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 (D_A , D_B ) as follows;

c²∂T² =  D_A²  - D_B²

Dimensional intervals have sub-intervals as follows;

D_A²  = D₃₄² - D₄₃² 

D_B²  = 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 – asin²_θ∂φ)²  - (p²/∆)∂r² - p²∂θ² - (sin²_θ/p²)([r²+a²]∂φ - ac∂t)²  

The geometry is;          a = J/mc

                                        r² = x² + y² + z²

                                        r_s = 2Gm/c²

r_Q² = Q²G/4πε₀c⁴

p² = r² + a² cos²_θ  

∆ = r² + a² - rr_s + r_Q²   

Where;                                   J is angular momentum

                                                m is dynamic mass

                                                Q is dynamic charge

                                                r_s is the Schwarzschild radius

        ε₀ is electric permittivity

       r_Q is an electric length scale

                                                c is the light constant

                                                G is the gravitational constant

Schwarzschild Forces;

The Schwarzschild force magnitudes (F_G , F_c , F₀) give the definition of the Schwarzschild radius (r_s), Plank force (F_P), the Einstien tensor (G_uv), and the stress-energy tensor (T_uv). These forces are related as;

                                                                F_G² = F₀F_c

The forces are defined as;                 F_G = mG^½/r_s

                                                                F_c = ½mc²/r_s

The magnitude of a unit vector (F₀) is;    F₀ = 1  (see Plank Units)

Substitution of force definitions gives the Schwarzschild radius;                 r_s = 2Gm/c²

The Schwarzschild forces are related to the Plank force as follows;           ¼F_PF_G² = F_c²F₀

Giving;                                                  F_P = c⁴/G

A magnitude of the Einstein tensor may be defined by forces;                   F₀F₁ = F_G²

Where;                                                 F₁ = |G_uv|/2πr_s

                                                                |G_uv| = 2πr_sF_G²/F₀

A magnitude of the stress-energy tensor may also be defined by forces;              F₀F₂ = F_c²

Where;                                                 F₂ = |T_uv|/r_s

                                                                |T_uv| = r_sF_c²/F₀

Giving a tensor ratio;                      |G_uv|/|T_uv| = G_uv/T_uv = 8πG/c⁴   

Energies;

The four energies (E₁ , E₂ , E_c , E_G) associated with Schwarzschild forces are defined as follows;

E₁ = |½G_uv/π|

E₂ = |T_uv|

E_c = ½mc² 

E_G = mv_G²  = mG^½ 

Velocity Interval;

A gravitational velocity (v_G) is defined as;              v_G = G^¼  

Where G is the gravitational constant.

The light constant (c) is assumed to be an invariant velocity. If a gravitational velocity (v_G) is also assumed to be an invariant velocity, then the invariant interval (∆v) is;

                                                                                                ∆v = c - G^¼  

Hadamard Interaction;

Two matrices (M_a , M_b) are the same size (m_xn) and have corresponding cells (a_ij , b_ij). They interact “directly” if corresponding cells interact exclusively. This rule is obvious for addition and subtraction of matrices.

The Hadamard product (◦) of two matrices (M_a◦M_b) is the product of all corresponding cells;  a_ijb_ij 

The Hadamard ratio (◦/◦) of two matrices (M_a◦/◦M_b) is the ratio of all corresponding cells;  a_ij/b_ij 

Frequency is represented as a Hadamard ratio of velocity and wavelength;  M_f = M_v◦/◦M_λ    

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; f_1j = v_1j/λ_1j

f_2j = v_2j/λ_2j

f_3j = v_3j/λ_3j

f_4j = v_4j/λ_4j

The frequency matrix (M_f) is;     M_f = M_v◦/◦M_λ

The average frequency (f_AJ) of any wave packet is the geometric mean of four frequencies;

                                                                f_AJ = (f_1jf_2jf_3jf_4j)^¼                  

The average temperature (T_AJ) of any packet is; T_AJ = hf_AJ/k

Where; h is the Plank constant

                k is the Boltzmann constant

The total energy (E_TJ) of any wave-packet is;       E_TJ = σT_AJ⁴ = σ(hf_AJ/k)⁴  

Where; σ is the Stephan-Boltzmann constant

The force of stress (F_j) associated with a wave-packet is;  F_j = E_TJ/r_s    

Where; r_s is the Schwarzschild radius

Giving;                                                                  F_j = (f_1jf_2jf_3jf_4j)(σh⁴/r_sk⁴)  

Fundamental Units;

