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On the Origin and Unification of Electromagnetism, Gravitation, and Quantum Mechanics

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Abstract

Physical action is a spacetime bivector. Based on this single assumption, this paper argues that spacetime itself is responsible for much of physics. The properties of its elements and their derivatives, especially the transformations of special relativity, allow us to successively derive and confirm major empirically found laws and equations. We exemplary derive equations for electromagnetism including: Biot-Savart Law, Electrostatic Force, Lorentz Force, Maxwell's Equations including charges, and also constants like the fine-structure constant, electric permittivity, and even additional terms that are related to spin. In strict analogy, we gain very similar equations for gravitation which, e.g., include Poisson's equation for gravity. A further derivation of these equations of gravitation leads to field equations which are comparable to the field equations of general relativity. Electromagnetism and gravitation unite into one set of equations. The fact that bivectors can be written as wave equations also unify classical physics with wave mechanics.
On the Origin and Unification of Electromagnetism, Gravitation,
and Quantum Mechanics
Ingo D. Mane
ingo.d.mane@gmail.com
February 21, 2024
Abstract
Physical action is a spacetime bivector. Based on this single assumption, this paper argues
that spacetime itself is responsible for much of physics. The properties of its elements and their
derivatives, especially the transformations of special relativity, allow us to successively derive
and confirm major empirically found laws and equations. We exemplary derive equations for
electromagnetism including: Biot-Savart Law, Electrostatic Force, Lorentz Force, Maxwell’s
Equations including charges, and also constants like the fine-structure constant, electric per-
mittivity, and even additional terms that are related to spin. In strict analogy, we gain very
similar equations for gravitation which, e.g., include Poisson’s equation for gravity. A further
derivation of these equations of gravitation leads to field equations which are comparable to
the field equations of general relativity. Electromagnetism and gravitation unite into one set of
equations. The fact that bivectors can be written as wave equations also unify classical physics
with wave mechanics.
1 Introduction
In physics, different theoretical frameworks exist to describe nature. Besides the classical theories,
some of the best known are probably quantum mechanics, the standard model of particle physics”
[2], or even string theory [14]. However, these theories mostly concentrate on describing particles
and the short range interactions between them but fail to include gravitation.
Both electromagnetism and gravitation can describe long scale interactions. To develop a new
model, we approach the problem by not starting from particles and their properties but by looking
at the long scale aspect of their interactions and therefore looking more in depth at spacetime and
its properties itself.
Spacetime already is the basis of the theories of special relativity [3] and of general relativity [4].
We will show the unifying and explanatory effect that spacetime, the transformations of special
relativity, and the use of the spacetime derivative ˚
, see (2), have by deriving the equations of
electromagnetism and gravitation from one of the most basic physical quantities: Action S. After
stating naming conventions, we begin with a very general overview before this paper then dives
right into spacetime, electromagnetism, and gravitation. Much of the necessary math resides in
the Appendix, not because it is less important, but because working step by step through a lot of
equations before arriving at the main topic seriously would obstruct the flow of this paper.
To simplify the equations and make connections and analogies more obvious, we use the notations
described below:
1
General:
SAction.
EEnergy
PMomentum
AAngular Momentum
FForce
Charges / charge operators:
q0j
u,qij
uSingle unit charges, affected by the respective action or force fields
m0j
u,mij
uSingle unit masses, affected by the respective action or force field
q0j,qij ,Q0j,Qij Multiple charges, affected by the respective action or force fields
m0j,mij ,M0j,Mij Multiple masses, affected by the respective action or force fields
q,mSimplified names without indices
Action fields:
BSMagnetic action field
ESElectric action field
GSGravitational action field
PSMomentum action field
Force fields:
EFor just EElectric force field
BFor just BMagnetic force field
GFor just GGravitational force field
PFor just PMomentum force field
Action:
SB=Q0jBSMagnetic action
SE=Qij ESElectric action
SG=M0jGSGravitational action
SP=Mij PSMomentum action
Force:
FB=Q0jBF=Q0jBMagnetic force
FE=Qij EF=Qij EElectric force
FG=M0jGF=M0jGGravitational force
FP=Mij PF=Mij PMomentum force
Momentum action field and force are new concepts that will be developed in this paper. They are
the gravitational equivalent to magnetic action and force. A static magnetic force as written here
FB=qB has not been observed yet. It is still useful for the math in this paper as seen below in e.g.
section 3.4 “Maxwell’s Equations of Static Charges” and we must include it in further derivatives.
In order to limit the complexity and allow for a mathematically manageable solution, some simpli-
fying assumptions are made. First, all equations describe either static scenarios in rest or scenarios
that can be described by the transformations of special relativity, e.g. two observer/rest frames
moving with a constant relative speed with regard to each other. Second, accelerations of observer
frames are not handled in this paper. Third, all spacetime geometry in a frame is described in a
fixed, extrinsic/global coordinate system like in special relativity. The intrinsic geometry of the
2
curvature of spacetime in general relativity (G.R.) will only be pointed out when comparing these
two approaches in section 4.9. However, this is typical for such analysis and therefore does not limit
the relevance of the results. Most of relativistic quantum mechanics follow the same approach.
Before presenting the concrete derivations of electromagnetism and gravitation, the following over-
view shows how applying the spacetime derivative to bivectors of action leads to energy, momentum,
and angular momentum. Successively applying the spacetime derivative ˚
, see (2), to each result
of the previous derivative then leads to forces and fields, which then leads to Maxwell’s equations
for electromagnetism and an equivalent for gravitation. The final derivative leads to equations of
energy-, stress- and momentum-density comparable to the field equations of general relativity:
SAction
˚
(S) Derivative level of energy, momentum, and angular momentum
˚
˚
(S)Derivative level of forces
˚
˚
˚
(S) Derivative level of Maxwell’s equations, Poisson’s equation for gravity
˚
˚
˚
˚
(S) Derivative level of energy-stress-momentum densities, G.R.
While we only look in detail at some of the steps described above, a summary of all the equations
gained in this paper either derived directly or in analogy to other equations can be found in
mostly bullet point style in section 5.
2 Spacetime
Since around the time that special relativity was first published by Einstein [3] and Minkowski held
his lecture about ”Space and Time” [16], it is assumed that we live in a four-dimensional spacetime.
Even before that time, four dimensional algebras were investigated by Hamilton [10], Gibbs [9],
Grassmann [7], Clifford [8] and others and later popularized e.g. as Geometric Algebra by Hestenes
[13] [12].
In general, a four-dimensional spacetime is described by scalars, vectors eα, bivectors eαβ, trivectors
eαβγ , quadvectors eαβγ δ , pseudovectors (= trivectors in 4D), and pseudoscalars (= quadvectors in
4D). As usual, Greek letters α,β,γ,δdenote dimensions 0, 1, 2, 3 (in time and space context),
while Latin letters i,j,kdenote only spatial dimensions 1, 2, 3. In this text, to shorten the
notation, we write the unit vectors and multivectors as follows:
eαunit vector
eαeβeαβ unit bivector
eαeβeγeαβγ unit trivector/pseudovector
eαeβeγeδeαβγδ unit quadvector/pseudoscalar
The basic mathematics are summarized in Appendix A.1.
