70th Birth Day Volume Year 2009 to 2010 of Prof Dr Kotcherlakota Lakshmi Narayana

ARTICLE No.1 by krj.srikanth@gmail.com dated 23rd July 2009

Birth Day Volume of the year 2009-2010 of trusciencetrutechnology@blogspot.com

ARTICLE No.1 Dated 23rd July 2009.

Respected,
The Pictures collected by me at early hours of am 6:30 to 7:30.It was to be the phenomenon of the century. But hordes geared up to view the celestial drama only to be let down by the thick cloud cover which played spoilsport. Only a lucky few were able to get a sneak peek of the eclipse for a few seconds. “I woke up at five am and drove to the temple veiw of sri venugopla located at Bobbili which is so ancient builted by Rajashab of bobbili, The most specturlarthing that it has seven slites , the sun rises up from those, splites in to rays. and waited for an hour in the hope of catching a glimpse of the eclipse, but in vain.” The clouds covered, even though i got spectrum of the sun from my Nookie express series with 2 MP camera, in different views which is very near to temple of Bobbili. What i heard Regarding this eclipse. VARANASI/WUHAN: A total solar eclipse on Wednesday swept across a narrow swathe of Asia, where hundreds of millions of people watched the skies darken, though in some places thick summer clouds blocked the sun. The longest total solar eclipse of the 21st century cut through the world’s most populous nations, India and China, as it travelled half the globe. It was visible along a roughly 250 km-wide corridor, U.S. space agency NASA said. In India, where eclipse superstitions are rife, people snaked through the narrow lanes of Varanasi and gathered for a dip in the Ganga, an act believed to bring release from the cycle of life and death. The eclipse next swept through Bangladesh, Nepal, Bhutan, Myanmar and over the crowded cities along China’s Yangtze River, before heading to the Pacific. In Hindu-majority Nepal, the government declared a public holiday and thousands headed for water. In central China, crowds gathered along the high dykes of the industrial city of Wuhan, roaring and waving goodbye as the last sliver of sun disappeared, plunging the city into darkness, although clouds cheated them of part of the spectacle. “As soon as the totality happened, the clouds closed in, so we couldn’t see the corona. That’s a pity,” said Zhen Jun, a man whose work unit had given him the day off to enjoy the spectacle. But eclipse viewers in central China were luckier than those in the coastal cities near Shanghai, where vercast skies and rain in some places blocked the view of the sun entirely. It lasted up to a maximum of 6 minutes, 39 seconds over the Pacific Ocean, according to NASA. The dark shadow of the moon — the conical shaped umbra —made its landfall in Gujarat at 6.30 a.m. and raced across the country within a span of a few minutes spreading darkness in the path of totality. VARANASI/WUHAN: A total solar eclipse on Wednesday swept across a narrow swathe of Asia, where hundreds of millions of people watched the skies darken, though in some places thick summer clouds blocked the sun. The longest total solar eclipse of the 21st century cut through the world’s most populous nations, India and China, as it travelled half the globe. It was visible along a roughly 250 km-wide corridor, U.S. space agency NASA said. In India, where eclipse superstitions are rife, people snaked through the narrow lanes of Varanasi and gathered for a dip in the Ganga, an act believed to bring release from the cycle of life and death. The eclipse next swept through Bangladesh, Nepal, Bhutan, Myanmar and over the crowded cities along China’s Yangtze River, before heading to the Pacific. In Hindu-majority Nepal, the government declared a public holiday and thousands headed for water. In central China, crowds gathered along the high dykes of the industrial city of Wuhan, roaring and waving goodbye as the last sliver of sun disappeared, plunging the city into darkness, although clouds cheated them of part of the spectacle. “As soon as the totality happened, the clouds closed in, so we couldn’t see the corona. That’s a pity,” said Zhen Jun, a man whose work unit had given him the day off to enjoy the spectacle. But eclipse viewers in central China were luckier than those in the coastal cities near Shanghai, where overcast skies and rain in some places blocked the view of the sun entirely. It lasted up to a maximum of 6 minutes, 39 seconds over the Pacific Ocean, according to NASA. The dark shadow of the moon — the conical shaped umbra —made its landfall in Gujarat at 6.30 a.m. and raced across the country within a span of a few minutes spreading darkness in the path of totality.(Soucrce from The hindu 23/july/09)

The entire work done by me under the guidance of Prof. Dr. Kotcherlakota Lakshmi Narayana

krj.srikanth, Assistant Professor of Engineering College, Bobilli.

