Unified Field Theory
Theory of everything, unifying all forces, matter & spacetime.
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1: Required Reading
The scientific notation I will be using throughout the rest of the document,
where we will be lower indices for covectors (rows) and upper indices for contravariant
vectors (columns) as per Einstein index notation. You create the covector by taking the
complex conjugate transpose of the vector. For in depth explanation, please expand the
various subjects below:
Wikipedia Articles:
In mathematics, especially in applications of linear algebra to physics, the Einstein notation
or Einstein summation convention is a notational convention that implies summation over a set of
indexed terms in a formula, thus achieving notational brevity. As part of mathematics it is a
notational subset of Ricci calculus; however, it is often used in applications in physics that
do not distinguish between tangent and cotangent spaces. It was introduced to physics by
Albert Einstein in 1916.
In quantum mechanics, bra–ket notation is a common notation for quantum states, i.e. vectors
in a \CC Hilbert space on which an algebra of observables acts. More generally
the notation uses the angle brackets (the \rangle and \langle
symbols) and a vertical bar (the | symbol), for a ket (for example, | A \rangle)
to denote a vector in an abstract usually \CC vector space A and a bra, (for
example, \langle f |) to denote a linear functional f on A.
The natural pairing of a linear function f = \langle f | with a vector
v = | v \rangle is then written as \langle f | v \rangle. On
Hilbert spaces, the scalar product (\ ,\ ) (with anti linear first argument)
given an (anti-linear) identification of a vector ket \phi = | \phi \rangle with
a linear functional bra (\phi,\ ) = \langle \phi |. Using this notation, the
scalar product (\phi,\psi) = \langle \phi | \psi \rangle. For the vector space
\CC^n, kets can be identified with column vectors, and bras with row
vectors.
Einstein notation will be what I will use throughout the rest of the document, which will
employ the standard of lower indices for covectors (rows) and upper indices for contravariant
vectors (columns) as per Einstein index notation. You get the covector by taking the complex
conjugate transpose of the vector.
Examples:
\begin{aligned}
A_{\mu} B^{\nu}
&= \langle A | B \rangle \\
&= A \cdot B
\end{aligned}
\begin{aligned}
A^{\mu} B_{\nu}
&= | A \rangle \langle B | \\
&= A \otimes B
\end{aligned}
Wikipedia Articles:
Geometrically, it is the product of the Euclidean magnitudes of the two vectors and
the cosine of the angle between them. In the case of vector spaces, the dot product
is used for defining lengths (the length of a vector is the square root of the dot
product of the vector by itself) and angles (the cosine of the angle of two vectors
is the quotient of their dot product by the product of their lengths).
An inner product space is a vector space with an additional structure called an inner
product. This additional structure associates each pair of vectors in the space with a
scalar quantity known as the inner product of the vectors. Inner products allow the
rigorous introduction of intuitive geometrical notions such as the length of a vector
or the angle between two vectors. They also provide the means of defining orthogonality
between vectors (zero inner product). Inner product spaces generalize Euclidean spaces
(in which the inner product is the dot product, also known as the scalar product) to
vector spaces of any (possibly infinite) dimension, and are studied in functional
analysis.
More precisely, for a real vector space, an inner product A_{\mu} B^{\nu}
satisfies the following properties shown.
The dot and inner product are commutative, meaning:
A \cdot B = \bar{B} \cdot \bar{A}
For all vectors A and B.
Figure 1(b): Dot Product.
Examples:
\begin{aligned}
A_{\mu} B^{\nu}
&= \begin{bmatrix} a & b \end{bmatrix} \begin{bmatrix} c \\ d \end{bmatrix} \\
&= ac + bd
\end{aligned}
\begin{aligned}
A_{\mu} B^{\nu}
&= \langle A | B \rangle \\
&= A \cdot B \\
&= {A}^\dagger{B}
\end{aligned}
\begin{aligned}
A_{\mu} B^{\nu}
&= B_{\nu} A^{\mu} \\
&= \overline{B \cdot A} \\
&= {B}^\dagger{A}
\end{aligned}
Wikipedia Articles:
The cross product or vector product (occasionally directed area product to emphasize the
geometric significance) is a binary operation on two vectors in three-dimensional space
(\RR^3) and is denoted by the symbol ×. Given two
linearly independent vectors A and B, the cross product A \times B is
defined as a vector C that is perpendicular (orthogonal) to both A and B, with a direction
given by the right-hand rule and a magnitude equal to the area of the parallelogram that
the vectors span.