2.1 Derivatives of Spacetime
For the analysis we must understand the concepts of contravariant and covariant vectors which will
be generalized in the following. An example for a covariant vector is the derivative dxexand for
a contravariant vector the velocity vxex. These covariant and contravariant types can be turned
into each other by raising or lowering the indices. This is done by multiplying the vector with the
metric tensor, dαeα=gµν dαeα. The covariant derivative accounts for changes in field strength but
3
also accounts for the changes of the local unit vectors. In general relativity, the changes of the local
unit vectors are described by Christoffel symbols.
As we will see later in this paper, equations for fields (E, B, G, P) describe how spacetime (=
fields) changes in a stationary frame of reference, flat spacetime. This frame of reference might be
called a stationary meta spacetime. As we mostly deal with a universal, flat meta spacetime, the
Minkowski metric can always be used in lieu of the metric tensor.
We can define the covariant meta-spacetime partial derivative as d0e0+d1e1+d2e2+d3e3with
d0=
c∂t , d1=
∂x , d2=
∂y , d3=
∂z or d0=
∂x0, d1=
∂x1, d2=
∂x2, d3=
∂x3.
By lowering the indices of the unit vectors, we get the partial contravariant spacetime derivative
d0e0d1e1d2e2d3e3.
In physics, the three-dimensional partial derivative / gradient is often written with the “nabla”
symbol, i.e., =
∂x +
∂y +
∂z . Furthermore, the four-dimensional partial derivative is sometimes
written as µ=
c∂t +. To distinguish this four-dimensional partial derivative from our partial
contravariant spacetime derivative, and to limit the use of indices, we define the symbol ˚
with
˚
def
=d0e0d1e1d2e2d3e3.(1)
To shorten the name “partial contravariant spacetime derivative”, we will simply refer to it as
“spacetime derivative”.
Next, we determine the second spacetime derivative:
˚
˚
= (d0e0d1e1d2e2d3e3)(d0e0d1e1d2e2d3e3)
=d0d0d1d1d2d2d3d3
Thus, the result is
˚
2=˚
· ˚
=d2
0d2
1d2
2d2
3=d2
c2dt2d2
dx2d2
dy2d2
dz2=d2
c2dt2 2=,(2)
where the -symbol is the d’Alembert operator (d’Alembertian). Besides other names, it is also
called the wave operator.
It should be noted that it is a property of spacetime itself or its second partial derivative that, no
matter what the argument is (scalar, vector, bivector, pseudovector or pseudoscalar), the second
spacetime derivative is always a wave equation. This does, however, not imply that every argument
can be quantized.
A complete step by step walk through of the first and second spacetime derivative of a bivector
field is gives as an example in appendix A.2.
2.2 Movements in Spacetime
Besides spacetime and the spacetime derivative, the most important building block of the model we
develop in this paper is the transformation of spacetime elements as described by special relativity.
Especially the transformations of bivectors will prove to be of particular interest and relevance.
To better understand the transformations and rotations in spacetime, we need to compare them
with transformations and rotations in euclidean space, respectively. In euclidean 2D or 3D space,
4
e1
e2
Figure 1: Invariant circle,
unit axes,
unit grid area
cos (α)e1
sin (α)e2
cos (α)e2
sin (α)e1
˜e1
˜e2
Figure 2: Invariant circle, rotated axes
˜e1=+ cos (α)e1+ sin (α)e2
˜e2=sin (α)e1+ cos (α)e2
a point
Ron a circle of radius r or on a sphere is an invariant. No matter the transformation of
the basis vectors, the value of r stays the same. To illustrate this for a circle, compare
Rwritten
with unit vectors,
R=xe1+ye2and written as absolute values, R2=x2(e1)2+y2(e2)2which
in euclidean space simplifies to r2=x2+y2. Similarly, a sphere written with unit vectors is
R=xe1+ye2+ze3and written as an absolute value, R2=x2(e1)2+y2(e2)2+z2(e3)2which again
simplifies to r2=x2+y2+z2.
Rotating the unit vectors eiaround the origin by an angle βleads to new unit vectors ˜ei, but the
spacetime invariant
Rdoes not change its absolute value, hence the name “Invariant”.
S=xe1+ye2
=e1e2x
y
=e1e21 0
0 1x
y
=e1e2+ cos αsin α
+ sin α+ cos α+ cos α+ sin α
sin α+ cos αx
y
= (transformation of unit vectors) (transformation of vector components)
=˜e1˜e2˜x
˜y
Together, the unit vectors span a unit grid area. The size of this area is 1 and does not change
under transformation of the basis vectors. The area of a rectangle is the determinant of the matrix
of the transformed vectors, in this case the unit area is always e1e2= (+ cos α)(+ cos α)
(+ sin α)(sin α) = cos α2+ sin α2= 1. To illustrate the rotation, figures 1 and 2 use the example
of α= 30, with cos α= 0.87 and sin α= 0.50.
In flat 4D spacetime, an invariant similar to
Ris often called
S. The equation corresponding to a
circle or sphere in unit vectors is:
S=cte0+xe1+ye2+ze3(3)
5
In absolute values we get S2=c2t2(e0)2+x2(e1)2+y2(e2)2+z2(e3)2or simplified S2=c2t2x2
y2z2. Note that, because of the Minkowski metric of flat 4D spacetime, the spatial parts now
show a negative sign.
Dynamic spacetime and special relativity are governed by the transformation of time and space
components and their unit vectors. The transformation of an invariant
Sin spacetime is called
Lorentz transformation if one of the involved unit vectors is e0(cet).
S=ctect +xex
=ect exct
x
=ect ex1 0
0 1ct
x
=ect exγ1β
β1γ1β
β1ct
x
= (transformation of unit vectors) (transformation of vector components)
=˜ect ˜exc˜
t
˜x
=c˜
t˜ect + ˜x˜ex
To simplify many equations of special relativity, one often uses the parameters
β=v
c(4)
and
γ=1
p1β2.(5)
The transformation of unit vectors and the transformation of vector components can be viewed
as two sides of a coin which complement each other. It is essential to point out that the “unit
grid area” covered by time and space unit vectors, both in rest and moving, remains the same and
makes vectors and components comparable in both cases.
Figures 3 and 4 are an example of the connections between the time- and space-axes in rest and
when moving, shown for values of β= 0.33 (b= 0.33 ·45), therefore γ= 1.06 and γβ = 0.35.
From the transformations of the unit vectors as described in A.3, we can directly conclude the
transformations of unit bivectors and their bivector components under movement. As unit bivectors
are formed by taking the wedge product of unit vectors, their transformations are obtained by taking
the wedge product of a transformed unit vector with another unit vector.
An example of a bivector transformation for a velocity v1(in direction 1) is
˜e02 = ˜e0˜e2
=γ1(e0+β1e1)e2
=γ1e02 +γ1β1e12
6
1 2 3
1
2
3
β
β
x
ct
Figure 3: moving time- and space-axes, unit grid
area
0.511.5
0.5
1
˜e1γβe0
γe1
˜e0
γβe1
γe0
x
ct
Figure 4: ˜e0=γe0+γβe1,˜e1=γ e1+γβe0
Just like a vector with the Lorentz transformation, the absolute (area) value of the bivector remains
the same. This makes bivectors and their components comparable when in rest and moving systems.
The area value of the transformed bivector is:
area = pγ1((e0)2+ (β1e1)2)
=qγ1(1 β2
1)
=1
= 1
The area value stays the same.
When dealing with movements in four dimension, it makes sense to assign indices to v,β, and γ
according to the spatial direction of the movement, i.e., vi,βi, and γiwhen the direction of the
movement is in direction i= 1,2,3.
v1,β1, and γ1when the direction of the movement is in direction 1.
v2,β2, and γ2when the direction of the movement is in direction 2.
v3,β3, and γ3when the direction of the movement is in direction 3.