On the possible Existence of Collapsing Gravitational Superconducting Currents of a Cosmic Entity and Generation of Bilinear Physical Entities

trusciencetrutechnology@blogspot.com,Vol.2009, No.7, Wednesday, 23rd July 2009: 1:06:13 PM : Pages 1-20
On the nature and possible existence of Collapsing Gravitational Superconducting Currents of a Cosmic Entity and the generation of Bilinear Physical Entities of Ghost Fields.
by Professor Dr. Kotcherlakota Lakshmi Narayana,
(Retd.Prof.of Phys, S. U), 17-11-10, Narasimha Ashram, Official Colony, Maharanipeta. P. O, Visakhapatnam-53002, AP, India.
trusciencetrutechnology@blogspot.com, Vol.2009, No.7, Wednesday, 23rd July 2009: 1:06:13 PM: Pages 1-20
Keywords: Astrophysics,Cosmology,Space science,ghost neutrino,superconductivity,bare charge,mixing angle,beta rotation,magnetic strengths,Dirac-like field equation,Einstein-Dirac-like equation,Spinor,killing vectors, current cone, current surface, coupling constants,Taub metric,3D phase space-time, bilinear physical entities,quasi-ghosts

ABSTRACT

The coupled Einstein-Dirac-like equations, have a richness of variety of solutions, by use of the detailed Energy Momentum Tensor quantities of free magnetic strengths, bare mass-less electrostatic ghost charges and the mass-less ghost neutrino, subject to pressure and other field contributions. In this paper, I have investigated the Taub space-time universes, with two different versions, of its exponent metric parameters, that admit a conformal Group of Motions. The Einstein-Dirac-like Field Equations and as well the Dirac-like field Equations have been solved and their relation with a full set of killing vector fields has been established. A new find is the set of non-vanishing partial derivative currents. Also the set of current densities of the bilinear physical entities have been obtained. In principle, these allow the observation of the bilinear physical entities. This opens a new vista of observability, as well of other types of bilinear physical entities of ghost fields. The associations of the ghosts allow the three symmetric Χ^± , Χ ⁰ and one pseudo-symmetric ζ⁰ state that constitute the 4- component Spinor Dirac-like wave field.

PAGE 2

Further it is found that Χ ⁰ and ζ⁰ could generate mixed bilinear physical entities, with a mixing angle designated as θ_csc where csc stands for collapsing superconducting currents. The General Theory of Relativity approach has been designed, to a fresh understanding of the infinite electrical conductivity, in terms of possible existence of collapsing gravitational superconducting currents. A significant finding is the generation of bilinear physical entities, near the defined “Current Surface”. A pairing mechanism, of the two ghost electrostatic charges, via the ghost neutrino interactions, has been formulated with a Lagrangean that is endowed with universal coupling constants. The theory is characterized by the possible ground states of 16 –band gaps resulting from the pair combinations of a Dirac-like 4-component Spinor function. Introduction of colour and flavour characterization adds to the enchanting variety of collapsing superconducting current densities. Of interest, is the characteristic wave patterns, produced by a string and membrane solutions of the Einstein-Dirac-like field equations, that exist projected in a 3D phase space over ” Current Cones”.

PAGE 3

CONTENTS
1. INTRODUCTION:----------------------------------------------4
2. SECTION I: THE AIM OF THE PRESENT WORK---------------------6
3. SECTION II : SOLUTIONS OF DIRAC-LIKE EQUATIONS:------8
THE CALCULATIONS: --- ----------------------------------------9
4. SECTION III: A SET OF CLASS OF EXACT SOLUTIONS FOR THE EINSTEIN-DIRAC-LIKE EQUATIONS FOR THE DISTINCT UNIVERSES OF THE TAUB METRIC IN THE NEW FORMALISM:-10
5. SECTION IV: THE KILLING VECTORS AND THE MAGNETIC STRENGTHS:------ 12
6. SECTION V: A. THE COLLAPSING COSMIC ENTITY GRAVITATIONAL SUPERCONDUCTING CURRENTS:----------------14
7. SECTION VI: B. HOW THE COLLAPSING COSMIC ENTITY GRAVITATIONAL SUPERCONDUCTIVITY CURRENTS ORIGINATE?------------------------------------------------15
8. SECTION VII: PARTICLES VERSUS VECTOR FIELDS:---------16
9. SECTION VIII: THE CHARACTERIC SURFACE CONES THAT DEPICT THE WAVE PATTERNS: --17
10. SECTION X :SUMMARY AND CONCLUSIONS:-----------------------18

ACKNOWLEDGEMENT-------------------------------------------- -----19

REFERENCES:--------------------------------------------------------19
OFF-LINE ADDENDUM:----- ---------------------------------20
================================================================
This is a

trusciencetrutechnology@blogspot.com, Vol.2009, No.7, Wednesday, 23rd July 2009: 1:06:13 PM published post and full paper may be obtained by Email to :
lakshminarayana.kotcherlakota@gmail.com
or kotcherlakota_l_n@hotmail.com
or
with the corresponding address:
Professor Dr. Kotcherlakota Lakshmi Narayana,
(Retd.Prof.of Physics, S. U),
17-11-10, Narasimha Ashram, Official Colony,
Maharanipeta. P. O, Visakhapatnam-53002, AP, India.
Mobile No. 9491902867 BSNL

PARTIAL SOLAR ECLIPSE 22nd JULY 2009, SPETACULAR EFFECTS ON THE LAND, SEA AND SKY OF COASTAL ENVIRONMENT, VISAKHAPATNAM, NARASINGARAO BAY

trusciencetrutechnology@blogspot.com Vol.2009, No.7, Dt.22nd July, 2009

THE “ NARASINGARAO BAY ” CLOUDED PARTIAL SOLAR ECLIPSE 22nd JULY 2009, SPETACULAR EFFECTS ON THE LAND, SEA AND SKY OF COASTAL ENVIRONMENT, VISAKHAPATNAM-INDIA


BY PROF. DR. Kotcherlakota Lakshmi Narayana,
17-11-10, Narasimha Ashram, Official Colony, Maharanipeta, Visakhapatnam-530002.

