. String theory,
loop quantum grav-ity, extra dimensions, non-symmetric connections,
com-plex metrics, Finsler spaces, etc., have not yet
worked.Theories of unification have become increasingly complexand
removed from verifiability and falsifiability such asthe GUT
theories. A much simpler formulation is possi-ble. What emerges is
a gravitational field as a relativisticgauge field along side the
other forces, and General Rel-ativity as a covariant gauge theory
of them all. In thisprocess the Einstein Field Equation (EFE) is
promotedto an identity.
II. METHODOLOGY
A subtle yet profound change to the ontological [3]basis of
physics can both lead the way to a unifiedfield theory and shed
light on the epistemological dif-ferences between General
Relativity(GR) and QuantumTheory(QT). There is a simple framework
for unification
-
that is testable, refutable, and leaves the
mathematicalstructure of general relativity and quantum field
theoryintact, namely.
Gµν + Λ gµν = −8π κ
c2Tµν , (1)
for the Einstein Field Equation (EFE) and
Fµν‖ν = jµ ∧
{Fµν|λ
}= 0, (2)
for the Maxwell Equations (ME).This already-unified gauge field
theory is an extremely
simple theory. It eliminates singularities from thetheory. It
reconciles the background-independent andbackground-dependent
foundations of general relativityand quantum theory, respectively.
It eliminates the needto ”put in by hand” the right hand side of
the EFE. It canalso provide the missing picture [4] for quantum
theory.It also solves several outstanding problems in physics
andnaturally includes Dark Matter. The following set of de-ductions
and inferences provides a compelling argumentfor this
already-unified gauge field theory. The reasoningis based upon two
principles that are not entirely inde-pendent. They are actually
inferred from the theory, notthe other way around.
A. Axioms
1. The only necessary field is the field of event
dis-placements.
Historically the field concept was introduced toavoid
action-at-a-distance [5]. A dynamical space-time accomplishes this
for gravity in GR throughthe metric tensor; matter curves
spacetime, space-time guides matter. There is no such view for
theEM forces. In that case the space is mitigated by afield. It is
necessary to have a dynamical spacetimeformulation of EM to fit it
into GR. The use of theword mitigated above is literal, to lessen.
Assum-ing matter at a distance cannot interact withoutfields yields
a contrapositive; if material particlesdo interact without fields,
then they must not be”at a distance”. There is only one way this
canhappen. Matter itself must extend through space-time, as part of
the same continuum, so that oneparticle can smoothly meld into
another. So in or-der to eliminate the field concept, a particle
mustbe made up of the same ”stuff” as spacetime withmost of it
fairly well localized to appear as a par-ticle. Two particles can
therefore interact using afield as an intermediary, or a dynamical
spacetime.The field concept becomes superfluous if it can beshown
that the event displacement field solves thesame equations.
2. The field should be free from singularities.
The presence of singularities in GR is unacceptableto many
general relativists [6]. This was also Ein-stein’s viewpoint [7];
an acceptable theory has towork everywhere. Infinities are
unmeasurable. Par-ticles must be represented by a finite matter
fieldin some finite region. The observables of spacetimeare
distance and duration. These are specified bythe metric tensor.
Therefore, for any real configu-ration of matter there must exist a
coordinate sys-tem that results in measurable intervals. As will
beshown, the removal of singularities is an automaticbyproduct of
unification.
B. The Gauge Field
The Maxwell equations, Eqs. (2) are analogous to theequations of
fluid flow, complete with sources, sinks andvortices. This was
noted early on and there were at-tempts to mechanize the field with
a quasi-elastic ethermodel [8]. Riemann attempted to unify gravity
and elec-tromagnetism with such a model [9]. The approach wasto
assume space contained some kind of substance thatcould flow or
spin. This idea was unworkable. Michelsonand Morely showed that
there is no lumeniferous ether[10]; such a substance would permeate
space and serve asa dynamical medium for EM kinematics. Therefore
theanalogy is either an accidental coincidence, or it repre-sents
some other kind of motion. There is only one otherpossibility for
such a displacement field. It is that theMaxwell equations
represent a transformation of space-time points themselves, rather
than some substance oc-cupying spacetime points. Just accepting it
as a ”field”admits that the structure of the equations is a
coinci-dence. Such an acceptance also introduces a new ele-mentary
object that requires its relationship to gravitybe separately
defined, complicating the ontology [11].
The transformation of events can be described mathe-matically in
the same way as that of a deformable phys-ical medium. Consider an
infinitesimal displacement, ξ,in the neighborhood of a small volume
element in a 3 di-mensional Euclidean space. It is composed of a
rotation,a compression (extension or shear), and a translation
[12].
ξi(xj)
= ξi(0) + ξi|jdxj +O
[dx2]
' ξi0 + gil[
1
2
(ξl|j + ξj|l
)dxj +
1
2
(ξl|j − ξj|l
)dxj],
{i, j, l} ∈ {1, 2, 3}. (3)
This can be generalized to a 4-dimensional pseudo-Euclidian base
space, having metric g, that is tangentto the Riemannian manifold
at some point. In that casetemporal displacements as well as
spatial displacements
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
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are both taken to be functions of space and time.
ξµ(x0, xj
)' ξµ(0, 0) + ξi|0dx
0 + ξ0|jdxj
+ gil[
1
2
(ξl|j + ξj|l
)dxj +
1
2
(ξl|j − ξj|l
)dxj],
{i, j, l} ∈ {1, 2, 3}, µ ∈ {0, 1, 2, 3}. (4)
For infinitesimal displacements this becomes
dξµ (xν) = gµλ[
1
2
(ξλ|ν + ξν|λ
)dxν
+1
2
(ξλ|ν − ξν|λ
)dxν
], {µ, ν, λ} ∈ {0, 1, 2, 3}. (5)
Putting this into covariant form allows for arbitrary
co-ordinate systems in the base space.
ξµ‖νdxν = gµλ
1
2
(ξλ‖ν + ξν‖λ
)dxν
+ gµλ1
2
(ξλ‖ν − ξλ‖µ
)dxν , {µ, ν, λ} ∈ {0, 1, 2, 3}. (6)
Expressing this in terms of the symmetry properties ofthe
displacement field,
ξµ‖νdxν =
1
2gµλ
(ξλ‖ν + ξν‖λ
)dxν +
1
2gµλ
(ξλ‖ν
−ξλ‖µ)dxν = gµλσλνdx
ν + gµλαλνdxν (7)
with σ the symmetric tensor and α antisymmetric. Therelationship
between the displacement and the metrictensor, g, then follows.
Assume a displacement field isintroduced into a locally flat region
of space with coordi-nates xµ and metric ḡ,
ḡ =
1 0 0 00 −1 0 00 0 −1 00 0 0 −1
, xµ = (x0, x, y, z) (8)so that
ds2 = ḡµνdxµdxν . (9)
An infinitesimal displacement of the points would causetwo
events with the coordinate separation dxµ to have anew coordinate
separation,
dx̄µ =(δµν + �ξ
µ‖ν
)dxν , � =
1
N
-
C. The Electromagnetic Field
The antisymmetric part of the displacement field inEq. (7)
represents rotations and flows of events with re-spect to the base
space coordinate system. It is also anexact tensor so automatically
satisfies 2 of the Maxwellequations. It is therefore taken to be
(proportional to)the electromagnetic field.
αµν =1
2
(ξµ|ν − ξν|µ
)={ξµ|ν
}≡ fµν ∝ Fµν = ηfµν ,
(14)with the Maxwell equations as in Eq. (2) and the Carte-sian
components of the microscopic electromagnetic fieldtensor and
4-vector potential represented as
Fµν =
0 Ex Ey Ez−Ex 0 Bz −By−Ey −Bz 0 Bx−Ez By −Bx 0
, (15)
Fµν = φµ|ν − φν|µ, φµ =(A0, Aι
), (16)
in Heaviside-Lorentz units. There is a problem with
in-terpreting the electromagnetic field as having flows. Thesource
equations seem to require sources (sinks) out of(into) which
whatever is flowing is appearing (disappear-ing) [8]. This can be
explained by imbuing spacetimewith a non-simply connected topology
[14, 15]. Mathe-matically that introduces a significant complexity.
