Unified Field Theory
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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:
Einstein Notation:
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.
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.
Bra-Ket Notation:
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.
Wikipedia Articles:
Example Math:
\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}
Inner, Dot & Scalar Product:
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 = \overline{B \cdot A}
For all vectors A and B.
Figure 1(b): Dot Product
Wikipedia Articles:
Example Math:
\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}
Cross Product:
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 \times. 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.
Exterior & Wedge Product:
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
Wikipedia Articles:
Example Math:
\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}
Outer & Tensor Product:
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.
Kronecker Product:
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.
Wikipedia Articles:
Example Math:
\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}
Hadamard Product:
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.
Wikipedia Articles:
Example Math:
\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}
Pauli Matrices:
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
Gamma Matrices:
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.
\gamma^1 = \gamma^x,
\gamma^2 = \gamma^y,
\gamma^3 = \gamma^z,
are space-like, anti-hermitian matrices.
Wikipedia Articles:
Example Math:
\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}
Complex Numbers:
A complex number is a number that can be expressed in the form
a + b\hat{i}, where a and b are \RR numbers, and i
represents the imaginary unit, satisfying the equation \hat{i}^2 = -1.
Because no real number satisfies this equation, i is called an imaginary number.
For the complex number a + b\hat{i}, a is called the real part, and
b is called the imaginary part. The set of complex numbers is denoted using the
symbol \CC. Despite the historical nomenclature "imaginary", complex
numbers are regarded in the mathematical sciences as just as "real" as the real
numbers, and are fundamental in many aspects of the scientific description of the
natural world.
Quaternion Numbers:
In mathematics, the quaternions are a number system that extends the compex
numbers denoted using the letter \HH. 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.
\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 \hat{i}, \hat{j}, \hat{k} are the
fundamental quaternion unit vectors.
Octonion Numbers:
In mathematics, the octonions represented by the letter \OO are a
normed division algebra over the real numbers, meaning it is a hypercomplex number
system. Octonions have eight dimensions; twice the number of dimensions of the
quaternions, of which they are an extension. They are noncommutative and
nonassociative, but satisfy a weaker form of associativity; namely, they are
alternative. They are also power associative.
Octonions are not as well known as the quaternions and complex numbers, which are much
more widely studied and used. Octonions are related to exceptional structures in
mathematics, among them the exceptional Lie groups. Octonions have applications in fields
such as string theory, special relativity and quantum logic. Applying the Cayley–Dickson
construction to the octonions produces the sedenions.
Wikipedia Articles:
- \CC :\rightarrow Complex
- \HH :\rightarrow Quaternion
- \OO :\rightarrow Octonion
- \SS :\rightarrow Sedenion
Example Math:
Complex Numbers
- \CC = \RR \oplus \star \RR
- \CC = \RR \oplus \RR \hat{i}
- \CC = a + b \hat{i}
- \hat{i}^2 = -1
\CC \otimes \CC^{\dagger} =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:c}
+1 & -\hat{i} \\
\hdashline
+\hat{i} & +1
\end{array}
\right]
\CC \otimes \CC^{\dagger} =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:c}
+\RR \hat{1} \otimes \RR^{\dagger} \hat{1} & -\RR \hat{1} \otimes \RR^T \hat{i} \\
\hdashline
+\RR \hat{i} \otimes \RR^{\dagger} \hat{1} & -\RR \hat{i} \otimes \RR^T \hat{i}
\end{array}
\right]
\CC \otimes \CC^T =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:c}
+1 & +\hat{i} \\
\hdashline
+\hat{i} & -1
\end{array}
\right]
\CC \otimes \CC^T =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:c}
+\RR \hat{1} \otimes \RR^T \hat{1} & +\RR \hat{1} \otimes \RR^T \hat{i} \\
\hdashline
+\RR \hat{i} \otimes \RR^T \hat{1} & +\RR \hat{i} \otimes \RR^T \hat{i}
\end{array}
\right]
Quaternion Numbers
- \HH = \CC \oplus \star \CC
- \HH = \CC \oplus \CC \hat{j}
- \HH = a + b \hat{i} + c \hat{j} + d \hat{k}
- \hat{i} = \hat{j} \hat{k} = -\hat{k} \hat{j}
- \hat{j} = \hat{k} \hat{i} = -\hat{i} \hat{k}
- \hat{k} = \hat{i} \hat{j} = -\hat{j} \hat{i}
- \hat{i}^2 = \hat{j}^2 = \hat{k}^2 = -1
