The Fifth Force: A scalar force that shows there is a relationship between Riemannian space, black body radiation, and physical constants

Thomas S. Kakovitch, Emeritus Professor, University of the District of Columbia

Join the Conversation #WolframTechConf

The scalar field equation shows that there is a relationship between Riemannian space; black body radiation; and physical constants.

Black body radiation laws, which are part of the quantum field, may be able to converge with the concept of the parallel displacement of directional vector in the Riemannian space which is the base of the principle of the general relativity. 
	Shortly after the event of the big bang, during the radiation dominated era and prior to the existence of matter or charge, small but finite regions of space were permeated with a scalar temperature field. As temperature decreased, space expanded. The four fundamental forces in nature, strong nucleic, electro-weak, and gravity depend either on charge or matter. 
	It is conceivable that the four fundamental forces are the components of an original invariant scalar force field belonging to Planck’s black body radiation family. This approach to proving that a scalar invariant force may exist is to transform Planck’s radiation law into a single, scalar field variable. Temperature is a scalar field. 
	One must be able to integrate over all frequencies and wavelengths of Planck’s radiation law, and transform the radiation law into a solely temperature dependent function. The result is given as: (*Text*)

(Subscript[u, i])(u^i) = - (c^2)

((([(Subscript[g, ik])(u^i)(u^k)])^2)/G)=[(∫_(c/p)^(c/u)){[((((ν^3)))/(Exp[hν/KT]))]-1}dν]/N{(∫_u^p)[(1/((-1+Exp[(hc/(KTλ))])(λ^5)))]dλ}(G)=(∏_(j=1)^4)icj/G						(Eq.1)

The Planck radiation law extends as far back as the radiation dominated, primordial universe. Eventually, the universe cooled and radiation was replaced by matter. The continuum theories, such as Einstein’s field equations, proceed to describe the space-time curvatures of a matter-dominated era. Our main objective is to reduce the Planck Radiation law into temperature field equations. We have been able to show that temperature field equations converge to produce a field that has units of force and is scalar. This is the force field that extends from the radiation to the presently matter dominated era and is referred to as the fifth force. The cosmic background radiation temperature of approximately three degrees Kelvin is predicted as it relates to the fifth force. 

See as reference: Thomas S.Kakovitch, The Fifth Force, Amherst : HRD Press, Inc., 2012. ISBN 978 - 1 - 59996 - 265 - 8.

Reducing the Planck Radiation law into temperature field equations

We have replaced frequency (Nu) with y and wavelength (Lambda) with x.We then integrate the Equation 1

Clear[x,a,T];

Clear[h,c,k];

chris[T_,x_]=(1/(x^5*(Exp[a/(x*T)]-1)))

nat[T_,y_]=((y^3)/(Exp[(h*y)/(k*T)]-1))

Clear[h,c,k];

The units are calculated in Gaussian units

h=6.62517*10^-27

c=2.99793*10^10

k=1.38044*10^-16

a=(h*c)/k

Reducing the Planck Radiation law into temperature field equations (cont.)

The integration is represented with only one variable, T

Clear[T,u,p,ss,qq];

G=6.673*10^-8

u=1
p=10

ss=c/p
qq=c/u

Int1 [T_]=Normal[Integrate[nat[T,y],{y,ss,qq}],ConditionalExpression]

Reducing the Planck Radiation law into temperature field equations (cont.)

The value attained for the (5^th) force, in units of dynes is:

Clear[ForceField]

ForceField[T_]=(1/G)*(N[Int1[T]]/N[Integrate[chris[T,x],{x,u,p}]])

We obtain the following:

ForceField[2]

Plot[Re[ForceField[T]],{T,1,10},PlotLabel->Style["Plot 1: For temperatures between 1 and 10 Kelvin the real values are plotted",FontSize->16,Bold],Axes->True,AxesLabel->{Style["T(Kelvin)",FontSize->16],Style["F(dynes)",FontSize->16]},Background->LightYellow]

Plot[Re[ForceField[T]],{T,0.1,3},PlotLabel->Style["Plot 2a: For temperatures between 0.1 and 3 the imaginary values are plotted",FontSize->16,Bold],Axes->True,AxesLabel->{Style["T(Kelvin)",FontSize->16],Style["F(dynes)",FontSize->16]},Background->LightYellow]

Einstein’s Field Equation in the vicinity of the Scalar Force Field

The fifth force equation is temperature dependent.This is what I refer to as "the fifth force," a temperature field that acts as an invariant scalar force field.The force can describe either the gravitational or the electromagnetic force fields at Planck' s scale of quantum perturbation.The spatial structure of the universe can either be a Euclidean three - space, or the three - dimensional sphere based on homogeneity or isotropy assumptions.The space - time geometry depends on a scale factor that sets the distance and also the magnitude of the spatial curvature and it is time - dependent (see book Thomas S.Kakovitch, The Fifth Force page 37).

Clear[u];

hbar=1.05443*10^-27
c=2.99793*10^10
G=6.673*10^-8

u=Sqrt[hbar*G/(c^3)]

(hbar*c)/(u^2)

(c^4)/G

Clear[hbar,c,G];

hbar=1
c=1
G=1

Einstein’s Field Equation in the vicinity of the Scalar Force Field (cont.)

Einstein' s equation in the vicinity of the fifth force can be transformed into :

sol=DSolve[{((((y'[x])^2+((A'[x])^2)*(Sin[y[x]])^2))*(hbar*c)+(c^4)/G)==0,((A'[x])^2)==((((-((c^4)/G)/(hbar*c))-y'[x])^2))/((Sin[y[x]])^2)},{y,A},x]

This is equation (Eq .2).