How-ever it is unnecessary. Consider the electric field compo-nent
of fµν .
f0i =1
2
(ξ0|i − ξi|0
)= −eee ∝ −EEE = ∇A0 +
∂AAA
∂t(17)
This ”flow” is a space-time rotation. It has two terms.The time
derivative represents a flow of spatial pointsthat has to go
somewhere for any steady state. How-ever for the electrostatic
case, with no other matter orenergy present, the vector potential,
AAA, or its divergence,is chosen to be zero (Coulomb gauge). In the
electrody-namic case the radiation has an oscillatory (or
transientin the case of non-periodic induction) vector potential.In
neither case is a novel topology required. For an elec-trostatic
field,
EEE = −∇A0 ∝ −ξ0|i, (18)
sources (electric charges) cause a gradient of the
timedisplacement, A0. It appears that a clock placed near
anelectric charge and then moved will be offset ahead orbehind
depending upon the sign of the charge. This maybe true on a
particular trajectory as determined by theconnections, which may
have factors linear in E, but arecoordinate dependent. The metric
ultimately determinesthe ”rate” at any event. The size of any such
offset orrate effect is determined by the proportionality
constant,η, between e and E. Magnetic fields correspond to
purelyspatial rotations.
This completely geometrized electromagnetic field is agauge
transformation on events. In general, gauge invari-ance is not to
be implied. Consider the particular casewhere the symmetric tensor
in Eq. (7) is zero. In thatparticular gauge the antisymmetric
electromagnetic fieldtensor is like a Lorentz transformation, with
two impor-tant differences:
1. It is a physical transformation of events, not coor-dinates
and
2. It is local not global. That is, it varies from event toevent
and is therefore a second order gauge trans-formation.
Also if
σλν =1
2
(ξλ‖ν + ξν‖λ
)= 0 (19)
then ξµ is a Killing vector field so the metric possessesa
hidden symmetry. This Killing field admits an electro-magnetic
field
gµν = ḡµν ⇒ αµν =1
2
(ξµ|ν − ξν|µ
)≡ fµν ∝ Fµν 6= 0
(20)In this particular gauge, since σµν = 0, and αµν is
antisymmetric,
gµν = (eααα)µλ ḡµν (e
ααα)ντ = ḡλτ , (21)
so that the electromagnetic field leaves the metric un-changed
and is therefore a gauge symmetry. This meansno curvature and the
electromagnetic field in this gaugeis not a source of gravity. So
although the metric tensorgives complete knowledge of the geometry
of spacetime,it does not necessarily provide complete knowledge of
thephysics of spacetime.
Consider the generalization of Eq. (13) where the gaugefield ζ
is a general second rank tensor. ζ can be decom-posed into a tensor
with zero divergence and one withzero curl (antisymmetrized
derivative).
ζµν = ξµν + fµν ξµν‖ν = 0 ∧{fµν|λ
}= 0. (22)
This means f is closed and therefore exact, admitting avector
potential
fµν = φ̄µ|ν − φ̄ν|µ (23)
and,
fµν‖ν = jµ 6= 0 ∧
{σµν|λ
}6= 0, (24)
in general. In addition, the vector field φ̄ can be de-composed
into one with zero divergence and one withzero curl.
φ̄µ = φµ + ḡµνην φµ‖µ = 0 ∧{ηµ|ν
}= 0⇒ ηµ = λ|µ.
(25)
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
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So any general tensor field ζζζµν can be derived from avector
field, φ, that satisfies all the Maxwell equationsEq. (2) and the
Lorentz condition,
fµν =1
2
(ξµ|ν − ξν|µ
)= φ̄µ|ν − φ̄ν|µ
= φµ|ν − φν|µ ∧ φµ‖µ = 0, (26)
plus a tensor field, ξ, derived from tensor, vector andscalar
elements;
ξµν = χµν +(φµ‖ν + φν‖µ
)+ λ|µ‖ν , (27)
where
χµν‖ν = 0. (28)
It is the main hypothesis here that these displacementfields,
ζ,
ζµν = χµν +(φµ‖ν + φν‖µ
)+ λ|µ‖ν + fµν , (29)
are what appear in the base space as force fields. Thisis the
implementation of AAA.1. Thus electromagnetismis incorporated into
GR simply by identifying it as thatpart of the displacement gauge
field solving the Maxwellequations. Electromagnetism remains
unchanged, as aflat space theory, and so does the formalism of GR,
as acovariant theory. The way electromagnetism enters intoGR
however is very different. The consequences of thiswill be dealt
with below.
Quantization of Charge
Since the EM field in (30) is a gauge field, it is a
gravi-tational potential. In the traditional
Reissner-Nordströmmetric [16] the contribution, Φg, from the
electric field,E, is proportional to q2 so that
g00 = 1−2m
r+ Φg = g00 = 1−
2m
r+Cq2
c2r2. (30)
Here Φg can be viewed as the EM self-energy, and as asource for
the gravitational field,
Cq2
c2r2=Cq2/
(rc2)
r=Cmer
, (31)
with methe mass equivalent of the EM energy. However,if the
gauge field is now proportional to the electric field,
Φg ∝Cq
r2=Cq/r
r, (32)
then the constant, C, must contain a factor of q, evenif only on
dimensional grounds. Since C is constant andthe energy is
proportional to q2, then there has to be
a universal q, equal to e, the basic electronic charge,contained
in C. In fact
C =κe
4π�0c4(33)
in SI units. So quantization of charge is a consequenceof
identifying the EM field as a GR gauge field.
D. The Gravitational Field
1. The Non-Relativistic (Exterior) Solution
The symmetric part of the displacement in Eq. (7) rep-resents
compressions (extensions or shears) of spacetimepoints. It is not
an exact tensor so its components in-volve the connections. This
means the tensor can betransformed away (locally) by a judicious
choice of coor-dinates. This is a key property of the gravitational
field.Also gravity is often depicted as stretches or compres-sions
of the ”fabric” of space and time. Identifying thesymmetric field
with gravity is therefore consistent withthe prevalent picture. The
exact connection between thesymmetric field, σµν , and gravity can
be deduced fromthe Schwarzchild solution to the EFE with Λ=0,
Gµν = −8πκ
c2Tµν , (34)
The spherically symmetric, static, free space solution ofEq.
(34),
Gµν = 0, (35)
in spherical coordinates(x0, r, θ, φ
), in the region exterior
to some mass, M, is the metric
gs =
1− 2m/r 0 0 0
0 −(1− 2m/r)−1 0 00 0 −r2 00 0 0 −r2 sin2(θ)
,m =
κM
c2. (36)
The EFE relates the metric tensor to matter content.However, the
general-relativistic matter density is dif-ferent from the
classical value for a given system. Thismeans the metric tensor
must be known in order to calcu-late the correct relativistic
value. While it is true that theEinstein tensor can be set to 0 and
the equation solved,this may not correspond to a physically
realizable con-figuration, at least not the one intended.
It is well established that there is Dark Matter (DM)everywhere.
Its signature is found in the Cosmic Mi-crowave Background (CMB),
galaxy rotations curves,and both stellar and galactic clusters.
That space is filledwith fluctuating zero-point fields is also
known. G = 0 iswrong for so-called empty space. Setting it equal to
zeroproduces singular solutions. It is also the cause of havingto
resort to imaginary ”fields”.
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
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If there is a methodology for determining a gaugefield instead
of a pre-specified T, these problems maybe solved. The requirements
are that the method shouldbe covariant, treat all forces the same,
pass the sameexperimental tests as the Schwarzchild solution and
cor-respond to the classical equation in the low-velocity
andweak-field limit [17]. It is easy to infer the appropriategauge
field.