- \hat{i} \hat{j} \hat{k} = -1
\HH \otimes \HH^{\dagger} =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc}
+1 & -\hat{i} & -\hat{j} & -\hat{k} \\
\hdashline
+\hat{i} & +1 & -\hat{k} & +\hat{j} \\
+\hat{j} & +\hat{k} & +1 & -\hat{i} \\
+\hat{k} & -\hat{j} & +\hat{i} & +1
\end{array}
\right]
\HH \otimes \HH^{\dagger} =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:c}
+\CC \hat{1} \otimes \CC^{\dagger} \hat{1} & -\CC \hat{1} \otimes \CC^T \hat{j} \\
\hdashline
+\CC \hat{j} \otimes \CC^{\dagger} \hat{1} & -\CC \hat{j} \otimes \CC^T \hat{j}
\end{array}
\right]
\HH \otimes \HH^T =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc}
+1 & +\hat{i} & +\hat{j} & +\hat{k} \\
\hdashline
+\hat{i} & -1 & +\hat{k} & -\hat{j} \\
+\hat{j} & -\hat{k} & -1 & +\hat{i} \\
+\hat{k} & +\hat{j} & -\hat{i} & -1
\end{array}
\right]
\HH \otimes \HH^T =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:c}
+\CC \hat{1} \otimes \CC^T \hat{1} & +\CC \hat{1} \otimes \CC^T \hat{j} \\
\hdashline
+\CC \hat{j} \otimes \CC^T \hat{1} & +\CC \hat{j} \otimes \CC^T \hat{j}
\end{array}
\right]
Octonion Numbers
- \OO = \HH \oplus \star \HH
- \OO = \HH \oplus \HH \hat{m}
- \OO = a + b \hat{i} + c \hat{j} + d \hat{k} + e \hat{m} + f \hat{I} + g \hat{J} + h \hat{K}
- \hat{i} = \hat{m} \hat{I} = \hat{K} \hat{J}
- \hat{j} = \hat{m} \hat{J} = \hat{I} \hat{K}
- \hat{k} = \hat{m} \hat{K} = \hat{J} \hat{I}
- \hat{m} = \hat{I} \hat{i} = \hat{J} \hat{j} = \hat{K} \hat{k}
- \hat{I} = \hat{i} \hat{m} = \hat{k} \hat{J} = \hat{K} \hat{j}
- \hat{J} = \hat{j} \hat{m} = \hat{i} \hat{K} = \hat{I} \hat{k}
- \hat{K} = \hat{k} \hat{m} = \hat{j} \hat{I} = \hat{J} \hat{i}
- \hat{m}^2 = \hat{I}^2 = \hat{J}^2 = \hat{K}^2 = -1
- \hat{I} \hat{J} \hat{K} = +\hat{m}
\OO \otimes \OO^{\dagger} =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc:c:ccc}
+1 & -\hat{i} & -\hat{j} & -\hat{k} & -\hat{m} & -\hat{I} & -\hat{J} & -\hat{K} \\
\hdashline
+\hat{i} & +1 & -\hat{k} & +\hat{j} & -\hat{I} & +\hat{m} & +\hat{K} & -\hat{J} \\
+\hat{j} & +\hat{k} & +1 & -\hat{i} & -\hat{J} & -\hat{K} & +\hat{m} & +\hat{I} \\
+\hat{k} & -\hat{j} & +\hat{i} & +1 & -\hat{K} & +\hat{J} & -\hat{I} & +\hat{m} \\
\hdashline
+\hat{m} & +\hat{I} & +\hat{J} & +\hat{K} & +1 & -\hat{i} & -\hat{j} & -\hat{k} \\
\hdashline
+\hat{I} & -\hat{m} & +\hat{K} & -\hat{J} & +\hat{i} & +1 & +\hat{k} & -\hat{j} \\
+\hat{J} & -\hat{K} & -\hat{m} & +\hat{I} & +\hat{j} & -\hat{k} & +1 & +\hat{i} \\
+\hat{K} & +\hat{J} & -\hat{I} & -\hat{m} & +\hat{k} & +\hat{j} & -\hat{i} & +1
\end{array}
\right]
\OO \otimes \OO^{\dagger} =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:c}
+\HH \hat{1} \otimes \HH^{\dagger} \hat{1} & -\HH \hat{1} \otimes \HH^T \hat{m} \\
\hdashline
+\HH \hat{m} \otimes \HH^{\dagger} \hat{1} & -\HH \hat{m} \otimes \HH^T \hat{m}
\end{array}
\right]
\OO \otimes \OO^T =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc:c:ccc}
+1 & +\hat{i} & +\hat{j} & +\hat{k} & +\hat{m} & +\hat{I} & +\hat{J} & +\hat{K} \\
\hdashline
+\hat{i} & -1 & +\hat{k} & -\hat{j} & +\hat{I} & -\hat{m} & -\hat{K} & +\hat{J} \\
+\hat{j} & -\hat{k} & -1 & +\hat{i} & +\hat{J} & +\hat{K} & -\hat{m} & -\hat{I} \\
+\hat{k} & +\hat{j} & -\hat{i} & -1 & +\hat{K} & -\hat{J} & +\hat{I} & -\hat{m} \\
\hdashline
+\hat{m} & -\hat{I} & -\hat{J} & -\hat{K} & -1 & +\hat{i} & +\hat{j} & +\hat{k} \\
\hdashline
+\hat{I} & +\hat{m} & -\hat{K} & +\hat{J} & -\hat{i} & -1 & -\hat{k} & +\hat{j} \\
+\hat{J} & +\hat{K} & +\hat{m} & -\hat{I} & -\hat{j} & +\hat{k} & -1 & -\hat{i} \\
+\hat{K} & -\hat{J} & +\hat{I} & +\hat{m} & -\hat{k} & -\hat{j} & +\hat{i} & -1
\end{array}
\right]
\OO \otimes \OO^T =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:c}
+\HH \hat{1} \otimes \HH^T \hat{1} & +\HH \hat{1} \otimes \HH^T \hat{m} \\
\hdashline
+\HH \hat{m} \otimes \HH^T \hat{1} & +\HH \hat{m} \otimes \HH^T \hat{m}
\end{array}
\right]
Penrose Diagram:
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.
One important point to make is that the light-like \infin are just the
same as space-like and time-like \infin, just rotated at an angle of 45°,
so just as we see space and time as being unique, we to need to see \RR
and \II light-like events as unique too. While the \RR
light-like events are gauge bosons we know, the \II light-like events
are an equivilent set of hyperbolic bosons (vacuum energy). So all time-like events
must lie between the walls defined by hyperbolic gauge bosons and gauge bosons. So
time and space are not real things as such, but projections of the difference
between light and vacuum energy, time being the outer product and space being the
inner product of the two. Spacetime special relativistic plane defines the balance
points between vacuum energy and light, so you can either define the axis as
coordinates of space and time or vacuum energy and light, difference being an overal
transformational rotation of 45°.