Also useful are the following factors:
11means this factor is 1 when the direction of the movement is in direction 1.
12means this factor is 1 when the direction of the movement is in direction 2.
13means this factor is 1 when the direction of the movement is in direction 3.
All parameters and factors that have other indices are 0 or ignored. Moreover, in the equations we
use ”, which means “logical or”. As an example, look at (γ1γ2γ3). This should be read as “if
the direction of movement is 1, then use γ1, parameters with other factors are 0 or ignored” or e.g.
“if the direction of movement is 2, then use γ2, parameters with other factors are 0 or ignored”.
In appendix A.3 we generalise the transformation of all unit bivectors under movement. Also in
A.3 we generalise the transformation of all bivector components under movement. To write down
these transformations we rely on the indexed parameters and factors above.
7
2.3 Spacetime Invariants, Bivectors and Wave Equations
Gibbs [9] showed that bivectors can be written as an ellipse xe1+ye2or a complex number x+iy
with vectors xe1and ye2and i2=1. The ellipse xe1+ye2can also be seen as a two dimensional
subspace of a spacetime invariant (3)
S=cte0+xe1+ye2+ze3.
Using Euler’s formula [6] - that states that eix = cos (x) + isin (x) for every real x- we see that
a bivector can be written in this exponential form. We also associate this formula with wave
equations.
As we see in 5.2 the spacetime derivative of action, S, is mainly the difference between energy, E,
and momentum, P(if we ignore for now angular momentum A),
˚
(S) = 0 = E P.
With this, we can also write for S
S(t) = S(t)
tt= +∆Et
S(x) = S(x)
xx=Px
S(x, t)=∆S(t)+∆S(x)=∆EtPx
Dropping the deltas, we can write the action bivector field as a wave equation. Because the
argument of the exponential has to be unitless, one has to divide it by a unit of action, in this
case. We also flip the signs of momentum and energy to adjust for our usual definition of direction
of travel of the wave. The wave equation of action now therefore is:
S(x, t) = ψ(x, t) = ei
S(x,t)=ei
(Px−Et)(6)
It is easy to see that this is a (non relativistic) wave equation as e.g. used in quantum mechanic.
On a side note: If we assumes that Sis quantised and one quantum of Sis h, then the minimum S
is S=h= Etor S=h= Px. These are the values of the original uncertainty principle
of Heisenberg [11]. Because of other considerations Heisenberg’s uncertainty principle has been
corrected to S=
2. An equivalent for the quantisation of the unexpected angular momentum -
which we will derive later - should be added but is more complicated because of the curl.
As shown in a general form in appendix A.2 “First Derivative of a Bivector Field” and the more
concrete section 5.2 “Summary Gravitational Energy, Momentum, and Angular Momentum” and
“Summary Electromagnetic Energy, Momentum, and Angular Momentum”, the actual internal
structure of the spacetime derivative of Sis more complicated than just the normally assumed E
and P, as it does not only contain the simple temporal and spatial derivatives of the action field
but also their curls, which we associate with angular momentum A- possibly related to spin and
magnetic moment.
To acquire an even more accurate equation, we would have to use energy, momentum and angular
momentum from the dynamic relativistic equations.
8
2.4 Operators, Charge and Mass
As will be shown below in 3.1, B-fields are induced by the relative movement of electric charges
or E-fields and have “hidden” components of the cross product between velocity- and E-field
components, both only scalar values. Take for example from (10) the term
B01e01 = (γ3β3˜
E31 γ2β2˜
E12)e01
The direction or orientation is given by the unit bivector e0j. We can multiply both sides with a
scalar value q. Therefore, because for electric charge qno value for qB that is not zero is observed
for now, electric charge cannot be a simple scalar value with units.
In current physics, electric charge is not considered to be fundamental and only one type of a
number of different charges (e.g. electric charge, color charge, etc.). Generally, electric charge is
considered in the context of symmetry groups (the purely spatial rotations of U(1)) and conserved
quantum numbers/ conserved currents.
As we have seen in 2.3, for our at first seemingly “classical” problem here, a solution is to define
a charge operator ˆqµν that acts on a bivector wave equation eµν and returns the charge qµν . Note
that this implies that the charge we use for action must be the same charge that we use for force,
as we use them on the same bivector. Moreover this charge operator acts a lot like the energy
operator on a wave equation, compare with ˆ
HΨ = ˆ
EΨ and similar equations. Also compare with
section 3.6, where an example for this “operator” is shown.
However, at least in a flat spacetime, the charge operator only appears to act on vectors or bivectors
that share common indices. When acting on other vectors and bivectors so far no observations were
made. After ˆqµν eµν acted on the bivector, we will just write the value qµν instead of ˆqµν .
To set up the indices for a good comparability to current physics, we could use the covariant form
of the well known existing electromagnetic field strength (Faraday) tensor (Here the factor cis
included in the tensor and the B-field)
Fem =
0E1E2E3
E10B3B2
E2B30B1
E3B2B10
(7)
to guess the indices of an electric charge q0jand a magnetic charge qij. However, because our matrix
components should be two dimensional and because of other considerations described immediately
below, we use the Hodge dual of this tensor which exchanges the indices / positions of the electric
and the magnetic field, so that this tensor looks like
Fem =
0B01 B02 B03
B01 0E12 E31
B02 E12 0 E23
B03 E31 E23 0
(8)
Later, in the chapter on gravitation 4, we will use a similar approach and use a hodge dual gravi-
tational tensor
Fem =
0G01 G02 G03
G01 0P12 P31
G02 P12 0 P23
G03 P31 P23 0
(9)
9
The above ordering of the indices is useful if one wanted to make a connection between the ”circles”
of bivectors and spacetime invariants of the form S= (ct)2x2y2z2. In this case, every bivector
e0jwhich includes an index 0 would represent a hyperbola, while every purely spatial bivector eij
represents a circle or ellipse. Derivatives of action hyperbolas and circles represent energies etc.,
second derivatives represent forces. The different resulting forces would therefore be attractive or
repulsive. Moreover circles represent a “periodic space” which is quantizable while hyperbolas are
not. It is beyond the scope of this paper to go into these details but the choice of the connection of
electric charge with purely spatial bivectors and mass with temporal-spatial bivectors stems from
here.
Taking this opportunity for a small detour into particle physics, this splitting might also suggests
that, assuming each bivector eij is linked to a electric charge ±1
3, then each bivector e0jmight be
linked to a fixed amount of different kind of charge (Compare with 3.5 for some thoughts about this).
The different rules of what makes up a whole, observable particle (quark, lepton, etc. ) should be
easily amendable and now possibly also explainable. Because of these ”observable particle” rules,
all charge bivectors should always be contained in a quark / electron etc. and not be observable as
an individual particle.
In an earlier, abandoned draft of this paper, it was assumed that mixed temporal-spatial bivectors
and charges were related to gravitation and purely spatial bivectors and charges were related to
electromagnetism. While this approach resulted in the same equations that are presented below
in this paper, ”magnetic charges” and ”momentum masses/charges” were not treated as virtual
(just a placeholder for the math) but treated more as an addition or something fundamental.
Also, there was the possibility of negative mass, which hasn’t been observed yet. Therefore, the
version presented in this paper assumes that electric charges directly work on / are linked to their
bivector. Because of this, a new idea for ”mass” had to be found. We now assume that mass
stems from the energy of the vibration of two bivectors against each other. The vibration of a
bivector eαβ against a bivector eβγ around the β-axis could now be described as a wave function /
bivector in the plane eαγ. Therefore we can also describe mass and gravitation with a bivector field.