The (0,0) and (1,1) components of the Schwarzchildmetric, gs,
look like the first two terms in a series expan-sion squared,
1− 2mr' (1 + �)2 ∧ (1− 2m
r)−1 ∼= (1 + �)−2 (37)
with � = −mr . Comparing with Eq. (11) this can be seento be an
aproximation of
ζζζµν =
Φ 0 0 00 −Φ 0 00 0 0 00 0 0 0
∧ ζζζµν = Φ 0 0 00 Φ 0 00 0 0 0
0 0 0 0
, (38)with ζµν symmetric and
Φ = Φ(r) = −mr. (39)
So Eq. (13) gives the new metric for this gauge trans-formation.
For a static, spherically symmetric mass it isgiven by
g =(eζζζ)µλḡµν
(eζζζ)ντ
=
e2Φ 0 0 00 −e−2Φ 0 00 0 −r2 00 0 0 −r2Sin2(θ)
. (40)So in this case the symmetric gauge field for gravita-
tion, ζζζµν , is simply the classical gravitational
potentialenergy per unit mass as measured in the base frame. Thisis
the solution of Poisson’s equation,
∇2Φ = −4πρ, (41)
in the region outside the matter distribution where ρ=0.This
however is still just a better approximation; thegauge field is not
relativistic. The coordinate singularityat r=2m (the event horizon)
is eliminated but the metricsingularity at r=0 still exists. The
solution of Eq. (41)for ρ=0 is only valid outside the matter
distribution. Asa gauge field there is no such requirement since ρ
is notthe correct (GR) density. The mass, m, now is only
aparameter, and the correct density (below) can now beintegrated
all the way to the origin. However, the gaugefield is still
metric-singular. For r>>2m this is the same
0.2 0.4 0.6 0.8 1.0
r
1
2
3
4
5
6
Ρ
FIG. 1. The invariant matter density (Laue scalar).
metric as the Schwarzchild solution, but now G6=0 ex-cept in the
limit r→∞. Putting Eq. (40) into the EFE,Eq. (34), yields
Tµν = −c2
8πκ×
Φ1(r) 0 0 00 Φ1(r) 0 00 0 Φ2(r) 00 0 0 Φ2(r)
, (42)where
Φ1(r) =−1 + e2Φ(r) + 2e2Φ(r)rΦ′(r)
r2,
Φ2(r) =e2Φ(r)
(2Φ′(r) + 2rΦ′(r)2 + rΦ′′(r)
)r
. (43)
The invariant matter density given by the Laue scalar is
ρ(r) = Tαα = −c2 (Φ1(r) + rΦ2(r))
4πr2κ(44)
and is shown in Fig. 1 for κ=c=m=1.Integrating the scalar
density, ρ(r)
√−g, over a spher-
ical volume with radius, r, gives the invariant mass con-tained
within the volume.
M(r) =
∫ r0
ρ (r′) 4πr′2dr′
=rc2(1− e2Φ(r) − e2Φ(r)rΦ′(r)
)κ
(45)
For Φ(r) as in Eq. (39) gives
M(r) =
(1− e− 2mr
(1 + mr
))rc2
κ(46)
The limit as r→∞ is
limr→∞
(M(r)) =mc2
κ= M. (47)
Fig. 2 shows that the mass increases smoothly from 0toward its
limit value quickly. The matter distributiongiven by Eq. (45)
completely accounts for the entire mass,
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
0 1 2 3 4
r
0.2
0.4
0.6
0.8
1.0
M
FIG. 2. The mass within sphere of radius, r.
M. This is only valid if at each point, r, the mass iscontained
in some sphere rs ≤ r according to the baseframe. That is, this is
an exterior solution. There still isthe singularity at the origin
to remove in order to satisfythe second axiom. So along with Eq.
(41) the gauge fieldis neither relativistic nor complete.
2. The Free Relativistic Solution
The mass is now just a parameter, but it has to bemodeled is
this simplified ontology. There is only onechoice. This is a single
component, i.e., scalar, field, Φ,as in Eq. (38). According to
relativity, its equation ofstate is the Klein-Gordon (K-G)
equation. Consider theK-G equation for a spherically symmetric
field.
�Φ + k2Φ = 0 (48)
The static solution is
Φ(r) = ±me−kr
r, (49)
and the corresponding dynamic solution for
�Φ +(ω2 + k2
)Φ = 0 (50)
is
Φ(t, r) = mCos(ωt+ α)e−kr
r. (51)
Any potential that avoids the singularity provides a pic-ture of
matter as a self-contained region of gravitationalenergy in a state
bound by the mass’s own gravita-tional attraction. The K-G field
assumes only the rel-ativistic relation between mass and energy,
and is there-fore ”generic”. Any field, regardless of its equation
ofstate, has to satisfy the K-G equation on a
component-by-component basis in addition to any other state
spe-cific conditions that are present. This is true regardlessof
scale from elementary particles to galactic clusters.Putting Eq.
(49) into Eq. (44) and Eq. (45) choosing the
1 2 3 4
r
-0.05
0.05
Ρ
1 2 3 4 5
r
-0.1
0.1
0.2
0.3
mass
FIG. 3. Laue scalar and mass profiles for a Yukawa
gaugefield.
minus sign, gives profiles for the Laue scalar and masscontained
within a radius, r as shown in Fig. 3.
Exactly half the mass is positive energy and half neg-ative
energy. Since this is the static solution, is it alsoa zero-energy
solution. The binding energy is equal andopposite to the
mass-energy. (For a dynamic solution,which needs a time-dependent
metric, the mass profileoscillates like a standing spherical wave,
but maintainsequal and opposite mass-energies at each moment.)
Thesame potential is the solution for the screening of an iso-lated
electric charge inside a dielectric. In that case op-positely
charged particles in the medium are shifted to-ward the isolated
charge having the effect of smearingthe charge out into a
Yukawa-field charge density profile.This is because the opposite
charges attract. What isshown in Fig. 3 is gravitational screening
due to oppositegravitational fields that repel. Apparently, as the
space-time fabric is concentrated toward the origin it does so
atthe expense of the surrounding space which is stretched.Both
regions are stabilized by their self gravitation, butrepelled from
each other. The net effect is a region ofenergy with zero total
gravitational charge, but inertialmass, k.
3. The Complete Relativistic Solution
Adding the free relativistic solution and the far fieldsolution
together the necessary potential is obtained:
Φ(r) = −m(1− e−kr
)r
. (52)
This is also obtained by solving the K-G equation in
theclassical gravitational potential of the mass by adding an
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
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5 6 7 8
r
-5.´10-6
5.´10-6
Ρ
FIG. 4. Negative Laue scalar and crossing points for the
com-plete potential in Eq. (52).
interaction term to the Lagrangian density for the K-Gequation
[18]. This is a material particle interacting withits own
gravitational field.
L = 12
Φ′(r)2 − 12k2Φ(r)2 + Lint (53)
with
Lint = V Φ(r), V = −k2m
r(54)
yielding
�Φ(r)+k2Φ(r)−V = 0⇒ Φ(r) = −m(1− e−kr
)r
(55)
Putting this potential,Eq. (52), into Eq. (40) and takingthe
limit shows that the metric tensor is finite at theorigin as it
should be.
Limitr→0
(g) =
e−2mk 0 0 0
0 −e2mk 0 00 0 0 00 0 0 0
. (56)It was stated above that AAA.1 and AAA.2 were not
entirely
independent. Even if the goal here were not to
eliminatesingularities, Eq. (52) is the inevitable result of
treatingthe classical potential as a GR gauge field. The Lauescalar
and mass profiles are similar to those shown inFigs. 1 and 2,
however the density is still screened andgoes through 0 and becomes
slightly negative for a rangeas shown in Fig. 4. The total mass
remains the samewhile the singularity is eliminated. This is
because theYukawa term adds equal amounts of positive and nega-tive
energy. This is scalar gravity returning [19], but nowin a form
that agrees with experiment, wrapped in GR.
