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Vol.2012, Issue No.10, dt: 30 October 2012 Time 17h16m P.M

A FIVE DIMENSIONAL THEORY OF GENERAL RELATIVITY

          AND PRODUCTION OF ELECTRON AND POSITRON PAIR.

By

Professor Dr Kotcherlakota Lakshmi Narayana,

{Retd.Prof.of Physics, SU} 17-11-10, Narasimha Ashram, Official Colony,

Maharanipeta.P.O. Visakhapatnam-530002. Cell: 9491902867

ABSTRACT   

        A solution of five dimensional equation of Einstein General Theory of Relativity obtained. The attempt made to understand the role of positron relative to its role in the new physics considerations of Dark Matter and Dark Energy.    

INTRODUCTION

How the scheme of gravitation be fitted into the stream of other observations made by Physicists. The paradoxical question of a Black Hole and its sisters do involve intricately the electron-positron scheme of Dirac is intriguing. However, people not paid attention to fit it to the scheme of electron-positron pairs that play a tremendous role in the Universe especially of the Black Holes. The expanding universe is an accepted fact measured due to the light-curves of Ia supernovae (1, 2, 3) and independently by WMAP satellite (4) and other CMB experiments (5, 6). The new gravitational physics seems to offer immense possibility of a cosmic acceleration.

Einstein's theory of gravity expected to break down at very short distances and when the curvature of space-time becomes large but our method of analysis presumes electron-positron pairs possibility within the framework of General Relativity but with five dimensions. The idea of Hawking that the electron and positron pairs within a black hole have a definite role is very encouraging and the five dimensional theory presented by me now would help to answer some of the fundamental problems of Physics of Gravitation relative to the black hole objects and especially the Dark Matter and the Dark Energy.

The Dark Plasma Theory asserts that dark matter may be present in our Solar system and specifically around the Earth. The influence of the invisible dark matter particles bombarding the Earth daily for the duration of its creation seems to be an inescapable idea. Earth gravitationally coupled to a Jupiter sized dark matter halo according to a scientist Jay Alfred.  The bio dark plasma cells systems date back to the creation of the Earth. The universe seems dominated by dark energy and dark matter.  Dark matter clumps may be responsible for galaxies, a thought process that not possible to rule out.

Hubble parameter H characterizes the expansion of the present universe as

H^2= (R˚/R) ^2 = 8π G_N ρ 3 – k / R^2  + Λ/3

where R(t) is cosmological scale factor and k takes either 0, +1 or -1 for spatially flat, closed or open universe. Λ is cosmological constant that contains all the vacuum energy density. For flat universe Hubble parameter gives the critical density as

ρc= 1.88E-29 h0^2 gcm(^^-3)

where                                     h0= H0/(100km Mpc^^-1 s^ ^-1)

The magnitude h0= 0.71± 0.01 obtained from the high red-shift supernovae Ia data (7).

Particle theorists believe that the cosmological constant has dimensions of (Mass)^4  where M is some characteristic mass scale in Physics namely M planck= mass of the Planck Mass, or M = Mp  the proton mass. Conventional estimates of Planck scale corrections which are of order (me / Mplanck )² ~ 10^−44..

A New force carrier φ is included, the possibility of a new annihilation channel  χχà φ φ  opens up. Absent couplings to the standard model, some of these particles could naturally be stable for kinematical reasons even small interactions with the standard model can then lead them to decay only into standard model states. If they decay dominantly into leptons, then a hard spectrum of positrons arises very naturally. Nima Arkani-Hamed (8).

“A scalar can couple with a dilatonlike coupling φ F^μν F_μν, which will produce photons and hadrons (via gluons). Such a possibility will generally fail to produce a hard e⁺ e^- spectrum. A more promising approach would be to mix φ  with the standard model Higgs with a term κ φ^² h^ϯh. Should φ acquire a  vev  <φ> ~  m _φ,  then we yield a small mixing with the standard model Higgs, and the φ will decay into the heaviest fermion pair available. For m _φ ≤  200 MeV it will decay directly to e⁺ e^-  while for 200 MeV ≤  m _φ ≤  250 MeV, φ will decay dominantly to muons. Above that hadronic states appear, and pion modes will dominate.   Both e⁺ e^-  and μ⁺ μ ^- give good fits to the PAMELA data, while e⁺ e^-  gives a better fit to PAMELA  + ATIC.” state Nima Arkani-Hamed, et al. (8). For quite perturbative values of α _Dark  it could be interesting for larger values and would lead to a greater multiplicity of softer e⁺ e^-   pairs in the final state.

