A CLASS OF EXACT SOLUTIONS FOR THE FLUID UNIVERSE FILLED WITH MAGNETOFLIUD –LIKE FIELDS

trusciencetrutechnology Vol.No.2009,Issue No.1,Dt.7th January, 2009A CLASS OF EXACT SOLUTIONS FOR THE FLUID UNIVERSE FILLED WITH MAGNETOFLIUD–LIKE FIELDS.
by
Professor Kotcherlakota Lakshmi Narayana,
http://trusciencetrutechnology.blogspot.com/
(Retd. Prof of Physics,Shivaji University,Kolhapur-416004}
17-11-10 Narasimha Ashram,Official Colony,Maharanipeta.P.O.
Visakhapatnam-530002
lakshminarayana.kotcherlakota@gmail.com
Dated: 3rd January 2009 12.30PM
KEY WORDS: FLUID FILLED UNIVERSE,RAO(NOVAE)STAR, Raychadhuri Equation,Expansion,Shear,Rotation,Relativistic Magnetohydrodynamics, General Relativity,Condensation point, Magnetic Monopoles,pole strengths,cosmic entities, Black Holes,Equation of Energy,Space-Time Model,Astrophysics,Cosmology,Relative Mass,test particle
ABSTRACT
A class of exact solutions of the Einstein Field Equation with Relativistic Magneto hydrodynamic aspects and as well of a Raychaudhuri-like equation for the evolution of the Universe has been obtained. The condensation point of the Universe has been found to be characterized by two types of magneto fluid-like field is remarkably different from the usual considerations of Lichernowicz [3] in terms of a magneto fluid flow vector. The implications of the Model are far-reaching to understand and asses the deeper Astrophysics structural features of several of cosmic entities and possibly the Novae Stars. The aspects of the dynamic equation and the Equation of Energy have been outlined with explicitly presenting the equation for determination of several thermodynamic parameters characterizing the space-time model termed the FLUID UNIVERSE.

Introduction:
The frequent findings on magnetic fields of cosmic entities(such as stars, galaxies, novae, pulsars, quarks, magnetic stars, Black Holes, dwarfs etc.) by Hubble telescope, ISS and other space probes of several countries it is thought it would be rewarding by a study of Relativistic Magneto hydrodynamics and would be of great influence in the realm of Astrophysics.
The recent experimental observations on just a few close by stars that are moving around the super massive Black Holes of our own Milky Way Galaxy, led to an understanding of its features. In this regard and also regarding the structure of multiversity of cosmic entities and stars, it is thought also to have a fresh look into the General Relativity and Magneto hydrodynamic aspects of Universe Evolution and structure of cosmic entities. [1, 2, 3]. The universe is currently undergoing an accelerated phase of expansion, a challenge that is being negotiated with effective negative pressure and of presence of a dynamical dark energy. This obviously gives a new insight of the subject of Astrophysics.[2]
Exact solutions of the Field Equations for a isotropic, isentropic, non-homogeneous universe with Magneto fluid-like fields with infinite Electrical Conductivity and a constant magnetic permeability have been deduced in the present work using the line element that has parameters laterally dependent, In terms of spherical coordinates the θ is regarded as the lateral coordinate. The energy-Momentum Tensor (Lichernowicz 1967)[3] has been left open to realize and determine the several of Thermodynamic variables viz., the density ρ , pressure p and Entropy S and the 4-velocity vector u ^α and the Magneto fluid-like fields.

The Method of Analysis:
This involves a fluid at rest relative to the world lines of constant r, θ, and φ and the co-ordinate system is co-moving system i.e.

u^μ= e ^(-Φ/2) δ^μ₄-----------(1)

ds²= e^Φdt²-e^Λ(dr² + r²dΩ²)/c²------(2)

with

dΩ ²=dθ ² + sin²dφ²

but explicitly treating the

Φ and Λ as functionsof r, θ, and t.


This is a departure from usual considerations of the spherically symmetric Cosmological Models [4]. The method uses the Einstein Field Equations given by

G^μ_υ = R^μ_υ - (1/2)g^μ_υ R= - Κc²T^μ_υ-----------(3)

where T^μ_υ

is the Energy –Momentum Tensor consisting both matter and field expressions and

R_μυ= R^τ_μτυ

is the Ricci Tensor (contraction of Curvature Tensor), and R is the curvature scalar given by R=R^μ_μ .Here Κ=8πG/c² with G as the Universal Gravitation constant

G=6.67E-11 N.m²/kg².

and c the velocity of light 3.0E-08 m/sec


Lateral Dependent Line Element Exact Solution:

The dynamic and lateral dependent yet another line element is given by

ds²=dt²-e^Λ(dθ²+r²dΩ²)/c²---------(4)

with

Λ=Λ(r, θ, φ) --------(5)

has been studied. This and the more general line element involving Φ that satisfy the Relativistic Magneto hydrodynamic and Einstein field Equations have been explicitly worked out. The results obtained for azimuthal dependence would be the subject matter for the next publication at this blogspot.
The solution for the metric Eq.4 gives

e^Λ=R²{1+g_m²/4r²R²}² cosec² θ ------------------------- (6)

The physical entity R used here is not the curvature scalar and is known as the scale factor, with dimensions of length and is a function of time only. The equation also depicts the evolution of the Fluid Universe, specified by the line element given by Eq.4

Λ_t=2R˙/R

The function R (t) in Robertson-Walker space-time is taken to be the expansion factor. Hubble parameter

H=R˙/R

and the deceleration parameter q is given by

q=-(H˙/H²)-1=-R˙˙R/R˙² ------------------(7)

The dot and suffix t, both imply differentiation with respect to the cosmological time parameter t. The physical entity e ^Λ involves obviously g _m only but allows the line element solution similar to both the De Sitter’s by putting θ=90⁰ and g_m=0, with the exception that Λ is a function of r, t and θ parameters.
THE FLUID UNIVERSE:
Another significant aspect of the present work is that a new Magneto fluid-like field, with field strength g_ad , gave rise to a new set of Relativistic Magneto-hydrodynamic equations for the total pressure p_T , the Energy density ρ_T and internal energy density ε_T.The equations found are

8πG ρ_T= 3R˙²/R²+[c²g_m²e^(-2Λ)cosec²θ]/r⁴+

[g_ad²e^(-Λ)cot²θ]/r²-------- (8)

8πG P_T/c²=-R˙²/R²- 2R˙˙²/R²+

[c²g_m²e^(-2Λ)cosec²θ]/r⁴+ [g_ad²e^(-Λ) cot²θ]/r²------(9)

8πG ε_T/c² =(1/ρ₀)[3R˙²/R² + [c²g_m²e^(-2Λ)cosec²θ]/r⁴+ [g_ad²e^(-Λ)cot²θ]/r²]------(10)

with

e^Λ=R²{1+g_m²/4r²R²}²cosec²θ

The formalism developed and reported herewith in this publication involves the Magneto fluid-like fields, with two different magnetic field strengths (supposedly) and the magnitude of the magneto fluid-like field given by:

h²=[2g_m²e^(-2Λ)cosec²θ/(kμr⁴) + g_ad²e^(-Λ)cot²θ/(kμr²)]

which may not be interpreted as Lichernowicz [3] has explicitly defined to be a product of the relevant ha vectors. The essential feature of the model presented here is that Einstein Equations involve on the right hand side the Geometric features of the space-time while the left hand side involves Energy-Momentum Tensor of the new Magneto fluid-like fields. The marriage between the new fields and the Geometry, of course, needs further digression that’s postponed at present.

THE KINEMATICAL PROPERTIES OF THE SOLUTION AND THE GAUZE CONDITION:
The Differential Geometric entity ω^φ_θ which is an element of the set of connection forms of the space-time is obtained by a proper choice of the orthonormal basis of the line element Eq.4 .for the Fluid Universe. The values of acceleration u ,expansion or the scale factor Θ , shear σ_αβ and rotation ω_αβ are obtained.
The gauze condition follows from the assertion that

ω^φ_θ=e^Λ/2(c/r){Λ_θ/2+cotθ)ω^φ = 0------(12)

where

ω^φ=e^Λ/2rsinθdφ------(13)

and Λ _θ is the derivative of Λ with respect to θ.
The scalar quantity Θ is found to be

Θ= Λ_t/2=R˙/R=u^α _;α ---------(14)

and the kinematical properties lead to

ω_αβ=σ_αβ=0--------(15)

The fluid universe obeys a Raychadhuri’s-like equation [9]

3[Θ,_β u^β]+3Θ² = u˙^α_;αδ^α₂-kc²{ρ_m + 3p_m/c²+μh²}=3(Ď²_uL)/L--------------(16)

and here

u˙^α

is not entirely zero and where it is defined that

Θ= (Ď_uL)/L -------(17)

and
Ď_uL=L_;βu^β -------(18)

