ON THE NATURE OF MAGNETO-STRANGE FIELD FLUID FILLED COLLAPSABLE COSMIC ENTITY, KILLING VECTORS AND A SET OF EXACT SOLUTIONS

trusciencetrutechnology Vol.2009, No.5, Dated: 4th May 2009: Time 10:12AM

trusciencetrutechnology@blogspot.com

ON THE NATURE OF MAGNETO-STRANGE FLUID FILLED COLLAPSABLE COSMIC ENTITY, WITH STRANGE QUARK MATTER MASS, KILLING VECTORS AND A SET OF EXACT SOLUTIONS
by Professor Kotcherlakota Lakshmi Narayana,
( Retd.Prof.of Phys,SU,Kolhapur-416004)
17-11-10, Narasimha Ashram, Official Colony,
Maharanipeta.P.O.Visakhapatnam-530002

ABSTRACT:

In this paper ,I have investigated the magneto-strange field fluid filled cosmic entity with strange quark matter (like a super star), in the lateral dependent space-time metric that admits a Group of Motions and collineations. The Einstein Field Equations have been solved and their relation with a full set of killing vectors has been established. The strange fluid mass( to be termed the strange charge) and the strange quark mass expressions,use of them in judicious way, led to the New Gravitational Red Shift formulas. Also the vertical lift in an embedded cylindrical symmetry has been obtained for the several set of the cases, with and without the cosmological constants.The numerical data and the graphical illustrations are given for a discussion on the features of the set of exact solutions obtained. These are presented to support the claim of existence of magneto-strange holes(Magstra holes) concomitant with the worm hole solutions. The General Relativity approach designed, to a fresh understanding of the Elementary Particle and High Energy Physics is given


INTRODUCTION:

A R Bodmer[1] has discussed and gave a descriptive presentation of collapsed
and in the context of collapsed process he stated that for the case of Q=o i.e.
equal number of quarks, the driving force is the mere difference
m_λ-m_n of the number of strange quarks, over the non-strange quarks. He mentioned that the average temperature,T _ave for a collapsed state relative to normal, one satisfies the ratio
T_ave/T_normal = 0.6.
He reports the values
y= hk_F/2πMc=2.3, R_N=r₀A^1/3
and the breathing mode υ as given by υ ^-1= 1.7E-22 sec.
Clear mathematical formalism of possible hyperon etc. stars has been given by
Naoki Itoh [2], with the idea that quarkian core to exist inside a super star
of large mass. He distinguished between para-fermion and fermion statistics to obtain expressions for macroscopic energy density of a quark star, the baryon number density, pressure and internal energy of gas, taking the magnitude of quark star mass as 10E-03 M_ө.

He discussed the Equation of State (EOS) and the unit of mass [7, 8, 9].In the year 1960, V.A. Ambantsumyan and G.S.Saakyan[3, 4, 5, 6]reviewed the neutron and hyperon stars. There are two ways of formation of quark matter [10,11] the quark-hadron phase transition in the creation stages of the universe and compression of neutron stars into strange ones at extremely high densities of the cosmic entities during their collapse. The idea of existence of strange matter, the exotic matter and the hyperon core matter, as constituents of the stars are being researched extensively.These are assumed to be endowed with magnetic fields of the order 10E+14 or even higher.

I have enunciated earlier that the existence of magnetic holes superimposed on wormholes bridges the fields of primordial magnetism and the presence of exotic matter [14]. This suggests that the information flow ideas have to be re-examined in the light of superimposed holes[15].

The presence or absence of cosmological constant and its spatial dependence have also been proposed to characterize the nature of Wormholes by several authors [13].
The later violates the null energy condition of the stress energy [16].The scalar field of scalar tensor theory of gravity may play the role of exotic matter. The energy density of vacuum thought to constitute about 70% of universe, and thought as related to the cosmological constant [17].