METHOD

The standard method of analysis adopted to simplify the approach to get a better feel of the Physics involved and not the mathematical details that would outweigh the formal theory developed here. The scientists globally captured by the e⁺ e^- pair creation and the analysis restricted to present its role in the formulation. The explicit calculations however, not detailed at present to see the creation of the e⁺ e^- pairs in the relativity model. However, the formalism presents an outcome of electron and positron annihilations and their roles as perhaps thought by Hawking radiation of Black Holes. The method adopted here distinctly differs. The metric tensor is anti-symmetric as per the choice made by Papapetrou but we differ in our choice of the metric tensor. We have an additional term of the g₅₅ and the anti-symmetric part of it.

            The metric tensor first adopted is

[ - α        0                      0     ω     ω’;

  0      - β                      0      0      0;

  0       - β * (sinθ)^2    0      0      0;

 - ω       0                        0      σ      0;

 - ω’      0                       0       0   - σ’]

with the determinant

g = [ α * σ * σ’ - ω ^2 * σ’ + ω’ ^2 *σ] *(sinθ) ^2 * β^2;

I get the expressions for the cofactors as

G’₁₁= - β² * σ * σ’ * (sinθ) ^2; 

G’₂₂= - ( α* σ* σ’ + ω ^2 * σ’ + ω’ ^2 *σ)*(sinθ)^2*β;

G’₃₃= - ( α *σ *σ’ + ω ^2 * σ’ + ω’^2 *σ) β;

G’₄₄ = (α*σ’ + ω’ ^2) *(sinθ) ^2 * β^2;

G’₅₅= (ω ^2   - α *σ) * (sinθ) ^2 * β^2:

G’₁₅= ω’ * β^2 * σ * (sinθ) ^2;  Note G’₁₅ = - G’₅₁;

G’₅₁= - ω’ * β^2 * σ * (sinθ) ^2;

G’₄₅= - ω * ω’ * β^2 * (sinθ) ^2;   Note G’₄₅= G’₅₄;

G’₅₄= - ω * ω’ * β^2 * (sinθ) ^2; 

We however make things easier to set up the equations that need to be solved by the choice of both ω and ω’ terms to vanish. And also choose the term - σ’ as simply exp(-2*ν*) where the term ν* represents the added term of our analysis.

RESULT

But, adopting the simple straightforward metric tensor without the non-diagonal elements and changing the expression for g₅₅ as from - σ’ to simply e ^ (-2ν*) with ν* as a variable and

α= e^2*Λ; β=r^2; g₃₃=- r^2*(sinθ) ^2; g44= e^2*v;

we obtain a standard fifth order metric tensor.

          This simplification yields the algebraic expressions for G₁₁ and G₄₄ components as

(dν/dr)^2 + d^2 ν /dr^2 – (2/r)*dΛ/dr – dΛ/dr*dν/dr + (dν * /dr)^2=0;

  And e^ 2*(ν- Λ)*[-d^2 ν /dr^2 - (dν/dr) ^2+ dν/dr*dν*/dr – (2/r)*dν/dr + dΛ/dr*dν/dr=0;  

 And with G ₂₂  the expression for the electron and positron yield the result

    e^(-2*Λ) *[ 1- r* dΛ/dr + r* (dν/dr  - dν*/dr) =1- 8*π* E₂₂+8*π* E’₂₂ ;

where we have E₂₂ and  E’₂₂ representing the electron and positron charge quantities.

         

          A similar expression for the G₅₅ leads to the positron equation  a special feature of the present work.

          Note: The change of the G₅₅ term to involve exp(-2ν*) is significant in the sense that the term of choice - σ’ leads to a non-solvable logarithmic expression.

ACKNOWLEDGMENT

I am greatly indebted to Prof. K. R. Rao D.Sc. (Madras). D. Sc. (London) for making it possible for me to study the Mathematical Physics at M.Sc. in Andhra University, Waltair during 1956-1960. I have little realized the ongoing political turmoil of Andhra University those days. I am indebted to his great patronage, research guidance and for the purchase of Mathematical Physics books obtained from the Moore market in Madras.

REFERENCES

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