Figure 1(h): Penrose Spacetime
Beyond 2D:
We can now take this penrose block, and stack them end to end in a similar way
to creating a jigsaw puzzle, to create the next level of penrose block, e.g. 2D
to 4D, 8D, 16D, etc. This is the equivalent of multiplying by \CC.
Making sure we rotate each block at 90° in either direction, creating two dualistic
copies of rotate left and rotate right as well as a negative reflection, ultimately
changing what was originally a time-like cone to time-like sphere. With all past
events being inside the sphere, all future events being outside and the surface
being created by four unique event horizons, two light-like and two hyperbolic
light-like \infin. In the same way, we will also create spherical
bubbles of space. This will create something akin to the hypercube or tesseract, with
all past time-like events in the center portion, and all space-like events in the
outer portions, with light-like events being the vertexes.
Think of the hyperbolic gauge bosons as defining the floor, and gauge bosons as
defining the ceiling. The difference is that hyperbolic gauge bosons have
\II frequency \hat{E} = \pm \hat{i} \hat{p} c and
light always has \RR frequency \hat{E} = \pm \hat{p} c.
We can now show the relation of up and down spin, with up spin towards \RR
light, and down spin towards \II light. So mass must lie on the
spacetime plane, representing spin 0 or resting frame of reference.
One thing not mentioned yet is vacuum energy, this would act on the same spacetime
plane as mass, and instead shows the net difference between \RR and
\II light-like events, so for example where vacuum energy is
positive, then there is more \RR light-like than \II
light-like events, skewing (rotating) the overall field towards \RR.
Mass and resting frame of reference is then a standing wave, between \RR
and \II light-like events with central minima on the spacetime plane, so it
must be connected to both light-like events. Virtual partical creation and annihilation
is more like a temporary rouge wave, caused by the overall fluctuation of vacuum energy.
If we think of vacuum energy as having spin \hat{i} and standard light
as having the usual spin \hat{1} then down spin is closer to vacuum energy
and up spin is closer to light, with mass as the resultant spin 0 standing wave
component between the two, with that mass oscillating between vacuum energy and light.
Formulation:
While normal Penrose diagrams are formulated using the transform
tan(u \pm v) = x \pm t which has roots of \pm \pi
and \infin at \pm \pi/2, I instead use
tan(u \pm \hat{i}v) = x \pm \hat{i}t with roots of
\pm \pi, \pm \hat{i} \pi and \infin at
\pm \pi/2, \pm \hat{i} \pi/2 as it combines both standard and
hyperbolic trigenometry into one diagram. I show the standard solutions using
\RR numbers and hyperbolic solutions using \star or
\II numbers to show it as a Hodge dual rotation, which is very
similar to varying levels of \II numbered rotations. As applying
the Hodge dual twice, is the same as double \II numbered rotation
that gives back the negative copy of the original function.
The primary reasoning is to show the difference between time-like and space-like
\infin as well as the two kinds of light-like \infin
possible. Vacuum energy would be one such thing that would be defined by
\hat{i} \infin light-like events, which makes sense as all time-like
events with real mass must lie between vacuum energy and light (which define the two
walls of the box if you will). Using this notation, both defines vacuum energy
and light as bosonic light-like events but importantly seperates them into
unique barriers.
While using this for concepts like black holes may make sense, the goal of my work
is to show that it is also fundamental into unifying the wave function of all
particles as well as showing how dark matter and energy can exist in orthogonal
spacetimes and what connectivity they share. For example, I hope to show how
spacetime is twisted my matter, with each fundamental particle seperated by
these two light-like events. This is similar, if not identical to theory of
pilot wave, and the reason atoms and larger break from quantum mechanics and
start relying on relativity is they essentially observe themselves. Special and
general relativity occur on two different levels of fold, so while special
relativity can be described using \HH math, you need to use
\OO math to describe curvature and general relativity. I.e. gravity
is not directly related to mass alone, but rather the 8D rotation based on
the different concentration of mass, antimatter, dark matter and dark energy in
any different area. E.g. the more mass an area holds, the less dark matter, dark
energy and antimatter it has, ergo the more it rotates or curves spacetime towards.
dark matter is then a form of positive curvature just like matter, dark energy and
antimatter similarly are examples of negative curvature. Both forms are made up of
their own rotated copies of standard model particles, only acting orthoganaly to the
spacetime we can see. Black holes are no longer endless pits, but mearly areas where
spacetime is rotated more than 45° (the event horizon), the light-like gravitational
event rotating anti-clockwise away from our special relaivistic plane of matter. The
reason the event horizon of black holes is not a solid edge is due to the mass eaither
side of that light-like event as having both \RR and \II
parts, outside the event horizon it is more \RR, inside it is more
\II. This way concept like hairs on black hole is the \RR
matter escaping out the event horizon and \II dark matter being
swallowed to add to the mass of the black hole. So matter is not swallowed by a
black hole, but rather converted to dark matter first. Harking radiation or hair
is the reverse where dark matter is converted back to matter in a round about similar
way that free neutrons decay to proton and electron, only the electron is replaced by
matter and the proton is the now charged black hole that remains. Black hole mass
gaps may be explained in a similar way to how protons, neitrons and electrons
can only exist with specific combinations and mass, but for black holes,
if they are low mass they attract matter, it they are slightly too high they would
radiate matter (like fissioning), so black hole mergers are the next level of
a sort of fusion reaction. We may possibly see this as a faint afterglow of
radiation after merger of two black holes leaving some excess mass above the
closest stable mass range (similar principle to electron orbitals).