The precision measurements of the geodetic effect byGravity
Probe B is in agreement with theory to betterthan 0.5% [20]. In
particular the traditional expressionfor the geodetic orbital
precession of the on-board gyrosis given by [21]
∆α = −2π
(−1 +
√1− 2m
r− m
r
)Sin[θ]
≈ 3mπSin[θ]r
+9m2πSin[θ]
4r2(57)
2 4 6 810-14
r HmL
-40
-20
20
40
acc Hms2L
´1012
Small mass Hk~mL
m=10-2810
-8HkgL
FIG. 5. Acceleration due to + and -(dashed) masses andasymmetry
for larger masses.
while here we have (since kr = r/m > 109 for the 650km high
orbit),
∆α = −2π
(−1 +
√e−
2(1−e−kr)mr − m
r
)Sin[θ]
≈ 3mπSin[θ]r
+m2πSin[θ]
4r2(58)
These expressions agree to first order in m/r. They dis-agree to
second order by a factor of 9, but the magnitudeof that term is one
part in 10−10 and 10−9, respectively,of the first term. These
differences are clearly beyondthe capabilities of that experiment.
So this theory agreeswith the traditional formulation in the case
of the weakgravitational field of the Earth. Below a comparison
ismade is for extremely strong gravitational fields.
The r-equation of geodesic motion gives
d2r
dt2= c2e−
4(1−e−kr)mr m
(−1− e
−kr
r2+e−krk
r
). (59)
If m is allowed to be negative, m < 0, then the accelera-tion
is in the positive r-direction, that is, repulsive. Thisis assuming
that a positive test mass travels in a positivetimelike direction
on the geodesics of m.
Also of note is the asymmetry in the exponential factorfor
larger masses which falls off rapidly. This is shownin Fig. 5. For
elementary particle-sized masses, |m| > 10−8kg with m < 0,the
acceleration is unlimited for small r. This will be
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
explained below and is connected to the apparent
baryonasymmetry.
For r >> 2m the small residual mass density as seenfrom
the laboratory frame is the mass due to the energyof what is
usually referred to as the external gravita-tional field. However,
it is the extension of the mass,through space, that interacts
directly with another mass;one mass melds smoothly into another. As
noted above,if they interact without fields, then they must not be
ata distance. Their gauge fields superpose being linear K-G
equation solutions. That energy appears in covariantform as the
Laue scalar, Tαα.
So the mass is therefore composed completely of mat-ter, or
gravitational field, or spacetime(displacements);they are all
equivalent satisfying Axiom 1.
The Gravitational Field Energy
This demonstrates the consequence of the theory. Pre-viously the
rule was that the right side of the EFE con-tains all
non-gravitational sources of energy, hypothe-sized in covariant
form. Now it contains only gravita-tional sources of energy,
directly proportional to the co-variant Einstein tensor.
This is in contrast to Φ(r) = 12Log(1−2mr ) which gives
the Schwarzchild solution. In that case the integratedmass is M,
but it does not depend on r by Eq. (45). TheLaue scalar is 0 and
the solution is only valid outsidethe mass, which must show up as a
discontinuity in theLaue scalar gradient at some point. That
solution alsoallows black holes, where the entire mass is in an
infinitedensity state at the origin and there is no covariant
ex-pression for the gravitational field. In fact, since Mc2 isthe
integrated energy, and it lies entirely interior to theSchwarzchild
solution region, there is no energy left overfor the gravitational
field outside, covariant or not.
It has been argued that such a solution state exists farin the
future for a collapsing body and is never actuallyattained in a
finite time, as referenced from the outside.However it is still a
solution, therefore allowed, and it isstill singular.
4. Black Hole vs Red Hole
Note that although spacetime is extremely distortedfor r ∼ 2m,
there is no event horizon and no black hole,just a very red hole
around such a very dense object.It should be stressed that all the
current observationalevidence for black holes appears to be
consistent withred holes as well, but they should be
distinguishable forlarge enough fields. Any phenomenon occurring at
rbaround a black hole occurs at rr around a red hole, asmaller
radius.
For example, Fig. 6 shows the differential redshift be-tween a
black hole with Schwarzchild radius, rs, and ared hole of the same
mass, rs/ 2. 400 nm light emitted
0 2 4 6 8 10
rHrsL400
500
600
700
800
ΛHnmL
rbrr
FIG. 6. ISCO red shift of 400 nm light.
2 4 6 8
rrs
-10
-1
1
10
% diff
ã-
2m
i
k
jjjjjjjjjjjjjj1-ã
-
r
m
y
{
zzzzzzzzzzzzzz
r
1 -
2m
r
FIG. 7. % difference between standard black hole metric
po-tential relative to red hole potential.
at the Inermost Stable Circular Orbit (ISCO), r = 3rs,would be
redshifted to about 490 nm for a black hole.For a red hole the same
490 nm red shift occurs at aboutr = 2.45rs, a somewhat smaller
radius. A model inde-pendent method of measuring the mass of the
hole andthe radius of the accretion disk at the ISCO should beable
to distinguish the black hole model from the red holemodel.
In terms of geodetic effects in high fields the two mod-els have
a relative difference of about 1% at r ∼ 7.5 rsand 10% at r ∼ 2.5
rs as shown in Fig. 7.
5. Other Interpretations
For another interpretation of Eq. (52) consider the nor-malized
scalar field ψ,
ψ(r) =
√k
4π
e−k2 r
r(60)
Then the expected value of m within a sphere of radius,r, in
such a state is
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
< m>=
∫ r0
(ψ(r)†mψ(r)
)4πr2dr = m
(1− e−kr
),
(61)so that the potential Φ as in Eq. (39) becomes
Φ(r) = −< m >r
= −m(1− e−kr
)r
. (62)
From this viewpoint the Yukawa part of Eq. (52) repre-sents the
nuclear force while the other term, the classicalgravitational
potential per unit mass, -m/r, is just theresidual. Note the
nuclear force part is repulsive; it pre-vents collapse.
For the Equivalence Principle (EP) discussion below,the
distinction will be made between the gravitationalmass, mg, and the
inertial mass, mi. In Eq. (49), mis clearly the gravitational mass.
For quantum-domainfields
k =mic
~, (63)
the inverse Compton length. It derives from the free K-G
equation so it is clearly (proportional to) the inertialmass. For
elementary particle-sized masses this can beput into a more
scalable form:
Φ(r) = −mgmimp2
(1− e−kr
)kr
,
mg =κM
c2,mi =
k~c,mp =
√~cκ, (64)
mp being the Planck mass.For macroscopic fields
k =c2
κ m, (65)
the inverse geometric mass. So the wavenumber, k, canbe put in a
form that covers all mass scales,
1
k=
(~mc
+κm
c2
). (66)
k is a maximum at the Planck mass. Below that thebehavior is
quantum and the wavelength increases withdecreasing mass. Above
that the characteristic lengthalso increases but the wave nature is
now hidden withinthe geometric mass and so the behavior is
classical.
Macroscopic fields may be realized in two ways. Macro-scopic
baryonic fields can arise from large masses likestars, gas and
planets. In this case the geometric massarises from the sum
(appropriate integral) of all the con-stituent elementary
particles. Their gauge fields can beadded due to the linearity of
the K-G equation. Alsothere is nothing to require that k comes from
elementaryparticles at all. As long as it is a solution of the
K-G
equation it can represent nothing more than a displace-ment of
events over some region specified by k. In this
case k 6= mc~ and k 6=c2
κm . This fits the properties of DarkMatter exactly. No dark
matter particles have yet beendetected [22]. It is likely that at
some critical densityDM condenses into elementary particles. This
may cor-respond to the well-documented acceleration scale foundin
the outer regions of spiral galaxies, the energy den-sity being
proportional to the square of the acceleration.Eq. (52) can also be
considered as the limiting case of thesuperposition of 2 free K-G
gauge solutions of this type- one of positive gravitational mass
and one of negativegravitational mass:
Φ(r) = −m
(e−k
−r − e−k+r)
r. (67)
As k− becomes arbitrarily small, Eq. (67) approachesEq. (52),
and all of its features. It looks like a non-singular mass at the
origin whose energy is entirely grav-itational and covariant and is
balanced at very large dis-tances by an equal and opposite mass of
negative gravita-tional energy. Such a configuration can be created
fromthe vacuum without energy. This will be taken to its log-ical
conclusion in the cosmology section below. This is acase of very
long wavelengths.
