ON THE NEW GRAVITATIONAL RED SHIFT FORMULAS AND A RELATIVISTIC LATERAL LAW OF GRAVITATION

TRUSCIENCETRUTECHNOLOGY VOL.2009, NO.4,DATED:7th APRIL 2009

On the New Gravitational Red shift formulas and a Relativistic lateral Law of Gravitation
by
Prof.Dr.Kotcherlakota Lakshmi Narayana, (Retd.Prof.of Phys, Shivaji Univ.) 17-11-10, Narasimham Ashram, Official Colony, Maharanipeta.P.O, Visakhapatnam-530002
Dated 9th March 2009 Time: 09:20AM
Keywords: Astrophysics, A new Law of Gravitation, cosmic entity, red shifts, cosmology, metric tensor, Einstein Equations, Relativistic magneto-hydrodynamics, Magneto fluids
trusciencetrutechnolgy@blogspot.com Vol.2009 No.4, Dated 4th April 2009: Time 16.15PM
ABSTRACT:
The exact solutions of the Einstein’s Equations for a magneto-fluid cosmic entity with an infinite electrical conductivity are obtained under the assumption that the space-time is laterally dependent. Results obtained with two different types of space-time metric tensors for fluid universe, Energy equation, Dynamic equations of motion and the red shifts formulas are detailed. For, the first line element, features of Energy and Dynamic equations of motion and their solutions are given. With the other line element, and by use of the boundary conditions, new mass functions, and the gravitational red shift formula are obtained. The new formulation of a Relativistic Lateral Law of Gravitation has been explicitly given.
INTRODUCTION:
Misner, Thorne and Wheeler [1] have treated computation of curvature for a pulsating or a collapsing star as an example in their book on “Gravitation”. The metric adopted by them has signature +2 and the exponential parameters of the metric Viz,Ф and Λ are considered as functions of r and t only. They have listed relevant quantities such as the Riemann-Tensor components, Connection 1-forms and the Einstein Tensor matrix elements in a Cartan’s orthonormal basis frame of reference.
It is yet unclear how fluid filled proto-stars can collapse as a whole. Departures from spherical symmetry might be responsible for gravitational collapse of a gas to lead to the appearance of proto-stars was thought also by Ginzburg [1]. He opinioned that role played by the magnetic field of the collapsing proto-star deserves particular attention. In the present work, the aim is to understand relevance of magnetic field of a collapsing superstars etc.
Cahill and McVittie [2] examined in curvature coordinates, the metric-tensor, that is continuous across a surface of discontinuity. They have defined
R²₂₃₂=2*M/r_b---------------------------------(1)
where the subscript “b” means evaluation at the boundary and M is mass function dependent exclusively on r and t.
They have outlined an analysis regarding existence of finite number of temporary negative-mass-shells of material. They have also treated a Schwarzschild solution with negative mass in the presence of charged distributions. They discussed the collapse without regard to the nature of
r₄= dr/dt=dx¹/dx⁴ ------------------------(2)
Earlier Hernandez and Misner [3] have also shown that if always r₄<0 the gravitational collapse occurs when x¹ ever becomes also negative. May and White [4] have considered adiabatic collapse and encountered situations where mass-energy does not increase monotonically with co-moving radial coordinate and asserted that when it happens then a negative mass-shell occurs.

SECTION I: LATERAL DEPENDENT METRIC-TENSOR AND ENERGY EQUATION.

The dynamical and lateral dependent, yet another line element is given below:
ds² = dt² – e^Λc^-2 [dr²+ r² dΩ²]-----------------------(3)
where
dΩ²=dθ²+ sin²θdφ²-----------------------------------(4)
with Λ= Λ (r, t, θ) which is investigated as a part of my series of articles, more general line element involving Ф that satisfies the relativistic magneto-hydrodynamic and Einstein field Equations have been explicitly worked out.
The solution for the metric given by Eq.3, yields
e^Λ=R²(1+ g_m²/(4r²R²))² cosec²θ------------------------------(5)
The physical entity e^Λ involves obviously g_m only but allows the line element solution similar to the DeSitter’s by putting θ=90⁰ and g_m=0, with the exception that Λ is a function of r, t and θ.

THE FLUID UNIVERSE:

Another significant result of the present work is that a new magneto-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 suffix T implies total values. I use R₄to denote the differentiation of R with respect to time.
The equations found are
8πGρ_T= 3R₄²/ R ² + (c² / r⁴ )g_m² e^–2Λ cosec² θ + ( g_ad² / r²)e^–Λ cot²θ------------------------------------------------------------------(6)
8πGP_T= -R₄²/R² – 2R₄₄ / R + (c² / r⁴)g_m²e^–2Λcosec²θ + (g_ad²/ r²) e^–Λ cot²θ------------------------------------------------------------(7)
8πG ε_T = (1/ρ₀) [3R₄²/R² + (c² / r⁴ )g_m² e^–2Λcosec²θ + ( g_ad² / r²)e^–Λ cot²θ]----------------------------------------------------------(8)
with e^Λ= R² (1+ g_m²/(4r² R²) ² cosec ² θ
The formalism developed, reported in this publication, involves the magneto-fluid like fields, with two different magnetic field strengths, given by
|h|²= (1/κμ)[ (g_m²/ r⁴ ) e^–2Λ cosec² θ - ( g_ad² / r²)e^–Λ cot²θ]-----(9)
This may be tried to resolve into the product of h vectors, as proposed, by Lichernowicz [5]
Consequently I got a class of new cosmological mode
ds²= dt²-R²(1+ g_m²/(4r²R²))²cosec²θ)*[dr²+ r² dΩ²]----------(10)
When g_m is zero this solution characterized by
ds²= dt²- R² cosec²θ * [dr²+ r² dΩ²]-----------------------------(11)
dose not corresponds to DeSitter’s universe since we have also g_ad non-vanishing.
The essential feature of the model presented here is that Einstein Equation involves on the Left-hand side the geometric features* and on the other side involves Energy-Momentum tensor of the new magneto-fluid like fields. The marriage between these two and the diagnostic is postponed at present*.
The significant result of this computation is that the factor F in
h¹= (F/r²)e^-α-γcosec θ ---------------------------(12)
has the value F= Hr₄/(cosec θ √κμ)----------------------------(13)
for κ and μ refer my publications[8]
This result is a consequence of the gauge conditions
r₄₁= r₁ r₄/r and r₁₁=2r₁²/r----------------------------------------------------(14)
and moreover with the price γ₁=0 and suffix “1” represents differentiation with respect to r. Both r₄₁ and r₁₁ seem to behave as physical entities, as it would become clear in the later part of this article.
We may introduce a conservation law as
8 π [T ^b _a];_b=-C,_b------------------------------------------------------(15)
with matter momentum tensor T ^b _a is given by (-ρ, p, p_t, p_t) and C as the cosmological constant, which varies spatially with r. Without any loss of generality, we take p_t= np, where n is the proportional multiplicative factor, and using restraint that
Λ/8π = ap----------------------------------------------------------(15a)
to obtain an expression for p as
p= Ar^–B, where A=e^{(1-n)Λ/(1-a)} and
B= 2(1-n)/(a-1)------------------------------------------(16)
THE ENERGY EQUATION:
Solving for Λ_tby using the Eq.5 we get
(Λ_t²/4 = e^–Λ/2 (R₄ – g_m ² R₄/(4r² R²)- g_m²/(2r³R)) ²cosec²θ ------------(17)
This expression is very useful to evaluate the initial value equation of the Einstein’s Field equations. Which the later is determined to be
(Λ_t²/4 c²) e^–Λ/2 =
[(1/3)*(Λ_rr + Λ_r²/4 + 2 Λ_r/r + Λ_θθ/r 2+ Λ_θcotθ /r²+ Λ_θ² /(4r²))
-(8πρ+C)/3]----------------------------------------------------------(18)
This equation relates R₄ to the initial moment of rate of expansion Λ_θ, Λ_r and ρ and the cosmological constant C at any initial moment of time. The equation may be referred as the Energy Equation.
From this I found, that treating that g_m=0,
R₄²= (c²/ cosec²θ)*
[(1/3)*(Λ_rr + Λ_r²/4 + 2 Λ_r/r + Λ_θθ/r 2+ Λ_θcotθ /r²+ Λ_θ² /(4r²))
-(8πρ+C)/3] -----------------------------------------------------------(19)
Inclusion of g_m allows the prescription of initial value of R₄ related to both R and R2.

