Wave Packets
A
wave has characteristics of frequency, wavelength (colour), or velocity. Velocity
is represented as a Hadamard product of wavelength and frequency. Four fundamental
wave-functions are assumed to “entangle” to form a wave packet. The
wave-lengths associated with a packet may be represented as edge-lengths of a
tetrahedron. Our perception of space-time is associated with fundamental waves.
Each wave-packet has an average frequency, and temperature, a total energy and
stress force. The stress forces of five packets are related as a force-field.
The metric solutions of a force-field equation depend upon twenty definitions
of frequency. Various metrics (Minkowski, Schwarzschild, Kerr, Reissner-Nordstrom,
etc.) are subsets of the Kerr-Newman (K-N) metric. The K-N metric may be
represented as a “tetradedral metric”.
Space-Time Interval;
Dimensions
(polar co-ordinates); r, θ, φ, t
Dimensions
(Cartesian co-ordinates); x, y, z, t
The
space-time interval (c∂T) is related to dimensional intervals (DA , DB )
as follows;
c2∂T2 = DA2 - DB2
Dimensional
intervals have sub-intervals as follows;
DA2 = D342 - D432
DB2 = D122 + D212
Dimensional
sub-intervals are related to fundamental waves.
The Kerr-Newman Metric;
The Kerr-Newman metric defines the geometry of space-time
near a homogenously massive, charged, rotating object. The metric is;
c2∂T2
= (∆/p2)(c∂t – asin2θ∂φ)2 - (p2/∆)∂r2 - p2∂θ2
- (sin2θ/p2)([r2+a2]∂φ - ac∂t)2
The geometry is; a
= J/mc
r2 =
x2 + y2 + z2
rs =
2Gm/c2
rQ2
= Q2G/4πε0c4
p2
= r2 + a2 cos2θ
∆ = r2
+ a2 - rrs + rQ2
Where; J
is angular momentum
m is dynamic
mass
Q is dynamic
charge
rs is
the Schwarzschild radius
ε0
is electric permittivity
rQ is
an electric length scale
c is the light
constant
G is the
gravitational constant
Schwarzschild Forces;
The Schwarzschild force magnitudes (FG , Fc
, F0) give the definition of the Schwarzschild radius (rs),
Plank force (FP), the Einstien tensor (Guv), and the
stress-energy tensor (Tuv). These forces are related as;
FG2 = F0Fc
The forces are defined as; FG
= mG½/rs
Fc
= ½mc2/rs
The magnitude of a unit vector (F0) is; F0 = 1 (see Plank Units)
Substitution of force definitions gives the Schwarzschild
radius; rs
= 2Gm/c2
The Schwarzschild forces are related to the Plank force as
follows; ¼FPFG2
= Fc2F0
Giving; FP
= c4/G
A magnitude of the Einstein tensor may be defined by forces; F0F1 = FG2
Where; F1
= |Guv|/2πrs
|Guv|
= 2πrsFG2/F0
A magnitude of the stress-energy tensor may also be defined by
forces; F0F2
= Fc2
Where; F2
= |Tuv|/rs
|Tuv|
= rsFc2/F0
Giving a tensor ratio; |Guv|/|Tuv|
= Guv/Tuv = 8πG/c4
Energies;
E1 = |½Guv/π|
E2 =
|Tuv|
Ec =
½mc2
EG =
mvG2 = mG½
Velocity Interval;
A gravitational velocity (vG) is defined as; vG = G¼
Where G is the gravitational constant.
The light constant (c) is assumed to be an invariant
velocity. If a gravitational velocity (vG) is also assumed to be an
invariant velocity, then the invariant interval (∆v) is;
∆v
= c - G¼
Hadamard Interaction;
Two
matrices (Ma , Mb) are the same size (mxn) and
have corresponding cells (aij
, bij). They interact
“directly” if corresponding cells interact exclusively. This rule is obvious
for addition and subtraction of matrices.