6. Higgs Field
The physical reason for the Yukawa potential Eq. (49)in Eq. (52)
is to prevent the singularity. That is whyit is attached to all
massive particles. This is the clas-sical analog of the Higgs
boson. It is a solution to thescalar field equation, with even
parity and it providesall massive particles with inertial mass, k.
It has equaland opposite positive and negative energy parts, like
thequark-antiquark pairs from Higgs decays [23]. Also ithas zero
energy, which is lower than the vacuum, like theHiggs. From a
quantum perspective this is an imaginary-mass field like the Higgs.
From (47)
�Φ + k2Φ = −∇2Φ + k2Φ = 0⇒ p̂2Φ = −k2Φ, (68)
so the field momentum is imaginary since k is real. Thereason
for this is that it is a momentum operator on anunmoving, static,
bound field.
Spacetime contains matter and therefore energy; it cor-responds
to the Higgs field in quantum terminology. Itprovides a ”picture”
for it. If enough energy density ispresent at some event, it will
collapse into a gravitation-ally bound structure. This structure
then shields itselffrom becoming a singularity by pulling a ”Higgs”
fromspacetime. In other words, it is more energetically favor-able
to form a gauge solution to the K-G equation thanit is to form a
singularity; the singularity does not solvethe K-G equation
locally, a nessessity of relativity. This”Higgs” is in fact the
source of the inertial mass of par-ticles, coupled as it is to
spacetime. The Higgs field that
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
permeates all space is simply spacetime itself, which isnot
empty; it contains solutions to the K-G equation asgauge
fields.
III. EXAMPLES
A. Gravitational Radiation
Take the tensor χµν in Eq. (29) to be symmetric andsolve Eq.
(28) with base metric and coordinates as inEq. (8). With
χµν =
0 0 0 00 0 0 00 0 ψ1(t, x) η1(t, x)0 0 η2(t, x) ψ2(t, x)
, (69)Eq. (28) is solved since the y- and z-derivatives are
ap-plied to functions of t and x only. These functions needto solve
the K-G equation,
�χµν + k2χµν = 0, (70)
with k=0 (mi=0) for a wave solution (below k is wavenumber),
�χµν = 0⇒ χµν = γSin (kµxµ + α) ,kµ = (ω, k, 0, 0), kµk
µ = 0⇒ ω = ck (71)
Choosing χµν traceless, and ψ and η the same phase,
χµµ = 0 ∧ α = 0⇒ ψ2 = −ψ1∧ η1 = η2 = γSin (kµxµ) (72)
giving
χµν =
0 0 0 00 0 0 00 0 ψ(t, x) η(t, x)0 0 η(t, x) −ψ(t, x)
,ψ(t, x) = η(t, x) = γSin (kµx
µ) . (73)
Using the polarization matrices
P1 =
0 0 0 00 0 0 00 0 1 00 0 0 −1
, P2 = 0 0 0 00 0 0 00 0 0 1
0 0 1 0
, (74)these two solutions can be written as
Φ1 = γP1Sin(ωt− kx) ∧ Φ2 = γP3Sin(ωt− kx). (75)
Since
P1 · kkk = P2 · kkk = 0, (76)
these tensor waves are transverse as well as
traceless(TT-gauge). Putting these TT-gauge fields into Eq.
(13)yields
g1 =
1 0 0 00 −1 0 00 0 −e2η 00 0 0 −e−2η
,
g2 =
1 0 0 00 −1 0 00 0 −Cosh(2η) −Sinh(2η)0 0 −Sinh(2η)
−Cosh(2η)
. (77)These metrics are very different from the ones
obtainedfrom the linearized theory, but agree to first order in
γ.Putting either of these two metrics into the EFE givesthe matter
tensors
Tµν =γ2ω2Cos2(ωt− kx)
4πκ
−1 1 0 01 −1 0 00 0 0 00 0 0 0
,
Tµν =γ2ω2Cos2(ωt− kx)
4πκ
−1 1 0 0−1 1 0 00 0 0 00 0 0 0
. (78)Interestingly, Tµν is the exact same result as the
TT-gauge Isaacson pseudotensor [24] obtained from the lin-earized
theory and the Landau-Lifshitz pseudotensor[25]. Some alternate
theories of gravity also give theIsaacson pseudotensor [26]. Now
however it is a truecovariant tensor and not limited to high
frequencies forconsistent interpretation. This is gravitational
energyexpressed in covariant form; no need for pseudotensors.This
is only possible because now gravity is treated onthe same footing
as the other forces, as a gauge field.Matter/energy is created in a
locale as the wave movesthrough compression/shear then destroyed.
Tµν is alsotraceless so that the negative energy density created
iscompensated for by an equal amount of negative pres-sure giving
zero rest mass. The distortion appears tomove but does not - only
the wave, so it can transportthe energy at the speed of light. This
exemplifies that factthat gravitational energy has been moved to
the right sideof the equation (identity actually). This is
achieved, inlike manner with the other forces, by putting their
gaugefields on the left side. This brings to an end the longhistory
of debate about whether gravitational radiationcan carry or
transfer energy. It does. Also it is a sourceof gravity as expected
(G 6=0). However EM radiation isnot, as shown below.
B. Electromagnetic Radiation
Proceeding as above, take the antisymmetric tensorfµν in Eq.
(29) only this time solve it for the vector equa-tions Eq. (26) in
charge- and current-free space where
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
fµν‖ν = jµ = 0. (79)
This gives
�φµ = 0, (80)
once again the K-G gauge equation for a massless field.Instead
of a polarization tensor as above, there existsa polarization
vector, eee, with 2 possible orientations fortransverse plane wave
solutions.
φ1 = eeeASin (kµxµ) , kµ =
(ωc, kx, ky, kz
),
kµkµ = 0⇒ ω = c|k| (81)
ey =
0010
, ez = 000
1
(82)Taking the z-polarization for example,
φµ = (0, 0, 0, aSin(ωt− kx)), (83)
the EM field is, with χ(t, x) = akCos(ωt− kx)),
Fµν =
0 0 0 χ(t, x)0 0 0 −χ(t, x)0 0 0 0−χ(t, x) χ(t, x) 0 0
, (84)and the gauge field is
φµ‖ν =
0 0 0 00 0 0 00 0 0 0χ(t, x) −χ(t, x) 0 0
. (85)Putting this gauge field into Eq. (13) yields the
metric,
g =
1− χ(t, x)2 χ(t, x)2 0 χ(t, x)
χ(t, x)2 −1− χ(t, x)2 0 χ(t, x)0 0 −1 0
−χ(t, x) χ(t, x) 0 −1
. (86)Putting this metric into the EFE gives the matter
tensor,
Gµν = Tµν = 0. (87)
The metric tensor shows changes in spacetime intervalsas the
wave passes, but the wave is not a source of gravity.Fig. (8) shows
the spatio-temporal distortions along the
Π
2
Π
3 Π
2 2 Π
x
-1
1
Amplitude
FIG. 8. Metric distortions of time (dotted) and space in
thepropagation direction (solid) and z (dashed).
plane-wave front for the eigenvalues of this metric. Theshape of
these curves indicate of a region of space rotat-ing clockwise,
then counterclockwise in the x-z plane asthe wave propagates along
the x-direction. The energychanges hand between the electric
field(rotating) and themagnetic field(rotated). There is no
curvature and nogravity because this is a traveling rotation -
unlike thecase for gravity waves where there is a traveling
shear.
This is a different gauge than in Eq. (21) where themetric was
unchanged, but the result is the same - nocurvature. This had to be
true, since it was determinedabove that anti-matter generates
anti-gravity and repelsmatter. If a system composed of an equal and
symmet-ric amount of both, like positronium, created no
gravity,then it would also be true for the photons that
resultedfrom its annihilation. Both systems however still
followspacetime geodesics, so this result does not exempt
EMradiation from the well documented redshift in a gravita-tional
field of another mass. What is interesting is thatthis result -
that had to be true for logical consistency -arose automatically
from the methodology. Although itis true for radiation fields of
the gauges considered above,it is not true in general.
Electromagnetic fields can bea source of gravity as shown below in
the section on thethe Aharonov-Bohm effect.