THE DYNAMICAL EQUATION:

The other significant equation is the one referred as the Dynamic equation of motion or the equation of the acceleration (akin to the Newtonian equation of acceleration).
It is found by me as:
Λ_tte^Λ/c²= (Λ_rr + Λ_r²/4 + 2 Λ_r/r + Λ_θθ/r²+ Λ_θcotθ /r²+ Λ_θ² /(4r²))
– 8 π(ρ+p) + 2C +( Λ_r/r+ Λ_r²/4- Λ_θcot θ /r²)-------------------(20)
The Hubble constant needs to be revised since H= R4/R which in my formulation depends on the magneto-fluids kerchiefed with the cosmological fluids and together with the cosmological constant C.
The three important physical quantities, namely, the density parameter σ, Hubble’s constant H, and the deceleration parameter q that are quite amenable for experimental determination, using the present formalism are determined and, are given below:
The deceleration parameter
q=-2(Λ_tt + Λ_t²)/ Λ_t²--------------------------------------------------(20a)
The Hubble constant
H= Λ_t / 2----------------------------------------------------------(20b)
And the density parameter,
σ= 16πGρ/(3 Λ_t²)-----------------------------------------------(20c)
Since Hughston [6] gave the theory of multiple-fluid cosmologies, it is not surprising to find an involved cosmology of both the cosmological fluid and the magneto-fluid.
THE REDSHIFT AND THE ARC PARAMETER:
The interval of time dt, during which a photon traveling a flat (θ= 90⁰) hypersurface of homogeneity [1] covers an arc measure given by
dη = dt/ R(t). --------------------------------------------------------(21)
Explicitly it is derived, in the present work, to be
dt= dΛ/ √[ 1/3)*(Λ_rr + Λ_r²/4 + 2 Λ_r/r + Λ_θθ/r 2+ Λ_θcotθ /r²+ Λ_θ² /(4r²))
-(8πρ+C)/3 * e^-Λ/c²]------------------------------(22)
The arc parameter[1] with the limits of integration from 0 to t is given by
η= ∫ dt/R(t).----------------------------------------------------(23)
The red shift has a complicated formula for its standing wave patterns, [7]
Zs= Δλ/λ= R(tr)/R(te) -1 ----------------------------------------(23a)
where in the notation of Misner, Thorne and Wheeler [1], R(tr) is the receiver time radius and R(te) is the emitter time radius. In terms of expansion parameter Λ, Red shift can be derived, in the present formalism as
R(tr)/R(te) = e (Λ_r - Λ_e)/2.----------------------------------------(23b)
My previous publications [8] are on the nature of a Universe filled with magneto-fluid and, the layered structure break up of a parent planet (termed MEL planet into three parts Viz. Mars, Earth and Lunar entities) and on possible klnholes superimposed on wormholes. So now, I have aimed at an understanding, of the intricacy of the lateral dependence of metric tensor and the nature of a collapsing cosmic entity. In relativistic magneto hydrodynamics, Yodzis [9] used kinematical parameters and obtained conditions for gravitational collapse of magneto-fluids.

SECTION II: MAGNETIC FLUID FILLED SPACE-TIME AND MAGNETIC MONOPOLE STRENGTH

According to General Theory of Relativity, the metric-tensor, dealt with in this paper describes the motion of a self-gravitating lateral dependent distribution of material. It therefore has only φ as a cyclic co-ordinate. Results of many aspects of the cosmic entities that are characterized by these metric forms, in an orthogonal and as well Cartan’s orthonormal frame of co-ordinate system, that have been investigated by me are presented in this article. The section thus outlines first, the metric given by Cahill and McVittie [2],
ds²=e^2γ(dx⁴)² – c^-2 [e^2α(dx¹)²+ r² dΩ²]-----------------------(24)
where γ , α and r are functions of x¹ and x⁴ only and
dΩ²=dθ²+ sin²θdφ²------------------------------------------------- (25)
This leads to the expressions for my metric flow vector
h¹= c*u/ (u¹* r²√κμ)-------------------------------------------------(26)
where u and u₁ are same as those defined in ref [2].
The magnetic flow vector h^α =( h¹,0,0,0) satisfies h^αU_α=0 where U^α =(0,0,0,U⁴) is the velocity flow vector of the fluid.
I have studied G₂²- G¹₁ difference that’s equated to κμ|h|² c² where |h|²=- h¹ h₁ and c is the velocity of light and κ= 8πG/ c².
The significant result of this computation is that the factor F in the expression
h¹= F e^–α-γ cosec θ-------------------------------------------------(27)
has the value
F= H r₄ / √κμ cosec θ ----------------------------------------------(28)
meaning there by that the parameter H of the theory represents possibly the magnetic monopole strength of the cosmic entity. This result is a consequence of the new Gauge conditions
r₄₁= r₁r₄/r and r₁₁= 2 r₁²/r---------------------------------------(29)
and moreover with the price γ₁=0. Both r₄₁ and r₁₁ seem to behave as physical entities.

SECTION III: A NEW CLASS OF EXACT SOLUTIONS FOR A LATERAL DEPENDENT SPACE-TIME OF A COLLAPSING COSMIC ENTITY.

In this section certain new formulas are given for the metric
ds²=e^2γ (dx⁴)²– c^-2[e^2α (dx⁴)²+ r²dΩ²]--------------- -----------(30)
with γ, α as functions of r, t and θ. The co-ordinate r is treated as a function of r and t only. The centre of symmetry is defined to be r=0, θ=0. The exponential function e ^2α >0 and e^2γ >0 for all r, t and θ.
The Cartan’s orthonormal basis frame of reference for the metric
ω^t = e^γdt, ω^r= - e^αdr/c, ω^θ = -rdθ/c, ω^φ = -rsinθ dφ/c-------(31)
is used to evaluate the Riemann-Tensor components, connection 1-forms and the Einstein Tensor. Typical affine connection turns out to be
ω^t_φ = (r ₄ e^-γ/ γ + cotθ θ₄ e^-γ)ωφ ---------------------------(32)
where ωφ is a basis function in an orthonormal basis frame of reference.
In the present work, the structured 2-forms are all evaluated, for the metric given in Eq.24. These yield R ^μν_αβ the Riemann-Tensor components that may be displayed in a matrix form. The mass function involves the typical R ²³₂₃ term given by

R²³₂₃ =(c/r)²[1+ (r₄²+ r r₄ θ₄cotθ) e ^-2γ - r₁² e^-2α)----------(33)

The lateral dependent term, is the new additional second power term (compare to a similar expression earlier given by Cahill and McVittie [2] without the lateral dependence). It is important to note that the boundary condition, as well in our formalism, i.e.
2M/r_b = ( R ³ ₂₃₂ )_b
holds good. The derived mass function is
m=M( r/ r_b)³ + 0.5 r [1 - ( r/ r_b)² + r² (r_u²/ r _b² )u² cosecθ- u²( r_u/r )²cosec θ]-----------------------------------------------------(34)
with the gauze condition
r/ r₄= (r_b/ r_4b)---------------------------------------------------------(35)
and that θ₄=1.
Certain typical Christoffel symbols of this metric are
Γ⁴₃₃=sin²[r₄²+ r r₄ θ₄ cotθ) e^-2γ ]; Γ³₃₄= r₄ + θ₄cotθ ; Γ¹₂₁= α_θ
Γ²₄₄= e^2γ c ² γ_θ/ r² ------------------------------------------------(36)
where subscript θ refers to differentiation with respect to θ.


SECTION IV: THE GRAVITATIONAL RED SHIFT:

The Hubble constant H, the deceleration parameter q, and the density parameter σ, are all determined by the measurement of red shift in spectral lines. It is observed, alike in all the spectral range. Large red shifts, from local bodies, can arise from relativistic velocities and from strong gravitational fields.
In my formalism the contributions due to the lateral dependence of the metric, due to the additional magnetic monopole strengths and the cosmological constant are explicitly determined.

The four quasi-stellar Red Shifts associated with NGC 520 are 0.77, 0.67, 2.11 and 0.72 from the galaxy to outward. It was thought that large Red Shift 2.11 is a superposition of gravitational contribution on a general velocity background [10]. In the present work, proper time intervals at the surface of a cosmic entity (like proto star, superstar or pulsating star etc as the case may be) are related to coordinate time intervals by
dτ= ( 1 - 2*M/ r_b )^0.5dt--------------------------------------------(37)
where r_b is the boundary radius of Schwarzschild like metric. For a distant observer 2M/r_b may be ignored and then dτ= dt. Number of waves emitted by the cosmic entity is ν ₀ Δτ and equating this with the number of waves received at the observer ν Δτ_∞, we get the velocity ratio
v₀/v = Δτ_∞ / Δτ = (1- 2M/r_b)^-0.5 --------------------------(38)
and in term of wavelengths it is
λ/λ₀ = Δτ_∞ / Δτ = (1- 2M/r_b)^-0.5 --------------------------------(38a)
or the Red Shift is
zs = (λ - λ₀)/ λ₀ = (1- 2M/r_b) ^-0.5-1 -----------------------(39)
Thus, the explicit expression obtained, in the present work, for the Red Shift is
zs = (r _b/ r_u) { 1+ r _b⁴( q_b + cot θ_b)/( 2 q_b r_u² H² c²) }sin ^0.5θ -1---------------------------------------------------------------------(40)
So
z_s =( r_b/r_u {1+ r_b⁴/(2r_u²H²c²)}---------------------(41)

Another significant result of the present work is that it implies that the solution asserts that the speed of a freely falling test particle at the boundary of the cosmic entity neither approaches velocity of light but also differs in velocity in a lateral accelerated motion. This result augments the finding made earlier by Jannis[11] that in the classical Schwarzschild line element with proper choice of units, the freely falling test particle may not achieve the velocity of light.

