trusciencetrutechnology, Vol.2009, No.7, Wednesday, 15 July 2009: 11:25:45 PM
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, Mobile: 9491902867, Email ID: kotcherlakota_l_n@hotmail.com
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 quantum observables, as well of other types of bilinear physical entities of ghost fields. The associations of the ghosts allow the three symmetric X ± , X 0 and one pseudo-symmetric X0 states that constitute the 4 component Spinor Dirac-like wave field. Further it is found that X0 and ζ0 could generate mixed bilinear physical entities, with a mixing angle designated as θcsc 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 the 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 formulation 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 color and flavor 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”.
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 now an edited trusciencetrutechnology@blogspot.com published post Dated: 23rd July 2009 and full paper in PDF also could 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 Phys, S.U), 17-11-10, Narasimha Ashram, Official Colony, Maharanipeta. P. O, Visakhapatnam-53002, AP, India. Mobile No. 9491902867
INTRODUCTION:
Interaction of neutrinos and gravitational field has been the subject of intensive research in the General Theory of Relativity, by Wheeler [1], who proposed neutrino Geon, an entity constituted entirely of neutrinos, and held together by their mutual gravitational attraction. He and his collaborators have emphasized that a collection of immaterial energy holds itself together by its own gravitational attraction. H. Bondi [2] also sought the concept of tangible energy from an intangible gravitational energy in the Special Theory of Relativity situation. Brill and Wheeler [3], of course, specify that in the absence of gravitational field, the spectrum of neutrino reduces similar to that of an electron, but with a zero rest-mass.
ν R (right handed circular polarisation neutrino) with wave number ( k x , k y , k z) is of positive energy, while ν L (left handed circular polarisation neutrino) is of negative energy. All these negative states have been occupied forming a “neutrino sea” and a hole in it (like Dirac electron hole in the negative sea of electrons) possess the negative momentum of the missing negative energy neutrino [3]. As per Lee and Yang, negative energy neutrino the momentum and spin angular momentum are opposite in direction, according to the equation, H=c (σ.p). Thus real physical antineutrino has also opposite momentum and spin-angular momentum.
Wheeler asserts that Neutrino in an Einstein gravitational potential satisfies radial metric quantity e ν = - g 44 , i. e. The “Geon metric “. The dimensionless effective potential for neutrino is obtained by him as
ξ (ρ) = k (k-1)/ 9 ρ2 where ρ = (c 2 /GM) r -------------- (1)
e ν = e -λ = (1- 2 G M/ c 2 r) for r > (9/4 GM/ c 2)
eν = 1/9 and e -λ= 1 for r < (9/4 GM/ c 2)- (2)
Geon have nothing to do with either elementary particles or the Astrophysics. Geon allows exploring the non-linear characteristics of the Gravitational field [3]. A Geon of toroidal form appears more stable than a spherical Geon, as analyzed also by Ernst F [4]. Properties of thermal Neutrino Geon can be obtained by a suitable scaling of properties of electromagnetic neutrino Geon. Former involves the neutrino energy density, while the later involve the photon energy density. In a spherical neutrino Geon, it is possible to conceive the orbits of neutrino, with energy of Geon concentrated in a thin spherical shell. If only states of single energy are occupied the neutrino would posses, higher energy to provide a given total mass [3]. The wave length of the neutrino is then very small, relative to the radius of Geon and so the quantum number k is very large.
Brill and Wheeler [3] consider an inelastic process of
(ν + ν’ =====> G+G) ---------------------------------------------------------------- (3)
where G is a graviton, just as two electrons would collide to produce an electron pair, while an elastic process would yield
ν +ν====> (via =>G+G) => ν’ + ν’------------------------------------------------ (4)
In the realm of true elementary particle physics, we have the processes
ν + ν’=====> μ+ + e- --------------------------------------------------------- (5)
and
ν + ν=====> (via => μ+ + e-) => ν’ +ν’-------------------------------------------- (6)
The testimonial Ratio cross-sections is
σ(ν + ν==> μ+ + e- ) : σ( ν + ν==> G+G) = (4 π 2 gβ c 2/ G h 2)~ 10E+34----------(7)
Here gβ is the beta coupling constant approximately ~ 10E-49 erg.cm 3.