Wikipedia Articles:
Introduction
One alternate form of standard model rather than showing it as U(1) \otimes
SU(2) \otimes SU(3) is in the form \RR \otimes \CC \otimes \HH
\otimes \OO, where \RR^2 represents the scalar energy field
that makes up all matter and light, U(1) = \CC^2 is the 2D field that
gives us separation of space and time as well as up, down spin.
SU(2) = \HH^2 is what then separates matter and antimatter as well as
separating matter into quark and lepton types, as well as Minkowski spacetime.
Finally SU(4) = \OO^2 which then splits matter further into matter,
antimatter, dark matter and dark energy spacetimes and finally giving us gravity
and the four points of curvature 0° representing the spacetime plane of matter we
can see, 90° being that of dark matter, 180° of the invisible antimatter and finally
270° being that of dark energy. The main reason dark matter and energy are dark
and antimatter is completely invisible.
This is to show the universe in the form of a binary counter. With
\RR, \CC, \HH, \OO representing the horizons we see that act as
either 0 or \infin barriers. In our case, we have the
\CC inner barrier the defines how all fields must act. \OO
outer barrier that provides matter dualistic base for matter and light to propagate
over, so vacuum energy, dark matter and energy replacing what was once called aether.
The \HH surface area between is what we refer to as special relativistic
plane for which all matter and light we can see must be in. Gravity being defined
by the concentration of matter in each of the four rotational copies of standard
model that make up matter, antimatter, dark matter and dark energy related to
\HH \otimes \CC = \OO, dark matter and dark energy being
the dual (imaginary \hat{m}) rotation of matter and antimatter.
Locally they would look no different to standard model physics and matter we
understand.
Figure 2(a): Standard Model
So each event horizon is defined by 2^{n}, where n is the level of
complexity of the special relativistic plane, in our case 4D. 2^{n-1}
defining complexity of fields, in our case 2D and inner barrier. Lastly
2^{n+1} defining the outer barrier and gravity at 8D by showing how
multiple groups of 4D spacetime interact. Each case is by taking the Hodge Star
operator to find the dual. This is why gravity requires curvatue, as it represents
the amount of rotation between the special relativistic plane and its dual. In
our case of dark matter (positive curvature), and dark energy (negative curvature)
This represents rotation r -90° \ge r \le +90°.
\star = \{\hat{i}, n=1\}, \{\hat{j}, n=2\}, \{\hat{m}, n=3\}.
As shown above, we now have the 4D construct to define special relativity and the 2D
construct we use to define the fields within it. Thus giving us the usual Minkowski
spacetime metric of [+,-,-,-], with [t,x,y,z]
representing \RR scalar values of length (using c=1 for clarity):
\begin{aligned}
(\hat{1} s)^2
&= \hat{1}^2 \times \left(
(\hat{1} t)^2 + (\hat{i} x)^2 + (\hat{j} y)^2 + (\hat{k} z)^2
\right) \\
&= t^2 - x^2 - y^2 - z^2
\end{aligned}
Using this metric and choice of unit vectors for time and space, spacetime event length
s is now expressible in \HH form, so a single particle or
wave described by one number. The event length being the radius of a 4D sphere, just
like r^2 = x^2 + y^2 + z^2 for a standard 3D sphere, only difference is
space and time are always imaginary orthogonal to each other. 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 (the \RR part we see outside the nucleus) and three colours of
quarks (the \II parts we can't see inside the nucleus). This can be seen
by looking at the \hat{1}, \hat{i}, \hat{j}, \hat{k} rotations of the
spacetime event interval s. One with metric [+,-,-,-] and
three with metric [-,+,+,+]:
\begin{aligned}
(\hat{i} s)^2
&= \hat{i}^2 \times \left(
(\hat{1} t)^2 + (\hat{i} x)^2 + (\hat{j} y)^2 + (\hat{k} z)^2
\right) \\
&= x^2 + y^2 + z^2 - t^2
\end{aligned}
So while the electrons are inside their own spacetime event cone s
and thus time-like causally connected events to each other, the quarks are always
space-like causally disconected events to the electrons a fundamental reason we see
them as seperate particles but quarks have time-like event cone \hat{i} s,
\hat{j} s, \hat{k} s with each other which is why we
can't seperate out quarks individually but the behave in a non-commutative way just as
multiplication of \II numbers do. 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. If we think of
electromagnetism and strong force as straight lines, they can only have momentum and
no mass, but weak force being two halves of a straight line at 90° must have
both momentum and mass.
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, non-commutative is when the order of operation changes
the result. I shown above we just extend that to the special relativity spacetime
intervals as well and this gives us the different properties of electrons and
quarks as well as electromagnetism, strong and weak forces.
This links the fact we see spacetime as four dimensional because we are made up of
matter that is inherently four dimensional, specifically three of one type of
non-commutative particle (quarks) giving us our connected view of three dimensional
space, and one further particle giving us one dimensional time (electron). Ergo
the observed spacetime is intrinsically linked to the dimensionality and connectivity
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 (rotational viewpoints),
leading to viewing real space with metric [-,+,+,+] or real time
with metric [+,-,-,-]. For this reason, we can never view a single
particle as both wave and point source as both these viewpoints are orthogonal to
each other, and we have to use both Minkowski spacetime metrics simultaneously,
\RR interval of electrons giving us [+,-,-,-],
and \II interval of quarks giving us [-,+,+,+],
my argument being 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 force interactions together in a single unified event interval
of the atom. Thereby creating a unified wavefunction of the atom encoding all
particles and forces into one theory.
Electromagnetic Force U(1) being that of the \RR commutative part
of time, SU(3) strong force being that of \II non-commutative
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
and electromagnetic forces, rather I postulate it is instead the rotational link
between the electromagnetic and strong forces. E.g. the real time-like event
s^2 \gt 0 for electrons and electrical charge,
s^2 \lt 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 (think straight line), 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 (think line with right angle in
centre). Aka momentum of strong force is converted to mass of electrons, or mass of quarks
is converted to momentum of electromagnetism.