SECTION V: A NEW CLASS OF EXACT SOLUTION GAUZE CONDITIONS AND MASS FUNCTION:

As mentioned in the introduction, the present paper deals with the new class of exact solutions for the Einstein Equations assuming that the space-time metric of the collapsing cosmic entity filled with a magnetic fluid. The fluid material constitutes the interior of the cosmic entity at any time and enveloped by empty space.
The co-moving co-ordinate system with U^αU_α=+1 where U^α is the velocity of the fluid [12]. We have the conditions that h4U 4=0 and |h|²=- h¹ h₁ where |h|² is now found from
G²₂ - G¹₁ = κμ|h|² c². ------------------------------------(42)
I have used the gauge condition r₁ = r^13/6 and by abbreviation that q=r₄/r we have the condition that α ₄= q + α_θθ₄ where α_θ=cotθ / 2 and which when integrated led to the relation
e^α = ru₁/u sin^1/2 θ -------------------------------------------(43)
Here u is a function of x¹ alone [2].
We obtained that γ_θ = α_θ = cot θ / 2.
The Eq.43 and the following expression
r₁₄–α₄ r₁- γ₁ r₄ + r r₁ θ₄ cotθ /(2 c²)=0---------------------(44)
yield
(ln r ₄),₁ = ( γ + ln r),₁ ----------------------------------------(45)
γ₁= r₄₁ / r₄ – r₁/ r = r₁/2r ------------------------------------(46)
Integration of Eq.46 yields
e^γ= H r₄sin ^0.5θ / r---------------------------------------------(47)
where H is a constant.
By means of Eq.47 and Eq.42 the mass function
m= 0.5 r[ 1+ e^-2γ( r₄ + r r₄ θ₄cotθ)- r₁² e^-2α )--------(48)
now becomes
m= 0.5 r [1+ (r/Hc) ² cosecθ - (u r_u/r)² cosec θ] -------(49)
The space-time metric-tensor therefore would be
ds² =(qH sin^0.5θ)² (dx⁴) ² – r² (du²sin θ+ u²dΩ²)/( c²u²)-----------------------------------------------------------------------(50)
This depends both on r and θ and their specific values would therefore yield the desired information needed for determination of the properties, of physical interest, of the cosmic entity.

THE NEW RELATIVISTIC LATERAL LAW OF GRAVITATION:

The relativistic dynamical equation, which is presented here is the new relativistic lateral law of gravitation analogous to the Newtonian law of gravitation and that given by Cahill and McVittie [2].The Law enunciated by me, here is subtly different from both the mentioned formulations and is given as follows:
e^-γ(e^-γ γ₄), ₄+ (c²/r r ₄) [e^-2γ(r r₄θ₄cot θ)],₄
= - m c² r² + c² e^-2α(γ₁r₁– r r₁² θ₄cotθ/ (c² r₄² )+
An Energy –Momentum Term).--------------------(51)
Apart from the inverse square law of attraction term, the equation involves terms containing cotθ and γ1 . The equation is significantly different from what Misner and Sharp [13], Bardeen [14] and May and White [4] have examined.

SECTION VI: KINEMATICAL CONSIDERATIONS OF THE NEW CLASS OF EXACT SOLUTIONS:

The acceleration is
Ů _α = c ² γ_θ δ^α₂+ e^-2αc ² γ₁δ^α₁-------------------------(52)
And the expansion parameter is
Θ = (1/3) [γ₄+ 2 q + cot θ θ ₄ ] e^-γ---------------------------(53)
For θ= 90 ⁰ we get, that in a frame of reference both α ₄ and q must vanish independently for θ =0.
A new relation involving the Kinematical factor Θ is formulated and is given below:
3 Θ,₄ U⁴ +(3Θ)² = R ₄₄ U⁴ U⁴ + U ^α;_α-c²e^-2α(2γ₁α₄ ) + e^-2γ[ α₄₄+ 2r₄₄/r + α ₄²+ 2r⁴2/r²+ 4r₄α₄/r+2 r₄ θ₄cotθ + 2 α₄ θ₄cot θ ]-------------------------------------------------(54)
This relation is different from that given by Raychaudhuri [15] establishes the fact that cot θ term arises related with r₄, α₄, and θ₄ which are all x⁴ dependent.

SECTION VII: DENSITY AND MAGNETIC ENERGY BALANCE:

In the present work, an attempt is made to relate the G44 component with the T44 term. The Energy-Momentum T⁴₄=-ρ. We have the density relation as given below:
ρ= 3/( κ c ² H²) + 3(u r₁/ u ₁ r )² ( κ r²) - 1/ r²) ---------(55)
And hence in the special case of
e ^2α= 3 r₁²-------------------------------------------------------(56)
We arrive at
κρ c² = 3/ H²--------------------------------------------------(57)
a balance of density energy with magnetic energy. The value of θ other than 90 0 would result in a different relation for the density of matter. This gives a reverse relation between the density and the magnetic field strtength. Heyl and Kulkarni [16] and Narayana r[17] surmised that in crab like nebula the ultramagnetized old neutron stars or magnetars seem to becom bright due to the magnetic decay of fields of strength ~ 10¹¹ to 10¹² G.They should have high density to be observable as SGR (Soft Gamma Repeaters).

SECTION VIII: RESULTS AND DISCUSSION

1. A new Gravitational Red Shift expression has been formulated. It relates to the boundary conditions of a magnetic fluid filled space-time of a collapsing cosmic entity with the Red Shift value.
2. The kinematical relation obtained emphasizes the occurrence of lateral dependence of the Space-Time in association with the r ₄, α₄ and θ₄, physical entities, which are all x4 dependent.
3. The mass function related to R²³ ₂₃ component of the Riemann-Tensor has an additional term, i.e. a contribution due to the θ dependence. Therefore, a test particle freely falling within this Space-Time on to the boundary of the Cosmic Entity can acquire different velocities in different directions and is unlikely to achieve the velocity of light.
4. An equation of balance between mass-density and the reciprocal of the square of magnetic monopole strength has been derived.
5. A new relativistic lateral Law of Gravitation has been enunciated that found to be subtly different from those forms given by earlier researchers, in the sense that present Law explicitly involves the lateral dependence of the Space-Time metric Tensor of the cosmic entity.
6. The new class of exact solutions obtained is useful to study the discrete sources of X-rays and Radio Waves emanating, from far away, collapsing cosmic entities.

ACKNOWLEDGMENT:

The author is deeply indebted to late Prof. K. Rangadhama Rao D.Sc. (Madras) D.Sc. (London) for his inspiration, guidance and constant encouragement to pursue my research and as well for financing my higher studies of education and for providing an enjoyable and fruitful environment at his research laboratories.