It’s interesting to note,according to Wheeler [1 ] that one electron and one neutrino attract when their propagation vectors are anti-parallel with twice Newtonian value and not at all, when their propagation vectors are parallel. Tolman and Ehrenfest [5] have investigated the metric due to a pencil-like concentration of Electromagnetic energy of identical character. In Gamow’s Urca process, neutrinos and anti-neutrinos escape in gravitational contraction more readily than photons and the rate of Energy liberation is decided by them [6]. In the stellar interiors, what is the equilibrium distribution of neutrinos? Is a question, which needs to specify the local neutrino temperature and the Fermi Energy?
Gamow et al [6] considered processes such as
Hot neutron==>cooler proton+ electron + neutrino
Hot proton==>cooler neutron+ positron + the neutrino------------------------------- (8)
With the expressions given for the energy –momentum tensor T ik for the Dirac field Ψ and the solutions offered for Einstein-Dirac Equations for the case of neutrino by Brill and Wheeler [3]. Davis and J.R .Ray [ 7 ] considered Ghost neutrinos, in General Theory of Relativity, adopting a Taub Metric[ 8]. They assert that metric field generates a gravitational field which is consistent with the geometry of gravitational field, but won’t generate it, because it yields zero Energy-Momentum tensor. Brill and Cohen [9] obtained solutions of Einstein-Dirac equations in Taub metric universe with diagonal Ricci Tensor that yields neutrino field to generate a gravitational field which is not consistent with the geometry of Taub universe. Since the currents do exist, Ghost neutrinos are likely to be observed and similarly other Ghost fields.
SECTION I: THE AIM OF THE PRESENT WORK:
The aim of the present work is to solve the Einstein-Dirac-like Equations for the case of bilinear associations of ghost neutrinos and ghost electrostatic charged spin entities viz., X 11, X 10, X 1 -1 and X 00. The field equations to solve are
Gik = Rik - g ik /2 R= -K T ik + T h ik - 8 π E ik --------------------------------------(9)
The term T h ik involves magnitude of the magnetic flow vector hα given by the expression,
h2 = - ( h1 h1 + h2 h2 + h3 h3 ) with h1 , h2 , h3 as the components of the four-vector magnetic flow vector , where hα =( h1 , h2 , h3 ,0).
The constant K=8πG/c, here G is universal gravitational constant and c is velocity of light and where T ik is the Energy-Momentum tensor for the Dirac-like field Ψ (with 4 components as the bilinear physical entities X 11 X 10 X 1 -1 and X 00 )
T ik = -(hc/8π) [Ψ† γ i Ψ ; k - Ψ†; k γ i Ψ + Ψ† γ k Ψ ; i - Ψ†; i γ k Ψ]-------------------(10)
Here h is Planck’s constant and c is velocity of light and where Ψ satisfies Dirac-like Equation
γ i Ψ; i =0 ---------------------------------------------------- (11)
with zero-rest mass. The electrostatic Energy Tensor is defined as
E ν μ= -F να F μα+ (1/4) g ν μ F αβ F αβ--------------- (11a)
where F αβ is an anti-symmetric second-rank tensor yielding Maxwell’s equations
F λμ; μ = J λ---------------------------------------- (11b)
and λis the four vector current density. For Dirac-like particles I expect the trace of the Energy-Momentum tensor to vanish but R the scalar curvature Tensor of the metric needn’t vanish. This is so, since in the Taub metric
ds 2= e 2u (dx 2 –dt2) + e 2v (dy2 +dz2) ----------------------------------- (12)
The exponential parameters u, v are assumed to be functions of (x, y, z, t). Restriction to (x, y, t) results in the case of an embedded 2D space, that has been dealt and led to a newer physical insight of the obtained solutions of the Einstein-Dirac-like field equations in the General Theory of Relativity.
I have retained, in the calculations, the Cartan orthonormal frame, defined by
ω 1= e u dx, ω 2= ev dy, ω 3= e v dz, ω 4= e u dt ------------------------------ (13)
with the said dependence of u, v on (x, y, z, t) in general.
The Dirac-like field function Ψ satisfies the covariant derivative of a Dirac-like Spinor, in the Cartan orthonormal frame of reference of the metric [Eq.9].
