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.