The exterior product or wedge product of vectors is an algebraic construction used in
geometry to study areas, volumes, and their higher-dimensional analogues. The exterior
product of two vectors A and B, denoted by A \wedge B, is called a bivector
and lives in a space called the exterior square, A vector space that is distinct from the
original space of vectors. The magnitude of A \wedge B can be interpreted
as the area of the parallelogram with sides A and B, which in three dimensions can also
be computed using the cross product of the two vectors.
Both the cross product and wedge product are anti-commutative, meaning:
A \times B = - B \times A
A \wedge B = - B \wedge A
For all vectors A and B. Unlike the cross product, the wedge product is associative.
Figure 1(c): Cross & Wedge Product.
Examples:
\begin{aligned}
A^{\mu} \times B^{\nu}
&= \begin{bmatrix} a \\ b \end{bmatrix} e_i \times \begin{bmatrix} c \\ d \end{bmatrix} e_j \\
&= det \begin{bmatrix} a & c \\ b & d \end{bmatrix} e_k \\
&= |ad - bc| e_k
\end{aligned}
\begin{aligned}
A^{\mu} \wedge B^{\nu}
&= \begin{bmatrix} a \\ b \end{bmatrix} e_i \wedge \begin{bmatrix} c \\ d \end{bmatrix} e_j \\
&= det \begin{bmatrix} a & c \\ b & d \end{bmatrix} \\
&= |ad - bc|
\end{aligned}
Wikipedia Articles:
The outer product of two coordinate vectors is a matrix. If the two vectors have dimensions n
and m, then their outer product is an n × m matrix. More generally, given two tensors
(multidimensional arrays of numbers), their outer product is a tensor. The outer product of
tensors is also referred to as their tensor product and can be used to define the tensor algebra.
The Kronecker product, sometimes denoted by \otimes is an operation on two
matrices of arbitrary size resulting in a block matrix. It is a generalization of the outer
product (which is denoted by the same symbol) from vectors to matrices, and gives the matrix
of the tensor product with respect to a standard choice of basis. The Kronecker product should
not be confused with the usual matrix multiplication, which is an entirely different operation.
Like the cross product, the outer product is anti-commutative, meaning that.
A \otimes B = - B \otimes A
For all vectors A and B.
Examples:
\begin{aligned}
A^{\mu} B_{\nu}
&= \begin{bmatrix} a \\ b \end{bmatrix} \begin{bmatrix} c & d \end{bmatrix} \\
&= \begin{bmatrix} ac & ad \\ bc & bd \end{bmatrix}
\end{aligned}
\begin{aligned}
A^{\mu} B_{\nu}
&= | A \rangle \langle B | \\
&= A \otimes B \\
&= {A}{B}^\dagger
\end{aligned}
\begin{aligned}
A^{\mu} B_{\nu}
&= - B^{\nu} A_{\mu} \\
&= - B \otimes A \\
&= - {B}{A}^\dagger
\end{aligned}
Wikipedia Articles:
The Hadamard product (also known as the element-wise, entrywise or Schur product) is
a binary operation that takes two matrices of the same dimensions and produces another
matrix of the same dimension as the operands where each element i, j is the product of
elements i, j of the original two matrices. It should not be confused with the more
common matrix product.
The Hadamard product is associative and distributive. Unlike the matrix product, it
is also commutative.
Examples:
\begin{aligned}
A^{\mu} \odot B^{\nu}
&= \begin{bmatrix} a \\ b \end{bmatrix} \odot \begin{bmatrix} c \\ d \end{bmatrix} \\
&= \begin{bmatrix} ac \\ bd \end{bmatrix}
\end{aligned}
\begin{aligned}
A^{\mu} \odot B^{\nu}
&= | A \rangle \odot | B \rangle \\
&= B^{\nu} \odot A^{\mu} \\
&= {A}{B}
\end{aligned}
\begin{aligned}
A_{\mu} \odot B_{\nu}
&= \begin{bmatrix} a & b \end{bmatrix} \odot \begin{bmatrix} c & d \end{bmatrix} \\
&= \begin{bmatrix} ac & bd \end{bmatrix}
\end{aligned}
\begin{aligned}
A_{\mu} \odot B_{\nu}
&= \langle A | \odot \langle B | \\
&= B_{\nu} \odot A_{\mu} \\
&= {B}{A}
\end{aligned}
Wikipedia Articles:
In mathematical physics and mathematics, the Pauli matrices are a set of three 2 × 2
\CC matrices which are Hermitian and unitary. Usually indicated by the Greek
letter sigma (\sigma), they are occasionally denoted by tau (\tau)
when used in connection with isospin symmetries.