Einstein’s Field Equation in the vicinity of the Scalar Force Field (cont.)

{y->Function[{x},-(1/2) I (-I+Sqrt[3]) x+C[1]],A->Function[{x},C[2]-(1/(-I+Sqrt[3]))(I+Sqrt[3]) (Log[2 Cosh[1/4 ((-I+Sqrt[3]) x+2 I C[1])]]-Log[2 Sinh[1/4 ((-I+Sqrt[3]) x+2 I C[1])]])]}

The particular solutions to the differential equation above (Eq.2) is given:

particularsols=Partition[Flatten[Table[{y[x],A[x]}/.sol/.{C[1]->1/i,C[2]->1/j},{i,-20,20,6},{j,-20,20,6}]],2];

Einstein’s Field Equation in the vicinity of the Scalar Force Field (cont.)

Parametric plot for the imaginary part of the particular solution:

ParametricPlot[Evaluate[Im[particularsols]],{x,-3,3},PlotRange->{-2,2}]

particularsols=Partition[Flatten[Table[{y[x],A[x]}/.sol/.{C[1]->1/i,C[2]->1/j},{i,-20,20,6},{j,-20,20,6}]],2];

Parametric plot for the real part of the particular solution:

ParametricPlot[Evaluate[Re[particularsols]],{x,-3,3},PlotRange->{-2,2}]

Einstein’s Field Equation in the vicinity of the Scalar Force Field(cont.)

particularsols=Partition[Flatten[Table[{y[x],A[x]}/.sol/.{C[1]->1/i,C[2]->1/j},{i,-20,20,6},{j,-20,20,6}]],2];

Parametric plot for the combined imaginary and real parts of the particular solution :

ParametricPlot[Evaluate[{Re[particularsols],Im[particularsols]}],{x,-3,3},PlotRange->{-2,2}]

Matrix analysis

We are interested to find a matrix formed by different dimensional units that could express not only the Fifth Force but also connect some of the universal constants.
Let us form a matrix such as, A = {h, 2 ∗ Pi ∗ d, Sqrt[alpha] ∗ d, m}, {h, 2 ∗ Pi ∗ d, 4 ∗ Pi ∗ d, m}, {J, 2 ∗ Pi ∗ h/(mj ∗ c), (G ∗ mj)/(c 2), mj}, {h, 2 ∗ Pi ∗ d, d, m}}.

Clear[hbar,c,G,alpha,mj];

A={{h,2*Pi*d,Sqrt[alpha]*d,m},{h,2*Pi*d,4*Pi*d,m},{J,2*Pi*h/(mj*c),(G*mj)/(c^2),mj},{h,2*Pi*d,d,m}}

MatrixForm[A]

Det[A]

Norm[A]

Clear[hbar,c,G,alpha];

hbar=1.05443*10^-27
c=2.99793*10^10
G=6.678*10^-8

e=4.80286*10^-10

alpha=(e^2)/(hbar*c)

h=hbar
d=Sqrt[hbar*G/(c^3)]
m=Sqrt[hbar*c/G]

J=(G*mj^2)/c

Norm[A]

P1=Plot[Norm[A],{mj,0.99 m,1.01 m},Axes->True,AxesLabel->{Style["(Subscript[m, j])(gm)",FontSize->16],Style["Norm(A)",FontSize->16]},Background->LightYellow]

The result shows that Planck' s Constant (hbar1); Newtonian Gravitational Constant (G1); Quantum Electronic Charge (e1); The fifth force (FJ) are all dependent on NormA.

Matrix analysis (cont.)

Clear[mj,d];

mj=m

d=Sqrt[hbar*G/(c^3)]

Norm[A]

Q=Norm[A]

M=2*Pi*Q

4*Pi*m

C1=G*M^2

(2*Pi)*(8*Pi*h*c)

MatrixRank[A]

Z=NullSpace[A]

MatrixForm[Z]

Y=Eigensystem[A]

MatrixForm[Y]

(9.1083*10^-28)/(4.662394388864052`*^-28)

(9.1083*10^-28)/(-4.661267614902187`*^-28)

(9.1083*10^-28)/(-4.66856389339687`*^-28)

hbar1=(G*(Norm[A])^2)/(4*c)

G1=(4*h*c)/(Norm[A])^2

e1=(Sqrt[alpha*G1])*Norm[A]/2

FJ=((2*Pi*Norm[A]*c^2))^2/(C1)

Conclusion

Physics searches for theories which can be described by laws of fields. Field theories have their advantages and shortcomings. Continuum theories are expressed by the differential equations. Differential equations reach singularities at world-points. However, the totality of physical point-events, including mass, are defined as point-like properties embedded in a four-dimensional continuum. 

In conformity with the quantum theory, a direct assertion about the state of a system is unattainable. A field becomes a good candidate as a distribution residue where even space-time cannot exist without it. Field inherently depends on temperature. Field ceases to exist in an absolute zero temperature domain. The dependence of black body radiation on temperature is fundamental to understand the properties of fields in general. One of the most fundamental fields is temperature field. The reduction of any thermal background radiation into a temperature function ascertains the existence of a temperature field. 

The Planck radiation law extends as far back as the radiation dominated, primordial universe. Eventually, the universe cooled and radiation was replaced by matter. The continuum theories, such as Einstein’s field equations, proceed to describe the space-time curvatures of a matter-dominated era. Our main objective is to reduce the Planck Radiation law into temperature field equations. We have been able to show that temperature field equations converge to produce a field that has units of force and is scalar. This is the force field that extends from the radiation to the presently matter dominated era and is referred to as the fifth force. The cosmic background radiation temperature of approximately three degrees Kelvin is predicted as it relates to the fifth force.