Mass however now isn’t an intrinsic fixed property of a bivector itself, but depends on the energy
levels of the vibrations of the various bivectors of the particles themselves (and then additionally
their interactions with other particles). With this assumption, we can still handle gravitation in
complete analogy to electromagnetism and additionally make the prediction that the energy/mass
of a particle is computable. Also, the ground states of these energies/wave functions are spherical
(giving particles a spherical appearance) but possible higher order energies/wave functions are also
of different shapes (compare with the solutions for electron shells in atoms). It is beyond the scope
of this paper to go into these details.
3 Electromagnetism
Most of the equations that describe electromagnetism start at the derivative level of forces and
(electric and magnetic) fields. It is therefore best to start exploring electromagnetism right there.
Electromagnetic Action, Energy and Momentum and Energy Densities can be found in the Sum-
mary 5.
3.1 Electric and Magnetic Field, Biot-Savart Law
When looking at the first spacetime derivative of a bivector field, as shown in appendix A.2, we see
that, except for the naming of the field components, the unusual unit vectors and pseudovectors and
10
some signs, this resembles Maxwell’s equations in vacuum. However, because we assume that the
purely spatial components Cij represent the electric field, the time and space related components
C0jcannot be original components of the magnetic field, as we will immediately below show that
the magnetic field is just a relativistic effect of a moving electric field. Also from this, we assume
that there are no magnetic charges.
To obtain the magnetic field, consider the following situation: An observer ˜
Ois at rest in a system
˜
S. Observer ˜
Oand system ˜
Sare moving with velocity vrelative to an observer 0, who is at rest
in system S. Observer ˜
Omeasures an ˜
E-field but no ˜
B-field. The magnetic field is understood to
be a relativistic effect of moving electric charges or fields. Utilising the reverse transformation of
components of bivectors under movement from appendix A.3
C01e01 = ((11γ2γ3)˜
C01 +γ3β3˜
C31 γ2β2˜
C12)e01
C02e02 = ((γ112γ3)˜
C02 +γ1β1˜
C12 γ3β3˜
C23)e02
C03e03 = ((γ1γ213)˜
C03 +γ2β2˜
C23 γ1β1˜
C31)e03
C12e12 = ((γ1γ213)˜
C12 +γ1β1˜
C02 γ2β2˜
C01)e12
C31e31 = ((γ112γ3)˜
C31 +γ3β3˜
C01 γ1β1˜
C03)e31
C23e23 = ((11γ2γ3)˜
C23 +γ2β2˜
C03 γ3β3˜
C02)e23
and assuming no stationary ˜
B-field exists ( ˜
C01 =˜
C02 =˜
C03 = 0), we get
C01e01 = (γ3β3˜
C31 γ2β2˜
C12)e01
C02e02 = (γ1β1˜
C12 γ3β3˜
C23)e02
C03e03 = (γ2β2˜
C23 γ1β1˜
C31)e03
C12e12 = (γ1γ213)˜
C12)e12
C31e31 = (γ112γ3)˜
C31e31
C23e23 = (11γ2γ3)˜
C23e23
Setting ˜
C12 =˜
E12,˜
C31 =˜
E31 and ˜
C23 =˜
E23, we identify the resulting space and time related
components as the magnetic field
B01e01 = (γ3β3˜
E31 γ2β2˜
E12)e01
B02e02 = (γ1β1˜
E12 γ3β3˜
E23)e02
B03e03 = (γ2β2˜
E23 γ1β1˜
E31)e03
E12e12 = (γ1γ213)˜
E12e12
E31e31 = (γ112γ3)˜
E31e31
E23e23 = (11γ2γ3)˜
E23e23
(10)
To justify the assumption that the C0jare the bivector components of a B-field compare the
equations to Biot-Savart law. Just by looking at the mixed time-space components of the above
equations, we can see that the C0j’s are indeed a magnetic field (with same units as the E-field
use factor cto convert to “normal” B-field units). Combining the time-space components of B
11
into a vector
B, multiplied with cto adjust for units, one can look at the amount d
Bfrom the
contribution of d
E(r):
d
Bc=γ
β×d
E(r)
With the substitution β=v
cand some rearranging this becomes
d
B=γv
c2×d
E(r)
Using 1
c2=ϵ0µ0and d
E(r) = QE
4πϵ0r2erwe get d
B=γϵ0µ0v ×QE
4πϵ0r2erand then
d
B=γµ0
4πr2QEv ×er
With the definition of current I=QE
tand v =d
L
dt , QEv can be rewritten as I
dL, so the whole
equation becomes
d
B=γµ0
4πr2I d
L×er
Except for the additional relativistic factor of γ(which can be set to 1 in non-relativistic scenarios),
this is Biot-Savart law:
d
B=µ0
4πr2I d
L×er(11)
3.2 Static Electromagnetic Force
The electric field Eis defined as E=force
electric unit charge . Multiplied with electric charge qe(= q), the
resulting force is
Fstatic = ˆqij Eij eij =qE
This is the static electric force (electrostatic force). As stated above in section 2.4, electric charges
do not seem to interact with the magnetic field B. However, we can use the idea of virtual “magnetic
charges” that interact with the B-field to fill in the math. Therefore, the electrostatic force can
for symmetry reasons be extended to and written as the static electromagnetic force. Compare
with the field strength tensor (8) for the correct plus and minus signs.
Fstatic =ˆq0jB0je0j+ ˆqij Eij eij (12)
3.3 Dynamic Electromagnetic Force and Lorentz Force
An observer at rest in its system ˜
Omeasures the static electromagnetic field ˜
Fin this system. An
observer at rest in another system O, which moves with velocity vwith respect to the system of
12
the charge, measures a different field F. Utilising the reverse transformation formulas for bivector
components under movement from appendix A.3
C01e01 =(11γ2γ3)˜
C01 +γ3β3˜
C31 γ2β2˜
C12e01
C02e02 =(γ112γ3)˜
C02 +γ1β1˜
C12 γ3β3˜
C23e02
C03e03 =(γ1γ213)˜
C03 +γ2β2˜
C23 γ1β1˜
C31e03
C12e12 =(γ1γ213)˜
C12 +γ1β1˜
C02 γ2β2˜
C01e12
C31e31 =(γ112γ3)˜
C31 +γ3β3˜
C01 γ1β1˜
C03e31
C23e23 =(11γ2γ3)˜
C23 +γ2β2˜
C03 γ3β3˜
C02e23
and setting ˜
C01 =˜
B01,˜
C02 =˜
B02,˜
C03 =˜
B03,˜
C12 =˜
E12,˜
C31 =˜
E31,˜
C23 =˜
E23, we get
C01e01 =(11γ2γ3)˜
B01 + (γ3β3˜
E31 γ2β2˜
E12)e01
C02e02 =(γ112γ3)˜
B02 + (γ1β1˜
E12 γ3β3˜
E23)e02
C03e03 =(γ1γ213)˜
B03 + (γ2β2˜
E23 γ1β1˜
E31)e03
C12e12 =(γ1γ213)˜
E12 + (γ1β1˜
B02 +γ2β2˜
B01)e12
C31e31 =(γ112γ3)˜
E31 + (γ3β3˜
B01 +γ1β1˜
B03)e31
C23e23 =(11γ2γ3)˜
E23 + (γ2β2˜
B03 +γ3β3˜
B02)e23
(13)
These are the full equations of the dynamic electromagnetic field. In the equations above, comparing
the second terms in parenthesis of all fields F0jwith section 3.1, we identify these terms as newly
induced B-fields and E-fields.