- \sigma_0 = \sigma_t = I_2
- \sigma_1 = \sigma_x
- \sigma_2 = \sigma_y
- \sigma_3 = \sigma_z
In mathematical physics, the gamma matrices, also known as the Dirac matrices, are a set
of conventional matrices with specific anti-commutation relations that ensure they generate
a matrix representation of the Clifford algebra Cℓ1,3(\RR). It
is also possible to define higher-dimensional gamma matrices. When interpreted as the matrices
of the action of a set of orthogonal basis vectors for contravariant vectors in Minkowski space,
the column vectors on which the matrices act become a space of spinors, on which the Clifford
algebra of spacetime acts. This in turn makes it possible to represent infinitesimal spatial
rotations and Lorentz boosts. Spinors facilitate spacetime computations in general, and in
particular are fundamental to the Dirac equation for relativistic spin-½ particles.
- \gamma^0 = \sigma_3 \otimes \sigma_0
- \gamma^j = i\sigma_2 \otimes \sigma_j
- \gamma^5 = \sigma_1 \otimes \sigma_0 = i \gamma^0 \gamma^1 \gamma^2 \gamma^3
\gamma^0 = \gamma^t is the time-like, hermitian matrix. The other three
\gamma^1 = \gamma^x,
\gamma^2 = \gamma^y,
\gamma^3 = \gamma^z,
are space-like, anti-hermitian matrices.
Examples:
\sigma_0 = \begin{bmatrix} +1 & 0 \\ 0 & +1 \end{bmatrix}
\sigma_1 = \begin{bmatrix} 0 & +1 \\ +1 & 0 \end{bmatrix}
\sigma_2 = \begin{bmatrix} 0 & -i \\ +i & 0 \end{bmatrix}
\sigma_3 = \begin{bmatrix} +1 & 0 \\ 0 & -1 \end{bmatrix}
\gamma^0 = \begin{bmatrix} +\sigma_0 & 0 \\ 0 & -\sigma_0 \end{bmatrix}
\gamma^1 = \begin{bmatrix} 0 & +\sigma_1 \\ -\sigma_1 & 0 \end{bmatrix}
\gamma^2 = \begin{bmatrix} 0 & +\sigma_2 \\ -\sigma_2 & 0 \end{bmatrix}
\gamma^3 = \begin{bmatrix} 0 & +\sigma_3 \\ -\sigma_3 & 0 \end{bmatrix}
\gamma^5 = \begin{bmatrix} 0 & +\sigma_0 \\ +\sigma_0 & 0 \end{bmatrix}
I_4 = \begin{bmatrix} +\sigma_0 & 0 \\ 0 & +\sigma_0 \end{bmatrix}
Wikipedia Articles:
- \CC: Complex
- \HH: Quaternion
- \OO: Octonion
- \SS: Sedenion
In mathematics, the \HH are a number system that extends the \CC
numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied
to mechanics in three-dimensional space. A feature of \HH is that multiplication of
two \HH is none-commutative. Hamilton defined a quaternion as the quotient of two
directed lines in a three-dimensional space or equivalently as the quotient of two vectors.
We get the following when we calculate \HH^2 or the inner product
between covariant conjugate vector and its contravariant form shown to the right:
\HH are generally represented in the form: a + \hat{i}b + \hat{j}c
+ \hat{k}d where a, b, c, and d are \RR numbers, and i, j, and k
are the fundamental quaternion unit vectors.
Examples:
- i = jk = -kj
- j = ki = -ik
- k = ij = -ji
- i^2 = j^2 = k^2 = -1
- ijk = -1
\HH_{\mu} \HH^{\mu} =
\begin{bmatrix}
+1 & -i & -j & -k \\
+i & +1 & -k & +j \\
+j & +k & +1 & -i \\
+k & -j & +i & +1
\end{bmatrix}
Wikipedia Articles:
In theoretical physics, a Penrose diagram (named after mathematical physicist Roger Penrose)
is a two-dimensional diagram capturing the causal relations between different points in spacetime
through a conformal treatment of infinity. It is an extension of a Minkowski diagram where the
vertical dimension represents time, and the horizontal dimension represents a space dimension,
and slanted lines at an angle of 45° correspond to light rays. The biggest difference is that
locally, the metric on a Penrose diagram is conformally equivalent to the actual metric in
spacetime. The conformal factor is chosen such that the entire infinite spacetime is transformed
into a Penrose diagram of finite size, with infinity on the boundary of the diagram. For
spherically symmetric spacetime, every point in the Penrose diagram corresponds to a
2-dimensional sphere.
Figure 1(h): Penrose Spacetime.