The strange stars add a new dimension to these ideas. [10]. Bag model of quark strange matter is nicely fitted to in the Einstein Equations, in spite, that it is strong interactions that are responsible for the breaking up of physical vacuum inside the hadrons. With flavours and colors included the number of quarks may exceed 48 and their 48 anti-quarks and possibly more due to characterization by cosmology, super-gravity, and parton structures etc. For a review of strange stars based on the Bag models and the earlier versions of the strange quark matter star models, refer the article by Aktas and Yilmaz [10]. Magnetized quark matter solutions of Einstein Equations admitting the conformal motions, they used the magnetic pressure given by p_mag=h ²/ 2 and the equation of state (EOS) for strange quark matter with p= (ρ-4B)/3 where B is the bag constant, with quark matter in a color superconducting phase. This way of bridging the strong interaction physics and the gravitational Einstein Equations is an interesting approach, to arrive at several forms of space-time geometries, depending on the types of EOS of strange quark matter. The results obtained by them seem to confirm that the candidate pulsars like Her X-I,4U 1820-30,SA XJ 180804.4-3658,4U 1728-34,PSR 0943+10 and RX J1 85635-3754 earlier thought to be neutron stars are indeed the good strange stars. To differentiate a neutron star with the one of a strange quark matter “core”, it is emphasized that color superconducting phase, results in an anti-Meissner effect that boosts the internal magnetic field to magnitudes higher than found in neutron stars. Neutron stars are, of course, presumed to have cores only of nuclear matter.

From the literature, it is obvious that strange matter is associated with strange
stars in the form of quarkian cores. Similar is the traditional concept of a hyperon star with the core material constituted by hyper nuclei and the exotic hypernucleii.
Dalitz [18] mentioned that, a hyper heavy nucleus has a shorter lifetime and
a candidate example of it is _ΛPb ²⁰⁹. Refer K.L.Narayana’s concept of resonating magic core model for hyperon nucleii[1].

Apart from the cores of pulsars and the magnetized neutron stars, Ginzburg [19], states that the nature of QUASARS and their cores and those of galactic nuclei is not clearly understood. He said that black (holes?) might not be conceived as the cores of galactic nuclei. Priority of studies, thus obviously shifted to formulate models of exotic, strange and hyperon quarkian cores of superstars. The dark energy brief history has been given by Sivaraman. [19]

From the above digression on strange stars (with strange quarkian cores) and hyperon stars of hyperon nuclei cores[1] it might be surmised that direct incorporation of the varied core matter to obtain the physics of the exotic matter cosmic entities is a promising and fruitful enterprise for both the relativists and particle physicists.


Therefore, I have aimed,in this paper, at an understanding of the intricacy of a lateral dependent metric tensor on the nature of a collapsing entity filled with both the magnetic and strange fluid fields and the quark strange matter and the cosmological constant.

The paper is presented as follows:

In Section II, Einstein Field Equations and their solutions are obtained for magnetic and strange fluid fluids and strange quark matter for the case of a lateral dependent space-time metric tensor by using Group of motions. The killing vectors have been determined and they are listed. Properties of these killing vectors relative to the parameters of the lateral dependent metric-tensor have been discussed.

In Section III solutions of the Einstein Equations are utilized to get the valid expressions for the Gravitational Red Shift for the two cases of strange and the strange quark matter.

Section IV discusses the embedded surface solution in terms of characteristic physical entities, strange field fluid, magnetic fluid, and the strange quark matter. Several graphs are illustrated to ascertain the numerical range of values for the said characteristic physical entities of the metric-tensor space-time universe. Also some formulas are explicitly evaluated and their numerical data is presented.

Section V. presents a discussion and concluding remarks giving a list of important results obtained by the present investigation.

SECTION I: A COSMIC ENTITY FILLED WITH MAGNETO-STRANGE FIELD FLUIDS AND STRANGE QUARK MATTER:

In my previous publication, I have sought the possibility of magnetic holes (kln holes[14D]) to co-exist superimposed on the Wormholes.

In this paper,I have used the line element:

ds² = - e ^ν(r,θ,t)dt² + e ^{λ(r,θ, t)} dr² + r² dΩ²-----------(1)

with
dΩ²= dθ² + sin²θ dφ² ,with x¹=r,x²= θ, x³= φ , x ⁴=t.--------------------(2)

The geometrized units have been adopted.