The Dirac-like Spinor Equation is
Ψ ; k = [ω j - Γ j] Ψ ------------------------------------------------------------ (14)
Where Γ j are the spin connections defined by
Γ ρ= - (1/4) γ abρ γ a γ b ------------------------------------------------- (15)
where γ abρ are the Ricci rotation coefficients. I adopt the Dirac matrices in the formalism given by Jauch and Rohrlich [10], to enable comparison with the results obtained earlier by authors in Refs. [1], [3] and [4].The essential difference lies in the fact that I don’t explicitly consider the neutrino (or the Ghost) field or its projections to get the left and Right circular polarisation of the neutrino field. Instead, I have retained (as per the original idea of Dirac) that Ψ is a 4-component field. I took that the 4-components are just the 4 projections labeled as
X 11, X 10, X 1 -1 and X 00 ------------------------------------(16)
Or in a simplified notation,
Χ + = Χ 11, Χ 0 = Χ 10, Χ- = Χ 1 -1 and ζ0 = Χ 00 ------------ (16a)
In this convention the Ψ is obtained as bilinear physical entities of two fermions i.e. the ghost neutrino and the ghost electrostatic bare charge spins, which are of course, both mass-less. Elaborate discussion on this is postponed to another section.
The solution aims at solving the Einstein-Dirac-like equations, for a test of the bilinear physical entity field in the given geometry. I have also obtained the solutions in the Energy-Momentum tensor to realize the Energy momentum tensor generated by these bilinear physical entities, as well, in both a constant magnetic permeability and infinite electrical conductivity. For the later I have used the electrostatic Energy Tensor
E ν μ= -F να F μα+ (1/4) g ν μ F αβ F αβ ------------ (17)
where Fαβ is an anti-symmetric second-rank tensor yielding Maxwell’s equations
F λμ; μ = J λ----------------------------------------------- (18)
and J λis the four vector current density.
The coupled Einstein-Dirac-like equations are defined to be
Gik = R ik - g ik /2 R= -K T ik -8π E ik + T h ik ------------------------------ (19)
And the T h ik is related with the magnetic field magnitude
h2 = - (h1 h1 + h2 h2 + h3 h3) -------------------------------------------------------- (20)
with h1 , h2 , h3 as the components of the four-vector magnetic flow vector ,
where hα =( h1 , h2 , h3 ,0)--------(21)
In the treatment of the solutions of Einstein-Dirac-like equations described in the subsequent sections I have considered two possible cases of the Taub metric Universes with either the
T h ik directly involving the entire magnitude of the h 2 or the individual spatial Gik involving the individual magnitudes -h1 h1 , -h2 h2, and -h3 h3 of the magnetic flow vector. Thus the source of gravitational field is not only the neutrino field but also, the free magnetic field strengths and the bare mass-less electrical charges.
The coupled Einstein-Dirac-like equations have a richness of variety of solutions, by use of the above detailed Energy Momentum Tensor quantities of free magnetic strengths, bare mass-less electrical charges and the mass-less neutrinos. Richness is obvious when contrasted with the solutions of Einstein-Dirac equations.
SECTION II: SOLUTIONS OF DIRAC-LIKE EQUATIONS:
The non-zero spin connection coefficients? Γ i are listed below:
Γ 1= - (hc/8π) (e – v u 2 γ 1 γ 2 + e – u u 4 γ 1 γ 4)
Γ 2= - (hc/8π) (e – v v 1 γ 1 γ 2 + e – u v 4 γ 2 γ 4)
Γ 3= -(hc/8π) (-e – u v 1 γ 1 γ 3 + e – v v 2 γ 2 γ 3+e – u v 4 γ 3 γ 4)
Γ 4= - (hc/8π) (e – v u 1 γ 1 γ 4 + e – v u 2 γ 2 γ 4)---------------- (22)
with no dependence of u and v on the x 3= z coordinate of the metric. Here, as in the usual convention a comma denotes partial differentiation and a semi-colon represents the covariant differentiation. With no loss of generality one may assume h/2π=c=1.
