Tensorial Structure of the Inhomogeneous Field Equations (IE)

Dr. Myron W. Evans

9/9/2004

Introduction (HE & IE)

The tensorial structure of the IE is derived from the Bianchi identity of differential geometry

D⋀(T^𝒶)=(Subscript[R, 𝒷]^𝒶)⋀(𝓆^𝒷)

d⋀(T^𝒶)=(Subscript[R, 𝒷]^𝒶)⋀(𝓆^𝒷)-(Subscript[ω, 𝒷]^𝒶)⋀(T^𝒷)

d⋀(T^𝒶)=-(𝓆^𝒷)⋀(Subscript[R, 𝒷]^𝒶)-(Subscript[ω, 𝒷]^𝒶)⋀(T^𝒷)

\(d⋀T\_μν\%𝒶\)=-((𝓆^𝒷)⋀(Subscript[R, 𝒷μν]^𝒶)+(Subscript[ω, 𝒷]^𝒶)⋀\(T\_μν\%𝒷\))

(Subscript[A, μ]^𝒶)=(A^((0))) (Subscript[𝓆, μ]^𝒶)

\(F\_μν\%𝒶\)=\(\(A\^\((0)\)\) T\_μν\%𝒶\)

({{\(d⋀F\_μν\%𝒶\), =-(A^((0)))((𝓆^𝒷)⋀(Subscript[R, 𝒷μν]^𝒶)+(Subscript[ω, 𝒷]^𝒶)⋀\(T\_μν\%𝒷\))}, {, \(\(∼\)\(0\ \((experimentally)\)\)\)}})

\(\(F\&~\)\_ρσ\%𝒶\)=(1/2)|ℊ\( | \^\(1\/2\)\)\(\(ϵ\_ρσ\^μν\) 
              F\_μν\%𝒶\)

\(\(R\&~\)\_𝒷ρσ\%𝒶\)=(1/2)|ℊ\( | \^\(1\/2\)\)(Subscript[ϵ, ρ])(σ^μ)ν\(R\_𝒷μν\%𝒶\)

\(TraditionalForm\`1\/2 | ℊ | ϵ\_ρσ\^μν\)

\(d⋀\(F\&~\)\_μν\%𝒶\)=-(A^((0)))((𝓆^𝒷)⋀(Subscript[\(R\&~\), 𝒷μν]^𝒶)+(Subscript[ω, 𝒷]^𝒶)⋀\(\(T\&~\)\_μν\%𝒷\))

(J^𝒶)=-((A^((0)))/(Subscript[μ, 0]))((𝓆^𝒷)⋀(Subscript[\(R\&~\), 𝒷]^𝒶)+(Subscript[ω, 𝒷]^𝒶)⋀\(\(T\&~\)\^𝒷\))

\(d⋀\(F\&~\)\^𝒶\  = \(μ\_0\) 
            J\^𝒶\)

({{(∂_μ) (F^𝒶)μν, =-(A^((0)))((𝓆^𝒷)(Subscript[R, 𝒷]^(𝒶μν))+(Subscript[ω, 𝒷]^𝒶)(T^𝒷)μν)}, {, =(Subscript[μ, 0]) 
                      (J^𝒶)ν}})

The Coulomb Law (ν = 0)

The Coulomb Law is the unified field theory is given by ν = 0, and is:

(∂_1)(F^(𝒶10))+(∂_2)(F^(𝒶20))+(∂_3)(F^(𝒶30))=(Subscript[μ, 0]) (J^𝒶)0

(J^𝒶)0=-((A^((0)))/(Subscript[μ, 0]))((Subscript[𝓆, 1]^𝒷)(Subscript[R, 𝒷]^(𝒶10))+(Subscript[𝓆, 2]^𝒷)(Subscript[R, 𝒷]^(𝒶20))+(Subscript[𝓆, 3]^𝒷)(Subscript[R, 𝒷]^(𝒶30))+(Subscript[ω, 𝒷1]^𝒶)(T^𝒷)10+(Subscript[ω, 𝒷2]^𝒶)(T^𝒷)20+(Subscript[ω, 𝒷3]^𝒶)(T^𝒷)30)

∇•(E^𝒶)=-(ϕ^((0)))((Subscript[𝓆, 1]^𝒷)(Subscript[R, 𝒷]^(𝒶10))+(Subscript[𝓆, 2]^𝒷)(Subscript[R, 𝒷]^(𝒶20))+(Subscript[𝓆, 3]^𝒷)(Subscript[R, 𝒷]^(𝒶30))+(Subscript[ω, 𝒷1]^𝒶)(T^𝒷)10+(Subscript[ω, 𝒷2]^𝒶)(T^𝒷)20+(Subscript[ω, 𝒷3]^𝒶)(T^𝒷)30)

Notes on the Coulomb Law

It can be seen from the familar vector notation in eqn. () that the origin of the Coulomb Law is spacetime.  Using the distributive equations:

(T^𝒶)=d⋀(𝓆^𝒷)+(Subscript[ω, 𝒷]^𝒶)⋀(𝓆^𝒷)

(Subscript[R, 𝒷]^𝒶)=d⋀(Subscript[ω, 𝒷]^𝒶)+(Subscript[ω, 𝒸]^𝒶)⋀(Subscript[ω, 𝒷]^𝒸)

D⋀(Subscript[R, 𝒷]^𝒶)=0

Therefore eqn () to () are constants on the variables on the right hand side of eqn. ().  Einstein field theory of gravitation is given by the limit:

(T^𝒶)=0

D⋀(T^𝒶)=(Subscript[R, 𝒷]^𝒶)⋀(𝓆^𝒷)=0

(Subscript[𝓆, 1]^𝒷)(Subscript[R, 𝒷]^(𝒶10))+(Subscript[𝓆, 2]^𝒷)(Subscript[R, 𝒷]^(𝒶20))+(Subscript[𝓆, 3]^𝒷)(Subscript[R, 𝒷]^(𝒶30))=0

The electromagnetic field is torsion of spacetime (eqn.()) so from eqn. () there is no electro-magnetism in the Einstein or Newton theories of gravitation.  This is of course a self-consistent result.  However, it is seen further from eqns () and () that Einstein or Newtonian gravitation does not affect the Coulumb Law.