The Hadamard
product (◦)
of two matrices (Ma◦Mb) is the product of all
corresponding cells; aijbij
The Hadamard
ratio (◦/◦)
of two matrices (Ma◦/◦Mb) is the ratio of all
corresponding cells; aij/bij
Frequency
is represented as a Hadamard ratio of velocity and wavelength; Mf = Mv◦/◦Mλ
Wave-Packets;
Waves
are assumed to combine (entangle) to form a wave packet. There are four waves in each packet. Five
wave packets (20 waves) are required to define a force field. A set of wave
characteristics (velocity, wavelength, frequency) is represented as a 4x5
matrix. Each cell represents a wave characteristic, and each column represents
a wave packet.
Where
‘j’ is the packet identifier (Matrix column identifier); j = 1,2,3,4,5
For
any wave-packet (j) the four wave-frequencies are; f1j = v1j/λ1j
f2j = v2j/λ2j
f3j = v3j/λ3j
f4j = v4j/λ4j
The
frequency matrix (Mf) is; Mf
= Mv◦/◦Mλ
The
average frequency (fAJ) of any wave packet is the geometric mean of four
frequencies;
fAJ =
(f1jf2jf3jf4j)¼
The
average temperature (TAJ) of any packet is; TAJ = hfAJ/k
Where; h is the Plank constant
k is the Boltzmann constant
The total energy (ETJ) of any wave-packet is; ETJ = σTAJ4 = σ(hfAJ/k)4
Where;
σ
is the Stephan-Boltzmann constant
The
force of stress (Fj) associated with a wave-packet is; Fj = ETJ/rs
Where;
rs is the Schwarzschild radius
Giving; Fj
= (f1jf2jf3jf4j)(σh4/rsk4)
Fundamental Units;
If;
fij = 1 Then; Fj = F0
A fundamental force (F0) is; F0
= σh4/rsk4
A
fundamental energy (E0) is; E0 = σh4/k4
Fundamental
frequencies; fc = σh3/k4
fG = (c3/ħG½)½
Fundamental
wavelengths; λc = ck4/σh3
λG = (ħG/c3)½
(Plank
Units)
Light velocity (vC) is; vC = c
Gravitational
velocity (vG) is; vG
= G¼
Mass (normal): mc
= σ(hfAJ)4/c2k4
Mass
(dark): mG
= σ(hfAJ)4/G½k4
Energy
(normal); Ec = mc2
Energy
(dark); EG = mvG2
= mG½
Field Equation;
A force-field
equation relates the forces of each wave packet;
F52 = F42
– F32 - F22 – F12
F12 + F22
+ F32 = F62
F42 = F52
+ F62
Fundamental Waves;
Frequencies
are represented as operators; f1
= ∂/∂t
, f2 = ∂/∂T
Where; t,T represent
time frames.
There
is assumed to be eight types of fundamental velocities;
c , ∂λc/∂T , ∂λ43/∂t , ∂λ34/∂t
vG , ∂λc/∂t , ∂λ21/∂t
, ∂λ12/∂t
The fundamental velocities are distributed (populated) over
the field table forming a velocity matrix. A matrix of fundamental velocity is;
Mv
=
|
c
|
∂λ12/∂t
|
vG
|
∂λc/∂T
|
c
|
|
∂λ21/∂t
|
c
|
∂λc/∂T
|
vG
|
∂λc/∂t
|
|
vG
|
∂λc/∂T
|
c
|
∂λ34/∂t
|
vG
|
|
∂λc/∂T
|
vG
|
∂λ43/∂t
|
c
|
c
|
The various types of velocity form diagonals within the
velocity table.
There is assumed to be four types of fundamental wavelength; λc
, p , b , q
A wavelength matrix (Mλ) is; Mλ
=
|
λc
|
b
|
p
|
p
|
λc
|
|
b
|
λc
|
q
|
p
|
b
|
|
q
|
p
|
λc
|
b
|
p
|
|
q
|
q
|
p
|
λc
|
q
|
Various types of wavelengths form parallel diagonals within
the wavelength table. The spatial dimensions (r , θ , φ) may be expressed as functions
of the fundamental wavelengths.