2: Introduction
I would like to start by clarifying how the dimensions of space and time are related
to one another in order to show how they link together in a four dimensional construct
most commonly seen in special relativity using [+,-,-,-] Minkowski metric, with
[ct,x,y,z] representing \RR scalar values of length:
\begin{aligned}
s^2
&= (ct)^2 + (\hat{i}x)^2 + (\hat{j}y)^2 + (\hat{k}z)^2 \\
&= (ct)^2 - x^2 - y^2 - z^2
\end{aligned}
Using this metric and choice of unit vectors for time and space, spacetime is now
expressible in one dimensional form, aka that of a single particle or wave. I would then like to
add that both space and time can also be defined using \HH numbers, to further
give us the view of spacetime from four unique particles, in our case one type of electron and
three colours of quarks. This can be seen by looking at the \hat{i}, \hat{j},
\hat{k} rotations of the spacetime interval s. One with metric [+,-,-,-] and three with
metric [-,+,+,+]:
\begin{aligned}
(\hat{i}s)^2
&= (\hat{i}ct)^2
+ (\hat{i}\hat{i}\hat{i}\hat{i})(x)^2
+ (\hat{i}\hat{i}\hat{j}\hat{j})(y)^2
+ (\hat{i}\hat{i}\hat{k}\hat{k})(z)^2 \\
&= x^2 + y^2 + z^2 - (ct)^2
\end{aligned}
So while the electrons are inside their own spacetime event cone s
and thus time-like events to each other, the quarks are always space-like events to
the electrons but time-like event cone \hat{i}s, \hat{j}s,
\hat{k}s to each other. In short the electrons and quarks are not
causally connected, except through rotations via the weak force, which is the
light-like event that is part electromagnetism and part strong force.
This is important as \RR numbers are commutative, but
\II part of \CC or \HH numbers are
anti-commutative. Commutative properties give the same result, regardless of the
order of operations, anti-commutative is when the order of operation changes
the result. I propose we just extend that to the special relativity spacetime
intervals as well.
This links the fact we see spacetime as four dimensional because, and only because we are
made up of matter that is inherently four dimensional, specifically three of one type of
particle giving us three dimensional space, and one further particle giving us one
dimensional time. Ergo the observed spacetime is intrinsically linked to the dimensionality
of the energy observing it. But not only that we have duality because we can only view
spacetime from the perspective of the quarks or electrons, leading to viewing real space
with metric [-,+,+,+] or real time with metric [+,-,-,-]. For this reason, we can never
view a particle as both wave and point source, and we have to use both Minkowski spacetime
metrics simultaneously (impossible as they are orthogonal to each other),
\RR interval of electrons giving us [+,-,-,-], and \II
interval of quarks giving us [-,+,+,+], my argument is we need to start using both and
thinking of the spacetime interval as fully \HH to unite both quark and
electron, strong, weak and electromagnetic forces into one construct or unified event.
Electromagnetic Force U(1) being that of the \RR part of time, SU(3)
strong force being that of \II parts of time and finally the weak force
being the rotations between \RR and \II. This is a
departure from normal standard model that teaches us that the weak force is independent
of the strong force, rather I postulate it is instead the rotational link between
the electromagnetic and strong forces. E.g. the real time-like event
s^2 > 0 for electrons and electrical charge,
s^2 < 0 for quarks and coloured charge, and
{\pm}s^2 = 0 for gluons and photons, finally with
{\pm}(\hat{i},\hat{j},\hat{k})s^2 = 0 for weak bosons.
So just as we have found a link between electromagnetism and weak force to give
electroweak theory U(1) x SU(2), we should also find a link between strong and weak
forces as SU(2) x SU(3). So rather than thinking of the nucleus as having electrical
charge, we instead need to think of how it is manipulating the weak force, and that
then manipulating the electromagnetic force.
Hence the concept of all dimensions being \HH, including the spacetime
interval s. Then the interval is not only in relation to any one particle, both up spin
\HH_0 and down spin \HH^0, but also how each particle
combines to create a compound event for either mesons with \CC^2
complexity, or atoms with \HH^2 complexity. Massless bosons being events
that stay within \RR or \II spacetime so energy and
momentum are equal, massive bosons being those that cross from \RR
to \II or vice-versa so energy and momentum are not equal, resulting
in them having mass.
3: Spacetime Topology
Previously we mentioned the special relativity interval. We now need to show all
four rotational copies relating to \pm ct, \pm \hat{i}ct,
\pm \hat{j}ct and \pm \hat{k}ct as we will define all
dimensions as \HH of which only the \pm ct and
\pm \hat{i}ct component is shown in Figure 1.