REFERENCES:

* Equivalence principle for macroscopic matter sets m_grav = m_inert where m_inert measures the resistance to the physical entity to acceleration and the m_grav occurs in the Law of Gravitational attraction enunciated by Newton and until to-date m the mass is regarded, as the gravitational charge of a particle. Observers in the elavators freely falling on towards the center of earth,do not sense a gravitational field,therefore the observer(within the restricted time) represents a local inertial frame of reference,which in contradiction to the Special Theory of Relativity, are accelerated relative to each other.Strong equivalence principle states that all local,freely falling, laboratories are fully equivalent for the performance of physical experiments.
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8. a. http://trusciencetrutechnology.blogspot.com/ 21 Saturday, January 3, 2009 A CLASS OF EXACT SOLUTIONS FOR THE FLUID UNIVERSE FILLED WITH MAGNETOFLIUD –LIKE FIELDS trusciencetrutechnology Vol.No.2009,Issue No.1, Dt.7th January, 2009 by Professor Kotcherlakota Lakshmi Narayana, (Retd. Prof of Physics, Shivaji Univ, Kolhapur-416004} 17-11-10 Narasimha Ashram, Official Colony, Maharanipeta.P.O.Visakhapatnam-530002
8b. trusciencetrutechnology, Kotcherlakota Lakshmi Narayana, Ind. Sci.Cong, Session X Ref.paper No.33,page 23,2008 Andhra University.
8c. Tuesday, March 3, 2009, The Nuclear Explosive Fission Break-up of Parent MEL-Planet into formation of present day Earth, Mars and Moon entities of the Solar system trusciencetrutechnology@blogspot.com, Paper No.3 Ind. Sci.Cong. Physical Sciences Section, 3rd Jan 2009
8d. trusciencetrutechnology@blogspot.comMonday, March 30, 2009. time 2:34PM A CLASSICAL SET OF WORMHOLE SOLUTIONS THAT POSSIBLY COEXIST SUERIMPOSED ON MAGHOLES. Professor Kotcherlakota Lakshmi Narayana, (Retd.Prof of Physics, Shivaji Univ) Narasimha Ashram, 17-11-10, Official Colony, Maharanipeta.P.O. Visakhapatnam-530002 dated 30th March 2009.
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13. C.W.Misner, Sharp D.H., Phys.Rev. Vol.136B, p.571,1964
14. J.M.Bardeen 2nd Texas Symp.on Rel.Astrophys. 1964
H.Denhen Inter.Theor. Phys Vol.3, p.509, 1970
15. A.Raychaudhuri Phys Rev Vol.98, p.1123, 1953.
16. J.s.Heyl and S.R.Kulkarni, Ap.J.,Vol.506,p.L61-64,1998
17. R.Narayana Astrophysics.J Vol.319,p.162,1987
18 S N Gupta,Proc Phys. Soc. Vol.65A,p.608,1952.
S N Gupta, Rev.Mod.Phys. Vol.29,p.334, 1957.
19 B.S.Madhava Rao, "Pauli's invariants in the Dirac's Algebra"
Ind. Acad.sci. Vol.xxii,No.6,Sec A,p.408-422,1965
20.s.Desser Gen.Rel.Grav. Vol.1, p.9-18, 1970.

OFFLINE ADDENDUM:
SANKHYATATVA KARIKA dates to 550 B.C and was propogated by Vindhyavasudu in 350B.C.
PROF.SAILESWAR SEN AND PROF.KOLLURU SOMASEKHARA SHASTRI
IN THE BOOK PUBLISHED date: ON SARVADHARI CHAITRA SHUKLA NAVAMI,1950,
by ANDHRA UNIVERSITY, ON SANKHYATATVA KARIKA, GAVE THE SLOKA NO.13
" Satvam Laghu prakashakmistam upashtambhkam chalam charaghha
Guruvaranak Mevathamha pradeepvachhardhatho vrusthiha"
Here Satvagunam is Laghuna,prakasaka
Laghva is the dharma responsible for vahni and other materials to get lifted up against gravity. The book was reviewed by also Duvvuru venkataramana shastri,
suryanarayana sashtri.
Prof. Saileswar Sen did experiments on Artificial Rain, i.e to make the water droplets to freely fall on to the Earth.Prof. Saileswar Sen was a colleague and friend of Late.Prof K.Rangadhama Rao.

A CLASSICAL SET OF WORMHOLE SOLUTIONS THAT POSSIBLY COEXIST SUERIMPOSED ON MAGHOLES.

A CLASSICAL SET OF WORMHOLE SOLUTIONS THAT POSSIBLY COEXIST SUERIMPOSED ON MAGHOLES.
by
Professor Kotcherlakota Lakshmi Narayana,
(Retd.Prof of Physics,Shivaji Univ)
Narasimha Ashram, 17-11-10, Official Colony,Maharanipeta.P.O
Visakhapatnam-530002
trusciencetrutechnology@blogspot.com
dated 30th March 2009. time 2:34PM
key words: Astrophysics,Mathematical Physics,Cosmolgy,Magholes,Wormholes,General Relativity,Redshifts,Throat,Bridge,lateral travel across Universes.
ABSTRACT:

A class of exact solutions of Einstein’s equations for the lateral dependent metric tensor of the form

ds² = e^2γ dx₄²– c^-2(e ^2αdr² +r² dΩ ²)

has been obtained for the cosmic entity filled with a magnetic fluid. In this article, specific attention is drawn to the set of possible wormhole solutions, with and without involving the cosmological constant( for ± values) and the mass of the cosmic entity. The new Red shift expression is given. As well possible new radial coordinates have been obtained to describe the metric in terms of space-time quadrant structure. A new mass function defined and it’s role to assert the Maghole formations is presented.The physical significance of the several findings, has been elaborated by extensive graphical illustrations. Co-existence of Magholes with Wormhole revolutionizes the concept of traversal across universes via novel lateral space-time quadrant structure surfaces.

INTRODUCTION:

The worm hole gas was defined by Hawking S.W. and Page D.N. [1] by a path integral formalism, overall possible asymptotic Euclidean 4-geometry and consideration of matter fields. They assumed that the Energy-Momentum tensor to vanish at infinity and with the excited states of the worm holes surmised to have their sources at infinity Wormholes are thought to produce shifts in effective masses of elementary particles and their interaction constants.[2] Wormholes can make the cosmological constant to be zero S.Coleman [3]. The effects of wormhole instantons have been reported by A. Hosoya and W.Ogura, [4] in the Einstein-Yang-Mills system and they asserted that wormhole instantons might be found both in the sub-planck physics and as well at all stages in lower energy physics. J. Ellis, E.Floreatos D.V.Nanopoulos[5] state that the Quantum gravitational effects might be significant at energies far below the Planck mass m_PL=10¹⁹GeV. Particularly they state that the large volume correction to the gravitational action due to spin-0 bosons and spin-1/2 fermions leads to the up and down quark mass ratio as 44/130.
S.Coleman[6] and S.B.Giddings & A.Strominger [6] gave a wormhole calculus suitable for a digression on quantum topological effects S.W.Hawking, D.N.Page, C.N.Pope[7] and G.V.Lavrelashvili, V.Rubakov and P.G. Tinyakov, [7] have conjectured that the wormholes calculations to modify the elementary particle masses and hence might lead to breakdown of quantum coherence.
A. Einstein, N.Rosen [8], C.Misner, and J.A.Wheeler [8], have given the topology of a bridge that exists between two asymptotically flat spaces. The bridge, similar to a throat, has a narrow region, the circumference of this region at time t=0(Schwarzschild) is given by 2*π*r_sch = 2*π*(2*G*M/c²) =4* π*m (m in units of cm). r would be a time like coordinate to an observer at this region of the throat, with r acquiring the lowest value. For him moving along the geodesic the throat shrinks to pinch-off effect at r=0. In the case of Reissner-Nordstrom solution, the stress of the electric field would keep off the throat from the pinch-off effect. Hence due to the flux of electric field the throat would never achieve the singularity at r=0. Actually, the radius of the throat pulsates between maximum and minimum values possible and for an observer sitting at the throat, in his proper time τ, accounts for the effect of the pinch effect at the throat. It is interesting to note that in the limit that the q²-->m² the Maxwell pressure through the wormhole just balances the gravitational forces. [9]
The special case of q²=m² for Reissner-Nordstrom solution to describe the exterior field of a static spherical charge dust cloud has been presented by Carter [10] adopting a Kerr solution. According to Carter[10] in Type-I ( r>r₊) , in Type-II ( r₊ > r> r_- ) or Type-III( r_- > r> 0) where r₊ = m ±√(m²-q²) both the Null forms of the Reissner-Nordstrom metric are incomplete, when q²=m². J. C. Graves and D. R. Brill [11] used two co-ordinate patches, in the neighbourhood of r₊ and r_- respectively to describe Reissner-Nordstrom metric, with the two patches superimposed and glued together along the boundary r_c .They give the new radial coordinate as
r*=r+ (r₊² * ln(r- r₊)- r_-² * ln(r-r_-))/ (r₊-r_-)--------------------------------------(1 )
Radius r of throat of the Reissner-Nordstrom wormhole as a function of proper time τ, results in the curves r Vs τ as various cycloids. Their period is 2*π*M is independent of charge q. If charge q=0 then the cycloid reaches the singularity r=0 at its cusp.
The quantum wormhole states i.e. solution of Wheeler-DeWitt equation has been researched by L.Garay [12]. P.V.Moriz questions a possible quantum wormhole in N=1 super gravity? [13]P.C.Vaidhya in a paper published on a Big Bang condition [14] gave a theory that smoothly links both the two simple initial solutions of Einstein Equations Viz. the one given by K. Schwarzschild, [15] and the other by Oppenheimer and H.Snyde [15].