Where L is the significant length gives the volume behavior of the Fluid. Here ρ_m and p_m are the matter density and pressure respectively. Noteworthy that u ¢a is not entirely zero, with the usual meaning of the term as the acceleration. Hence the class of exact solutions found out here distinct.
It is not surprising to investigate the Relativistic Cosmology from an unfamiliar point of view of anisotropy etc. features of a space-time. In literature, the treatment of initial conditions, in this context, has been stressed, for example in Hot Big-Bang Models by including the anisotropy.
Once the time evolution of expansion factor R(t) is known, the density and pressure and in turn the Magneto fluid-like fields determine how the expansion proceeds in Cosmological time t of the Fluid Universe.
The Eq.16 given above would be thus helpful to gain an insight into the closed loop logic [ 5 ] of the Field Equations and also to understand the dynamic Equation that involves the second derivative of the expansion factor. The initial value equation i.e. he Energy Equation asserts regarding the Cosmological Fluid Universe Model envisaged and apparently which is a Field Dominated Universe.
MAGNETOFLUID-LIKE FIELD STRENGTHS:
The idea of point source or the existence of a condensation centre of the magneto fluid filled Universes seems to be centric with certain kind of static solutions.
The constant gm is usually interpreted as the magnetic field pole strength of the Condensation centre of a magnetic fluid. I retained this concept partly in the present model of FLUID UNIVERSE considerations but with the caution that the surrounding Universe is not a smoothed out version by the “ambient” perfect fluid.
This is so since although the centre of condensation of the magneto fluid-like fields occurs through the vanishing of the strength gm, the affects of gad would persist. This outweighs the present day believes of a collapse to a central condensation region of a Universe or as a matter of fact Collapse leading to formation of a Black Hole or Super massive Black Holes etc.
It may be remarked that the constant gad occurring in the Fluid Universe Model line element is similar to the one given by Nordstrom-Jeffery solution for gravitational field of an electron. This is somewhat surprising. A similar constant according to Das [7] cited already above implies the gad type of magnetic pole strength of a point source at r=0.
The relative mass of a test particle depends therefore as well on the additional magneto fluid-like field pole strength gad. [8, 9, 10]
Discussion:
In the co-moving frame of reference the time of collapse τ is of the same order of magnitude as the time required for the free fall of a particle in a field. The present study hints that presence of an additional Magneto Fluid-like field inside the collapsing entity that would give rise to possible certain belts or rings consisting of the Relativistic and non-relativistic particles. These rings (belts) would be the immense source of EM waves in radio, optical and X-ray regions of the spectrum. The total energy of particles contained in these rings (belts) may be comparable to the Magneto Fluid –like fields of energy of order 10E+56 ergs or more by several orders of magnitude. Whether these rings (belts) be regarded as magnetic fluid-like rings is an open problem. Best example, in this context, of course may be those occurring in Crab Nebula of the supernovae 1054(or in the region of Novae star termed Rao star)
ACKNOWLEDGEMENT:
The author is deeply indebted to Professor K. Rangadhama Rao D.Sc.(Madras).D.Sc.(London) who inspired me for research work and provided me constant guidance and encouragement. He also presented me with several books in the subjects of Theoretical Physics and Mathematical Physics.
REFERENCES:

1.A.H.Taub Phys.Rev D Vol.103,p.1454 (1971)
2.P.Yodzis Phys.Rev D vol.3 No.12, p.2941 (1971)and Padmanabhan. T. Roy Choudhury T Mon.Not.R.Astronomy Soc astroph/10212573(2003)
3.Andre Lichernowicz Relativistic Hydrodynamics and Magneto Hydrodynamics, Benjamin Inc. NY,1967
4.L.C.McVitte General Relativity and Cosmology, Chapman and Hill Ltd, London, p.140, 1965.
5.C.W.Misner, K.S.Thorne, J.A.Wheeler Gravitation, W.H.Freeman & Co. pages 566,72 (1970)
6.W. Israel, Differential Forms in General Relativity
Dublin Adv.Studies series A No.19 (1970)
7.A.Das, Prog.Theor.Phys Vol.24, p.915 (1960).
8. The negative mass shells were first encountered by M.M.May and R.H.White Phys Rev.141, p.1232 (1966) the negative mass was used by the present author to observe the Rao Stars (a publication made at this blogger and reported to Ind.Sci.Cong. held at Andhra University in 2008) see also M.E.Cahill and G.C. McVittie J.Math.Phys.vol.11, No.4 p.1382, 1970.
9.A. Ray Chaudhuri Relativistic Cosmology Phy.Rev Vol.98, p.1123 and GFR Ellis Relativistic Cosmology Ed by RKSachs, AP, NY. Also T.G.W.Misner Astro J.,vol.151,Feb p.431,(1965)
10.Kotcherlakota Lakshmi Narayana, Ind. Sci.Cong, Session X Ref.paper No.33,page 23,2008 Andhra University.