Historically, the EM stress-energy tensor is taken to bethe
matter tensor, T, asserting that the mass equivalentof the field
energy is a gravitational source and thereforebelongs on the right
hand side of the EFE. The reasonfor this is the ansatz that T
should contain all sources ofenergy, in an assumed covariant form,
excepting of coursegravitational energy. This is an unconfirmed
assumption.All tests of GR involve gravitational fields created
bymasses. One implication of this is that the radiationcomponent of
cosmic expansion in ΛCDM models shouldset Ωγ = 0, affecting
cosmological time scales.
Equivalence Principle
The equivalence principle would obviously need mod-ification if
matter and antimatter repel each other. Theratio between
gravitational mass and inertial mass would
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
then take one of three values,
mgmi
= (−1, 0, 1) (88)
for antimatter, EM radiation and matter,
respectively.Measurements of antihydrogen in Earth’s
gravitationalfield are ongoing at the LEAR project and hopefully
thiswill be determined soon [27]. The sign changes wouldgo with the
gravitational mass; the inertial mass of theantiproton has already
been measured to high accuracyand is in agreement with that of the
proton [28].
C. Charged Particle
Combining a static antisymmetric E-M field with asymmetric
gravity field gives according to Eq. (29)
ζζζµν = χχχµν + fffµν . (89)
In sperical coordinates
ζζζµν =
Φ E 0 0−E Φ 0 00 0 0 00 0 0 0
∧ ζζζµν = Φ E 0 0E −Φ 0 00 0 0 0
0 0 0 0
, (90)is the appropriate gauge for a spherically
symmetriccharged particle. It also corresponds to a complex
scalarfield as in Eq. (124) below. Putting this into Eq. (13)yields
the metric tensor, g, which in turn gives the Ein-stein tensor, G.
For E(r) = 0 the density and mass arethe same as in Fig. 3. For
Φ(r) = 0 however, G = 0.This explains the observational fact that
charged parti-cles without gravitational mass do not exist.
D. The Aharonov-Bohm Effect
The above methodology can now be used to calcu-late the
gravitational effects of electromagnetic fieldsand show their
relationship to quantum theory. TheAharonov-Bohm Effect is a good
example. It has beenstated that this effect causes the vacuum to
have struc-ture [29] in that region which is free of magnetic
field,but has a non-zero vector potential, AAA. It will be
shownthat this ”structure” is a displacement of events givenby AAA,
whose unit is the meter as in Table 1. This dis-placement results
in a spacetime shear and an associatedgravitational field. The
idealized experimental setup re-sults in a clean separation between
regions of space withEM fields and regions with a vector potential
but no EMfield.
Consider a long cylinder of radius, ρ, uniformly mag-netized
with magnetic field, BBB, in the z-direction. Thecylinder’s mass,
and gravitational field due to its mass,are ignored. The
cylindrical coordinates, x̄µ, and metric,
ḡ, in the laboratory frame are
x̄µ = (t, r, θ, z) ∧ ḡ =
1 0 0 00 −1 0 00 0 −r2 00 0 0 −1
. (91)The EM field and vector potential, Φ are [29]Outside the
cylinder
Fµν =
0 0 0 00 0 0 00 0 0 00 0 0 0
∧ Φµ = (0, 0, xBρ22r2
, 0
)
∧Φµ =(
0, 0,−Bρ2
2, 0
). (92)
The physical components of the vector potential are
AAA =
(0, 0,
Bρ2
2r, 0
). (93)
It has dimensions of length, and since its curl is zerooutside
it can be written as the gradient of some scalarfunction, χ:
∇∇∇×AAA =∇∇∇×∇∇∇χ = 0⇒ χ = B ρ2
2θ. (94)
Picture for Quantum Theory
Quantum fields now have a ”picture”. AAA is a
physicaldisplacement of points in space. In the covariant
deriva-tive
p̂→ (p̂− eAAA) , (95)
the momentum operator, p̂, is the generator of trans-lations.
The covariant deriative indicates that the trans-lation has to be
shifted to compensate for the physicaldisplacement of spacetime by
the vector potential, AAA, toobtain the net translation [30].
Incidentally, the zerothcomponent of the covariant form of the
covariant deriva-tive,
p̂0 →(p̂0 − e φ0
), (96)
indicates the energy operator as the generator of tempo-ral
translations with the shift due to the scalar potential.This is
consistent with the above description, Eq. (18),of the electric
field as the gradient of a time translation.Concomitantly the
quantum field undergoes a local gaugetransformation,
Ψ→ eiΛ Ψ = ei e χ~ Ψ, (97)
so that the phase angle, Λ, is
Λ =e B ρ2
2 ~θ =
B π ρ2
h/eθ =
n
2θ, (98)
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
where n is the number of quanta of magentic flux throughthe
cylinder.
It is usually stated that Λ is a rotation in the ”internalspace”
of the field. Here it is clear that the displacementsare spatial
and these are real, physical rotations in ther-θ plane. Until now,
solutions to the K-G equation wereused above for event
displacements regardless of whetheror not they were macroscopic
fields or quantum fields.That is because it does not matter. This
point of view isa consequence of AAA.1. This example shows that
quantumfields are therefore made of the same ”stuff” as
spacetime,or gravitational fields, etc. The only difference is that
forsmall scales the fields need to be treated as operatorsand
measurement theory comes into play. Besides, sincethere is no
prevailing ”picture” of quantum fields, viewingthem as spacetime
amplitudes, densities, displacements,etc. cannot matter as long as
they obey the same equa-tions. However, this can provide a great
insight into theirnature and connection to classical theory and
unification.The key foundational point is that the equations are
ongravitational gauge fields (in the sense of Eq. (29)); theyare
flat space equations. Their incorporation into GR isthrough Eq.
(13). Just as predicted, EM has served as abridge between GR and
QT.
Of course the gauge transformation, Eq. (97) is on acomplex
function, Ψ. There is nothing special about us-ing the complex
numbers in quantum theory. Eq. (97) isexactly equivalent to
ψ =
(φχ
)→(
Cos(Λ) −Sin(Λ)Sin(Λ) Cos(Λ)
)(φχ
)= eiΛΨ
(99)separating the complex equation into coupled real
equa-tions.
Inside the cylinder
Fµν =
0 0 0 00 0 Br 00 −Br 0 00 0 0 0
∧ Φµ = (0, 0, B2, 0
)
∧Φµ =(
0, 0,−Br2
2, 0
). (100)
Using Eq. (14) to change to the dimensionless field, b,outside
the gauge field is given by,
φµ‖ν = ζµν =
0 0 0 0
0 0 − bρ2
2r 0
0 − bρ2
2r3 0 00 0 0 0
, (101)and the metric tensor is given by Eq. (13),
gµν =(eζζζ)µλḡµν
(eζζζ)ντ
=
1 0 0 0
0 −Cosh(bρ2
r2
)rSinh
(bρ2
r2
)0
0 rSinh(bρ2
r2
)−r2Cosh
(bρ2
r2
)0
0 0 0 −1
. (102)
This is the metric for a shear in the r-θ plane with amaximum
value, b, at the cylinder boundary, r=ρ.
From the EFE, Eq. (34),
Tµν =
−b2ρ4c2Cosh
(bρ2
r2
)4πr6κ 0 0 00 0 0 00 0 0 0
0 0 0 −b2ρ4c2Cosh
(bρ2
r2
)4πr6κ
.(103)
There is therefore gravitational energy outside themagnet due to
the magnetic field inside the magnet.There is a negative energy
density, and positive pressurein the z-direction equal in magnitude
to the density. Tis not traceless, and the trace is negative. This
meansthere would be a repulsive gravitational force in the
r-direction in the region that is outside the magnet andoutside
most of the gravitational field. Integrating theLaue scalar over
the exterior region, and multiplying byc2 gives the total energy:∫
∞
ρ
Tµµc2√−‖g‖2πzdr
=z c4(−1 + Cosh(b)− bSinh(b))
2κ. (104)
(1− Cosh(b) + bSinh(b)) = b2
2+O(b)4 (105)
Although the gravitational energy stored is roughlyproportional
to the energy density of the magnetic field,it does not depend at
all on ρ, the radius of the magnet.This is because the space of a
cross-sectional disk of themagnet is rotated as a unit with no
shears within. Thespace outside has a higher energy density Eq.