We treat the coordinate φ as a cyclic.
The metric form is same as that used by me earlier but with difference now that
the physical parameters are endowed (as it would become clear at a later stage
of this article) with properties of strange quark matter and the strange field fluid.

The famous Einstein Equations adopted are,

R_ij – ½ R g_ij=8πT_ij-----(3)
The total stress-energy tensor for the strange star with magnetic fields become

T_ij= (ρ+ p + h ²) u_i u_j + (p+h² /2) g_ij – h_i h_j----(4)
where h ² /2 and
u ⁱ=δⁱ₄e-^ν(r,θ,t)/2
with the condition that
h_i u ⁱ =0.-----------(5)

Earlier I have attempted to derive the modifications that would arise due to the
lateral dependence of the metric tensor. An insight into new physics can be possible.
It might be necessary to use homothetic motions that allow one to examine the distribution
fluids in non-spherical symmetry cosmic entities.

I skip all the notation details of the metric-tensor used (since they are explained in my previous publication) and this section directly deals with the characteristics Group of motions and collineations.


SECTION II: KILLING VECTORS AND A SET OF EXACT SOLUTIONS OF EINSTEIN EQUATIONS

I have developed the model of the cosmic entity, based on the physical feature that, the parameter H of the theory(see below) represents, not only the magnetic monopole strength, but as well is related to the mass of strange field fluid M_F.

The magnetic fluid flow vector
h¹= F/ r²e^{–ν/2–λ/2}cosec θ
involves the factor F= H r₄ / √ μκcosec θ,
but now that, H is related with strange field fluid mass M_F.

I have obtained the following Einstein Tensor Equations:

G ¹ ₁=e ^–ν(-2 r₄₄/r – r ₄ ²/r ²+ r₄ν₄/r)+ e^–λ(r₁² /r ²+ r₁ν₁/r - 1/r²)+ e ^{– ν}(θ₄² – cotθ θ₄₄ -3 r₄ θ ₄ cot θr + cot θθ ₄ν₄ / 2)+ ( ν_θθ/ 2 + ν_θ ² / 4 + _θ/2 cot θ )/ r²
= 8 πp-Λ --------------------------------(6)

G²₂-G¹₁=e^–ν (λ₄ν₄/4 - λ₄₄/2 - λ₄²/4 + r₄₄/r - r₄ ν ₄/ 2r + r₄ ²/ r²)+ e^–ν( -λ_θ4cotθ+r₄θ ₄cot θ+ 1/r² + e ^{– λ}( ν₁²/ 4r₁ / 2r + r ₁₁/r) –λ r ₁/ 2 r - λ ₁ν₁/ 4 + ν₁₁ /2 - r₁² /r ²)+( ν_θλ_θ/4+cotθλ_θ/2 - ν_θ ²/4-ν_θθ/2)/ r ²
= κμh ²+ 8π(p_t – p)-Λ+ a possible term due to the choice of the cosmological constant+ another term to account for bag constant.-------------------------------------------(7)

In this work, attempt has been made to obtain the results when the space-time admits the group of motions.

L_ξ g_ij = ξ _{i ; j} + ξ _{j ; i}=(Aδ ⁱ ₁, Bδⁱ ₂,Cδⁱ ₃,Dδⁱ ₄)ξ_i;j
with g _{i j}=0 i≠j--------(8)

where L_ξ is the Lie derivative, semi-colon indicates covariant differentiation, and here A,B, C, D are determined for the metric tensor given by Eq.1,and the following expressions for the killing vectors have been obtained:

ξ ¹ = ψr/2r₁, ξ²= -ξ⁴θ₄ , ξ ³ =- ξ⁴θ ₄, ξ⁴= ψr/2r₄------------------(9)

The expressions

e ^ν = r ₄ ² H ² sinθ / ψ ² and e^λ =ψ ² sin θu ₁²/ u ²-------(10)

where ψ is an aribitrary function to be ascertained, (depending on the form of the metric-tensor space-time to be investigated) and the killing vectors given in the Eq.9 contain all the implications from the existence of Group of motions and the concomitant collineations.