A frequency matrix (Mf) is; Mf
= Mv◦/◦Mλ
Mf
=
|
c/λc
|
∂λ12/b∂t
|
vG/p
|
∂λc/p∂T
|
c/λc
|
|
∂λ21/b∂t
|
c/λc
|
∂λc/q∂T
|
vG/p
|
∂λc/b∂t
|
|
vG/q
|
∂λc/p∂T
|
c/λc
|
∂λ34/b∂t
|
vG/p
|
|
∂λc/q∂T
|
vG/q
|
∂λ43/p∂t
|
c/λc
|
c/q
|
Each cell
of the frequency matrix represents a wave. Commonality is represented as a
diagonal within the matrix.
Dimensional Components;
The
space-time interval (c∂T) is; c2∂T2
= DA2 - DB2
Dimensional
components have sub-components as follows;
DA2 = D342 - D432
DB2 = D122 + D212
The
dimensional sub-components are also associated with matter (mass), electric
charge, spin, and rotation. Rotation may include orbital rotation.
The
sub-components for DA (functions of φ,c,a) are associated with spin
and are defined as follows;
D34 = (q/p)∂λ34
∂λ34/∂t = c – (b2/a)∂φ/∂t
D342 = (q/p)2(c∂t – b2∂φ/a)2
= (q2/p2)(c∂t – asin2θ∂φ)2
D43 = (b/p)∂λ43
∂λ43/∂t = (R2/a)∂φ/∂t – c
D432 = (b/p)2(R2∂φ/a –
c∂t)2 = (sin2θ/p2)(R2∂φ – ac∂t)2
Giving; DA2 = (q2/p2)(c∂t – asin2θ∂φ)2 - (sin2θ/p2)(R2∂φ – ac∂t)2
Where; b = asinθ
R2 = r2 + a2
a = J/mc
J
is angular momentum for spin
The
sub-components for DB (functions of θ,v,a2) are
associated with rotation and are defined as follows;
D12 = ∂λ12
= (b2/u2)∂λ10
∂λ10/∂t = (R22/a2)∂θ/∂t
– v
D122 = (b2/u2)2(R22∂θ/a2
– v∂t)2 = (R22∂θ/p
– a2∂r/p)2
D21 = (p/q)∂λ21
∂λ21/∂t = v – (b22/a2)∂θ/∂t
D212 =
(p/q)2(v∂t – b22∂θ/a2)2 = (p2/q2)(∂r – a2sin2β∂θ)2
Where; b2 =
a2sinβ
u2 = psinβ
R22 = p2 + a22
∂r
= v∂t
a2
= J2/mv
J2
is angular momentum for rotation
Giving; DB2 = (R22∂θ/p – a2∂r/p)2 + (p2/q2)(∂r – a2sin2β∂θ)2
Packet Forces;
A packet stress force is; Fj
= (σh4/rsk4)(f1jf2jf3jf4j)
Fj = F0(v1j/λ1j)(v2j/λ2j)(v3j/λ3j)(v4j/λ4j)
The packet forces are;
F1
= F0(c/λc)(∂λ 21/b∂t)(vG/q)(∂λc/q∂T)
F2
= F0(∂λ12/b∂t)(c/λc)(∂λc/p∂T)(vG/q)
F3
= F0(vG/p)(∂λc/q∂T)(c/λc)(∂λ43/p∂t)
F4
= F0(∂λc/p∂T)(vG/p)(∂λ34/b∂t)(c/λc)
F5
= F0(c/λc)(∂λc/b∂t)(vG/p)(c/q)
The Field Equation is; F52
= F42 – F32 - F22
– F12
Giving; c2∂T2 = (q/p)2∂λ342 - (b/p)2∂λ432 - ∂r122 - (p/q)2∂λ212
c2∂T2 = D342 - D432 - D122 - D212
c2∂T2 = DA2 - DB2
Kerr-Newman Metric;
The
K-N metric includes spin but not rotation, therefore; a2 = 0 and R2 = p
DA2 = (q2/p2)(c∂t – asin2θ∂φ)2 - (sin2θ/p2)(R2∂φ – ac∂t)2
DB2 = p2∂θ2 + (p2/q2)∂r2
c2∂T2 = DA2 - DB2
Where; R2
= r2+a2
q2
= ∆
Giving the K-N metric;
c2∂T2