(103) withincreasing ρ, but that is exactly offset by the fact
thatthere is less of it. That is, the bottom limit of the in-tegral
increases. All of the energy of the magnetic fielddoes not serve as
a source of gravity; T is 0 inside. Theoutside energy must arise as
some of the (negative) workneeded to establish the field.
E. Cosmology
Dark Matter
The potential in Eq. (52) is matter whether baryonicor not. The
K-G equation is linear in the gauge fields sothey can be added for
aggregates of baryonic matter. Fornon-baryonic matter, it is just
pure uncondensed matter,Dark Matter. Consider a solution
corresponding to thebaryonic matter in a typical galaxy, embedded
in a largermass of non-baryonic matter (Dark Matter):
Φ(r) = −m+
(1− e−k+r
)r
−m−
(1− e−k−r
)r
. (106)
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
0 10000 20000 30000 40000 50000rHLightYearsL0
20
40
60
80
100
120
vHKmsL
Rotation Curve for M33
0 20000 40000 60000 80000rHLightYearsL0
50
100
150
200
vHKmsL
Rotation Curve for NGC 4157
FIG. 9. The potential in Eq. (67) provides excellent fit for
actual data from M33 and NGC 4157.
Assuming circular orbits [31], the r equation of motionfor the
metric in Eq. (40) is
d2r
ds2=e2Φ(r)rω2
c2− e4Φ(r) Φ′(r) = 0. (107)
giving
v2 = r2ω2 = c2re2Φ(r)Φ′(r). (108)
With the potential in Eq. (67), Fig. 9 shows
excellentqualitative fits for this gauge field to actual rotation
curvedata for the galaxies M33 and NGC 4157, assuming abaryonic
mass content of 20 × 109M�and a diameter of60,000 LightYears for
M33. The NGC 4157 curve is fit.
5 10 15 20 25 30kpc
1.´10-7
2.´10-7
3.´10-7
4.´10-7
5.´10-7
6.´10-7
7.´10-7
kms
FIG. 10. Rotation curve showing flatness due to
offsettingcontributions from baryonic (dotted) and DM (dashed)
com-ponents.
The typical flat velocity profile is seen farther out onthe
spiral. It is common that galactic rotation curves aresimilar to
the one for NGC 4157 in that there is a dipafter the first peak
before the slope increases again. It isdue to the dark matter
taking over from the more cen-tral baryonic matter. From simple
Newtonian mechanicsany matter distribution that increases linearly
from theorigin (inverse square density) gives circular orbits
withconstant speed. The mass distribution in Fig. 2, for ex-ample,
is approximately linear over a wide range of radii.However, this is
not the source of linearity here.
Fig. 10 shows rotation curves for the baryonic matteralone, the
DM alone and their combination. The flatvelocity profile is seen to
be due to an increasing con-tribution from DM which precisely
offsets a decreasing
10 20 30 40radius
1
2
3
4
mass
FIG. 11. Comparison of integrated density profiles for
theory(solid) and NFW (dashed).
profile from the baryonic matter. This fine-tuning canbe
obtained by adjusting the curvature parameter, k, aswell as the
dark matter ratio. Fine-tuning is also used inthe parameters of the
NFW density profiles usually usedto obtain matches also [32]. There
is a big difference herethough. The NFW is a purely
phenomenological profileused in simulations to achieve the
necessary density pro-files. Here, though, there is no choice of
model; it isdictated by theory.
Fig. 11 shows a comparision of the integrated massfrom this
theory-based density profile to one based onNFW. Unlike the NFW
model, no arbitrary cutoff atsome high virial radius is needed
because the densityproceeds exponentially to zero on its own. In
additionat low radii the increasing density, as shown above, canbe
integrated all the way to the origin so that no low
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
cutoff is needed either. These facts make the theory-based
density profiles superior to NFW. It should alsobe mentioned that
since the NFW model is used exten-sively in DM gravitational
simulations, the potentials ofEq. (52) or Eq. (67) is superior
there as well. They arethe actual theory-based gravitational
potentials and theybehave very well in simulations, having finite
values at ar-bitrarily small separations, overlaps and
superpositions.
Boundary Condition
Until now it has been assumed that fields came fromsmall
displacements of spacetime from its value with-out the disturbance,
that is, far from the disturbance.Nothing has been said about the
boundary condition,assuming space was Lorentzian at large
distances. Asmentioned above Gµν = 0 is wrong, at least near
thesource. Although G=0 still means the space is empty,this
condition is not to be found, except at the bound-ary between equal
and symmetric distributions of matterand antimatter beyond the
observational horizon of ouruniverse. As a solution of gauge type
Eq. (52) from astar decays, it blends into the dark matter of the
galaxyas in Eq. (67), which blends into the dark matter of thelocal
group, and so on. The metric decreases exponen-tially from the
source never exactly reaching zero. Thissuggests that the boundary
condition on the metric fora localized matter distribution
approaches a small non-zero ”vacuum” value corresponding to Eq.
(67) for itsencompassing distribution. Proof of this is that all
ag-gregations of matter measured contain DM. There aresome dark
galaxies but recent re-calibrations show thatat least one galaxy
thought to be devoid of DM does in-deed contain it. So the metric
for any distribution ofmatter will tend at large distances to the
gauge field so-lution, Eq. (52) for the larger distribution of
matter inwhich it participates, that is, its dark matter.
ΛCDM models estimate about 16% of matter is bary-onic, the rest
being Dark Matter. It is an obvious spec-ulation to consider that
the baryonic matter condensedfrom the DM the same way a cloud forms
from water va-por at a critical pressure/energy density. From Eq.
(42)
T 00 = ρ = T11 = −P (109)
so that the ratio of radial pressure to rest density is
-1,giving these solutions the same property as a Cosmolog-ical
Constant, Λ, except that it can vary in both spaceand time. These
solutions however occur at any scale, sothey can represent the DM
as shown above for galaxies,galaxy clusters, superclusters, etc.
They have negativepressure and have regions of negative energy
density intheir outer regions, either of which may appear as
DarkEnergy fueling the accelerated expansion of the cosmos.In
addition, time dependent solutions like Eq. (51) maymimic
Quintessence. So at once these ”scalar fields” mayprovide the seeds
for large structure formation while ac-counting for both DM and DE
obviating the need for
Λ, inflation or other heretofore unobserved phenomena.That is,
these are just normal gravitational fields ex-pressed as gauge
fields.
Critical Energy Density (Mass
Discrepancy-AccelerationRelation)
The pattern of Eq. (52) repeats itself at all scales.At some
point it appears that spacetime collapses intoa gravitationally
bound structure. The classical energydensity near the ”edge”, R for
a mass M is
E = 18πκ
(κM
R2
)2∝(M
R2
)2. (110)
Very roughly, M/R2 in SI units for a neutral meson,
a galaxy and the observable universe might be
10−28
(10−14)2 ∼
1042
(1021)2 ∼
1052
(1026)2 ∼ 1 (111)
Although this is a crude estimate, it is interesting thatover
such an enormous scale the ratio is about the same.That leads to
the speculation that structure formed fromthe outside in. That
gives an acceleration at the bound-ary of
a = κM
R2∼ .667× 10−10m
/s2 (112)
which is less than a factor of 2 from the value 1.2 ×10−10m
/s2 which is the small acceleration cutoff value
[33] for the MOND model. The curvature parameter,k, in these
potentials determines where the zero energy,zero scalar curvature
radii are located. It is at these radiiwhere the Mass Discrepancy
problem begins; it is wherethe rotation curve profiles flatten out.
k is proportionalto the mass enclosed. It is possible that when
these met-rics for matter distributions are made dynamic they
willprovide a calculation for both critical energy density andthe
low acceleration threshold.