Spacetime Topology
Previously we mentioned the special relativity interval. The main reason for doing this
is so we can now show all four rotational copies relating to \pm t_0,
\mp \hat{i} t^1, \mp \hat{j} t^2 and
\mp \hat{k} t^3 as we will define all dimensions as \HH
of which only the \pm t_0 and \mp \hat{i} t^1 component
is shown in Figure 3(a).
We can see in Figure 3(a) the light-light infinite singularity event
T^{0}, and the ever increasing event horizon (shown as the first
black square from the centre), followed by the infinite future light like
event horizon. The concept of now is redefined from a point (as would be seen by a
single particle with standard special relativity), to a surface with everything
inside being a previous event which was causally connected (time-like) to the event
horizon in some way. In our case the big bang event which is the start of the causal
connection for all matter. Different particles now represent a fraction of the total
surface area with light-like events having two possible start and end points, fist is
the \pm \infin edges \gamma, g as they stay within a
block, second is the W^{\pm}, Z^0 that directly connect the electron
and quarks event position.
Assuming we start with \HH_{0} = t_0 +\hat{i}x^0 +\hat{j}y^0 +\hat{k}z^0,
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.
This will give us a central single \HH complexity 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 and electrons,
our observable domain and the \II time parts will represent the three
colours of the strong force and quarks. Thus not only showing the connectivity between
electrons and quarks, electromagnetism, weak and strong forces, but also between
matter and antimatter.
Figure 3(a): Penrose Spacetime (2D)
One other thing to note is all \RR dimensions are parallel to each
other as are all \II dimensions (regardless if they are related to
space or time). Giving us S^{\alpha}, S^{\gamma} as \RR
light-like infinities, and S^{\beta}, S^{\delta} as \II
light-like infinities. We can further say all electromagnetic force light-like events
\gamma are with \RR time \II space and
strong force light-like events g with \II time and
\RR space. Weak force being rotations being half \RR
and \II light-like events.
Next I will switch from t_0 +\hat{i}t^1 +\hat{j}t^2 +\hat{k}t^3 to
\color{#E619E6}{\bar{w}} +\color{#FF0000}{r} +\color{#00B050}{g}
+\color{#0080FF}{b} and from t^0 -\hat{i}t_1 -\hat{j}t_2 -\hat{k}t_3
to \color{#E619E6}{w} +\color{#FF0000}{\bar{r}} +\color{#00B050}{\bar{g}}
+\color{#0080FF}{\bar{b}}, aka we are just using the time components of
\HH_{\mu}(t) above, obviously we will also have another three matrixes
representing the spatial connectivity, giving us a total 4x4x4 matrix.
The time based quantities of electromagnetic and colour charges now give us quantum
chromodynamics, quantum electroweak and quantum electrodynamics connectivity.
From Figure 3(a) all the contravariant light-like vectors denoted by
\color{#E619E6}{w}, \color{#FF0000}{r}, \color{#00B050}{g},
\color{#0080FF}{b} all flow from the light-like \RR
\pm \infin (S^{\alpha}, S^{\gamma})
and the covariant light-like vectors denoted by \color{#E619E6}{\bar{w}},
\color{#FF0000}{\bar{r}}, \color{#00B050}{\bar{g}}, \color{#0080FF}{\bar{b}}
all flow from the light-like \II \pm \infin
(S^{\beta}, S^{\delta}). So \color{#E619E6}{w}
\color{#E619E6}{\bar{w}} represents U(1) of the electromagnetic force
\gamma with time-like vectors being the electrons, rotation
\color{#FF0000}{r} +\color{#00B050}{g} +\color{#0080FF}{b} \leftrightarrow
\color{#E619E6}{w} would be the W^{+} weak boson,
rotation \color{#FF0000}{\bar{r}} +\color{#00B050}{\bar{g}}
+\color{#0080FF}{\bar{b}} \leftrightarrow \color{#E619E6}{\bar{w}} would be the
W^{-} weak boson, and diagonal translation \color{#FF0000}{r}
\color{#FF0000}{\bar{r}} +\color{#00B050}{g} \color{#00B050}{\bar{g}} +
\color{#0080FF}{b} \color{#0080FF}{\bar{b}} \leftrightarrow \color{#E619E6}{w}
\color{#E619E6}{\bar{w}} being the Z^{0} weak boson. The
remainder is SU(3) of the strong nuclear force with time-like vectors
\color{#FF0000}{r} \color{#FF0000}{\bar{r}}
\color{#00B050}{g} \color{#00B050}{\bar{g}} and
\color{#0080FF}{b} \color{#0080FF}{\bar{b}} representing the three coloured
quarks. Strong and electromagnetic force carriers \gamma, g being
light-like events that pass through particle events, weak force carrier bosons pass
between particle events.
Figure 3(b): Light Rays
As you can see, the outer T^1 := \mp \{ 1,i \} \infin
square is best seen as the event horizon of a black hole, and the inner
T^0 := \pm \{ 1,i \} \infin as the singularity, both
are \infin far away from the moment of here and now, being
the surface area defined by the middle square. The points
\{ T^\alpha, T^\gamma \} = iT^1 and
\{ T_\beta, T_\delta \} = T_1, and as both points
T^0, T^1 are singularities, defining the structure of the
universe between them. One singularity T^0 being 2D,
defining the structure of all fields withing the universe, and the other
T^1 being 4D defining how those fields are folded around
each other to give us the different particle properties and spatial structure.