The resulting observable force for an electric charge qij therefore is
F12e12 = ˆq12 (γ1γ213)˜
E12 (γ1β1˜
B02 γ2β2˜
B01)e12
F31e31 = ˆq31 (γ112γ3)˜
E31 (γ3β3˜
B01 γ1β1˜
B03)e31
F23e23 = ˆq23 (11γ2γ3)˜
E23 (γ2β2˜
B03 γ3β3˜
B02)e23
These are the components of the dynamic electromagnetic force.
In the non-relativistic limit we can set γi= 1. We can also set B01 =˜
B01,B02 =˜
B02,B03 =˜
B03,
E12 =˜
E12,E31 =˜
E31 and E23 =˜
E23. Observing that βi=vi
cand that the B-fields in our unit
system have a factor of cincluded compared to the “normally” used B-fields, we can rewrite the
equations above as
F12e12 = ˆq12 E12 (v1
ccB02
v2
ccB01
)e12
F31e31 = ˆq31 E31 (v3
ccB01
v1
ccB03
)e31
F23e23 = ˆq23 E23 (v2
ccB03
v3
ccB02
)e23
13
Using (8) for the correct plus and minus signs, the dynamic electromagnetic force in the non-
relativistic limit is thus identified as the Lorentz force
FLorentz =q(E+v×B) (14)
3.4 Maxwell’s Equations for Static Charges
Some of the most important equations in electromagnetism and physics are Maxwell’s equations.
Instead of just using the result from the general derivative in Appendix A.2, we will now derive the
equations explicitly from the electrostatic force.
Looking at the static electromagnetic force in section 3.2 and assuming that forces are conserved,
we can write
˚
∇F =0=˚
(ˆq0jB0je0j+ ˆqij Eij eij )) = ˚
(q0j)B0je0jq0j˚
(B)e0j+˚
(qij )Eij eij +qij ˚
(Eij eij )
Because the components are separable by their unit bivectors, we can look at each bivector com-
ponent independently, e.g. for a magnetic field component
q03˚
(B03)e03 =˚
(q03)B03e03 results in ˚
B03e03 =˚
(q03)B03
q03 e03
and e.g. for an electric field component
q12˚
(E12)e12 =˚
(q12)E12e12 results in ˚
(E12)e12 =˚
(q12)E12
q12 e12
All components combined, we now have the equation
˚
(B0je0j+Eij eij ) = ˚
(q0j)B0j
q0je0j+˚
(qij )Eij
qij eij
The left side of the equation can be directly compared with ˚
(B+E) = M.E.s in vacuum from
appendix A.2
˚
(B0je0j+Eij eij ) = (d1B01 +d2B02 +d3B03 )e0
(d0B01 (d2E12 d3E31))e1
(d0B02 (d3E23 d1E12))e2
(d0B03 (d1E31 d2E23))e3
(d0E12 (d1B02 d2B01))e012
(d0E31 (d3B01 d1B03))e031
(d0E23 (d2B03 d3B02))e023
(d1E23 d2E31 d3E12)e231
The other, right side of the equation depends on the strengths of the B- and E-fields. With a unit
electric field strength Euand a unit magnetic field strength Bu, one can write B0j=S0jBuand
Eij =Sij Eu.
14
Therefore we can write
˚
(q0j)B0j
q0je0j+˚
(qij )Eij
qij eij =˚
(q0j)S0jBu
q0je0j+ (˚
qij )Sij Eu
qij eij
=S0j˚
(q0j)Bu
q0je0j+Sij ˚
(qij )Bu
qij eij
This “unit” E-field Euand an unit electric charge qij
ucan be used to define a constant ϵ0as
ϵ0=qij
u
Eu
(15)
This constant is called vacuum electric permittivity ϵ0. For its value and units see appendix C.
Note that the units of charge are [A·s] and here an electric field has hkg·m
A·s3i, which combines to
hA2·s4
kg·mi. This discrepancy to the “normal” units of hA2·s4
kg·m3istems from the fact that our resulting
separate equations are one dimensional, not three-dimensional densities.
With the well-known equivalence
ϵ0µ0=1
c2(16)
we can also get the constant µ0, the vacuum magnetic permeability. For the values of these constants
also refer to section C.
In analogy to (15) we can also define a constant ϵbfor magnetic unit charge and field
ϵb=q0j
u
Bu
(17)
even though we assume that magnetic charges don’t exist.
Combining it all, we now can write
˚
(q0j)B0j
q0je0j+˚
(qij )Eij
qij eij =˚
(q0j
u)S0j
ϵb
e0j+˚
(qij
u)Sij
ϵ0
eij
=1
ϵb
˚
(Sij q0j
u)e0j+1
ϵ0
˚
(Sijqij
u)eij
Because S0jand Sij are only scalar numbers, we can define Qij =S0jq0j
uand Qij =Sij qij
u. Using
these definitions, we now have 1
ϵb
˚
(Q0j)e0j+1
ϵ0
˚
(Qij )eij . It becomes clear that we can use the
same formula from Appendix A.2 that led to ˚
(B+E) = M.E.’s in vacuum. In summary, the
derivative of the (static) electromagnetic force ˚
∇FEB =0=˚
(q0jB0je0j+qij Eij eij ) leads to
˚
(B0je0jEij eij ) = 1
ϵb
˚
(Q0j)e0j+1
ϵ0
˚
(Qij)eij
This result will be used in 3.6 “Fine-Structure Constant”
15
Components related to 1
ϵb
˚
Q0j:
1
ϵb
(d1Q01 +d2Q02 +d3Q03)e0
1
ϵb
(d0Q01)e1
1
ϵb
(d0Q02)e2
1
ϵb
(d0Q03)e3
1
ϵb
(d1Q02 d2Q01)e012
1
ϵb
(d3Q01 d1Q03)e031
1
ϵb
(d2Q03 d3Q02)e023
All of these components contain virtual “magnetic charges” and are normally not included in
Maxwell’s equations.