The gauge conditions r₄₁= r₁r₄/r
and r₁₁= 2r₁²/r
can be retained to study the physical characteristics of both the killing vectors and the different formulas of experimental interest and possible verification of the present theory.
It is noteworthy, that the second gauge condition mentioned above, would effect the radial variation of the radial component of the killing vector to vanish. This would lead to a restricted model.

A rough comparison with the constants given in ref[10] implies

C₃ ² = r² sin θ and
1/ψ = u₁/u and

C₂ ² = r² H₄ ²sin θ / r ⁴

But I have used ψ= r to simplify and get a reasonable set of killing vectors with θ₄ =-1.

For the case of θ= 90 ⁰ the values of θ₄ plus or minus 1 do not affect the set of Exact Solutions and other formulas derived in the present work.

There is a possibility to choose ψ= u / u₁ and that
would result in another of exact solutions of Einstein Equations.


The essential difference, in the present work, from my previous results, is that I define the existence of strange field fluid mass, related to R ³ ₂₃₂ a component of Riemann-Christofell Tensor.

2M_F/r²_b =[(1+ e ^-νr₄²+ r₄ θ₄ r cot θ)-e ^{– λ} r ₁ ²] _b------------------------(11)

where M_F is the=================mass,”b” subscript is an abbreviation for the boundary. This definition adopted by me radically differs from the one given by Cahill & McVittie[20]

The strange mass function may be defined in the usual manner as well, I have adopted [14e, 20]

i.e. m = 0.5 r R ³ ₂₃₂----------------------------(12)

with R ³ ₂₃₂ as a Riemannian-Christoffel Tensor component.

Section III:A NEW CLASS OF EXACT SOLUTIONS AND THE NEW RED SHIFT FORMULAS:

In a cursory treatment of the ground state energy E₀ of a strange gas, one can take, using velocity of light c, and the Planck’s constant h,
E₀= 4V m_sc² / h² *(p_F³/ 3) ------------------------(13)
where p_F is the Fermi energy level.Using
M=3m_sN with N number of strange particles, we get for the pressure exerted on the boundary

p₀= E₀/V= 3G M ²/(20R⁴π).------(14)

The G is gravitational constant 6.67E-08 g^-1cm ³ sec^-2 and R is the radius of the gas sphere. Adopting the expression for the ratio of N/V we have M = 5R/3=1.6666R with geometric units. Thus a relation between the of the fermion gas in a volume of radius R can be conceived.

The significant feature of the present model formulation is to distinguish between the strange field fluid mass M _F and the strange quark matter mass m_s.
The strange quark matter mass m_s is found to be

m_s= M_F r ⁴+ 0.5[r² r_b⁴– r⁴ r_b²+ r⁴ r _u²r_b²-u ² r_u² r_b⁴] / r r_b⁴--------(15)

And involves the strange field fluid mass M_F and is decided by the boundary of magneto-strange field fluid.

The physical quantity u is an arbitrary function of r and r_u
=d_Fpect to u. The boundary is decided by r= r_b when u=1.
[20] Otherwise the u can assume different values other than unity. Hence m_s depend primarily on r_u and M_F With r=3, u=2 we have the range of [r_u,M_F] = (1:1:3) values from 1 to 3 in step of 1 giving the set of nine m_s values ,which might be regarded as the representative values of strange quark matter mass.


1.8242 2.8867 3.9492
1.9297 2.9922 4.0547
2.0352 3.0977 4.1602













The new class of exact solutions obtained for the metric given in Eq.1 and with the determined killing vector Group of motions collineations, can be readily adopted to derive the RED SHIFT formulas.

The proper time intervals at the surface of the magneto-strange field fluid boundary (since they exist superimposed) are related to coordinate time intervals by

cdτ=H r₄/ r sin ^1/2 θcdt= e^νcdt-------(16)

where the H the most significant parameter of the model formulation is
found to be

H ^-1 = (2M_F – r_b ²+ r_4b² r_b²)^1/2/r_b²-------------------------(17)

(2*M_F-rb²+(rub²)*rb²)-----------(18)
It is significant that the magneto strange fluid filled cosmic entity with the strange quark matter also exhibits certain wormhole like features.More appropriately I term them as the magneto-strange holes(Magstra holes)They are illustrated in the Fig.2 given below[14d].