T.R.Choudhari and T.Padmanabhan [16] gave tortoise coordinate modified as
r*= ∫ dr (1-2*m/r- H²*r²)^-1 with U=t-r* ,V= t+r* ----------------------------(2)
and r*=∫ dr/f ( r ) where ds²*= f( r ) dt²–f( r ) ^-1 – r² dΩ ² .-------------------(3)

Wesson I.M.T. and M. Israelit [17] have discussed creation of fundamental particles. The cosmological constant space dependence is related to the energy density of electron [18] and as well it influences the effective gravitational mass of astrophysical system [19].
An important feature of the wormhole description is its shape and is determined by the radial metric coefficient dependence on the radial coordinate r. The embedded surface z=z(r) shaping the wormhole spatial geometry is a solution of the equation for the rate of change of vertical distance with r. The solutions for the embedded surface are given by the function z(r), and actual realization of the wormhole is by the surface of revolution of the curve given by z(r), about the z-axis. [20].Graves and Brill [11] stated that the curve needs to be rotated in the θ and φ in case of Reissner-Nordstrom solution in the R-r plane.

SECTION I: A COSMIC ENTITY FILLED WITH MAGNETO-FLUID:

In my previous publications, I have discussed [21], on the nature of Universe filled with magneto fluid and the layered structure and break up of a parent planet {termed the MEL planet into three parts, Viz., Mars, Earth and Lunar entities}. Therefore, I have aimed, in this paper, at an understanding of the intricacy of the lateral dependence of metric tensor and the nature of a collapsing magnetic fluid filled cosmic object. In relativistic magneto-hydrodynamic approach, Yodzis [22] used kinematical parameter and obtained conditions for gravitational collapse of magneto fluids. I elaborate and present new findings of this collapse with regard to novel hole[Maghole might be refered as klnholes] formations.
According to General Theory of Relativity, the metric-tensor, dealt with in this paper describes the motion of a self-gravitating lateral dependent distribution of material. It therefore has only φ as a cyclic co-ordinate. Many aspects of the cosmic entities that posses this metric have been examined in this article. The section thus outlines first, the metric given by Cahill and McVittie [28]
ds²=[(r₄H/r)²dx 4²–(r²/(c² u²)du ²]sinθ - (r/c)² dΩ² ---------------------( 4)
dΩ² = dθ² + sin²θ dφ² ------------------------------------------------------(5)
with u as a function of r alone and r _u adopted as the derivative of r with regard to u( refer [28]) and where
H^-2= (1 - 2M/r_b – (r _u /r)² _b) /[ ( r_b / c)²(1+ 2M/r_b )]----------------------( 6)
The Mass function is defined by me as
(1-2*M/r_b) (1+ r_b² / (Hc)²)= (r _u /r) ² _b------------------------------------(7 )

The significant result of the computation is that the factor F in the expression
h1= F/r ²e ^–α-γ cosec θ-------------------------------------------------------(8)

has the value
F= H r₄ / √ (κμ) cosec θ ------------------------------------------------------(9)
refer my previous publication [21] for the definitions of κ and μ.
Meaning there by that the parameter H of the theory represents possibly the magnetic monopole strength of the cosmic entity. This result is a consequence of the Gauge conditions
r ₄₁ = r₁ r₄ /r and r₁₁= 2*r₁²/r -----------------------------------------------(10)
moreover, with the price that γ₁=0.

SECTION II: A NEW CLASS OF EXACT SOLUTIONS AND THE NEW REDSSHIFT FORMULAS:

A new class of exact solutions obtained for the above metric will be detailed in my forthcoming publication. In this section, I would present the new Redshift formula obtained using the defined mass function. Recent experimental findings on Redshifts necessitates a re-examination of plausible new formula to account for possible novel universes existence [25] and cycles of creation of universes and their destruction[26]. I retain the metric tensor given in Eq.4 & Eq.5

The derived Riemann-Tensor components, Connection 1-forms, the Einstein Tensor matrix elements etc. are all listed[23]. The typical affain connection form turns out to be,
ω^t_φ= (r₄* e^{– γ}/r + cotθ θ₄ e ^{– γ})ω^φ------------------------------------------------(11)
where ω^φ is the Cartan’s orthonormal basis function for the metric given by Eqs.4 & 5. Here r₄ and θ₄ are the derivatives of r and θ with respect to time respectively.
Large redshifts from cosmic entities could arise from relativistic velocities, from strong gravitational fields and, now I state that due to the magnetic filled physical systems. Redshifts of order of 7 etc. are known[24]. In the present article, proper time intervals dτ at the surface of a magnetic fluid entity are related to coordinate time intervals dt by
c*dτ= r₄ H Sin^1/2θ(c*dt)----------------------------------------------------------(12)
where H^-2=(c/r_b)²[(2*M/r_b -1) +( r_u/r)²_b]----------------------------------------(13)
with θ_b=90 ⁰ chosen and, the subscript b implies evaluation at the boundary[see foot note *] of a magnetic fluid cosmic entity. The u function is same as that used by Cahill and McVittie[28].
In terms of wavelengths the Redshift is given by
λ/λ₀ = Δτ_∞/Δτ = ( r/ r₄H) sin ^-1/2θ ------------------------------------------------(14)
And so new Zs formula for the Red-shift obtained is as follows:
Zs=(r/r_b)*(c/r₄)[(2*M/r_b -1) +(r_u/r)²]_b^(1/2)-1-----------------------------(15)
and is about 6.746 for r ₄ =1 ,r _u=1,c=1, r_b =0.5 and M=3.

Fig.1: The graph depicting the variation of Redshift Zs with the distance r. M=3;r_b=0.5;c=1;r₄=1;

SECTION III: THE POTENTIAL ENERGY EXPRESSION AND RADIUS OF APASTRON:

S. Chandhrasekhar in his book on the Mathematical Theory of Black holes [27] has nicely given the methodology of Lagrangian and Hamiltonian formulations of a metric tensor. Using that treatment, I have obtained expression for the Potential Energy for the metric given in Eq.4 & Eq.5.
p_t= E,------------------------------------------------------------------------(16)
p_θ= r² dθ/dτ,----------------------------------------------------------------(17 )
p_φ= r² sin²θ dφ/dτ, and p_r= -d(lagrangian)/d ŕ ------------------------------(18 )
(E²- V²(r )) = e^(2γ-2α)/ c² = (dr/dτ) ² =Ē²------------------------------------(19)
The effective potential relativistic equation for the radial part of motion is given by
V²(r)= (r₄²/r²)*[(c/r)² *(1+L²)- (μ²-1)]/ [(c/r_b)²*(2*M/ r_b²-1 + (r_u /r)_b ²]->(20)
with θ = 90⁰.
The shapes of the potential function are illustrated, below for different values of the variable parameters, listed in the graphs.

Fig.2 The Potential function with the parameter values r_ub=2;M=3;μ=2;L=14;r_b=3;c=1;r₄=2

Fig.3 The Potential function with the parameter values
r_ub=2; μ=2;c=1;r₄=2;M=2;L=10;

Fig.4:The Potential function with the parameter values μ=2;c=1;L=0;r₄=3;r_b=2;r_ub=2;M=2;

The proper time τ that augments the Schwarzschild coordinate by the infinitesimal amount Δr is given by
τ= ∫ dr(uc/u ₁)/[E²r₄/ r₄² + ((μ²-1)] ^1/2----------------------------------------(21)
The radius at which the particle has zero velocity, R_ap gives i.e. apastron position, is as follows:
R_ap²=r₄²*H²(1- μ²)-------------------------------------------------------------(22)
Using this expression, we get, for the proper time τ
τ= (Hc r₄u/ (u₁E) )* 1/R_ap* (sec ^-1(r/R_ap))--------------------------------------(23)
The energy exhibits an average behaviour while the proper time τ it has a pulsating character. This is a curious formula unlike the cycloid motion. It is found to depend on the ratio H to E and as well on the quantity μ used here which the later is same as that defined by S.Chandhrasekhar. The time of fall by the cosmic entity fluid filled mass, slows down with decrease in r, especially when μ=0.

SECTION IV:A NEW MASS FUNCTION AND A NEW RADIAL COORDINATE:

Similar to the method of introducing the tortoise co-ordinate wheeler[27] and Regge[27], and Regge and wheeler[27], we may find an equation for the new coordinate as
r*= ∫ dr*= ∫ dr (c/ r₄)²)*(2*M/r -1 + u r_u/r]---------------------------------(24)
r*=-(c/r₄)²)*(2*M/r²+r+r_u/r²)----------------------------------------------(25)
But of more interest is the situation when a new mass function is adopted Viz.,
(2*M/r_b- 1)(1+ r_b²/a²)=-(r_u/r) ² _b-------------------------------------------(26)
where a=Hc and b prescribes the boundary when u=1.
What about the case when mass=0? Then
a²+r²_b = ((r_uu/r)² _b
yields
r⁴_b- r_ub2+ r _b²=0------------------------------------------------------------(27)
and it yields an equation
a²=( r_uu/r)²_b- r²_b -----------------------------------------------------------(28)
The expression is similar to the one given by S.Chandhrasekhar for a charged space-time metric involving the r³ term and solved by him in terms of JacobiElliptic functional form.