Assuming we start with \HH_{0} = t + \hat{i}x + \hat{j}y + \hat{k}z, and
\HH_{1} = \hat{i} \HH_{0}, \HH_{2} = \hat{j} \HH_{0},
\HH_{3} = \hat{k} \HH_{0}, aka looking at spacetime as one dimensional
number where t, x, y, z are all \RR numbers defining scale, but
the \RR and \II unit vector parts defining uniqueness
and orthogonality:
\HH^{0} \HH_{0} =
\begin{bmatrix}
\color{cyan}{+t^2} & \color{orange}{-\hat{i}tx} & \color{orange}{-\hat{j}ty} & \color{orange}{-\hat{k}tz} \\
\color{orange}{+\hat{i}tx} & \color{red}{+x^2} & \color{green}{-\hat{k}xy} & \color{green}{+\hat{j}xz} \\
\color{orange}{+\hat{j}ty} & \color{green}{+\hat{k}xy} & \color{red}{+y^2} & \color{green}{-\hat{i}yz} \\
\color{orange}{+\hat{k}tz} & \color{green}{-\hat{j}xz} & \color{green}{+\hat{i}yz} & \color{red}{+z^2}
\end{bmatrix}
\HH^{j} \HH_{j} =
\begin{bmatrix}
\color{cyan}{+t^2} & \color{orange}{+\hat{i}tx} & \color{orange}{+\hat{j}ty} & \color{orange}{+\hat{k}tz} \\
\color{orange}{-\hat{i}tx} & \color{red}{+x^2} & \color{green}{-\hat{k}xy} & \color{green}{+\hat{j}xz} \\
\color{orange}{-\hat{j}ty} & \color{green}{+\hat{k}xy} & \color{red}{+y^2} & \color{green}{-\hat{i}yz} \\
\color{orange}{-\hat{k}tz} & \color{green}{-\hat{j}xz} & \color{green}{+\hat{i}yz} & \color{red}{+z^2}
\end{bmatrix}
This will give us a central single time dimension as being complete and still one dimensional
with one \RR and three \II parts. The reason to do
this is the \RR time part will represent spacetime from the view of
electromagnetism, our observable domain and the \II time parts will
represent the three colours of the strong force. Thus not only showing the connectivity
between electrons and quarks, electromagnetism and strong forces, but also between matter
and antimatter.
Figure 3(a): Penrose Spacetime (2D)
The only thing that differentiates the view of spacetime \HH^2
of electrons and that of quarks, is the polarity of the charges is reversed. They both share
the same identity matrix, aka they exist in the same spacetime framework, and they both spin
in the same direction. The \RR time gives us the commutative properties of
electromagnetism, and \II time gives us the non-commutative properties
of the strong force. The weak force now being described as rotations between electromagnetism
and strong force, which means it to must be commutative in nature as combination of
\RR and \II is always commutative.
The concept of the big bang is now obvious to see as it represents a moment when
T^{0} had zero radius, just like a ripple in a pond doesn't exist before
a stone is thrown in, as there is no concept of before or negative radius. Expanding spacetime
is when the trough of T^{0} and peaks of S^{\alpha} and
S^{\beta} are increasing in separation which would have the effect of
driving left-handed and right-handed matter as well as antimatter apart, forcing
electromagnetism and strong forces apart and giving us the concept of Higgs mechanism.
This greater the separation of left and right-handed matter (diameter of T^0),
the greater the Higgs field coupling and greater the mass of the particles. This should
have the effect of coalescing quarks into mesons then baryons and atoms as the universe
cools down, shown as reducing frequency, or increasing wavelength defined by the
radius of T^0, S^{\alpha} and S^{\beta} which equates to
\lambda{/}{4}. So the universe of matter we see is only quarter of the
total universe, the interaction between positive and negative \RR parts
gives us time-like \RR matter and antimatter events as seen by the interaction
of electrons and electromagnetism, the interaction between \II gives us
time-like \II matter and antimatter of quarks and strong force, which is
simultaneously space-like event to electromagnetism and electrons.
We also have the concept of spacetime rotation, where space and time swap roles. If
you can see the wave-like nature of spacetime in Figure 1 you can imaging this
rotation as the wave moving in the direction of S^{\alpha} or
S^{\beta}, or phase shifting. If there is no movement then we could
classify the universe as a standing wave, and just as mass can have both rest mass and
momentum, so too can the universe. This possibly could explain how mass lost to black
holes contributes to the rotation of the universe, and our universe is but one of an
infinite series. A the universe we see is only the standing wave portion, energy phase
shifted out of our universe and into another adjacent universe, see as energy being rotated
a full wavelength away from us. So mass and energy are never lost in a black hole, but
rotated \lambda{/}{4} to another universe, likewise matter entering our
universe from a black hole is rotated in from another universe. The event horizon of the
black hole being where events change from time-like to space-like through light-like.
