Schwarzschild energy (E_S) applies to a massive field. It is related to the Schwarzschild radius (r_S) and to an associated Schwarzschild force (F_S). This energy appears in the literature (usually in partial form) and may be associated with; the Schwarzschild metric, Plank length, Plank energy, Newtonian gravitational force, and the Einstein Field Equation. An equivalent form of Schwarzschild energy applies to the electric field.

The Schwarzschild radius is;              r_S = 2mG/c² 

Where;                        m is the mass of an object

                                                                G is the gravitational constant

                                                                c is the light constant

An “object”(possessing mass) is assumed to cause stress to the local continuum. The object is assumed to be spherical and homogenous. At the Schwarzschild radius, the area (A_S) of a spherical region of space under stress is;                   

                                                                A_S = 4πr_S² 

The stress (σ_S) acting on the continuum at the Schwarzschild radius is;    σ_S = F_S/A_S 

Where;                 F_S is a force which is distributed over the stressed area; F_S = E_S/r_S

                                E_S is the Schwarzschild energy

Schwarzschild energy may be derived from Newton’s law of reciprocating forces. The force magnitude acting upon the continuum (F_S) at a distance (r_S) has a reciprocating force (F_R). According to Newton;

                                                           F_S + F_R = 0

Force may be represented as a vector. A unit vector of force (F₀) acting from the center of the object in any direction has a magnitude;  

     |F₀| = F₀ = 1  

The force magnitude acting on the object (F_c) is;  F_c = E_c/r_S

Where;   E_c is the stress energy acting on the object;  E_c = ½mc²  

Reciprocating force shall be;       F_R = (F₀F_c)^½  

According to Newton;                    F_S = - F_R

                                                           F_S² = F_R² = F₀F_c 

                                                           (E_S/r_S)² = (1)(E_c/r_S)

E_S² = r_SE_c  = (2mG/c²)(½mc²)  = m²G

Giving Schwarzschild energy;      E_S = mG^½  

Plank Length;

Schwarzschild energy is related to Plank Length (r_P) by a balance of forces.

F_S occurs at the Schwarzschild radius;     F_S = E_S/r_S

At the Plank radius (r_P) the force acting on the continuum (F_P) is;

                                                            F_P = ½E_S/r_P = (mG^½)/2r_P   

According to Newton;                    F_P = - F_R

                                                           F_P² = F_R² = F₀F_c 

Where;                                              F_c = E_c²/ħc

                                                            ħ = h/2π

                                                            E_c = ½mc²

Substitution gives;                           F_P² = F₀F_c 

(mG^½)²/4r_P²   = (1)(½mc²)²/ħc 

m²G/4r_P²   = m²c³/4ħ 

Giving Plank Length;                       r_P = (ħG/c³)^½ 

Plank Energy;

Plank energy may be obtained from a balance of forces;       F_PF_S = F₀F_C

Substitution gives;                    (E_P²/4ħc)(E_S²/ħc) = (1)(½mc²)²/ħc

E_P²E_S² = ħm²c⁵

E_P = (ħc⁵/G)^½ 

Einstein Field Equation;

Schwarzschild energy is included in the EFE. A ratio of tensors is equal to a ratio of force magnitudes;

F₁ = E_S²/ħc = 2π(mG^½)²/hc 

F₂ = E_c²/hc = (½mc²)²/hc 

G_μν /T_μν = F₁/F₂ = 8πG/c⁴ 

Binary Gravitational Interaction;

Two objects with mass (m₁,m₂) interact through a separation distance (r) according to their Schwarzschild energies. The force of interaction (F₁₂) is related to the continuum force of each object (F₁,F₂) at radius “r”;

F₀F₁₂ = F₁F₂ = (E_S1/r) (E_S2/r) = (m₁G^½/r)( m₂G^½/r)  

Giving Newton’s gravitational equation;                F₁₂ = m₁m₂G/r² 

Electric Field energy;

A Schwarzschild similar energy (E_Se) applies to the electric field;   E_Se = Qk_e^½ 

Where;        Q is electric charge

k_e is the electric field constant  k_e = 1/(4πε₀) 

                                          ε₀ is the electric permeability of free space

A Plank similar energy (E_Pe) also applies to the electric field;   E_Pe = ħc⁵/k_e 

Wave Particle duality;

The Schwarzschild force is;          F_S = E_S²/hc = E_S/r_S

Where;                                 c = ωλ

Giving a particle to wave relationship;  r_SE_S = λ(hω) 

This may be written as forces;    F₀F₁F₂ = F₃³

                                                     (1)(½mc²/λ)(hω/λ) = (mG^½/λ)³                                 

Conclusion;

Schwarzschild energy is well documented in the literature (usually in partial form), it is a fundamental energy.