E.g the inner singualrity T^0 \in \CC^2, and outer singularity
T^1 \in \HH^2.
So overall the progression of the universe, start from a singular field
and gradually gets more and more complex as it folds. E.g.
\RR^2 \mapsto \CC^2, \CC^2 \mapsto \HH^2,
\HH^2 \mapsto \OO^2, then \OO^2 \mapsto \SS^2
and so on. Each visible universe mapped between two singularities, in our
case \CC^2 \mapsto \HH^2.
Combining the above, we now get:
\left[ \HH^{\mu} \HH_{\mu} \right] (t) =
\left[
\def
\arraystretch{1.2}
\begin{array}{cccc}
\color{#E619E6}{\bar{w}} \\
\hdashline
\color{#FF0000}{r} \\
\color{#00B050}{g} \\
\color{#0080FF}{b}
\end{array}
\right]
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc}
\color{#E619E6}{w} &
\color{#FF0000}{\bar{r}} &
\color{#00B050}{\bar{g}} &
\color{#0080FF}{\bar{b}}
\end{array}
\right]
\left[ \HH^{\mu} \HH_{\mu} \right] (t) =
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc}
\color{#E619E6}{\bar{w}} \color{#E619E6}{w} &
\color{#E619E6}{\bar{w}} \color{#FF0000}{\bar{r}} &
\color{#E619E6}{\bar{w}} \color{#00B050}{\bar{g}} &
\color{#E619E6}{\bar{w}} \color{#0080FF}{\bar{b}} \\
\hdashline
\color{#FF0000}{r} \color{#E619E6}{w} &
\color{#FF0000}{r} \color{#FF0000}{\bar{r}} &
\color{#FF0000}{r} \color{#00B050}{\bar{g}} &
\color{#FF0000}{r} \color{#0080FF}{\bar{b}} \\
\color{#00B050}{g} \color{#E619E6}{w} &
\color{#00B050}{g} \color{#FF0000}{\bar{r}} &
\color{#00B050}{g} \color{#00B050}{\bar{g}} &
\color{#00B050}{g} \color{#0080FF}{\bar{b}} \\
\color{#0080FF}{b} \color{#E619E6}{w} &
\color{#0080FF}{b} \color{#FF0000}{\bar{r}} &
\color{#0080FF}{b} \color{#00B050}{\bar{g}} &
\color{#0080FF}{b} \color{#0080FF}{\bar{b}}
\end{array}
\right]
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 but had \HH complexity, a
point when electrons and quarks were unified just like a ripple in a pond doesn't
exist before a stone is thrown in, there is no concept of before time or negative
radius. Expanding spacetime is when the trough of T^{0} and peaks of
S^{\alpha}, S^{\gamma} and S^{\beta}, S^{\delta}
are increasing in separation which would have the effect of driving matter and
antimatter apart, as well as forcing electromagnetism and strong forces apart and giving
us the concept of Higgs mechanism. This greater this separation relating to the 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. 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 3(a) you can imagine this
rotation as the wave moving in the various combinations of
S^{\alpha}, S^{\gamma} and S^{\beta}, S^{\delta}
seen as phase shifting. If there is no movement then we could classify the universe as
being 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. 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 90° to another universe,
likewise matter entering our universe from a black hole or vacuum 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 3(a) 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} :\rightarrow \color{#E619E6}{w} \color{#E619E6}{\bar{w}} .
- e^{-}_{R}, e^{+}_{L} :\rightarrow \color{#E619E6}{\bar{w}} \color{#E619E6}{w} .
- q_{L}, \bar{q}_{R} :\rightarrow \color{#FF0000}{r} \color{#FF0000}{\bar{r}}, \color{#00B050}{g} \color{#00B050}{\bar{g}}, \color{#0080FF}{b} \color{#0080FF}{\bar{b}} .
- q_{R}, \bar{q}_{L} :\rightarrow \color{#FF0000}{\bar{r}} \color{#FF0000}{r}, \color{#00B050}{\bar{g}} \color{#00B050}{g}, \color{#0080FF}{\bar{b}} \color{#0080FF}{b} .
- H^0 :\rightarrow light-like \pm \infin (with \HH complexity).
You may also see that when time T^{0} is created we must also have the
separation of matter and antimatter spaces S^{\alpha}, S^{\gamma} and
S^{\beta}, S^{\delta}. We can also say this is the point where
the Higgs field H^0 centred around T^{0} became none
zero with \HH complexity and is increasing, while H^0
centred around T^{\alpha}, T^{\gamma} and
T^{\beta}, T^{\delta} must be decreasing. Think of this as ever decaying
number of possible future events, but ever expanding number of past events. Very similar
to how we would see the flow of time in an hour glass, with each particle of sand
representing a unique event in spacetime.
So in our example, the Higgs field H^0 that defines the seperation of
all the matter and light we can see in our universe, must be itself based on
\HH complexity number with \RR portion giving us electrons,
electromagnetism and the light we see, and \II parts the quarks,
strong force we see within the nucleus. It is both source of all light, and seperation of
mass (aka Higgs effect). The secondary decaying negative Higgs field as described above
along with the incresing T^0 volume is most likely what we call dark energy,
and related to the collection of all past events (T^0 volume) no matter if
they were light-like, space-like or time-like as well as the reduction in volume of
T^{\alpha}, T^{\gamma} and T^{\beta}, T^{\delta} representing
the total life left and all possible future events.
So as T^0 volume is increasing to the positive, we see entropy and flow of
time as positive, eventually it will get to a point where it either stops or reverses and
entropy and flow of time will similarly either stop pr turn negative. Giving us not only
the possability of heat death but also cyclical universe, it also gives us a connection
to a continuous wave like multiverse of universes as well, where each universe relates to
90° for its expansion phase and 90° for its collapsing phase, making expansion, entropy
and the overal flow of time cyclical in nature as well.