In order for gravitational forces to change the Coulomb Law the condition is:

(Subscript[𝓆, 1]^𝒷)(Subscript[R, 𝒷]^(𝒶10))+(Subscript[𝓆, 2]^𝒷)(Subscript[R, 𝒷]^(𝒶20))+(Subscript[𝓆, 3]^𝒷)(Subscript[R, 𝒷]^(𝒶30))≠0

(Subscript[R, 𝒷]^𝒶)⋀(𝓆^𝒷)≠0

(Subscript[R, μ])νρσ + 
        (Subscript[R, ρ])μνσ + (Subscript[R, ν])ρμσ ≠ 
      0

\(TraditionalForm\`Γ\_μν\%𝓀 ≠ Γ\_νμ\%𝓀\)

Eqn () means, self-consistently, that torsion tensor must be non-zero.

\(T\_μν\%𝓀\)=(Subscript[𝓆, 𝒶]^𝓀)(Subscript[T, μν]^𝒶)=\(Γ\_μν\%𝓀 - Γ\_νμ\%𝓀 ≠ 0\)

Therefore in order for gravitation to influence the Coulomb Law, gravitational torsion must be present experimentally.

This means that there must be spin present and forces such as Coriolis and Centripetal accelerations.  Centrally diverted gravitation does not affect the Coulomb Law.  This is in agreement with experimenttal data, a charge (Subscript[e, 1]) on (Subscript[𝓂, 1])interacting with charge (Subscript[e, 2]) on (Subscript[𝓂, 2])obey the Coulomb Law to very high precision.  Changing mass (Subscript[𝓂, 2]), keeping the other three variables ((Subscript[𝓂, 1]),(Subscript[e, 1]),(Subscript[e, 2])) the same, will not change the Coulomb Law.

However, if the masses (Subscript[𝓂, 1])and (Subscript[𝓂, 2])are not point masses, and spinning, the Coulomb Law is changed in general.

The Ampère Maxwell Law (ν = 1,2,3)

The Ampère Maxwell Law is:

(∂_0)(F^(𝒶01))+(∂_2)(F^(𝒶21))+(∂_3)(F^(𝒶31))=(Subscript[μ, 0]) 
          (J^𝒶)1 (ν = 1)

(∂_0)(F^(𝒶02))+(∂_1)(F^(𝒶12))+(∂_3)(F^(𝒶32))=(Subscript[μ, 0]) 
          (J^𝒶)2 (ν = 2)

(∂_0)(F^(𝒶03))+(∂_1)(F^(𝒶13))+(∂_2)(F^(𝒶21))=(Subscript[μ, 0]) 
          (J^𝒶)2 (ν = 3)

(J^𝒶)1=-((A^((0)))/(Subscript[μ, 0]))((Subscript[𝓆, 0]^𝒷)(Subscript[R, 𝒷]^(𝒶01))+(Subscript[𝓆, 2]^𝒷)(Subscript[R, 𝒷]^(𝒶21))+(Subscript[𝓆, 3]^𝒷)(Subscript[R, 𝒷]^(𝒶31))+(Subscript[ω, 𝒷0]^𝒶)(T^𝒷)01+(Subscript[ω, 𝒷2]^𝒶)(T^𝒷)21+(Subscript[ω, 𝒷3]^𝒶)(T^𝒷)31)

(J^𝒶)2=-((A^((0)))/(Subscript[μ, 0]))((Subscript[𝓆, 0]^𝒷)(Subscript[R, 𝒷]^(𝒶02))+(Subscript[𝓆, 1]^𝒷)(Subscript[R, 𝒷]^(𝒶12))+(Subscript[𝓆, 3]^𝒷)(Subscript[R, 𝒷]^(𝒶32))+(Subscript[ω, 𝒷0]^𝒶)(T^𝒷)02+(Subscript[ω, 𝒷1]^𝒶)(T^𝒷)12+(Subscript[ω, 𝒷3]^𝒶)(T^𝒷)32)

(J^𝒶)3=-((A^((0)))/(Subscript[μ, 0]))((Subscript[𝓆, 0]^𝒷)(Subscript[R, 𝒷]^(𝒶03))+(Subscript[𝓆, 1]^𝒷)(Subscript[R, 𝒷]^(𝒶13))+(Subscript[𝓆, 2]^𝒷)(Subscript[R, 𝒷]^(𝒶23))+(Subscript[ω, 𝒷0]^𝒶)(T^𝒷)03+(Subscript[ω, 𝒷1]^𝒶)(T^𝒷)13+(Subscript[ω, 𝒷2]^𝒶)(T^𝒷)23)

Notes on the Ampère Maxwell Law

It is seen from eqns () to () that electric current is spacetime.  The same consideration apply as to the Coulomb Law, because both laws are part of the IE.  Therefore gravitational torsion is needed to generate electric current and electric power.