Fig.5:The novel parabolas of the expression given Eq.27
The graph of the identity Eq.28, shows that with r_b, a over a rang -5 to 5,gives parabolas for r_ub=6.This behaviour holds even when M=0.Yet times M is used for the mass, especially when boundary (u=1) specifications are involved. Field function thus is given by
a²=( r_uu/r)²_b- r²_b
This implies possible existence of Magholes realizable by rotation of the parabolas about the field axis.
A new radial coordinate r* arise if we use
2m/r(1+r²/a²)=e^-2α/c²--------------------------------------------------(29)
r*= -2*m * log r+m (r/a)²+constant.-------------------------------------(30)
If m=0 then r*=0 hence constant. The significance of this new radial coordinate is that it is proportional to the mass and depends on the ratio (r/a)² involving the field expression of the magnetic fluid of the cosmic entity.
SECTION V: MAGHOLES & WORMHOLES AND THE COSMOLOGICAL CONSTANT:
In my formulation relaxation of the condition Λ=p/(8*π) results in several wormhole solutions, depending on the presence of tidal forces,with no tidal forces and finally without the presence of cosmological constant. I have used Λ= p/(4*π) for an aesthetic beauty of the formalism.
The expression for vertical lift is given by
Z=∫ [3a²*r₄² *r₁² – a²* r₄² + r⁴–Λ*r² *a²* r₄²)/( a²* r₄² – r⁴+ r²*a²* r₄²*Λ]^1/2dr-------------------------------------------------------------------------------------(31)
The graphs of the lift i.e. shapes of the wormholes are presented below:
a=2;r₁=2;r₄=2;

For the case of a tidal force γ ₁= r₁/(2*r)

and in the absence of it,
Z=∫[a²*r₄² *r₁² – a²* r₄² + r⁴–Λ*r² *a²* r₄²)/( a²* r₄² – r⁴+ r²*a²* r₄²*Λ]^1/2dr-----------------------------------------------------------------(32)
a=2;r₁=2; r₄=2;

And in the absence of Λ and r₁=1, I have obtained a simple wormhole expression
Z= ∫ [r⁴ /( a²* r₄² – r⁴]^1/2 dr------------------------------------------------(33)
a=2;r₄=2;

Rotation about the vertical axis would result in the explicit appearance of the throat or the bridge connecting the flat universes.

SUMMARY:

To summarize I have found a classical set of Wormhole solutions that might exist superimposed on Magholes. The possible role of both the negative and positive cosmological constant Λ in determining the shapes of Magholes and Wormhole has been outlined. Results are easily extended to a discussion on the magholes that exist even in the absence of vanishing mass of the cosmic entity.The curves that chartacterize the Magholes and the Wormholes have been presented for ease of visualization of the physics behind the collapsing Cosmic entity. The present article, has a fruitful contribution to extend the Wormhole calculus for other aspects of quantum gravity and fundamental particle physics, especially in the presence of Magholes. The redshift expression with and without the mass and cosmological constant considerations, for cosmic boundaey radiations has been explicitly given.
ACKNOWLEDGEMENT:

The author is deeply indebted to late Professor K. Rangadhama Rao D.Sc.(Madras). D.Sc.(London) for his inspiring guidance, constant encouragement, for providing me financial support for my higher education and for providing an extremely congenial and fruitful environment at his laboratories in India and scattered world over.
REFERENCES:
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2. Lyons A 1989 Nucl.Phys B324, p.253
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3. S. Coleman Nucl.Phys. B310, p.643, 1988
4. A.Hosoya and W.Ogura, Phys.Letts. Vol.B225, No.1, 2, p.117, 1989
B.J.Keay and R.Laflamme,Phys.Rev Vol.D40, p.2118, 1989
5. J. Ellis, E.Floreatos D.V.Nanopoulos, Phys.Letts.Vol.B 225, No.1, 2, p.121, 1989
6. S.Coleman Nucl.Phys.Vol.307B, p.867, 1988
SB Giddings & A.Strominger Nucl.Phys.Vol.307B, p.854, 1988
7. S.W.Hawking D.N.Page, C.N.Pope, Nucl.Phys B170, p.283, 1980
G.V.Lavrelashvili V.Rubakov, P.G. Tinyakov, JETP Lett.Vol.46, p.164, 1987
8. A. Einstein and N.Rosen Phys.Rev Vol.48, p.73, 1935
C.Misner and J.A.Wheeler Ann.Phys.Vol.2, 525, 1957
9. B.Bertotti Phys.Rev. Vol.116, p. 1331, 1959.
10. B.Carter Phys.Lett Vol.21, No.4, p.423, 1st June 1966.
11. J.C.Graves and D.R.Brill “Oscillatory character of Reissner-Nordstrom Metric for
an Ideal Charged Wormhole”, Phys.Rev. Vol.120, No.4, p.1507, 15th Nov.1960.
12. L. Garay Phys.Rev.Vol.D48, p.1710, 1990
L. Garay Phys.Rev.Vol.D44, p.1059, 1992
13. P.V.Moriz Grav.Gen.Rel. Vol.28, 1996
14. P.C.Vaidhya Current Science Vol.37, p.191, 1968
15. K.Schwarzschild, Berlin Res. Vol.424, p.245-247, 1916.
J.Oppenheimer and H.Snyde Phys.Rev, Vol.56, p.455, 1939
16. T.R.Choudhari and T.Padmanabhan General Rel.Grav. Vol.39, p.1789-1811, 2007
17 I.M.T.Wesson and M.Israelit, General Rel. Gravitation Vol.40, p.2469-2492, 2008
18. Ray S, Bhadhra S. Phys.Lett.A322, p.150, 2004
Ray S, Bhadhra S. Sengupta gr-qc/103103,
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The "cosmod" transformation proposed by the present author (KLN) in 1976-1977 relates the graviton mass with the rest mass energy of a Spin 2 elementary particle ( f-meson) and I have also classified it as an element under unitary transformations of a nonet.As a result I conjectured the existence of a second kind of gravitational force and possible existence of strange quark stars etc. The conjecture was widely reported in national news papers of 1977.
20 H.Weyl in Space,Time, Matter Dover Pub., NY. P.259-260, 1952.
L.Flamm Physik Z., Vol.17. p.448,1916
21. (a)

Saturday, January 3, 2009 A CLASS OF EXACT SOLUTIONS FOR THE FLUID UNIVERSE FILLED WITH MAGNETOFLIUD –LIKE FIELDS trusciencetrutechnology Vol.No.2009,Issue No.1,Dt.7th January, 2009 by Professor Kotcherlakota Lakshmi Narayana, http://trusciencetrutechnology.blogspot.com/
(b)Tuesday, March 3, 2009The Nuclear Explosive Fission Break-up of Parent MEL-Planet into formation of present day Earth, Mars and Moon entities of the Solar system : trusciencetrutechnology@blogspot.com

Ind Sci.Cong. Physical Sciences Section, Paper No.3 Ind Sci.Cong. Physical Sciences Section , 3rd-7th Jan 2009.
22 P.Yodzis ,Phy.Rev Vol.D3, p.2941,1971
23 refer forth coming article on this space-time to be published at this blogspot.
24. A.I.Lilley,M.S.Robewrts, EEEpstein, Dieter N.H. Astron.J Vol.67,p.270-271,2008
25 L. Klebanov,L Susskind, T.Banks-SLAC preprint SLAC-PUB-4705,1988
26 "Scientific American" Number of Articles on Cosmology published during July 08-Feb.2009
27 "The Mathematical Theory of Black Hole" by S. Chandhrasekhar,
Clarendon Press,Oxford, 1983.
J.A.Wheeler "Geons", Phy.Rev Vol.97,p.51-536,1955
T.Regge, "General Relativity without coordinates"
Nuovo Cimento Vol.19,p.558-571,1961
T.Regge and J.A.Wheeler, Phy.Rev. Vol.108,p.1063-1069,1957
28 M.E.Cahill and G.C.McVittie J.Math.Phy.Vol.11, No.4, p.1382, April 1970
*Shocks at boundary discussed by Von Neumann in J.Appl.Phys. Vol.21, p.232 ,1950 and Collapse kills its own Schwarzschild radius i.e. Rsch= 2*m(μ)G/c 2 > R(μ)
@ Formulation of wormhole instanton is given by A.Hosoya and W.Ogura [4] with a classical action as
S_cl=-3/(126a²*Λ) + 3*pi/(4*α) ln (4*sqrt(e)/r₀H) +O(H) where H= 8/3*pi*G* Λ and α₀= f²/(4*pi) with f as fine structure constant, a=(r^2+ r₀²)^1/2) with r₀ as the wormhole radius., and Λ the cosmological constant.