Hence the Red-Shift is
zs= r/( r₄ r_b²) [(2M_F -– r_b ²+ r_ub²r_b² )^1/2]-1----(19)

The set of numerical data for zs is as follows.

r=2; r₄ =1;rb=2;M_F=2;rub=2;zs=63 ;
r=2; r₄ =1;rb=2;M_F=1;rub=1.5;zs=27
r=1; r₄ =1;rb=1;M_F=0.5;rub=1.5;zs=0.1250

Even if M_F=0 the expression would not correspond to the red-shift formula for the case of a neutron star or simply a magneto-fluid filled super star. Here the lateral dependence of the metric is considered by taking that θ= 90⁰.

An alternate expression for the Red-Shift zo of magneto-strange field fluid filled cosmic entity with the strange quark matter is obtained.

z0=r_b/r_u[ 1+ (r_b²/ √2 r_u H) ²]-1--------------------------------(20)

The set of numerical values are


H=3;rb=3;ru=3;z0=1;
H=1;rb=3;ru=3;z0=9;
H=1;rb=2;ru=3;z0=0.8519;
rb=2;ru=2;z0=4

Here H involves M_F. It may be noted that this expression differs from the
previous a similar expression obtained for the case of a magnetic hole(kln hole[14d]).

SECTION IV: THE VERTICAL LIFT:

The restriction of the metric to a plane passing through the origin yields,
dg= e^λ dr ² + r² dθ ²
with t=constant and φ =constant. As the metric on a surface in 3D. If it is
taken in cylindrical coordinates then the coordinate z ( r ), we get

dz²+dr²+ r d θ²
= dr² [1+(dz/dr) ²]+r dθ² = e^λ dr ² + r ² dθ ²------------------(21)

(dz/dr)= √(e^λ-1)---------------(22)

As a consequence, the intersection with φ =constant results in the generation of a paraboloid of revolutions of a complex nature.

The expressions for the vertical lift z of the magneto-strange field fluid filled cosmic entity is given by

z= ∫ { [r₁²r_b ² H² - r_b ² H²+ 2 M_F H² – r²r_b ²]/[r_b ² H²- 2 M_F H² + r² r_b ²]}^1/2dr

r₁=r;M_F=2;rb=2; H=2;











For the case of absence of r₁ variation, the vertical lift expression becomes,

z= ∫ { [2 M_FH² – r² r_b ²]/[1- 2 M_F H² + r² r_b ²]}^1/2 dr-----(23)

For the case of r₁ = r ^13/6 the vertical lift may be evaluated numerically. But this would be only an academic interest.

Thus a set of exact solutions for MAGNETO-STRANGE FLUID FILLED COLLAPSABLE COSMIC ENTITY, WITH STRANGE QUARK MATTER MASS, that co-exist with the wormholes are found for the metric-tensor given in the Eq.1,but with the distinct feature that the parameters of the metric tensor involve MAGNETO-STRANGE FIELD FLUID MASS M_F, strange quark mass m_s and the magnetic monopole strength H.

The presence of the cosmological constant and bag constant does considerably change the numerical values listed for the Red Shift as well the vertical lift.

SECTION V: A DISCUSSION AND THE CONCLUDING REMARKS:

1. The present work explicitly defines the magneto-strange field fluid mass and the strange quark matter mass of a cosmic entity.
2. The present study brought out the salient feature associated with role of magneto-strange field fluid mass in the determination of Red Shift formulas.
3. The magneto-strange field fluid mass exists associated with the strange quark matter mass and hence influences the wormhole physics and their calculus of the metric-tensor.
4. Three different sets of relations for the vertical lift are obtained from an embedded 2D surface of cylindrical symmetry.
5. The presence of cosmological constant does considerably changes the numerical data of both the Red Shifts and the Vertical Lifts.
6. Graphical presentation is given wherever they have become feasible due to the physical reasonability of the numerical data.

The considerations of magneto-strange field fluid apart from the strange quark matter mass is an important finding in that it establishes a novel the General theory of Relativity approach, to the study of Elementary Particle and High Energy Physics. The consideration of magneto-strange field fluid as a part of the metric-tensor space-time allows the possibility of experimental verification, by an analysis and observation of radio-astrophysical scintillations. New technologies in the study of Astrophysics are thus suggested.