Components related to 1
ϵ0
˚
Qij :
1
ϵ0
(d2Q12 d3Q31)e1
1
ϵ0
(d3Q23 d1Q12)e2
1
ϵ0
(d1Q31 d2Q23)e3
+1
ϵ0
(d0Q12)e012
+1
ϵ0
(d0Q31)e031
+1
ϵ0
(d0Q23)e023
1
ϵ0
(d1Q23 +d2Q31 +d3Q12)e123
There are some previously unobserved vector components:
1
ϵ0
(d2Q12 d3Q31)e1,1
ϵ0
(d3Q23 d1Q12)e2,and 1
ϵ0
(d1Q31 d2Q23)e3
With the electric current Qij
x0
=jij
E(18)
and the electric charge density
Qij
xk
=ρij
E(19)
16
the previously observed trivector components are
j12
E
ϵ0
e012,j31
E
ϵ0
e031,j23
E
ϵ0
e023,and ρE
ϵ0
e123
Remembering that we started from
˚
∇F =0=˚
(q0j)B0je0j+˚
(qij )Eij eij q0j˚
(B0j)eij +qij ˚
(E)eij
˚
(B0je0jEij eij ) = 1
ϵb(Q0j)e0j+1
ϵ0(Qij )eij
we can recombine the equations and components from above side by side:
(d1B01 +d2B02 +d3B03)e0=1
ϵb
(d1Q01 +d2Q02 +d3Q03)e0
(d0B01 (d2E12 d3E31))e1= (1
ϵb
d0Q01 1
ϵ0
(d2Q12 d3Q31))e1
(d0B02 (d3E23 d1E12))e2= (1
ϵb
d0Q02 1
ϵ0
(d3Q23 d1Q12))e2
(d0B03 (d1E31 d2E23))e3= (1
ϵb
d0Q03 1
ϵ0
(d1Q31 d2Q23))e3
(d0E12 (d1B02 d2B01))e012 = ( 1
ϵ0
d0Q12 1
ϵb
(d1Q02 d2Q01))e012
(d0E31 (d3B01 d1B03))e031 = ( 1
ϵ0
d0Q31 1
ϵb
(d3Q01 d1Q03))e031
(d0E23 (d2B03 d3B02))e023 = ( 1
ϵ0
d0Q23 1
ϵb
(d2Q03 d3Q02))e023
(d1E23 d2E31 d3E12)e123 =1
ϵ0
(d1Q23 +d2Q31 +d3Q12)e123
(20)
While the new and unobserved terms might seem surprising at first, there are engineering applica-
tions where it is helpful and customary to introduce terms like “magnetic charge”, e.g. in Antenna
theory [1]. However, we will have a critical look at the validity of all parts of these equations below
in 3.5.
If we substitute B0j:= Bj(No change of sign! We included this step in the beginning), Q0j:= Qj
B,
Eij := Ek, and Qij := Qk
Eand rearrange slightly we can write in short notation (with terms that
are so far not included in Maxwell’s Equations marked in the color gray)
· B=1
ϵb
( · QB)
× E=d0B1
ϵb
(d0QB)1
ϵ0
( × QE)
× B=d0E+1
ϵ0
Jq1
ϵb
( × QB)
· E=1
ϵ0
ρq
(21)
17
Setting the unobserved terms to 0, we get these four lines, which represent all four of Maxwell’s
equations:
· B= 0
× E=d0B
× B=d0E+1
ϵ0
Jq
· E=1
ϵ0
ρq
(22)
Gauss’s Law for Magnetism
Look at (20) and at the real temporal component e0:
(d1B01 +d2B02 +d3B03) = 0
This can be written as Gauss’s law for magnetism
· B= 0 (23)
Maxwell-Faraday Equation (Faraday’s Law of Induction)
Look at e.g. the real spatial components e1(and at e2and e3):
d2E12 d3E31 =d0B01
Up to here, Bwas measured in the same units as E. With the substitutions B=cBand
d0=1
c
∂t and eliminating c, these can be combined into Maxwell-Faraday equation (Faraday’s law
of induction)
× E=
∂t B(24)
Amp`ere’s Circuital Law (with Maxwell’s Addition)
Look at the imaginary component e012 or similar at components e031 and e023: The familiar com-
ponents here are
ϵ0(d1B02 d2B01) = J12
q+ϵ0d0E12
With B=cBand Jq=d0Qe=1
c
∂t Qe=1
cjqthis becomes
0(cd1B02
cd2B01
)=(j12
q+ϵ0
∂t E12)
1
µ0
(d1B02
d2B01
)=(j12
q+ϵ0
∂t E12)
(d1B02
d2B01
) = µ0(j12
q+ϵ0
∂t E12)
These components can be combined into Amp`ere’s circuital law (with Maxwell’s addition)
× B=µ0(jq+ϵ0
∂t E) (25)
Gauss’s Law
Look at the imaginary temporal component e123:
(d1E23 +d2E31 +d3E12)e123 = ( ρq
ϵ0
)e123
This can be written as Gauss’s law
· E=ρq
ϵ0
(26)
18
3.5 Interpretation of the Equations Leading to Maxwell’s Equations
Looking at equation (20) and comparing them with the known form of Maxwell’s equations (22) the
most obvious thing to notice is that no terms including Q0jhave been observed. As the magnetic
field Bis only a relativistic effect of the electric field E, maybe the terms including Q0jare really
zero or non-existent. Perhaps they are some other kind of charge which work on original, non
derived action and force fields F ield0j
Sand F ield0j.
Before we take a guess about these new kind of charges and fields, there is one more unknown term
in (21) which contain the definitely existing Qij, this is 1
ϵ0( × QE). Looking at the units and
the components, this is a spatial derivative - which makes it equivalent to a momentum (at the
derivative level of energy, momentum ...). However, it also adds a rotation, which, when combined,
makes this equivalent to an angular momentum. Looking a bit further back in the history of the
derivation of this term, we see that it comes actually from the (internal) rotation of a unit charge.
A reasonable assumption therefore seems to be that this term is connected to some kind of spin
and should be included in extended Maxwell’s equations:
· B= 0
× E=d0B1
ϵ0
( × QE)
× B=d0E+1
ϵ0
Jq
· E=1
ϵ0
ρq
(27)
This leaves the question of what Charge0jrepresents and to which fields they connect. A possibly
far fetched idea that would need serious research might be the following: For symmetry reasons,
these charges should show similar behaviour to electric charges. There should be two or three
different fractional values of them which can be summed up to a whole or zero and there should
be a positive and a negative value to each (charge / anti-charge). Thinking about quarks and their
properties, the idea presents itself that these charges Charge0jmight represent color charges which
work on their own color field (which of course must have a complementary derived field in analogy
to the B-field of the electric E-field. Also equations equivalent to Maxwell’s equations must exist,
including some kind of color spin).
All in all, from this we now could assume that quarks and leptons are made up of combinations of
up to six independent bivectors. These are acted upon by color and electric charge operators.
If one wanted to continue to leave the realm of ”reasonable assumptions” and go a bit further then
one could come up with the idea that, when viewing these bivectors as spacetime invariants, it
might also be possible to view them as strings (which are described by the equation of spacetime
invariants). One step further might lead to the assumption that these are not strings of some kind
in spacetime, but that these strings actually represent spacetime itself. However, when thinking
about the field equations of general relativity and how they are interpreted as ”mass tells spacetime
how to bend” then one could also actually take the equality sign to literally mean ”mass is bent
spacetime”. In this way, the idea that these bivectors represent/are spacetime does not seem to be
too unlikely. The superposition of all the bivectors/wave functions would make up all of spacetime.
Particle/wave duality could also easily be explained this way. Energy or force singularities at r= 0
19
would no longer be problematic (Derivative/slope of a circle at angle α= 0. αstands in for radius
/ distance from origin)
3.6 Fine-Structure Constant
Generally speaking, a bivector field Cµν eµν can be written as qµν Fµν eµν .
