A CLASSICAL SET OF WORMHOLE SOLUTIONS THAT POSSIBLY COEXIST SUERIMPOSED ON MAGHOLES.
by
Professor Kotcherlakota Lakshmi Narayana,
(Retd.Prof of Physics,Shivaji Univ)
Narasimha Ashram, 17-11-10, Official Colony,Maharanipeta.P.O
Visakhapatnam-530002
trusciencetrutechnology@blogspot.com
dated 30th March 2009. time 2:34PM
key words: Astrophysics,Mathematical Physics,Cosmolgy,Magholes,Wormholes,General Relativity,Redshifts,Throat,Bridge,lateral travel across Universes.
ABSTRACT:
A class of exact solutions of Einstein’s equations for the lateral dependent metric tensor of the form
ds2 = e2γ dx42– c-2(e 2αdr2 +r2 dΩ 2)
INTRODUCTION:
The worm hole gas was defined by Hawking S.W. and Page D.N. [1] by a path integral formalism, overall possible asymptotic Euclidean 4-geometry and consideration of matter fields. They assumed that the Energy-Momentum tensor to vanish at infinity and with the excited states of the worm holes surmised to have their sources at infinity Wormholes are thought to produce shifts in effective masses of elementary particles and their interaction constants.[2] Wormholes can make the cosmological constant to be zero S.Coleman [3]. The effects of wormhole instantons have been reported by A. Hosoya and W.Ogura, [4] in the Einstein-Yang-Mills system and they asserted that wormhole instantons might be found both in the sub-planck physics and as well at all stages in lower energy physics. J. Ellis, E.Floreatos D.V.Nanopoulos[5] state that the Quantum gravitational effects might be significant at energies far below the Planck mass mPL=1019GeV. Particularly they state that the large volume correction to the gravitational action due to spin-0 bosons and spin-1/2 fermions leads to the up and down quark mass ratio as 44/130.
S.Coleman[6] and S.B.Giddings & A.Strominger [6] gave a wormhole calculus suitable for a digression on quantum topological effects S.W.Hawking, D.N.Page, C.N.Pope[7] and G.V.Lavrelashvili, V.Rubakov and P.G. Tinyakov, [7] have conjectured that the wormholes calculations to modify the elementary particle masses and hence might lead to breakdown of quantum coherence.
A. Einstein, N.Rosen [8], C.Misner, and J.A.Wheeler [8], have given the topology of a bridge that exists between two asymptotically flat spaces. The bridge, similar to a throat, has a narrow region, the circumference of this region at time t=0(Schwarzschild) is given by 2*π*rsch = 2*π*(2*G*M/c2) =4* π*m (m in units of cm). r would be a time like coordinate to an observer at this region of the throat, with r acquiring the lowest value. For him moving along the geodesic the throat shrinks to pinch-off effect at r=0. In the case of Reissner-Nordstrom solution, the stress of the electric field would keep off the throat from the pinch-off effect. Hence due to the flux of electric field the throat would never achieve the singularity at r=0. Actually, the radius of the throat pulsates between maximum and minimum values possible and for an observer sitting at the throat, in his proper time τ, accounts for the effect of the pinch effect at the throat. It is interesting to note that in the limit that the q2-->m2 the Maxwell pressure through the wormhole just balances the gravitational forces. [9]
The special case of q2=m2 for Reissner-Nordstrom solution to describe the exterior field of a static spherical charge dust cloud has been presented by Carter [10] adopting a Kerr solution. According to Carter[10] in Type-I ( r>r+) , in Type-II ( r+ > r> r- ) or Type-III( r- > r> 0) where r+ = m ±√(m2-q2) both the Null forms of the Reissner-Nordstrom metric are incomplete, when q2=m2. J. C. Graves and D. R. Brill [11] used two co-ordinate patches, in the neighbourhood of r+ and r- respectively to describe Reissner-Nordstrom metric, with the two patches superimposed and glued together along the boundary rc .They give the new radial coordinate as
r*=r+ (r+2 * ln(r- r+)- r-2 * ln(r-r-))/ (r+-r-)--------------------------------------(1 )
Radius r of throat of the Reissner-Nordstrom wormhole as a function of proper time τ, results in the curves r Vs τ as various cycloids. Their period is 2*π*M is independent of charge q. If charge q=0 then the cycloid reaches the singularity r=0 at its cusp.
The quantum wormhole states i.e. solution of Wheeler-DeWitt equation has been researched by L.Garay [12]. P.V.Moriz questions a possible quantum wormhole in N=1 super gravity? [13]P.C.Vaidhya in a paper published on a Big Bang condition [14] gave a theory that smoothly links both the two simple initial solutions of Einstein Equations Viz. the one given by K. Schwarzschild, [15] and the other by Oppenheimer and H.Snyde [15].