400+ old yrs Steatite rock soup pot from Tamilnadu by Prof. E.V. Gangadharam.avi

The Study Meeting on LINAC establishments at Visakhapatnam

trusciencetrutechnology Vol.2009 No.2 Dated 1st Feb 2009 11.30AM
THE STUDY MEETING on LINAC establishments at Visakhapatnam .
By
Prof. Kotcherlakota Lakshmi Narayana and Prof.R.V.S. Sita Ram
trusciencetrutechnology@blogspot.com
The new technology of LINACas a sealed off ultrahigh Vacuum High power device is coming at a fast rate. What you can do about it? There are several channels, to which you can talk. The idea is to create LOCAL neutron source facility and isotopes availability. How to posses the maximum equipment? Here is a capability to promote, of course, with the objective of which accelerator to be used and for what end use? Man power training in high technology and develop suitable formulations for isotopes utility.
Prof. R.V.Sita Ram,Professor in T.I.F.R at time of Transfer to SAMEER as its first, founding Director SAMEER, Powai, Mumbai has outlined the on the 29th January 2009 between 5.00PM to 6.45 PM, the strategy to establish the LINAC. The study meeting was held at the residence of Late Gorthi Venkata Rao (former Professor of History, AU, Visakhapatnam) in the first lane of Official Colony, Maharanipeta, attended by Prof E.V.Gangadharam, Dr.D.V.N.Sarma, Dr. Gorthi Padmavathi and Prof. K. Lakshmi Narayana. The meeting was initiated by the persistent effort of Shri Dr. Sudha Shankar Bhide of SAMEER who has developed advanced technology for Sealed off compact high energy, high power LINAC tubes and systems. Actually the research on accelerators was an outcome of Dr. Homi. J. Bhaba’s concern of development of clean radiation sources for High Energy studies and radioactive prospecting in India . Shri Sita Ram studied at Narasannapeta in Vizag and did his Intermediate course study in a college in Parlkamidi Town bordering Orissa state. He complete his B.Sc. M.Sc. degree courses in Banaras Hindu University , and joined the TIFR in 1947 after having been interviewed by Dr Homi J Bhaba. Prof R V. S. Sita Ram He worked and drew the preliminary designs of accelerator designs in Mumbai and worked in the environment of T.I.F.R Mumbai in its successive locations at Peddar Road ,Old Yatch Club, and present T.I.F.R Building .

During 1947-1960 Prof. R.V. S. Sita Ram worked on the properties of cosmic rays, particle detectors, high speed electronics, Radio Astronomy and other various programmes of T.I.F.R. In 1960, T.I.F.R used an accelerator for a long time obtaining Electronics and Gamma Rays of energy 3.5 MeV. Acceleration was achieved in the field of Microwaves.
But H.J.Bhaba was more concerned of development of country and to apply the scientific knowledge. The first step was to have Geological Prospecting and portable GM counters were designed and developed. Mostly U and Th sources underground were detected at several places in India . Shri Sita Ram has provided the electronic circuits for such equipment. The idea was to quantify the actual Radioactivity mines.
Prof. R.V. S. Sita Ram had opportunity to work in UK accelerator Laboratories,during 1950-1952 when their Theory and Technology was just being discovered and applied.
When Prof. R.V. S. Sita Ram took over the Directorship of SAMEER he concentrated on electron linear Accelerators that use of high power Microwaves while Cyclotrons and Synchrotrons use relatively lower radiofrequencies. SAMEER developed LINACS for specific applications for cancer therapy in hospitals and for Non-Destructive testing Radiography for Space Applications and Industry. They can also be used to produce Isotopes and Neutron beams.
Some of the Radar Applications contributed by SAMEER are MST Radar at Tirupati, Radiotheodolites used at several Meteorological stations,Tracking Radars used in ISRO, weather Doppler radars now in experimental stage.

Man power training in proposed high technology areas of Isotopes and neutron beams, helps developing countries to catch up with developed countries like China, Japan, Europe,USA and Russia

The Nuclear Explosive Fission Break-up of Parent MEL-Planet into formation of present day Earth, Mars and Moon entities of the Solar system

PAPER SUBMITTED TO MATHEMATICAL SCIENCES on 06th September 2008, of THE 96TH INDIAN SCIENCE CONGRESS HELD DURING JANUARY 3 TO 7, 2009‏
From: Lakshmi Narayana Kotcherlakota (lakshminarayana.kotcherlakota@gmail.com)
Sent: 06 September 2008 16:01PM
Accepted on 10th November 2009 by the SECTION OF PHYSICAL SCIENCES
The Nuclear explosive Fission break-up of parent MEL-Planet into formation of present day Earth, Mars and Moon entities of the Solar system.By
Kotcherlakota Lakshmi narayana
Tru-Science & Tru-Technology Organization,
17-11-10,Official Colony,Maharanipeta.P.O.Viskahapatnam
Email ID: Kotcherlakota_l_n@hotmail.com
Key words: Nuclear Fission, Core Model, Earth, Moon, Mars, MEL-Planet, Moment of Inertia, debris of explosion, primary and secondary processes.
ABSTRACT:
The Earth, Mars and Moon were thought and modeled to have existed as a single planet at about 1.26AU from the Sun. The known facts and physical data of Earth, Mars and Moon have been neatly incorporated in the parent MEL-Planet to get its Moment of Inertia, Surface Tension and Nuclear Fission Explosion event parameters. The primary and secondary processes of Fission were envisaged to arrive at sequential formation of firstly the Mars planet and secondly the Moon planet from a single parent planet termed the MEL-Planet. Simple considerations of deformation theory of the MEL-core was found to suffice the explanation of the orbital parameters of present day Earth, mars and Moon entities of the Solar system.
Email ID: Kotcherlakota_l_n@hotmail.com
INTRODUCTION:
The ancient scriptures of India have given hymns describing the formation of the planet Mars (ANGARAKA) from a parent Earth-like planet. It would be an interesting exercise and a model development to account for the formation of Mars as a daughter planet of a parent planet, which I term as the MEL-Planet. The essential features of Earth, Mars and Moon are necessarily to be endowed as well in the description of MEL-Planet. The aim of the present article is to project an account of the quantitative results obtained from my model considerations and simulation studies.
The recent and the past Moon Voyager space vehicles and man-made flights(the exact number unknown)have all concluded in general that Moon is constituted by a Solar or extra-Solar genesis. The main reason behind such an argument is that the composition of soil, rocks, grains and the lunar dust of the Moon interior and surface do not bear a relationship with Terrestrial or other Solar planetary material. The model envisaged here in this paper holds a contrary view.
THE MODEL OF EML-PLANET:
The EML-Planet was existing eons ago constituted by the material that now exists in the Earth, Mars and Lunar entities except for the fact that over millions of years(if not billions) of ageing certain morphological and physical manifestations have changed them drastically, quite radically and beyond even an easy and ready comprehension.
The core of the EML-Planet, existing at that time presumably at a distance of 1.26AU was not that much different from the material of present day Earth, Mars and Moon core materials. The present day Mars and Moon core sizes are so small that fit into neatly as a package of the core of the MEL-Planet. This is a significant aspect of my model of MEL-Planet core composition.
The neat fit of the same core material of course, need to be understood only in terms of the medium properties of the MEL-Planet core that was existing at 1.26AU some eons ago. The exact date when the MEL-Planet disappeared giving rise to the daughter planets viz. Earth) at 1AU), Mars (1.52AU) and Moon (a few km away from earth) is not mentioned in the available scriptures in India. It may not be that much difficult to ascertain it from the present-day technological measurements. The process of the break-up of the MEL-Planet is also not elaborated in the scriptures available in India. It is my firm conviction that MEL-Planet existed at 1.26AU with Bodes’ law validity as a questionable.
The core-mass Matrix is given below. It leads to the possible eigenvalues in which it the core can manifest itself.
Table No.1 The Mass-Core Matrix
4.308057 -3.84772 -0.41308 -0.04725
-3.84772 -3.43657 -0.36894 -0.0422
-0.41308 -0.36894 -0.19707 -0.00453
-0.04725 -0.0422 -0.00453 -0.00052
This matrix gives the levels in which the possible break up of the MEL-Planet could take place. The Nuclear Explosive Fission break up of the MEL-Planet led to the formation of the Earth, Mars and Moon daughter planets is what my model asserts.
CERTAIN PHYSICAL FEATURES AND RELEVANT DATA:
Obviously the mass of MEL-Planet should satisfy the equation
Mmel = (4/3) *π*R 3 mel * ρ mel ………………………..eq.1
where ρ mel is the density, R mel is the core radius and the general equation of pressure is given by
-p0 (r) *4 *π*R 2mel +(3/20)* G * M2 mel / R 2mel =0. ……..eq.2

where p0 (r) is the pressure. It is well known that the core of the Earth is about 2898km. By the same arguments I have used the core sizes of Mars as about 1533.83km and that of Moon about 789.87km. I have derived a value of R mel = 6716.246km which is not much different from the mean Earth’s radius of about 6376.65km.