So we should think of the interior of all black holes to be space-like
\II matter that connects universes (aka outside of universe),
and \RR matter what connects black holes (inside of universe), with
light mapping the spacetime framework between the two. As shown in Figure 1 as
the expanding circumference of T^0, or cosmic microwave background (CMB)
where differences in red shift will indicate greater quantities of either time-like
\RR or space-like \II matter.
- e^{-}_{L}, e^{+}_{R} is in + \HH .
- e^{-}_{R}, e^{+}_{L} is in - \HH .
- q_{L}, \bar{q}_{R} is in + \hat{i} \HH, + \hat{j} \HH, + \hat{k} \HH .
- q_{R}, \bar{q}_{L} is in - \hat{i} \HH, - \hat{j} \HH, - \hat{k} \HH .
- H^0 separate each block.
You may also see that when time T^{0} is created we must also have the
separation of matter and antimatter space S^{\alpha} and
S^{\beta}. We can now also define three categories of matter:
- Matter: Share both common space and time dimensions (positive time and space).
- Dark Matter: Share only common space dimension (negative space).
- Dark Energy: Share only common time dimension (negative time).
Mesons being the only solution using \CC^2 (2D) spacetime, atoms of
one colour electron and three coloured quarks the only solution using
\HH^2 (4D) spacetime. The idea of negative space and time may be
hard to grasp, but if you think of them as hidden within the spacetime framework as we do
for vacuum energy, and spacetime framework as being the walls we can't see through. This
way spacetime is both moulded on the positive side with matter, and negative side with
dark matter, dark energy and vacuum energy. Particle creation from vacuum energy or
dark matter annihilation is based on a rotation between negative and positive spacetime.
It is also important to note that S^{\alpha} and S^{\beta}
do not wrap around but instead connect to a different universe (as defined by its own central
T^{0}), as we must define the overall multiverse as continuous with no hard
edges. This is the wave-like nature mentioned previously.
Last point to note is that with Penrose diagrams the origin or destination for light must always
be the diagonal edges, and matter moves from the central time area outward in one direction.
For time-like \RR mass must stay within the blue areas, and for space-like
\II mass within the red areas. Electromagnetic force (γ) must
stay within the electron area, and the strong force (g) in the quark areas. Only the weak
bosons (W±, Z0) can cross between quark and electron areas and
will be made of part \RR momentum and \II momentum,
which is why they must have mass and be commutative in nature.
4: Energy Equivalence
Now while using the top most Penrose spacetime block, I propose to now look at the rates of
change of time and space instead, by doing the following two transforms:
- Temporal: t :\rightarrow \hbar \gamma^{0} \partial_{0}.
- Spatial: x :\rightarrow \hbar \gamma^{j} \partial_{j}.
We must also add the factor of c to the spatial axis to change what would be units of
momentum to units of energy, so it matches the units used along the temporal axis. You
should notice this is still one dimensional as one axis is always \RR while
the other is \II, mass being defined as the radius or hypotenuse
created by what is now energy and momentum axis.
This then gives us the image as seen in Figure 2.
We have now highlighted the four quadrants for space-like and time-like events, and the black
lines represents light-like events. The blue and red curved surfaces now represent the particles
equal energy and momentum levels respectively. One thing that is now shown is that for each
energy level, the gap between each level must decrease as they bunch up towards the corners of
infinite energy and zero energy and momentum represented by the cross of the black axis in
the center of the diagram. We are using the Minkowski metric [+,-,-,-,] here to show the
time-like energy is always positive (usually depicted by the colour blue) and
space-like energy is always negative (usually depicted by the colour red).
| \hat{E} = +i{\hbar}c \gamma^{0} \partial_{0} | ||
|
i \hat{p} c = - \hbar c \gamma^j \partial_j
|
|
i \hat{p} c = + \hbar c \gamma^j \partial_j
|
| \hat{E} = -i{\hbar}c \gamma^{0} \partial_{0} | ||
Figure 4(a): Energy Equivalence
Using Pythagorean Theorem:
- x^2 + y^2 = r^2
Transposing:
- x :\rightarrow i\hat{p}c = \hbar c \gamma^j \partial_j
- y :\rightarrow \hat{E} = i \hbar c \gamma^0 \partial_0
- r :\rightarrow mc^2
Giving Us:
(i\hat{p}c)^2 + (\hat{E})^2 = (mc^2)^2
(\hbar c \gamma^j \partial_j)^2 - (\hbar c \gamma^0 \partial_0)^2 = (mc^2)^2
Normalizing:
(\hat{E})^2 = (mc^2)^2 + (\hat{p}c)^2
\nabla^2 - \frac{1}{c^2} \frac{\partial^2}{\partial{t^2}} = (\frac{mc}{\hbar})^2
Resulting In Einstein & Klein-Gordon Equations.