Cµν eµν =qµν Cµν
qµν eµν =qµν Fµν eµν
For simplification, we will not write out eµν or any unit vectors below. The first spacetime derivative
yields
˚
Cµν =˚
(qµν Fµν ) = Fµν ˚
(qµν ) + qµν ˚
(Fµν )
With the condition ˚
Cµν = 0, we can write
0 = Fµν ˚
(qµν ) + qµν ˚
(Fµν )
qµν ˚
(Fµν ) = Fµν ˚
(qµν )
˚
(Fµν ) = Fµν
qµν ˚
(qµν )
With qµν =Sµν
qqµν
uand Fµν =Sµν
FFµν
uthis leads to
˚
(Fµν ) = Fµν
u
qµν
u
˚
(Sµν
qqµν
u)Sµν
F
Sµν
q
and finally, using a constant proportionality factor 1
ϵ=Fµν
u
qµν
u
˚
(Fµν ) = 1
ϵ˚
(Sµν
Fqµν
u)
˚
(Fµν ) = 1
ϵ˚
(Qµν
F) (28)
An example of such a proportionality factor can be seen above in 3.4 when deriving the electric
permittivity ϵ0from the electromagnetic force:
Fµν
EM =qµν Fµν
EM
qµν =qµν Eµν
Taking the derivative leads to the combined form of Maxwell’s Equations
˚
(Eµν ) = 1
ϵ0
˚
(Sµν qµν
u) = 1
ϵ0
˚
(Qµν )
After this and according to equation (61) the second derivative of electromagnetic force leads
to
(˚
· ˚
· Eµν ) = 1
ϵ0
(˚
· ˚
· Qµν )
20
and more general, the second derivative of a bivector field Cµν leads to
(˚
· ˚
· Fµν ) = 1
ϵ(˚
· ˚
· Sµν qµν
u) (29)
Remembering from 2.3 that bivectors can be written as wave functions and taking the second
derivative of a complex wave function, we get
dx·dx·eikx =ikdx·eikx =k2eikx
Because each side of equation (29) is such a wave equation, this can also be written as
(˚
· ˚
· Fµν ) = k2
ϵ(Sµν qµν
u) = 2α
ϵ(Sµν qµν
u) (30)
Here, we use the constant αinstead of the constant k2for reasons that will become clear below
in 3.6.
In analogy, the second derivative of the general bivector field from 28 and above is
(˚
· ˚
· Fµν ) = α
ϵQµν (31)
Another concrete example of this can be seen later in 4.8 “Field Equations Of Gravitation”.
After looking at this general derivation, we can now look at a concrete example. Just above, as an
example, we looked at the first and second derivative of a general bivector field. Looking at the
concrete bivector field of electric action S, we know that action, with the right scaling, is
Su=h= 2π(32)
we can also write
Su=qSu
q=q2π
q=qES
With ˚
∇S =0=˚
(qES) = ˚
(q)ES+q˚
(ES) the first derivative becomes
˚
(ES) = 2π
q2˚
(q) (33)
Note that this is the full spacetime derivative that not only includes energy but also momentum
and angular momentum. We chose to ignore Heisenberg’s uncertainty principle here which does
apply to each single term of the derivative. The second derivative, a force, becomes
(˚
· ˚
· ES) = 2π
q2(˚
· ˚
· q) (34)
We contract the terms of the second derivative to gain a single term for force. This of course intro-
duces a proportianality factor. As we have seen in 3.4 “Maxwell’s Equations for Static Charges”,
we can assume that the static electric force is F=qijEij = 0. We can therefore write the first
derivative of this force as
˚
E=1
ϵ0
˚
q=˚
(1
ϵ0
q)
21
The factor 1
ϵ0was also derived while deriving Maxwell’s equations. We can compare this with the
derivative of (34), the third derivative of electromagnetic action
˚
(˚
· ˚
· ES) = ˚
2π
q2(˚
· ˚
· q)
By comparing the arguments in the parentheses of the derivatives with charge, we see that
2π
q2(˚
· ˚
· q) = 1
ϵ0
q
and
˚
· ˚
· q=q2
2πϵ0q(35)
With the fine-structure constant α
α=q2
4πϵ0c(36)
this leads to
˚
· ˚
· q=2
cαq (37)
The somewhat unexpected factor of 2
cis explainable by only looking at energy and observing
Heisenberg’s uncertainty principle. Using
2instead of just in the first derivative of action equation
(33) and c, which often is the conversion factor between tand x0, lets us derive the fine-structure
constant α.
The above is actually the equation of a bivector component qeµν . We therefore might assume that
qcan be written as a wave equation
q=qeikx (38)
and ˚
· ˚
· q=k2qeikx =k2q(39)
Comparing equations (37) and (39) and removing the factor 2
cas described above, we can see that
the wave number kof electromagnetic action charge would be k=α=qq2
4πϵ0c=q
2ϵ0hc
k=q
2ϵ0hc (40)
While the above only shows the example of electromagnetic action, the same should be true for all
kinds of “charge”, electromagnetic, gravitational, and others.
4 Gravitation
4.1 Action and Angular Momentum, Gravitational and Momentum Action
Field
The second derivative of action is force. As shown in (2) and equation (61), the second derivative
of a bivector field is also a bivector field. Therefore, in analogy to electric forces and electric
force fields, e.g. FE=qE and E=FE
q, we can define a gravitational action SG=mGSwith a
22
gravitational action field GS=SG
mwith units action
mass . In analogy to Eand the motion induced
field Bβ×Ewe introduce another field PSβ×GS. This leads to similar equations to equation
(10) and (11), “Biot-Savart law”.
G01
Se01 =(11γ2γ3)˜
GS
01e01
G02
Se02 =(γ112γ3)˜
GS
02e02
G03
Se03 =(γ1γ213)˜
GS
03e03
P12
Se12 =(γ2β2˜
GS
23 γ1β1˜
GS
31)e12
P31
Se31 =(γ1β1˜
GS
12 γ3β3˜
GS
23)e31
P23
Se23 =(γ3β3˜
GS
31 γ2β2˜
GS
12)e23
(41)
In analogy to the static electromagnetic action and force, and with electromagnetic charge replaced
be mass mµν we can write down a static gravitational action
Sstatic =m0jG0j
Se0j+mij Pij
Seij (42)
4.2 Dynamic Gravitational Action
In analogy to 3.3 “Dynamic Electromagnetic Force” and equation (13) we can write
S01e01 =m01 (11γ2γ3)˜
GS
01 + (γ3β3˜
PS
31 γ2β2˜
PS
12)e01
S02e02 =m02 (γ112γ3)˜
GS
02 + (γ1β1˜
PS
12 γ3β3˜
PS
23)e02
S03e03 =m03 (γ1γ213)˜
GS
03 + (γ2β2˜
PS
23 γ1β1˜
PS
31)e03
S12e12 =m12 (γ1γ213)˜
PS
12 (γ1β1˜
GS
02 γ2β2˜
GS
01)e12
S31e31 =m31 (γ112γ3)˜
PS
31 (γ3β3˜
GS
01 γ1β1˜
GS
03)e31
S23e23 =m23 (11γ2γ3)˜
PS
23 (γ2β2˜
GS
03 γ3β3˜
GS
02)e23
(43)
In analogy to 3.3 “Dynamic Electomagnetic Force and Lorentz Force” and equation (14) we can
write in the non-relativistic limit the dynamic gravitational action
Sdynamic =m(GS+v×PS) (44)
4.3 Energy, Mass, Momentum, and Angular Momentum
Assuming action is conserved, the spacetime derivative of the static gravitational action is
0 = ˚
∇S =˚
(m0jG0j
Se0j+mij Pij
Seij ) (45)
With G0j
S=S0jG0j
Su (scalar strength of G multiplied by unit action field), Pij
S=Sij Pij
Su (scalar
strength of P multiplied by unit action field), 1
αG=G0j
Su
m0j
u
, and 1
αP=Pij
Su
mij
u
, equation (45) can be split
into
˚
(G0j
Se0j+Pij
Seij ) = S0j
αG
˚
(m0j
u)e0j+Sij
αP
˚
(mij
u)eij
˚
(G0j
Se0j+Pij
Seij ) = 1
αG
˚
M0j+1
αG
˚
Mij
23
Even though the equations above describe derivatives of action and Maxwell’s equations describe
derivatives of force, both sides transform like Maxwell’s equations, with MGand GStaking the
place of components related to the B-field and MPand PStaking the place of components related
to the E-field. Comparing with the short form of Maxwell’s Equations (21) gravitation has its
analogy in
· GS=1
αG
( · MG)
× PS=d0GS1
αG
(d0MG)1
αP
( × MP)
× GS=d0PS+1
αP
(d0MP)1
αG
( × MG)
· PS=1
αP
( · MP)
When applying the same logic as we did when interpreting the equations leading to Maxwell’s
equations in 3.5, we assume that all terms with MPcan be dropped and that the likely equations
for gravitation are
· GS=1
αG
( · MG)
× PS=d0GS1
αG
(d0MG)
× GS=d0PS+1
αG
( × MG)
· PS= 0
(46)
All the terms that are derivatives of action with respect to time, d0, are related to energy. All the
terms that are derivatives of action with respect to spatial directions, di, are related to momentum.