T.R.Choudhari and T.Padmanabhan [16] gave tortoise coordinate modified as
r*= ∫ dr (1-2*m/r- H2*r2)-1 with U=t-r* ,V= t+r* ----------------------------(2)
and r*=∫ dr/f ( r ) where ds2*= f( r ) dt2–f( r ) -1 – r2 dΩ 2 .-------------------(3)
Wesson I.M.T. and M. Israelit [17] have discussed creation of fundamental particles. The cosmological constant space dependence is related to the energy density of electron [18] and as well it influences the effective gravitational mass of astrophysical system [19].
An important feature of the wormhole description is its shape and is determined by the radial metric coefficient dependence on the radial coordinate r. The embedded surface z=z(r) shaping the wormhole spatial geometry is a solution of the equation for the rate of change of vertical distance with r. The solutions for the embedded surface are given by the function z(r), and actual realization of the wormhole is by the surface of revolution of the curve given by z(r), about the z-axis. [20].Graves and Brill [11] stated that the curve needs to be rotated in the θ and φ in case of Reissner-Nordstrom solution in the R-r plane.
SECTION I: A COSMIC ENTITY FILLED WITH MAGNETO-FLUID:
In my previous publications, I have discussed [21], on the nature of Universe filled with magneto fluid and the layered structure and break up of a parent planet {termed the MEL planet into three parts, Viz., Mars, Earth and Lunar entities}. Therefore, I have aimed, in this paper, at an understanding of the intricacy of the lateral dependence of metric tensor and the nature of a collapsing magnetic fluid filled cosmic object. In relativistic magneto-hydrodynamic approach, Yodzis [22] used kinematical parameter and obtained conditions for gravitational collapse of magneto fluids. I elaborate and present new findings of this collapse with regard to novel hole[Maghole might be refered as klnholes] formations.
According to General Theory of Relativity, the metric-tensor, dealt with in this paper describes the motion of a self-gravitating lateral dependent distribution of material. It therefore has only φ as a cyclic co-ordinate. Many aspects of the cosmic entities that posses this metric have been examined in this article. The section thus outlines first, the metric given by Cahill and McVittie [28]
ds2=[(r4H/r)2dx 42–(r2/(c2 u2)du 2]sinθ - (r/c)2 dΩ2 ---------------------( 4)
dΩ2 = dθ2 + sin2θ dφ2 ------------------------------------------------------(5)
with u as a function of r alone and r u adopted as the derivative of r with regard to u( refer [28]) and where
H-2= (1 - 2M/rb – (r u /r)2 b) /[ ( rb / c)2(1+ 2M/rb )]----------------------( 6)
The Mass function is defined by me as
(1-2*M/rb) (1+ rb2 / (Hc)2)= (r u /r) 2 b------------------------------------(7 )
The significant result of the computation is that the factor F in the expression
h1= F/r 2e –α-γ cosec θ-------------------------------------------------------(8)
has the value
F= H r4 / √ (κμ) cosec θ ------------------------------------------------------(9)
refer my previous publication [21] for the definitions of κ and μ.
Meaning there by that the parameter H of the theory represents possibly the magnetic monopole strength of the cosmic entity. This result is a consequence of the Gauge conditions
r 41 = r1 r4 /r and r11= 2*r12/r -----------------------------------------------(10)
moreover, with the price that γ1=0.
SECTION II: A NEW CLASS OF EXACT SOLUTIONS AND THE NEW REDSSHIFT FORMULAS:
A new class of exact solutions obtained for the above metric will be detailed in my forthcoming publication. In this section, I would present the new Redshift formula obtained using the defined mass function. Recent experimental findings on Redshifts necessitates a re-examination of plausible new formula to account for possible novel universes existence [25] and cycles of creation of universes and their destruction[26]. I retain the metric tensor given in Eq.4 & Eq.5
The derived Riemann-Tensor components, Connection 1-forms, the Einstein Tensor matrix elements etc. are all listed[23]. The typical affain connection form turns out to be,
ωtφ= (r4* e– γ/r + cotθ θ4 e – γ)ωφ------------------------------------------------(11)
where ωφ is the Cartan’s orthonormal basis function for the metric given by Eqs.4 & 5. Here r4 and θ4 are the derivatives of r and θ with respect to time respectively.