= (∆/p2)(c∂t – asin2θ∂φ)2
- (sin2θ/p2)([r2+a2]∂φ - ac∂t)2 - p2∂θ2
- (p2/∆)∂r2
Dimensional Perception;
Our perception of spatial dimension (r, θ, φ) is
assumed to be based on fundamental waves. The “wave-structure” of φ is determined from two
velocities;
∂λ34/∂t = c – (b2/a)∂φ/∂t
∂λ43/∂t = (R2/a)∂φ/∂t – c
Removing
‘c’ gives; ∂φ = a(∂λ34 + ∂λ43)/(R2
- b2)
The “wave-structure”
of θ
is also determined from two velocities;
∂λ10/∂t = (R22/a2)∂θ/∂t
– v
∂λ21/∂t = v – (b22/a2)∂θ/∂t
Removing
‘v’ gives; ∂θ = a2(∂λ21
+ ∂λ10)/(R22
- b22)
The “wave-structure” of r is; ∂r(b2-2 - R2-2) = ∂λ21/b22
+ ∂λ10/R22
Tetrahedral Geometry;
A tetrahedron
(n) has orthogonal edges or “legs” (xn , yn , zn).
Where; ‘n’ is the tetrahedron identifier.
The
longest edge (Rn) is; Rn2
= xn2 + yn2 + zn2
Also; wn = un + zn
The
tetrahedron encloses a spatial volume (Vn) with four triangular
plane surfaces (faces).
The
volume is; Vn = xnynzn/6
Each
plane surface may be represented by three edge lengths (vector magnitudes). The
four faces are;
(Rn,Pn,zn)
, (Pn,xn,yn) , (Qn,yn,zn) ,
(Rn,xn,Qn)
The edge lengths of each surface are related as follows;
Rn2 = Pn2
+ zn2
Pn2 = xn2
+ yn2
Qn2 = yn2
+ zn2
Rn2 = xn2
+ Qn2
Tetrahedral Interaction;
Two
tetrahedra (n = 1,2) interact along one common edge. The interaction shall be
defined as; Q2 = R1
Giving; Q22
= R12
y22 + z22
= x12 + y12
+ z12
(w2-u2)2 - x12
- y12 - (w1-u1)2 = -y22
The
tetrahedral metric is; (∂w2
- ∂u2)2
- ∂x12
- ∂y12
- (∂u1
- ∂w1)2 = -∂y22
The tetrahedral metric is equivalent to the Kerr-Newman
metric if;
q2∂x12
= p2∂r2
∂y12 = p2∂θ2
-∂y22 = c2∂T2
p∂w1 = bc∂t
p∂w2 = qc∂t
p∂u1 = Rk∂φ
ap∂u2
= b2q∂φ
Where; q2 = ∆
R2 =
r2 + a2
b = asinθ
k = Rsinθ
Conclusion;
A
tensor may be represented by the geometry and interaction of tetrahedra. The
magnitude of a tensor may be represented as energy. Four wave functions are
assumed to entangle, forming a wave packet. Each wave packet is represented as
a column within a 4x5 frequency field matrix. A force field equation relates
the stress forces of all wave packets. Five wave packets are required to
represent a force field.
A
frequency matrix is the Hadamard ratio of a velocity matrix and a wavelength
matrix. The Kerr-Newman metric may be represented as a matrix of frequencies.
Our
perception of spatial dimension is based on fundamental waves.