Dirac Equation By Summing Vectors:
\hat{E} - \hat{p}c - mc^2 = 0
Normalizing:
i \hbar \gamma^{\mu} \partial_{\mu} - mc = 0
As:
\left[
\partial_{0}^{2} =
\frac{1}{c^2} \frac{\partial^2}{\partial t^2}
\right]
\left[
\partial_{j}^{2} = \nabla^2 =
\frac{\partial^2}{\partial x^2} +
\frac{\partial^2}{\partial y^2} +
\frac{\partial^2}{\partial z^2}
\right]
\left[
\gamma_{\mu} \gamma^{\mu} =
I_4
\right]
5: Special Relativity
This explains the need for two Minkowski metrics which must be used together, in our case as
we have the following:
- Electron, Positron: [ct, ix, iy, iz], Metric: [+,−,−,−]
- Red, Anti-red Quark: [ict, x, y, z], Metric: [−,+,+,+]
- Green, Anti-green Quark: [jct, x, y, z], Metric: [−,+,+,+]
- Blue, Anti-blue Quark: [kct, x, y, z], Metric: [−,+,+,+]
In terms of momentum and energy, electrons and quarks must all be moving at right angles to
each other (defining that they are unique particles). The event horizon between the electrons
and quarks is the gap between the electron cloud and quark lowest energy level. The higher the
energy level of the electron, the further out it goes. The higher energy level of the quark,
the more the electron is pushed away. This stretching of the energy and momentum between
the electron and quarks, gives us the effects of charge separation and gravity.
-
\Delta \approx \partial_{\mu}Linear (first order) shear, relates to momentum and charge like attraction.
-
\Delta \approx \partial_{\mu}^{2}Curved (second order) shear, relates to acceleration and gravitational attraction.
This shows charge between electrons and quarks acts as a special relativistic boost between the
two, and the rate of change of these boosts act like curvature and gravity. This is how quarks
and electrons alone do not hold gravity and why it has been so difficult to unify special and
general relativity or strong and electromagnetic forces using only a single real valued metric
of spacetime.
However the combination of quarks and electrons that make mesons, baryons or atoms as well as the
use of a fully connected \HH model of spacetime, it is possible to unify both
special, general relativity and all forces together in one unifying model. It also shows the
fundamental energy and motion of quarks and electrons as well as the electromagnetic and strong
forces are all the same, it is only the differences in location with respect to each other that
define the differences in properties.
Treating the separate particles as standing waves, would only give the rest mass and special
relativistic and charge connectivity. Adding in the further rate of vibrational fluctuation of
each particle will now also show the general relativistic connection. In the case of the atom,
the quarks are changing energy far more often than the electrons as they not only change in energy
level the same as electrons, but are also constantly swapping colour charge as well. This net
difference in rates of change in energy levels will create a gravitational attraction towards the
center of mass defined by the net movement of quarks in the nucleus.
Special relativity is where we define the axis as \hat{E} and
\hat{p}c and general relativity is where the axis becomes
\hat{P} = \partial_0 \hat{E} and \hat{F} = \partial_j \hat{p}c.
So while special relativity has axis of energy and momentum (both in momentum units),
general relativity has axis of power and force (both in acceleration units). So general
relativity is just the gradient of special relativity, if we use the triangle of energy, mass
and momentum of special relativity, general relativity defines how that triangle is deformed.
With positive de-Sitter curvature increasing the angles to greater than 180 degrees, negative
anti de-Sitter curvature decreasing the angles to less than 180 degrees and flat Minkowski
spacetime being the special case where the angle comes to exactly 180 degrees. So special
relativity will be the relation between electromagnetism, weak and strong forces, electrons
and quarks, specifically through charge and momentum and general relativity is how the
momentum and energy changes over space and time. So just as special relativity has bosons
(light-like events) of photon (γ), gluon (g), W± and Z0,
So too general relativity should have tachyon (τ) being rate of change of electromagnetism,
graviton (G) as rate of change of strong force, and V± and Y0
as the rate of change of the weak force, which again is a rotation between tachyon and
graviton acceleration. Now we also have the concept of space-like, light-like and time-like
curvature, time-like de-Sitter curvature gives us entropy and dark energy, space-like
anti de-Sitter curvature gives us gravity and dark matter and finally light-like curvature
gives us special relativity and flat Minkowski spacetime.