MObile :9491902867 BSNL

trusciencetrutechnology@blogspot.com Vol.2009, No.7, Dt.22nd July, 2009

PRELIMINARY REPORT

The Narasingarao bay extends from the old light house ( Palm Beach area) to the foothills region of Yeraddakonda (Dolphin’s Nose Hill, the new light house situated on the top) of Visakhapatnam Eastern Naval Coastal command. The author chose the vantage position for observations, the height of the Doctor’s Arcade complex of 6 floors, (built on the site originally belonging to Dr. Kotcherlakota Ramaswamy) in Maharanipeta, official colony, Do.No.17-11-9. The research study and investigations of the said partial solar eclipse effects on the environment has been made from 5:32 AM to 7.15AM.

Preparation has been made by the author, in view, of previous observations of ‘FRACTAL AURORA AND AIR-GLOW” produced by solar eclipses exhaustively reported at this blogspot published articles by the author. (See the References listed below)

Nearly Rs. 50k has been invested, for the present research from his own pocket by the author, to get himself properly equipped with suitable experimental set-ups. Though simple, but sophisticated electronic Video and Imaging facility, helped the author to capture the environmental changes( in a modern style of scientific and technological research investigations.). I have also used a Nikkon F (presented to me in 1950's by Dr.KVNRao of 1, Old Billerica street, Bedford (Mass),USA) camera with Kodak 35 film of ASA400 speed with aperture 11 and time of exposure 1/125 sec.The lens is Nikkor-5, Auto ,1:2.8, f=3.5cm, No.922563,the lens made in Japan. About 20 photos have been taken.

The ACER SYSTEM 5738G OF ASPIRE MODEL NO.M52264, ID:89578-OEM-7332157-0021, WITH CHARGED BATTERIES BY USING ADT AP0650102691705B76P100 AND nVidia GeForce 512MB graphic card and 39.62cm(15.6)HD LED-backlit screen with 16:9 aspect ratio has been used. The Crystal Ball Web Camera ideal for low ambient light, has been found to be excellent in caputering both the Camera and Video shots.
The details of observations would be published in due course of time after the full analysis. Contact may be made with me at:

lakshminarayana.kotcherlakota@gmail.com or

kotcherlakota_l_n@hotmail.com

The time of partial solar eclipse initiation observed 5:30AM(approx) and the maximum around 6:22AM respectively. But I found that atmospheric changes started to significantly to change by 6:00 o'clock. Especially at 6:15 AM they became some what striking. At 6:22AM the ambient light and, all around in the coastal sky there was low-brightness. Many special effects have been found at between 6:00 AM to 6:44AM. But enchanting splendour of the environmental changes could be noted and using various methods they have been successfully recorded. About a full time observation between 5.32AM to 7:21AM has been made. Initially the reddish hue of the sky was observed around 5:21AM.
The results are obtained, dealing with
a. Winds
b. Dust(special effects were noticed)
c. Fractal Aurora like air-glows.
d. Spectral changes
e. Sun rays other effects on the clouds

f. Other features

Several novel effects that would throw light on the gravitational changes and also the coastal environment have been noticed and recorded. The analysis would of course, requires lot of money and time.

==========================================================
A NOTE ON INDIAN CALCULATIONS ON PARTIAL SOLAR ECLIPSE IN SOUTH INDIA:
Surprisingly the TTD Tirupati people published an Indian Almanc(called the Panchanga) that gave more or less very accurate timings for observation of the partial solar eclipse. Their details are as follows: Ashada Bhahula Amavasya Wednesday "Ketu Grasta" "ardhadhika(more than 50%) solar eclipse Initial touch 5:28AM Middle time: 6:22AM Mokshakala(i.e withdrawl) 7-21AM Adhyantha(start to end) time 1hr:53mins with Sun rise at 5-58AM.Punarvasu and Pushyami onstellations and so dangerous for the people with that birth star, Also karkataka Rasi (Zodaic sign)people are forbidden to observe it.Others can watch the eclipse.

=======================================================

REFERENCES TO EARLIER BLOGPOSTS made at this blogspot by the present author.
Sunday, February 17, 2008
Solar Eclipse 2008 sunspot 982 and Visakhapatnam
The Second and third week of February 2008 weather conditionsat Visakhapatnam

by Professor Kotcherlakota Lakshmi Narayana

http://trusciencetrutechnology.blogspot.com/
Friday, August 8, 2008
The Fractal Aurora/Airglow Dance Observation during the Partial Solar Eclipse of 1st Aug 2008 at Visakhapatnam(India)
by Professor Kotcherlakota Lakshmi Narayana

Dated: 1st August 2008 http://trusciencetrutechnology.blogspot.com/

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.
============================================================================
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.