If; f_ij = 1                 Then; F_j = F₀

 A fundamental force (F₀) is;                       F₀ = σh⁴/r_sk⁴ 

A fundamental energy (E₀) is;                     E₀ = σh⁴/k⁴ 

Fundamental frequencies;                          f_c = σh³/k⁴ 

       f_G = (c³/ħG^½)^½ 

Fundamental wavelengths;                         λ_c = ck⁴/σh³   

                                                                          λ_G = (ħG/c³)^½    (Plank Units)

Light velocity (v_C) is;                                        v_C = c

Gravitational velocity (v_G) is;                       v_G = G^¼ 

Mass (normal):                                                 m_c = σ(hf_AJ)⁴/c²k⁴ 

Mass (dark):                                                       m_G = σ(hf_AJ)⁴/G^½k⁴

Energy (normal);                                              E_c = mc²

Energy (dark);                                                   E_G = mv_G² = mG^½ 

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 , ∂λ_c/∂T , ∂λ₄₃/∂t  , ∂λ₃₄/∂t

v_G , ∂λ_c/∂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;                M_v = 

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;                λ_c , p , b , q 

A wavelength matrix (M_λ) is;                      M_λ =

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 (M_f) is;                         M_f = M_v◦/◦M_λ    

                                                                                M_f =

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 =  D_A²  - D_B²

Dimensional components have sub-components as follows;

D_A²  = D₃₄² - D₄₃²   

D_B²  = 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 D_A (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 – asin²_θ∂φ)² 

D₄₃  = (b/p)∂λ₄₃   

∂λ₄₃/∂t  = (R²/a)∂φ/∂t – c  

D₄₃²  = (b/p)²(R²∂φ/a – c∂t)²  =  (sin²_θ/p²)(R²∂φ – ac∂t)² 

Giving;                           D_A²  = (q²/p²)(c∂t – asin²_θ∂φ)²  - (sin²_θ/p²)(R²∂φ – ac∂t)² 

Where;                          b = asin_θ

R² = r² + a²

a = J/mc

                                         J is angular momentum for spin

The sub-components for D_B (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 – a₂sin²_β∂θ)² 

Where;                          b₂ = a₂sin_β

u₂ = psin_β

R₂² = p² + a₂²

∂r  = v∂t 

a₂ = J₂/mv 

                                         J₂ is angular momentum for rotation

Giving;                             D_B²  =  (R₂²∂θ/p – a₂∂r/p)²  + (p²/q²)(∂r – a₂sin²_β∂θ)² 

Packet Forces;

A packet stress force is;                F_j = (σh⁴/r_sk⁴)(f_1jf_2jf_3jf_4j)  

                   F_j = F₀(v_1j/λ_1j)(v_2j/λ_2j)(v_3j/λ_3j)(v_4j/λ_4j)

The packet forces are;

       F₁ = F₀(c/λ_c)(∂λ ₂₁/b∂t)(v_G/q)(∂λ_c/q∂T)

                                                F₂ = F₀(∂λ₁₂/b∂t)(c/λ_c)(∂λ_c/p∂T)(v_G/q)

                                                F₃ = F₀(v_G/p)(∂λ_c/q∂T)(c/λ_c)(∂λ₄₃/p∂t)

                                                F₄ = F₀(∂λ_c/p∂T)(v_G/p)(∂λ₃₄/b∂t)(c/λ_c)

                                                F₅ = F₀(c/λ_c)(∂λ_c/b∂t)(v_G/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  = D_A² - D_B² 

Kerr-Newman Metric;

The K-N metric includes spin but not rotation, therefore;   a₂ = 0 and R₂ = p

D_A²  = (q²/p²)(c∂t – asin²_θ∂φ)²  - (sin²_θ/p²)(R²∂φ – ac∂t)² 

D_B²  =  p²∂θ²  + (p²/q²)∂r²

                                        c²∂T²  = D_A²  - D_B² 

Where;                           R² =  r²+a²

q² =  ∆

Giving the K-N metric;  

       c²∂T² =  (∆/p²)(c∂t – asin²_θ∂φ)² - (sin²_θ/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” (x_n , y_n , z_n). Where; ‘n’ is the tetrahedron identifier.

The longest edge (R_n) is;  R_n² = x_n² + y_n² + z_n²

Also;  w_n = u_n + z_n    

The tetrahedron encloses a spatial volume (V_n) with four triangular plane surfaces (faces). 

The volume is;   V_n = x_ny_nz_n/6 

Each plane surface may be represented by three edge lengths (vector magnitudes). The four faces are;

                                (R_n,P_n,z_n) , (P_n,x_n,y_n) , (Q_n,y_n,z_n) , (R_n,x_n,Q_n)  

The edge lengths of each surface are related as follows;

R_n² = P_n² + z_n²

P_n² = x_n² + y_n²

Q_n² = y_n² + z_n²

R_n² = x_n² + Q_n²

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 = asin_θ 

k = Rsin_θ 

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.