Large redshifts from cosmic entities could arise from relativistic velocities, from strong gravitational fields and, now I state that due to the magnetic filled physical systems. Redshifts of order of 7 etc. are known[24]. In the present article, proper time intervals dτ at the surface of a magnetic fluid entity are related to coordinate time intervals dt by
c*dτ= r4 H Sin1/2θ(c*dt)----------------------------------------------------------(12)
where H-2=(c/rb)2[(2*M/rb -1) +( ru/r)2b]----------------------------------------(13)
with θb=90 0 chosen and, the subscript b implies evaluation at the boundary[see foot note *] of a magnetic fluid cosmic entity. The u function is same as that used by Cahill and McVittie[28].
In terms of wavelengths the Redshift is given by
λ/λ0 = Δτ∞/Δτ = ( r/ r4H) sin -1/2θ ------------------------------------------------(14)
And so new Zs formula for the Red-shift obtained is as follows:
Zs=(r/rb)*(c/r4)[(2*M/rb -1) +(ru/r)2]b(1/2)-1-----------------------------(15)
and is about 6.746 for r 4 =1 ,r u=1,c=1, rb =0.5 and M=3.
Fig.1: The graph depicting the variation of Redshift Zs with the distance r. M=3;rb=0.5;c=1;r4=1;
SECTION III: THE POTENTIAL ENERGY EXPRESSION AND RADIUS OF APASTRON:S. Chandhrasekhar in his book on the Mathematical Theory of Black holes [27] has nicely given the methodology of Lagrangian and Hamiltonian formulations of a metric tensor. Using that treatment, I have obtained expression for the Potential Energy for the metric given in Eq.4 & Eq.5. Fig.2 The Potential function with the parameter values rub=2;M=3;μ=2;L=14;rb=3;c=1;r4=2 Fig.4:The Potential function with the parameter values μ=2;c=1;L=0;r4=3;rb=2;rub=2;M=2; The proper time τ that augments the Schwarzschild coordinate by the infinitesimal amount Δr is given by Similar to the method of introducing the tortoise co-ordinate wheeler[27] and Regge[27], and Regge and wheeler[27], we may find an equation for the new coordinate as Fig.5:The novel parabolas of the expression given Eq.27 For the case of a tidal force γ 1= r1/(2*r) To summarize I have found a classical set of Wormhole solutions that might exist superimposed on Magholes. The possible role of both the negative and positive cosmological constant Λ in determining the shapes of Magholes and Wormhole has been outlined. Results are easily extended to a discussion on the magholes that exist even in the absence of vanishing mass of the cosmic entity.The curves that chartacterize the Magholes and the Wormholes have been presented for ease of visualization of the physics behind the collapsing Cosmic entity. The present article, has a fruitful contribution to extend the Wormhole calculus for other aspects of quantum gravity and fundamental particle physics, especially in the presence of Magholes. The redshift expression with and without the mass and cosmological constant considerations, for cosmic boundaey radiations has been explicitly given. The author is deeply indebted to late Professor K. Rangadhama Rao D.Sc.(Madras). D.Sc.(London) for his inspiring guidance, constant encouragement, for providing me financial support for my higher education and for providing an extremely congenial and fruitful environment at his laboratories in India and scattered world over. Saturday, January 3, 2009 A CLASS OF EXACT SOLUTIONS FOR THE FLUID UNIVERSE FILLED WITH MAGNETOFLIUD –LIKE FIELDS trusciencetrutechnology Vol.No.2009,Issue No.1,Dt.7th January, 2009 by Professor Kotcherlakota Lakshmi Narayana, http://trusciencetrutechnology.blogspot.com/ Ind Sci.Cong. Physical Sciences Section, Paper No.3 Ind Sci.Cong. Physical Sciences Section , 3rd-7th Jan 2009.
pt= E,------------------------------------------------------------------------(16)
pθ= r2 dθ/dτ,----------------------------------------------------------------(17 )
pφ= r2 sin2θ dφ/dτ, and pr= -d(lagrangian)/d ŕ ------------------------------(18 )
(E2- V2(r )) = e(2γ-2α)/ c2 = (dr/dτ) 2 =Ē2------------------------------------(19)
The effective potential relativistic equation for the radial part of motion is given by
V2(r)= (r42/r2)*[(c/r)2 *(1+L2)- (μ2-1)]/ [(c/rb)2*(2*M/ rb2-1 + (ru /r)b 2]->(20)
with θ = 900.
The shapes of the potential function are illustrated, below for different values of the variable parameters, listed in the graphs.