ACKNOWLEDGMENT:

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 books by Courant, Vol.1 & 2, Barnard & Child Modern Algebra, G.H.Hardy,Pure Mathematics,R.C.Tolman Cosmolgy & Reativity and many other and for providing an extremely congenial and fruitful environment at his laboratories to carry out my research.

REFERENCES:

1. a. A. R Bodmer, Phys.Rev. Vol.D3, No.6, p.1601, Sept 1971
b. In the year 1969 I have discussed the hyper nucleus _ΛB¹⁰ on the basis of a resonating magic core model for B¹⁰ nucleus.
[K. L. Narayana, Curr.Sci. Vol.38, p.487, 1969]
K. L. Narayana, Vol.38, p.261, 1969.] The idea was that the last odd neutron of B¹⁰ nucleus gets replaced by a hyperon Λ. The strength of Λ-N interaction was taken to be less than 100MeV.
c. The _ΛB⁹ has binding energy 6.4MeV.
See M. W. Holland, H. G Miller, J. P. Roalsvia Phys.Rev Vol.161, p.911, 1967
2. Naoki Itoh, Prog.Theor.Phys.Vol.44, No.1, p.291, 1970
3. V.A. Ambantsumyan and G.S.Saakyan, Soviet Astron. A J. Vol.4, p.187, 1960
4. AGW Cameron Ann Rev.Astron Astrophysics Vol.8, p.179, 1970
5. R.J.Oakes, Phys.Rev. Vol.131, 2239, 1963 and
6. F.J.Dyson and N.-H.Xuong Phys.Rev. Letters Vol.13, p.865, 1964.
7. J.P. Oppenheimer and G.Volkoff, Phy.Rev Vol.55, p.374, 1970.
8. G.S.Saakyan and Yu.L.Vartanyan, Nuovo Cimento, Vol.30, p.829, 1963.
And see Soviet Astronomy Vol.8, p.147, 1964.
9. A.G.W. Cameron, S.Tsuruta, Can. J. Phys. Vol.44, p.1895, 1966.
10. C.Aktas, I. Yilmaz, Gen.Rel.Gravit, Vol.39, p.849-862, 2007.
Our set of killing vectors from those reported in this paper.
11. Witten Phys.Rev, Vol.55, 1983
12 K.S.Thorne” on primordial elements formation, primordial magnetic
field and the isotropy of universe” Astrophys.J.Vol.148, p.51, 1967
13. F.Rahman et al, Gen.Rel.Gravit Vol.39, p.145-151, 2007
See also e-print Archives gr-qc/0605095 and gr-qc10512113
14. 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 University, Kolhapur-416004} 17-11-10 Narasimha Ashram, Official Colony, Maharanipeta.P.O.Visakhapatnam-530002
14 b.trusciencetrutechnology, Kotcherlakota Lakshmi Narayana, Ind. Sci.Cong, Session X Ref.paper No.33,page 23,2008 Andhra University.
14 c. 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
14 d. 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.
14 e. trusciencetrutechnolgy@blogspot.com Vol.2009 No.4, Dated 4th April 2009: On the New Gravitational Red shift formulas and a Relativistic lateral Law of Gravitation by Prof.Dr.Kotcherlakota Lakshmi Narayana,
15. a. N.Panchapakesan, “Does a Hole sequeal? Quantum Black Holes and Information loss”,IAGRG, Feb. 18, 1994. He mentions stretched horizon with two kinds of entropy 1. Entropy of Entaglement 2.Entropy of ignorance (or thermal entropy).
b. Prof. E. C. G. Sudharsan of USA, replied, during the 2008
Ind. Sci. Cong. Session, to the author, very vehemently regarding
the existence of spooky interaction.
16. Morris M., Thorne K.S. Am.J.Phys Vol.56, p.39, 1988
17. Peebles P, Ratra B, Rev.Mod.Phys Vol.75, p.559, 2003
18. R.H. Dalitz “Nuclear interactions of the Hyperons” Editor H.J.Bhabha,
TIFR studies in Physics, Vol.2, p.93, 1965
19.a. V.L.Ginzburg, Book “Key problems of Physics & Astrophysics”
Mir publishers, Feb., p.147, 1978.
b. A. Sivaraman, “A brief history of Dark Energy”,
Astrophys.Space Sci. Vol.319, p.3-4, 2009.
c. M. J. Rees Mon.Not Roy Soc. Vol.98,p.210,1978 “On Quasars”
d. Summaries of Galactic Center: Mon.Not Roy Soc. Vol.98,p.196-294,1978
e. For Laplacian Black Holes refer
G.C. McVittie Mon. Not.Roy. Soc. Vol.98,p.272- 274,1978
20. M.E.Cahill, G.C.McVittie, J.Math.Phys, Vol.11, N0.4, p.1382, April 1970.