The core-mass Matrix is given below. It leads to the
possible eigenvalues in which it the core can manifest itself.

Table No.2 The Moment of Inertia and the Rotational Energies
Planet/ Satellite Moment of Inertia Rotational Energy
Units of Kg.m2 Joules
MEL-Planet 1.934E+37 5.11E+28
Earth Planet 1,55E+37 4.14E+28
Mars Planet 4.6823E+35 1.18E+27
Moon Satellite 1.42036E34 4.31E+22

CORE-MANTLE SIZES:
The Earth body has been divided into Six zones A, b, C, D, E, F, G with respective distances mentioned as about 33km, 413km, 984km for A, B and (C,D) and E at about 2898 km and the remaining distance of the radius of Earth, for the F and G zones.
The core size adopted in the present article of course, refer to the core sizes that were existing inside the EML-Planet at about 1.26AU distance from the Sun. The values adopted by me for the core-sizes are physically plausible magnitudes, since they (should) reflect finally the formation of Earth, mars and Moon entities. The sizes do differ in view of the eons age of evolution of from the date of their formation, through which evolution they have evolved to the present day core-mantle structures.
THE NUCLEAR FISSION BREAK-UP MODEL:
We need to know the surface energy (larger than surface tension)
That holds the planet MEL in a stable condition. The model envisaged here readily offers the surface tension value of 5.4987E+53 N/m. The core has a characteristic oscillation period of about 1.65E-17sec. For the formation of the Mars, Moon and Earth system in the break-up process of the MEL, I have obtained the Energy magnitudes of ejection of Mars, Moon and Earth from the MEL-Planet as about 1.93E+32 J, 3.84E+28J and 2.65E33J respectively.
(a)Primary Explosion:
This is a break through finding of the author’s model. These energies essentially give the total Nuclear Fission Explosion energy of the system.
The magnitude of the explosive energy corresponds to the trapped Hydrogen gas of the MEL-Planet as about 4.5E+18Kg. Thus the MEL model gives an insight into the Physics of the core material of the MEL-Planet. There are several papers that think of a solid crystalline Hydrogen core of the Earth Planet and some others contend it is only consists of the Nickel Iron in Liquid /solid form. The primary might not have anyway exhausted the entire hydrogen content of the MEL-Planet and so solace is for the scientists who argue about the presence of the crystalline Hydrogen inside the core of present day Earth.
The Nuclear (thermo) explosion fission led to the fission fragmentary components of the MEL-Planet and they eventually formed the cores of the Earth, mars and Lunar entities ,in a sequence of primary and secondary explosive processes.
(b) THE MATERIAL OF MOON (secondary explosive process):
At first Mars was created and then the secondary explosion led to the creation of Moon using the remnants of the primary process of thermonuclear explosion of MEL-Planet.
This is another significant finding of the present model that the material of the Moon differs essentially from the material of the Earth or the mars, due to the fact, that Moon is constituted by the debris of the primary Nuclear Fission explosion of the MEL-Planet core years ago.
CONCLUSIONS AND RESULTS:
1.MEL-Planet existed at a distance of 1.26 AU from the Sun.
2.MEL-Planet broke up into fission fragments of present day Earth, Mars and Moon
3.The material of moon was the debris of primary nuclear fission process break-up of the MEL-Planet.
4.The less energetic secondary explosion has thrown off the Moon material from the Earth’s core, immediately after the primary explosion.
5.The energy levels of the MEL-Planet core provide an interesting insight into the gravitational and temperature profiles of the planetary systems during the early stages of their formation and as well before the break-up into fission fragments.
6.The known facts and physical data of Earth, Mars and Moon neatly fit into the MEL-Planet constitution and structure.
7.The nuclear Fission Explosion break-up of the core of MEL-Planet leads to the present orbital parameters of the Earth, Moon and Mars entities effectively and with reasonable accuracy.
8.The core oscillations frequency of MEL-Planet was found to be of the order of 1.65E-17sec, which is a physically reasonable value.
ACKNOWLEDGMENT:
The author wishes to record his deep sense of gratitude and indebtedness to late Professor K. Rangadhama Rao D.Sc. (Madras) D.Sc. (London) for his constant encouragement and guidance in my research work. He is also grateful to IAEA, SAE and UNESCO for grant of an International Fellowship to him during the year 1964-1965 which made it possible for me to visit various European and Scandinavian research laboratories.
REFERENCES:
1.The Atomic Nucleus by RD Evans McGraw-Hill Co.NY, p.387 (1955).
2. W.J.Christensen Jr. Gen Rel Grav Vol.39, p.105-110, (2007)
3. an article by Neven Bilic et al p.46-52. on Thermonuclear Instability
*Ancient Sanskrit scriptures of India mentioning some hymns of creation of Angaraka (Mars) planet from Earth. Bhakti monthly magazine June/July/Aug 2008.

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I. On the origin of Moon-Earth system:
Doc for trusciencetrutechnology.blogspot.com dt3rd March 2009 copyright held
Alfven and Arrhenius have stated that Moon could not have been generated in the environment of the earth by a normal accretion process as claimed by some authors. They proposed that Moon was also a captured satellite with several aspects of tides evolution and capture orbit not understood completely. Of course, they point out that Earth-Moon system with a satellite-to-planet ratio of 1:80 is different from normal systems the general approach proposed by them is the transfer of angular momentum from the central body to prograde secondary bodies. This can take place by hydro magnetic processes occurring in the plasma environment of the central body. A simple formula of gravitation potential κΦ has specific values where Φ =Mc/R orb with R orb is orbital radius of secondary body and κ the gravitational constant. Mars has satellites Deimos at 34.2 and Phobos at 13.5 earth radii. Their analysis stresses that Luna planet accreted in a jet stream near Earth’s orbit. The jet streams of formation of Moon and that of Earth cannot be asserted whether Luna jet stream was located inside or outside Earth’s jet stream.
Chemical compositions of the two jet streams are different. Moon accreted with cool interior while Earth formed due to a hot core. Original lunar orbit intersected Earth’s orbit and due to frequent Earth-Moon encounters resulted eventually to its capture by Earth.
H. Alfven ICARUS vol.8, No.3, p.215-226, 1964
H. Alfven and G. Arrhenius the Moon vol.5 p.210-230(all rights reserved)
Tides make the Moon’s distance increase, means that when extrapolated backward Moon was closer to Earth. Darwin was suggested by H. Alfven to have calculated the variations of Moon’s Orbit and half century later Gerstenkorn solved its orbit elements. During 1950-60. See ref L H Alfven, Origin of Moon, Science Vol.148. p.3669, 1965.

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II. Chadrayaan mission meant to explore Lunar surface by simultaneous photo geological and chemical mapping to identify different geological parameters. The great Impact hypothesis is what the Editorial of Dream2047 vol.11, No.2, p.21, 2008 of India feels the currently accepted hypothesis on the Orgin of Moon. It proposes that a body of the size of Mars slammed into Earth shortly after formation of Iron core interior. The impact took place with the blasting off a big chunk of Earth into space that ejecta began orbiting Earth. The ejecta coalesced into Moon. It seems that Impact Hypothesis explains missing Iron of the Moon, similarity of rocks of Earth and the Moon and the tilts of Moon and Earth orbits respectively.

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III. Note on Mars:
The Vedic hymn on Mars is as follows:
“ Angaraka Mahiputra Bhagvan! Bhaktavatsala
Namaste Mamashesha Runamashu Vimochaya”
Pedda Bala Shiksha of Hindu schools has this sloka to be learnt by school children. Once upon a time the Mars was a fiery Planet. Also mentioned by Devarakonda Sheshagiri Rao in Kumudam Bhakti Special, page 43, July 2008

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A new Model by J.J.A. Getino and A. Escapa from Universities of Valladolid and Aliconta explains movements of Moon according to ScienceDaily News dt. Dec 20, 2008. They have new formulas to study rotation of Moon, based on its structure of a solid external layer and a fluid internal core. Their model proposes among other things the study of multi-layer theories to study rotation and movements of the Moon based on the small changes that take place within the Earth’s rotational axis. Kopeikic et al., Adv. Space Res., Vol.42, No.8, p.1378, 2008. They claim that Laser ranging would allow Earth-Moon distances to be determined to the accuracy of millimeter.

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.