Baryon Asymmetry
As shown above, there is some asymmetry betweenmatter and
anitmatter, although their masses are thesame. Consider the g00
component of the Schwarzchildmetric (which is still valid for T=0)
with M replaced by-M.
g00 = 1−2κM
c2r→ 1 + 2κM
c2r(113)
This is obviously not symmetric - one expression canapproach
zero and the other cannot. However the re-placement
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
1−2κMc2r
→ −1+2κMc2r
= −(
1− 2κMc2r
)= −g00. (114)
restores symmetry if g→-g. This amounts to switchingfrom the
Mostly Minus (MM) convention to the MostlyPlus (MP) convention for
the metric. The prevalent wayto do this is to concomitantly change
the matter tensorT→-T in the EFE. In this already-unified field
theorythe EFE is postulated as an identity so the actual
mattertensor changes sign, not the equation.
Since matter and antimatter are mutually repulsive itis obvious
where the missing antimatter is. It separatedfrom matter in the
early epochs of the universe and isstill out there beyond the
horizon This suggests the in-terpretation that a change in metric
signature changesfrom a region of space dominated by matter to one
dom-inated by antimatter. That would also imply that g=0at the
boundary. The boundary then is the region wherethere is no matter
and no spacetime. This restores thesound philosophical principle
that space and time rely onmatter for their existence. This was
Einstein’s belief asa consequence of Mach’s principle [7].
For example, the static, spherically symmetric gaugefield in Eq.
(38) changes
ζζζµν =
Φ 0 0 00 Φ 0 00 0 0 00 0 0 0
→ −Φ 0 0 00 −Φ 0 00 0 0 0
0 0 0 0
∧
ζζζµν =
Φ 0 0 00 −Φ 0 00 0 0 00 0 0 0
→ Φ 0 0 00 −Φ 0 00 0 0 0
0 0 0 0
. (115)since the base metric also changes. So Eq. (40)
becomes
ga =(eζζζ)ᵀ·
−1 0 0 00 1 0 00 0 r2 00 0 0 r2Sin2(θ)
· eζζζ
=
−e2Φ 0 0 0
0 e−2Φ 0 00 0 r2 00 0 0 r2Sin2(θ)
. (116)As shown in Fig. (12) the mass profiles for
potentials
Eq. (49) and Eq. (52) have perfect symmetry under signa-ture
reflection. Since antihydrogen has been considereda CPT conjugate
of hydrogen, symmetry now requiresCPTg as the new conjugacy.
Big Bang
Matter must have separated from antimatter in theearly epochs of
the universe. Assume the universe started
1 2 3 4
r
-0.3
-0.2
-0.1
0.1
0.2
0.3
M
1 2 3 4
r
-1.0
-0.5
0.5
1.0
M
FIG. 12. The mass profiles for potentials in Eq. (49) andEq.
(52) under signature symmetry (+m solid,-m dashed).
out symmetric. Consider matter and antimatter evenlydispersed.
As an example consider a simple cubic lat-tice like salt with
matter at the sodium sites and anti-matter at the chloride sites.
Such a configuration wouldhave zero energy. It would also be highly
unstable. Ifthe lattice spacing was very small the metric would
bezero, which challenges the notion of lattice spacing.
Avanishingly small perturbation would start the separa-tion with
matter and antimatter segregating as space andtime come into being.
An elementary simulation of thiscan be found at
https://thematterofspace.com/. Eachspecie begins to implode to a
high density as they con-tinue to separate. The implosion imparts
kinetic energyto each specie which then causes an expansion. Of
coursea detailed cosmology needs to be built on this, but
some-thing like this must have happened. It also has someadvantages
over current cosmologies. The prevalent pic-ture is that all matter
comes into existence instantly atan infinitely high temperature
singularity. Once againsingularities and infinities are unphysical.
This modelhas the initial condition of nothingness. Although
thereis initially no space or time, it might be said that it
allstarted with an infinitesimally small perturbation an
in-finitely long time ago, just as a manner of speaking.
The mathematical model of this is obtained fromEq. (51).
However, at t=0 no perturbation of the metricis small so Eq. (11)
would now be
dx̄µ = ζζζµνdxν (117)
making Eq. (13)
gµν = ζζζµλḡµνζζζ
ντ . (118)
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
So using the solution Eq. (51) with α=-π/2 in Eq. (68)
Φ(r) = −mSin[ωt]
(e−k
−r − e−k+r)
r. (119)
The metric is zero on the boundary between the mat-ter and
antimatter ”universes”. This means there is nospacetime separation
between them. If the CosmologicalPrinciple holds, each event being
equivalent, this impliesa certain topology. The antimatter part
would be beyondthe horizon for all events.
Methodology Summary
Everything herein follows solely from the two axioms.
Ontology
AAA.1 is really all that is needed. AAA.2 should be a
re-quirement of any theory. It was needed to help deducethe
implementation of AAA.1. AAA.1 is an enormous ontolog-ical
simplification of physics. What can be said to existis spacetime
and its distortions.
Epistemology
This entire ”theory” is mainly just the hypothesis thatEFE is an
identity rather than an equation:
Gµν + Λgµν ≡ −8πκ
c2Tµν , (120)
which completely changes the epistemology of GR. His-torically T
has been taken to contain all sources of energy- except
gravitational; gravity was ”accounted for” on theleft side of the
equation. This has not been satisfactoryfrom either a unification
or a quantization point of view.Now all the forces are on the left
as gauge fields, treatedin the same way. This results in the
observable, T, as allthe energy that is the source of the
gravitational field.
As mentioned above, the metric tensor must be knownin order to
calculate the correct relativistic value for T.So to solve for the
metric tensor in this circular conun-drum, both the metric and the
matter density must bedetermined together along with symmetry
conditions andan equation of state. An equation of state is
providedapart from the field equation and therefore apart fromthe
general theory of relativity, even if it is expressed incovariant
form. The Correspondence Principle histori-cally has been used by
posulating that the limit of theleft hand side of the EFE is equal
to the classical limit ofthe right hand side in the weak-slow
approximation. Inthis way
Gµν = −8πκ
c2Tµν −→ ∇2Φ = −4πρ (121)
This methodology uses the historical approach in reverse.The EFE
is now a wrapper for the base space fields. Thatencapsulation is
what allows unification without modifi-cation of the mathematical
structure of GR.
∇2Φ = −4πρ, Fµν‖ν = jµ, etc.,−→
Gµν (gαβ (ζστ )) + Λgµν (ζ
στ ) ≡ −
8πκ
c2Tµν . (122)
Complex quantum fields need to be represented as realfunctions
of the coordinates to make the mapping. Forexample, a complex
scalar field, Ψ can be expressed as
Ψ̄ = e−iΛΨ→(φχ
)=
(Cos[Λ] Sin[Λ]−Sin[Λ] Cos[Λ]
)(φχ
)(123)
for a gauge transformation using coupled real fields or
Ψ = φ+ iχ =
(1 00 1
)φ+
(0 1−1 0
)χ =
(φ χ−χ φ
),
(124)
using matrix representations of 1 and i. In this casethe result
is the sum of a symmetric field representing themass and an
antisymmetric field representing the chargeas expected for a
complex scalar field. This can be ex-tended to spinor fields, etc.
There is much more work tobe done to turn this into a complete
theory.
Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
-
TABLE I. A sample of physical quantities in these new
MSIunits.
Quantity Symbol UnitElectric field E m/sMagnetic field B 1Vector
potential A mCharge q kg/sPermittivity � kg/m3
Permeability µ m2/NMagnetic flux Φ m2
Momentum q A kgm/s
Appendix: Geometrized Units
The electric field components of the EM tensor in SIunits is
E/c. This has units of kg/(s C). So this is the unit
of the proportionality constant, η, in Eq. (14) betweenthe EM
field and its dimensionless displacement field.Table 1 contains a
sample of physical quantities in thesenew MSI units.
This provides a mechanical picture of spacetime suchthat the
speed of light is
c =
√1 /µ0�0∼
√B
ρ, (A.1)
where B is the bulk modulus and ρ the density as istypical for
materials. So �0 has the role of density and µ0has units of
compressibility in this mechanized spacetime.
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Copyright c©2019 Brian Lee Scipioni. All Rights Reserved.