Offline Addendum:

Late Dr. Arka Somayaji, a Sanskrit Scholar and Project officer of creative work and research TTd,Tirupati, states in his book on What a Wonderful Universe”,p.43,1985:

Avirbhuta prakasaanam avipuplutacetasam
Atitanogatajanam pratyaksat na virudhyate,
Atitudriyan asamvedyan pasyanti arsenar caksusa,
Ye bhavam vacanam tesava nanumancna badhyate.


"Faculty of looking into past and future, with seers of yore, to whom eye of intuition had opened, whose minds not tainted by senses go by the name supersensual knowledge………."
Dr AS admired late Professor K.Rangadhama Rao as a man of intuitive intellect. Dr Arka Somayaji has guided about six students and wrote several treatises on Surya Siddhanta and Tidal Force(see Wheeler’s book) published by TTD, Tirupati,but most are out of print and or unavailable.

FELICITATIONS TO Dr. DUVVURI NARASIMHA SINHA

TRUSCIENCETRUTECHNOLOGY
RESEARCH FORUM MEETING
DATED 3RD MAY 2009


trusciencetrutechnology Vol. 2009, No.5, Dated 3rd May 2009

The meeting was attended by
1. Rachakonda Prabhakar 30-15-179, Dabagardens, Visakhapatnam 0891 2525088, 9848365029,
2. Shri Goda Jaganadha Rao D.No 9-35-7/1 Pitapuram Colony, 0891 250243; 9292905134;
3. D.Soma Sunddara Rao Flat No. 208, Sri Sai Gardens, Nacharam, Hyderabad 76 ;27171479;
4. Bayya Mahadeva Shastri, Farmer, s/o Shri B.N.Sarma, Maharanipeta, vsp-2 ; 0891 2706040
5. Prof. E.V.Gangadharam, Bhagyasai Residency, opp.century club,
6. Prof.K.L.Narayana, 17-11-10 official colony, Visakhapatnam-530002, 9885542291
7. Dr.D.N.Sinha, B2- Flat Dolphins Apts, next to taj residency hotel, Vizag
8. Dr.D. V. N Sarma, A1 Flat, Dolphins Apts, Visakhapatnam
============================================================================

The Research Forum meeting felicitated Dr. Duvvuri Narasimha Sinha
For the Award of Doctor of Philosophy of Magadha University, Buddha Gaya, Bihar (Founder Sir S.Radha Krishnan) for his thesis on

“PEEP INTO THE PAST” – (Architecture).

Dr.D.N.Sinha spoke about the motivation and what prompted him to write this Thesis. He also gave a vivid account of the various chapters into which it has been formulated. The study would be a great boost others to follow his suit to excel in Academic and Scientific presentations.
His effort of visiting Calcutta National Library, Royal Asiatic Society library etc for his thesis work has been elaborated.
The thesis is soon to be published as a book.
Dr. D. Narasimha Sinha is travelling abroad for four months from the 25th May, 2009 till October 2009.
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trusciencetrutechnology Vol. 2009, No.5, Dated 3rd May 2009

AGRICULTURAL EFFORT IN INDIA:
A perspective review and fruitful suggestions:

Another high light of the meeting is the lively dialogue and discussions about the Seed Development (past and present trends), Quality control measures by Govt. Agencies, Methods of tagging the seed sales to Farmers, Plant protection methods and use of recommended pesticides and fertilizers. Emphasis has been made to review the disappearance of old verities of crops and failure to introduce verities from Japan etc. foreign countries. Farm management seems to a very important exercise.
Experiences of problems of famine in southern states, Andhra, Kerala, parts of Maharashtra in 1948-1952 were reviewed. Scientists are very concerned of new type of famine in Andhra during last four years, i.e. heavy rain scattered in isolated regions and no rain in some other areas of the state.
Farmers of these days have become more aware and convincing them to introduce verities of seeds or grow crops is being questioned and challenged by the primary class educated ryots in rural areas.
Special mention has been made of BAM-3(Orissa Berhampur) Thoka Sannalu, GEB-24 King variety, (that has good reflection of light from the seed due its constituents useful to puliohara,etc), EG Akulu,Krishna katukulu, Nellore Sannalu,Pula -1100, SR-26 for saline lands of low lying areas of delta areas, Basmati, Massori(becomes very pulpy on cooking, more broken rice drawbacks)
Etc. several cross verities is being suggested by research scientists and Govt. and Non-Govt. agencies.
All the agricultural scientists have enjoyed their jobs of FTC efforts of training the knowledge and skills of the farmers. At least Prabhakar has been asked by Bobbili farmers to stand as an MLA from Bobbili constituency for his free and expert advice and teaching to the local farmers.
Financing the farmer on a timely basis is very crucial to have more productivity.
Need of the hour is that more results of research to be passed on to the farmers.
Create seed villages.
Govt. interference led farmers to abandon their traditional method of seed generation and reserve seed cultivation of crops.
Previous concepts validity, experience and methods of transformation of knowledge, farmer’s acumen ship to have a personal evaluation, their intelligence and especially political awareness are some aspects of Modern India.
======================================================================
Dr. D. V. N Sarma hosted the meeting at his residency and arranged refreshments to the pariticipants.A vote of thanks has been proposed to him.
======================================================================

FELICITATIONS
TO
Dr. DUVVURI NARASIMHA SINHA
BY
His Friends and
Well Wishers
For
The Award of Doctor of
Philosophy
Of Magadha University,
Buddha Gaya, Bihar, Founded by
Sir S. Radha Krishnan
For the Thesis on
“peep in to the Past”
-Architecture-
3rd May 2009
Visakhapatnam.

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.
Denhen [14] gives conditions for rigid frames of reference and as well for inertial reference frames using the kinematical parameters.Spin[18] and interaction of particles[19] seem not to effect the principle of equivalence of masses. S.Desser sought the principle of equivalence in terms of universal coupling,in a nongeometrical approach and, surmised Einstein's Equations as curvature without curvature and flat spacetime without flat spacetime.[20]

1. C.W.Misner, K.S.Thorne, J.A.Wheeler Gravitation, W.H.Freeman & Co. pages 566, 72 (1970), also sections 27.8, 27.9, and 29.2 for arc parameter.
2. M.E.Cahill, G.C.McVittie, J.Math.Phys, Vol.11, N0.4, p.1382, April 1970.
3. W.C.Hernandez, C.W.Misner Astrophys.J, Vol.143, p.452, 1966

4. M.M.May and R.H.White, Phys. Rev. Vol.141, p.1232, 1966
5. Andre Lichernowicz Relativistic Hydrodynamics and
Magneto-Hydrodynamics, Benjamin Inc. NY, 1967
6. L.Hughston, Astrophys. J., Vol.158, p.987-989, 1969
7. D.J.Baylis, Vol.Mon.Not.R.Soc.astr.soc, Vol.151, p.93-107, 1970.
Communicated by W.H.McCrea.
D.K.Ross, Il Nuvo Cimento Vol.23, No.1, p.55-62,11th March 1971
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.
9. P.Yodzis Phys.Rev D vol.3 No.12,p.2941 (1971)
10. A.H.Taub Phys.Rev D Vol.103, p.1454 (1971)
11. A.I.Jannis Phy.Rev. D8, p.8, 1973
12. G.F.R.Ellis J.Math, Phys. Vol.8, No.5, p.1171-1194, 1961.
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.
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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.
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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