Fig.3 The Potential function with the parameter values
rub=2; μ=2;c=1;r4=2;M=2;L=10;
τ= ∫ dr(uc/u 1)/[E2r4/ r42 + ((μ2-1)] 1/2----------------------------------------(21)
The radius at which the particle has zero velocity, Rap gives i.e. apastron position, is as follows:
Rap2=r42*H2(1- μ2)-------------------------------------------------------------(22)
Using this expression, we get, for the proper time τ
τ= (Hc r4u/ (u1E) )* 1/Rap* (sec -1(r/Rap))--------------------------------------(23)
The energy exhibits an average behaviour while the proper time τ it has a pulsating character. This is a curious formula unlike the cycloid motion. It is found to depend on the ratio H to E and as well on the quantity μ used here which the later is same as that defined by S.Chandhrasekhar. The time of fall by the cosmic entity fluid filled mass, slows down with decrease in r, especially when μ=0.
SECTION IV:A NEW MASS FUNCTION AND A NEW RADIAL COORDINATE:
r*= ∫ dr*= ∫ dr (c/ r4)2)*(2*M/r -1 + u ru/r]---------------------------------(24)
r*=-(c/r4)2)*(2*M/r2+r+ru/r2)----------------------------------------------(25)
But of more interest is the situation when a new mass function is adopted Viz.,
(2*M/rb- 1)(1+ rb2/a2)=-(ru/r) 2 b-------------------------------------------(26)
where a=Hc and b prescribes the boundary when u=1.
What about the case when mass=0? Then
a2+r2b = ((ruu/r)2 b
yields
r4b- rub
and it yields an equation
a2=( ruu/r)2b- r2b -----------------------------------------------------------(28)
The expression is similar to the one given by S.Chandhrasekhar for a charged space-time metric involving the r3 term and solved by him in terms of JacobiElliptic functional form.
The graph of the identity Eq.28, shows that with rb, a over a rang -5 to 5,gives parabolas for rub=6.This behaviour holds even when M=0.Yet times M is used for the mass, especially when boundary (u=1) specifications are involved. Field function thus is given by
a2=( ruu/r)2b- r2b
This implies possible existence of Magholes realizable by rotation of the parabolas about the field axis.
A new radial coordinate r* arise if we use
2m/r(1+r2/a2)=e-2α/c2--------------------------------------------------(29)
r*= -2*m * log r+m (r/a)2+constant.-------------------------------------(30)
If m=0 then r*=0 hence constant. The significance of this new radial coordinate is that it is proportional to the mass and depends on the ratio (r/a)2 involving the field expression of the magnetic fluid of the cosmic entity.
SECTION V: MAGHOLES & WORMHOLES AND THE COSMOLOGICAL CONSTANT:
In my formulation relaxation of the condition Λ=p/(8*π) results in several wormhole solutions, depending on the presence of tidal forces,with no tidal forces and finally without the presence of cosmological constant. I have used Λ= p/(4*π) for an aesthetic beauty of the formalism.
The expression for vertical lift is given by
Z=∫ [3a2*r42 *r12 – a2* r42 + r4–Λ*r2 *a2* r42)/( a2* r42 – r4+ r2*a2* r42*Λ]1/2dr-------------------------------------------------------------------------------------(31)
The graphs of the lift i.e. shapes of the wormholes are presented below:
a=2;r1=2;r4=2; 
and in the absence of it,
Z=∫[a2*r42 *r12 – a2* r42 + r4–Λ*r2 *a2* r42)/( a2* r42 – r4+ r2*a2* r42*Λ]1/2dr-----------------------------------------------------------------(32)
a=2;r1=2; r4=2; 
And in the absence of Λ and r1=1, I have obtained a simple wormhole expression
Z= ∫ [r4 /( a2* r42 – r4]1/2 dr------------------------------------------------(33)
a=2;r4=2; 
Rotation about the vertical axis would result in the explicit appearance of the throat or the bridge connecting the flat universes.
SUMMARY:
ACKNOWLEDGEMENT:
REFERENCES:
1. Hakwing S.W., Page D.N. Phys Rev. Vol.D42, p.2655, 1990
2. Lyons A 1989 Nucl.Phys B324, p.253
Dowker H.F. Nuclear phys B331, p.194, 1990
Dowker H.F Laflamme R, Nucl.Phys B366, p.201, 1991
3. S. Coleman Nucl.Phys. B310, p.643, 1988
4. A.Hosoya and W.Ogura, Phys.Letts. Vol.B225, No.1, 2, p.117, 1989
B.J.Keay and R.Laflamme,Phys.Rev Vol.D40, p.2118, 1989
5. J. Ellis, E.Floreatos D.V.Nanopoulos, Phys.Letts.Vol.B 225, No.1, 2, p.121, 1989
6. S.Coleman Nucl.Phys.Vol.307B, p.867, 1988
SB Giddings & A.Strominger Nucl.Phys.Vol.307B, p.854, 1988
7. S.W.Hawking D.N.Page, C.N.Pope, Nucl.Phys B170, p.283, 1980
G.V.Lavrelashvili V.Rubakov, P.G. Tinyakov, JETP Lett.Vol.46, p.164, 1987
8. A. Einstein and N.Rosen Phys.Rev Vol.48, p.73, 1935
C.Misner and J.A.Wheeler Ann.Phys.Vol.2, 525, 1957
9. B.Bertotti Phys.Rev. Vol.116, p. 1331, 1959.
10. B.Carter Phys.Lett Vol.21, No.4, p.423, 1st June 1966.
11. J.C.Graves and D.R.Brill “Oscillatory character of Reissner-Nordstrom Metric for
an Ideal Charged Wormhole”, Phys.Rev. Vol.120, No.4, p.1507, 15th Nov.1960.
12. L. Garay Phys.Rev.Vol.D48, p.1710, 1990
L. Garay Phys.Rev.Vol.D44, p.1059, 1992
13. P.V.Moriz Grav.Gen.Rel. Vol.28, 1996
14. P.C.Vaidhya Current Science Vol.37, p.191, 1968
15. K.Schwarzschild, Berlin Res. Vol.424, p.245-247, 1916.
J.Oppenheimer and H.Snyde Phys.Rev, Vol.56, p.455, 1939
16. T.R.Choudhari and T.Padmanabhan General Rel.Grav. Vol.39, p.1789-1811, 2007
17 I.M.T.Wesson and M.Israelit, General Rel. Gravitation Vol.40, p.2469-2492, 2008
18. Ray S, Bhadhra S. Phys.Lett.A322, p.150, 2004
Ray S, Bhadhra S. Sengupta gr-qc/103103,
Tiwari R.N. Ray S Ind.J. Pure and Appl.Maths.Vol.27, p.207, 1966
Tiwari R.N. Ray S, Bhadhra, Ind.J. Pure and Appl.Maths.Vol.31, p.1017, 2000
19. L.Dymnikova Class. Quantum Grav. Vol.19, p.725, 2002
L.Dymnikova Grav.Cosmol.Suppl. Vol.8, No.1, p.131, 2002
The "cosmod" transformation proposed by the present author (KLN) in 1976-1977 relates the graviton mass with the rest mass energy of a Spin 2 elementary particle ( f-meson) and I have also classified it as an element under unitary transformations of a nonet.As a result I conjectured the existence of a second kind of gravitational force and possible existence of strange quark stars etc. The conjecture was widely reported in national news papers of 1977.
20 H.Weyl in Space,Time, Matter Dover Pub., NY. P.259-260, 1952.
L.Flamm Physik Z., Vol.17. p.448,1916
21. (a)
(b)Tuesday, March 3, 2009The Nuclear Explosive Fission Break-up of Parent MEL-Planet into formation of present day Earth, Mars and Moon entities of the Solar system : trusciencetrutechnology@blogspot.com
22 P.Yodzis ,Phy.Rev Vol.D3, p.2941,1971
23 refer forth coming article on this space-time to be published at this blogspot.
24. A.I.Lilley,M.S.Robewrts, EEEpstein, Dieter N.H. Astron.J Vol.67,p.270-271,2008
25 L. Klebanov,L Susskind, T.Banks-SLAC preprint SLAC-PUB-4705,1988
26 "Scientific American" Number of Articles on Cosmology published during July 08-Feb.2009
27 "The Mathematical Theory of Black Hole" by S. Chandhrasekhar,
Clarendon Press,Oxford, 1983.
J.A.Wheeler "Geons", Phy.Rev Vol.97,p.51-536,1955
T.Regge, "General Relativity without coordinates"
Nuovo Cimento Vol.19,p.558-571,1961
T.Regge and J.A.Wheeler, Phy.Rev. Vol.108,p.1063-1069,1957
28 M.E.Cahill and G.C.McVittie J.Math.Phy.Vol.11, No.4, p.1382, April 1970
*Shocks at boundary discussed by Von Neumann in J.Appl.Phys. Vol.21, p.232 ,1950 and Collapse kills its own Schwarzschild radius i.e. Rsch= 2*m(μ)G/c 2 > R(μ)
@ Formulation of wormhole instanton is given by A.Hosoya and W.Ogura [4] with a classical action as
Scl=-3/(126a2*Λ) + 3*pi/(4*α) ln (4*sqrt(e)/r0H) +O(H) where H= 8/3*pi*G* Λ and α0= f2/(4*pi) with f as fine structure constant, a=(r^2+ r02)1/2) with r0 as the wormhole radius., and Λ the cosmological constant.