We can now also define four categories of matter:
- Matter: (0°) Common positive spacetime (positive time and space).
- Dark Matter: (90°) Partially rotated spacetime (negative time, positive space).
- Antimatter: (180°) Common negative spacetime (negative time and space).
- Dark Energy: (270°) Partially rotated spacetime (positive time, negative space).
Mesons being the only solution using \CC^2 (2D) complexity, atoms of
one colour electron and three coloured quarks the only solution using \HH^2
(4D) complexity. 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 (null vectors). 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. As you can see, we are
separated from antimatter by both dark matter and energy.
It is also important to note that while S^{\alpha}, S^{\gamma} and
S^{\beta}, S^{\delta} are all internally connected they do not wrap
around but instead connect to a different universe (as defined by its own central
time T^{\alpha}, T^{\gamma} and T^{\beta}, T^{\delta},
are better viewed as universes that are seperated by unique H^0 fields
so as we must define the overall multiverse as continuous with no hard edges.
This is the wave-like nature mentioned previously. We must also not that while the inner
H^0 has \HH complexity, the outer H^0
field does not have to be the same, it could have \OO or
\SS complexity and we only see its 4D shadow interacting with
ours, think of it as 8D or 16D universe, with one or more 4D bubbles of which we
are in one.
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 back to the spatial and temporal axis to change what
would be units of momentum to units of energy. 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 4(a).
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:
- \partial_{0}^{2} = \frac{1}{c^2} \frac{\partial^2}{\partial t^2}
- \partial_{j}^{2} = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} = \nabla^2
- \gamma_{\mu} \gamma^{\mu} = I_4
Special Relativity
We now need to show the four rotational copies of the Minkowski metric which
must be used together, in our case we have the following:
Particle: \color{#FF0000}{q}, \color{#FF0000}{\bar{q}}
Vector: [-\hat{i}t^1, -x_1, +\hat{k}y_1, -\hat{j}z_1]
Metric: [-,+,-,-], [+,-,+,+]
Vector: [-\hat{i}t^1, -x_1, +\hat{k}y_1, -\hat{j}z_1]
Metric: [-,+,-,-], [+,-,+,+]
Particle: \color{#00B050}{q}, \color{#00B050}{\bar{q}}
Vector: [-\hat{j}t^2, -\hat{k}x_2, -y_2, +\hat{i}z_2]
Metric: [-,-,+,-], [+,+,-,+]
Vector: [-\hat{j}t^2, -\hat{k}x_2, -y_2, +\hat{i}z_2]
Metric: [-,-,+,-], [+,+,-,+]
Particle: \color{#0080FF}{q}, \color{#0080FF}{\bar{q}}
Vector: [-\hat{k}t^3, +\hat{j}x_3, -\hat{i}y_3, -z_3]
Metric: [-,-,-,+], [+,+,+,-]
Vector: [-\hat{k}t^3, +\hat{j}x_3, -\hat{i}y_3, -z_3]
Metric: [-,-,-,+], [+,+,+,-]
Particle: \color{#E619E6}{e}, \color{#E619E6}{\bar{e}}
Vector: [+t_0, +\hat{i}x^0, +\hat{j}y^0, +\hat{k}z^0]
Metric: [+,-,-,-], [-,+,+,+]
Vector: [+t_0, +\hat{i}x^0, +\hat{j}y^0, +\hat{k}z^0]
Metric: [+,-,-,-], [-,+,+,+]
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc}
\color{#E619E6}{+t_0} &
\color{#E619E6}{+\hat{i}x^0} &
\color{#E619E6}{+\hat{j}y^0} &
\color{#E619E6}{+\hat{k}z^0} \\
\hdashline
\color{#FF0000}{-\hat{i}t^1} &
\color{#FF0000}{-x_1} &
\color{#FF0000}{+\hat{k}y_1} &
\color{#FF0000}{-\hat{j}z_1} \\
\color{#00B050}{-\hat{j}t^2} &
\color{#00B050}{-\hat{k}x_2} &
\color{#00B050}{-y_2} &
\color{#00B050}{+\hat{i}z_2} \\
\color{#0080FF}{-\hat{k}t^3} &
\color{#0080FF}{+\hat{j}x_3} &
\color{#0080FF}{-\hat{i}y_3} &
\color{#0080FF}{-z_3}
\end{array}
\right]
=
\left[
\def
\arraystretch{1.2}
\begin{array}{cccc}
\color{#E619E6}{\bar{e}} \\
\hdashline
\color{#FF0000}{q} \\
\color{#00B050}{q} \\
\color{#0080FF}{q}
\end{array}
\right]
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc}
\color{#E619E6}{+t^0} &
\color{#FF0000}{+\hat{i}t_1} &
\color{#00B050}{+\hat{j}t_2} &
\color{#0080FF}{+\hat{k}t_3} \\
\hdashline
\color{#E619E6}{-\hat{i}x_0} &
\color{#FF0000}{-x^1} &
\color{#00B050}{+\hat{k}x^2} &
\color{#0080FF}{-\hat{j}x^3} \\
\color{#E619E6}{-\hat{j}y_0} &
\color{#FF0000}{-\hat{k}y^1} &
\color{#00B050}{-y^2} &
\color{#0080FF}{+\hat{i}y^3} \\
\color{#E619E6}{-\hat{k}z_0} &
\color{#FF0000}{+\hat{j}z^1} &
\color{#00B050}{-\hat{i}z^2} &
\color{#0080FF}{-z^3}
\end{array}
\right]
=
\left[
\def
\arraystretch{1.2}
\begin{array}{c:ccc}
\color{#E619E6}{e} &
\color{#FF0000}{\bar{q}} &
\color{#00B050}{\bar{q}} &
\color{#0080FF}{\bar{q}}
\end{array}
\right]
The reasoning to show the two lines above is to highlight the two light-like
barriers, in the first matrix, the vertical line denotes the \RR
\pm \infin that gives us \gamma and the colours
\color{#E619E6}{w}, \color{#FF0000}{r}, \color{#00B050}{g},
\color{#0080FF}{b}, acting as the barrier between time and space and
therefor the light we see. While the horizontal like denotes the \II
\pm \infin that gives us the colours \color{#E619E6}{\bar{w}},
\color{#FF0000}{\bar{r}}, \color{#00B050}{\bar{g}}, \color{#0080FF}{\bar{b}}
acting as the barrier between particles (E.g. the gap between the electron cloud
and nucleus), these are the virtual bosons we can't see. For the conjugate matrix
form, these two barriers are reversed with \RR bosons being the
horizontal line, and \II virtual bosons the vertical line.