The crossproduct terms are related to angular momentum (spin?).
Mass does not act like charges (eg. it is only positive and only a measure of the energy levels of the
vibrations of the bivectors), so there might be some differences in the logic. However, there should
be an equivalent to the color charge and the color fields, just like mass is the equivalent to the
electric charge.ON the other hand, if mass acts differently than charge and the masses connected
with the momentum filed cannot be dropped, then this might play a role in the different energy
levels caused by spin that should be investigated.
Effectively, the sum of these energy, momentum, and angular momentum terms represent the
gravitational/mass related Lagrangian of this system. If there is an equivalent to the color field, this
adds another Lagrangian. Combined with the electromagnetic Lagrangian and the color Lagrangian
we then have a total of four Lagrangian describing the whole system. This might be closely related
to the Standard Model of Particle Physics.
4.4 Perihelion Shift of Mercury
One of the earliest confirmations of general relativity is the successful prediction of the Perihelion
shift of Mercury. To see if our approach to gravitation can yield the same results, we can look at
the equations of energies in the different approaches to gravitation.
24
Orbital Energies in Newtonian Gravitation
total energy = rest energy + kinetic energy + potential energy
Et=mc2+1
2mv2+mVG(r)
With (63), the velocity vin polar coordinates, this becomes
Et=mc2+1
2m( ˙r2+r2˙
ϕ2) + mVG(r)
Et=mc2+1
2m˙r2+m1
2r2˙
ϕ2+mVG(r)
With equation (64), angular velocity and angular momentum, this becomes
Et=mc2+1
2m˙r2+m1
2r2(L
mr2)2+mVG(r)
Et=mc2+1
2m˙r2+mL2
2m2r2GnmM
r
Etmc2=1
2m˙r2+mL2
2m2r2GnM
r(47)
Orbital Energies in General Relativity: Looking at energy from the viewpoint of gravitation
and general relativity, as described e.g. in [17], one finds the following equation. With eigentime τ
and angular momentum per unit mass L=L
mwe have
1
2(E2
mmc2) = 1
2m(dr
)2+mL
2r2GnM
rGnM
c2L2
r3
1
2(E2
mmc2) = 1
2m(dr
)2+mL2
2m2r2GnM
rGnM
c2
L2
m2r3
1
2(E2
mmc2) = 1
2m(dr
)2+mL2
2m2r2GnM
rGnM
c2
L2
mr3(48)
Comparing this equation with equation (47) from Newtonian gravitation, we find that the significant
term responsible for the precession of the perihelion of Mercury is the last term,
GnM
c2
L2
mr3(49)
Derivation of the additional energy term With our assumptions about relativistic effects, we
can look again at the energies of Newtonian gravitation. We also assume the simplification that
for the instant of our observation Mercury moves on a circular geodesic around the sun. Circular
and geodesic here means that ris constant and we can still apply special relativity because no
acceleration takes place. The velocity vof Mercury is parallel to the tangent of movement to the
circle, vis perpendicular to r.
Static case / energies: Estatic =mc2mGnM
r
Dynamic case / total energies: Just like rest mass/energy gets an additional dynamic term the
kinetic energy the potential energy VGbecomes VG+β×VP. Remembering that VPoriginated
25
from β×VG, we get VG+β×β×VG. Because vis perpendicular to VG,γis only applied to the
first term of the energy, the rest mass.
Et=γmc2VGβ×β×VG
Et=γmc2mGnM
rβ×β×mGnM
r
β×β×VGis a vector triple product, see B, but because v(and therefore β) is perpendicular to
VG, we get β×β×VG=β2VG=VGv2
c2. With γmc2=mc2+1
2
v2
c2mc2=mc2+1
2mv2the total
energy becomes
Et=mc2+1
2mv2mGnM
rmGnM
rc2v2
The last term, mGnM
rc2v2, is a new term compared with Newtonian gravitation. It is a relativistic
effect of potential energy.
Transforming the new energy term Using equations (63) and (64) we can write |v|2= ˙r2+
˙
ϕ2r2=|v|2= ˙r2+L2
m2r2. Using this, we write
mGnM
rc2v2
=mGnM
rc2( ˙r2+L2
m2r2)
=mGnM
rc2( ˙r)2GnM
c2
L2
mr3
Because we assumed that Mercury in the instant of our observation is moving on the tangent
of a circular orbit and therefore r in that instant is constant, the first term vanishes and we are
left with the term
GnM
c2
L2
mr3(50)
This matches with the last term from (48) “Orbital Energies in General Relativity” above, the
significant term for the perihelion shift of Mercury.
4.5 Static Gravitational Force
In analogy to section 3.2 “Static Electromagnetic Force”, the gravitational force field Gis defined
as G=force
mass . Multiplied with a mass m, the resulting force is
Fstatic =mG =ˆm0jG0je0j
This is the static gravitational force.
The static gravitational force can also be extended as
Fstatic =ˆm0jG0je0j+ ˆmij Pij eij .
26
4.6 Dynamic Gravitational Force
In analogy to section 3.3 “Dynamic Electromagnetic Force” we can derive the dynamic gravitational
force. Utilising the reverse transformation formulas for bivector components under movement from
appendix A.3
C01e01 =(11γ2γ3)˜
C01 +γ3β3˜
C31 γ2β2˜
C12e01
C02e02 =(γ112γ3)˜
C02 +γ1β1˜
C12 γ3β3˜
C23e02
C03e03 =(γ1γ213)˜
C03 +γ2β2˜
C23 γ1β1˜
C31e03
C12e12 =(γ1γ213)˜
C12 +γ1β1˜
C02 γ2β2˜
C01e12
C31e31 =(γ112γ3)˜
C31 +γ3β3˜
C01 γ1β1˜
C03e31
C23e23 =(11γ2γ3)˜
C23 +γ2β2˜
C03 γ3β3˜
C02e23
Setting C01 =G01,C02 =G02,C03 =G03,C12 =P12 ,C31 =P31, and C23 =P23, we get
G01e01 =(11γ2γ3)G01 + (γ3β3P31 γ2β2P12)e01
G02e02 =(γ112γ3)G02 + (γ1β1P12 γ3β3P23)e02
G03e03 =(γ1γ213)G03 + (γ2β2P23 γ1β1P31)e03
P12e12 =(γ1γ213)P12 (γ1β1G02 γ2β2G01)e12
P31e31 =(γ112γ3)P31 (γ3β3G01