6: General Relativity
We now have defined what dark matter and energy could be, as well as how they both must transfer
energy and momentum to normal matter by further defining how they are connected to matter
through space and time, yet still being physically separated so as not to directly interact with
matter. Causality is now the net direction of all of the various spacetime
elements combined. So we see time moving forward, and entropy as always being positive because
there is more dark energy pulling on the outer time-like connections (four corners of
Figure 1). The has the effect of increasing charge separation of electrons and quarks,
decaying free neutrons. While the smaller amount of dark matter helps pull matter together,
increasing the chances of nuclear fusion.
Now putting this all together we create a wave front moving in one direction as shown in
Figure 3. However from a stationary point, it will look as if the universe is expanding
to a maximum point, then contracting back to nothing. Matter now has three states being minimum
when energy or momentum are time-like (shown as blue), maximum when they are space-like (shown in
red), connected by light-like type (horizontal plane along z=50, center of the green areas).
Depending what part you are in, spacetime is being continuously created and destroyed, think of
this as the big bang and crunch happening continuously, but constantly moving location with
the overall energy staying constant. If stationary you would see energy changing
from space-like to time-like through light-like and back again continuously, that change would show
as changes to entropy, and causality, e.g. time flowing backwards, heat going from hot to cold,
etc. This resolves the paradoxical answer to how we can both have a big bang creating spacetime,
but energy can neither be created or destroyed. Nor is the idea the point of creation as being a
static point, or singular; like everything else it is wave-like and continuous.
Figure 6(a): General Relativity
7: Calculated Tensor
NOTE: This is currently a work in progress to show how the \HH
math can be shown in n x n format for 3 x \HH for space dimensions and 1 x
\HH for time dimension. While this normally would be shown as a 4 x 4 tensor
array, this is designed for \CC mathematics instead.
+ ℑ
− ℜ
− ℑ
+ ℜ
+ ℑ
− ℜ
− ℑ
Dark Matter
Matter
Dark Energy
+ ℑ
− ℜ
− ℑ
+ ℜ
+ ℑ
− ℜ
− ℑ
Dark Matter
Matter
Dark Energy
The X/Y rotation will rotate space to time, given the Minkowski relationship which requires one
to be imaginary when the other is real. The Z rotation does the same, only it is in and out of the
page. This shows how each time and space component consists of bubbles (waves) which repeat
infinitely in both X/Y and Z planes.
8: Summary
This framework shows that for any spacetime domain, for example that of the electrons must
connect to both normal matter and dark matter through spatial connectivity and similarly normal
matter and dark energy through temporal connectivity. Matter is then based on the complexity of
the temporal and spatial connectivity as shown:
-
\CC:1D + 1T (Meson, Particle Pairs).
-
\HH:3D + 1T (Atom, Baryon)
-
\OO:7D + 1T (Beyond Standard Model)
-
\SS:15D + 1T (Beyond Standard Model)
As there are no definable numbers between \CC and \HH, we will
not see any type of compound particles that are purely 3D, say a meson with an electron as it is
not allowed. Below \CC is also not possible as you must create space-like dimension
as well as time-like dimensions in order to satisfy the wave-like nature and continuous topology
overall.
While we have good proof to determine our own universe as being 4D, the universes that connect
externally to ours don't necessarily have to be 4D as well. We also can't assume that the amount
of fermions and quarks are equal, as dark matter is connected spatially it may have greater
pull to either electrons, or one of the colours of quarks in different areas of space. Likewise
for dark energy, only it will vary over time; which has pretty much been determined already in
cosmology shows a growth spurt earlier in the evolution of our universe.
The graph shown in Figure 3 is how rotation of the wave-like nature of the universe could
be perceived, as space is converted to time, and vice versa. From any stationary point, this would
look like spacetime expanding then contracting as one half of the wave form, then inverting
to become timespace expanding and contracting for the second half of the wave form. If we add to
this the time-like matter, we would first experience spacetime as matter then timespace as
imaginary time-like matter, then spacetime as antimatter, then timespace as imaginary
antimatter, before repeating a new cycle. However as we can only see like matter, there really
is no way to tell which quadrant we are in currently, they will always look like spacetime and
matter locally.
One important conclusion is rather than matter being four dimensional because of spacetime,
but rather the inverse is true, that we see the universe as four dimensional because of the
complexity of matter we are made of. This both allows for matter that is not causally connected
to the big bang event and that event is only relevant to the matter we are made of. This renders
the idea the universe has one beginning or end as meaningless, as even at the universal scale,
time has only local relevance. Concepts such as dark matter and dark energy do not need to be
fully connected to all the same dimensions, but rather overlap partially. The fact we haven't
been able to find any connectivity between dark matter and electromagnetism would suggest as
much. If this theory is correct, we may also need to check connectivity of things like dark
matter with the strong and weak forces as well as quarks, not just electrons and the
electromagnetic force. Light is not the only yard stick we can use to probe the universe.