However one thing to note, is the diagonal of both matricies are always positive
and \RR, for this reason the electron energy (time component) is
commutative allowing free electrons, but the momentum (space component) is
non-commutative giving us the variation of orbitals, greater the momentum,
the more complex the orbital paths and the more electrons that can fill that
shell, no different than higher harmonics. However the opposite is true for
quarks, as the energy is non-commutative which is why they can never be free,
and just like the momentum of the electrons, the energy levels of quarks will
show harmonic structure too. The momentum however is commutative and free to
move about, so colour swapping of quarks is no similar to the photoelectric
effect of electrons, only instead of changing energy levels as in the case of
electrons, quarks move location by swappping colour charges. So everything we
have learned for electromagnetism can be used for quarks, only we need to change
energy and momentum terms around, so colour charge is more like magnetism and
colour momentum is more like electric charge. Thus showing both the duality and
universality of the particles and forces that make up atoms.
The spatial and temporal components (momentum and energy) of all particles are
seperated by these two light-like inner barriers, regardless if they are matter
or antimatter, and furthermore by two light-like \pm \infin
barriers that define begining and end of causal connection and what ultimately
seperates matter and antimatter. Thus showing the walled garden each particle
must play within. With the energy and momentum of both electron and quarks
being unique, but also interconnected in very specific ways giving us the
properties we see. So we can either look at the universe in terms of spacetime
as with walls of 0, \pm \infin space and time to denote here, now (0)
and the \infin future, past and distance, or as \RR
and \II light-like walls having space and time dimensions as the
diagonal lines and light as the verical and horizontal axis. Thus concepts
such as positive curvature can be imagined as greater \II barrier
between electrons and quarks giving us the decay of free neutrons, and greater
\RR as the production and stretching of light. Thus giving us
the properties of fusion and fission as well as effects of gravity we see.
However if the curvature was negative, the opposite happens. Light is blue
shifted and absorbed, neutrons would gain stability as the barrier between
electrons and nucleons would also decrease and fission of heavier elements
would cease. Thus light emiting stars show areas of net positive curvature,
neutron stars and black holes show net negative curvature. The extreme being black
holes where the light-like lengths turn negative and time and space, \RR
and \II light swap places, so spacetime becomes timespace and
the visible light of black holes is trapped within - which if you were inside
the black hole would look exactly the same as we see the cosmic microwave
background. The event horizon of the black hole being the T^0
moment of causal connection for all matter and light within it. Oddly this also
will change electrons to quarks and quarks to electrons as spacetime is literally
turned inside out. This folding of spacetime will create higher complexity on
the inside of black holes (8D, 16D, etc) and lower complexity outside. So our
universe is inside a 4D black hole, with 8D black holes inside it, and 2D black
hole outside, it is the 2D black hole that gives us the duality of energy and momentum
and why all fields within our universe have 2D base structure.
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, shown above
as when the electron is contravariant vector, the quarks must always be covariant
vectors no matter if we are talking about matter or antimatter and this is the
case for both the time-like energy of the particles and space-like momentum.
The \II light-like 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. Charge seperation being the \II,
\RR light-like linear gradient aka massless transfer, and
gravity being where the gradient is curved, so accelerating it. Think of
the graviton as \RR gravity, being where space is more compressed
than time, where the tachyon is \II gravity where time is more
compressed than space.
-
\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°, negative anti de-Sitter curvature decreasing the angles to less than
180° and flat Minkowski spacetime being the special case where the angle comes
to exactly 180°. 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
\gamma, gluon g, W^{\pm} and
Z^0, So too general relativity should have tachyon \Tau
being \delta{^2}_t \gt \delta{^2}_s, graviton G as
\delta{^2}_s > \delta{^2}_t, the acceleration equivalents of
\gamma, g and V^{\pm} and Y^0 as the
acceleration equivalents 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.
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 shown as four corners of Figure 3(a). This 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 6(a) to show a rotating universe, expansion would be
shown as the wave reducing in frequency. However looking at the rotating view
from a stationary point, it may also 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). Heat death senario would look like the wave
form stretching until it reaches 0 Hz (aka longitudinal stretch), cyclical would
show the overal phase shift as shown in Figure 6(a).
Figure 6(a): General Relativity
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.
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 x 4 tensor array.
+ 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.
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 :\rightarrow1D + 1T (Meson, Particle Pairs).
-
\HH :\rightarrow3D + 1T (Atom, Baryon)
-
\OO :\rightarrow7D + 1T (Beyond Standard Model)
-
\SS :\rightarrow15D + 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 leptons and quarks are equal, as dark matter is
connected spatially it may have greater pull to either leptons, 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(a) 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.