For the condition that
Ψ ; k=0---------------------------------------------------------- (23)
Using the spin connections listed above I have obtained the following set of new equations, for the case u = -v
Ψ; 1=-1/2e2u u2γ1γ 2Ψ-1/2u4 γ1γ4Ψ
Ψ ; 2=-1/2e-2uu 1γ 1 γ 2Ψ+1/4e-2u u4 γ2 γ4Ψ
Ψ ;3=-1/2e-2uu1γ1 γ 3Ψ-1/2u2 γ2γ3Ψ + 1/2u 4 γ3 γ4Ψ
Ψ ; 4=-1/2e2uu 1γ1 γ 4Ψ -1/2u2γ 2γ4Ψ ----------------------- (24)
And for the case v=0 the set of new equations are
Ψ ;1=-1/2u2γ1γ2Ψ - 1/2 e-u u 4 γ1γ4 Ψ
Ψ;2=0
Ψ;3=0
Ψ ;4=-1/2u1γ 1 γ 4Ψ -1/2 u 2γ 2 γ 4Ψ ------------------------------ (25)
These imply the existence of partial derivative currents, a novel feature of the present formulation and these involve remarkably the ordinary currents (without imaginary number i) given by
S j=Ψ†γ jΨ--------------------------------- (26)
For the u=-v the diagonal
Ψ†γ1 Ψ, 1=-1/2e2u u2S2-1/2 u4S4
Ψ†γ2Ψ , 2=-1/2 e –2u (u1S1-u4S 4)
Ψ†γ3Ψ ,3=-1/2 e –2u( u1S1)+1/2u2S2+ 1/2u4S4
Ψ†γ4Ψ , 4=1/2 e 2u (u1S1-u2S2)-------------- (27)
And for the case of v=0 only two diagonal partial derivative currents exist. These are
Ψ†γ1 Ψ, 1=-1/2[u2S2+ e–u u4S4]
and Ψ†γ4Ψ , 4=1/2 (u1S1-u2S2)--------------------------------- (28)
Other many non-diagonal currents do exist in both the situations of u = -v or v=0 cases. For example, for u= -v we have
Ψ†γ3Ψ ,2=-1/2e–2u( u1Ψ† γ1γ2γ3Ψ+u4Ψ† γ2γ3γ4Ψ )---------------(29)
And for v = 0 case
Ψ†γ2Ψ ,1 = 1/2 ( e –vu 2 S1 + e –v u 3 Ψ† γ1γ2γ3Ψ-e–uu4Ψ† γ1γ2γ4Ψ)--------(30)
which are non-vanishing?
These expressions for the partial derivative currents of the present formulation have a far reaching significance in the description of gravitational collapse of a cosmic entity.
The zero rest - mass neutrino equation for Dirac field [1, 3, 9] Viz,
γkΨ; k=0 --------------------------------- (31)
has no direct analogue in the present formulation, described herein, because, at the outset, I haven’t allowed at all, the possibility of existence of massive particles. This is a radical departure, from conventional theories of Ghost neutrinos, Geons, and Ghost fields, studied earlier by several authors, in the subject of General Theory of Relativity, who have used only the coupled Einstein-Dirac Equations.
THE CALCULATIONS:
Continuing my presentation of the new formalism enunciated herein, it may be noted that the other Energy Momentum Tensors are those that involve the presence of Magnetic strengths, and the Electrostatic charge. These are respectively, designated by T h ik and E ik, where h stands for the magnetic contribution to the Energy Momentum Tensor, to distinguish it from the Spin contribution T ik [refer Eq.10].
Thus the Einstein-Dirac-like Equations may be written down as, [refer Eq.19]
Gik=Rik-g ik/2R = -KT ik + T h ik-8πEik --------------------- (32)
The term T h ik involves magnitude of the magnetic flow vector h α given by the expression,
h2 = - ( h1 h1 + h2 h2 + h3 h3 ) with h1 , h2 , h3 as the components of the four-vector magnetic flow vector [refer Eq.21] where hα=( h1 , h2, h3, 0).
I have obtained
h1 = F 1e -2v, h2 = F 2 e-2u, h3= F 3 e -2u,-------------- (33)
So, h2=-(F21e-4v-2u+ 2 F22e-4u-2v) ------------------------------------------------- (34)
I have assumed, for an aesthetic beauty that F 2=F 3 -------------------------- (35)
for the magnetic strengths, in the x2 = y and x3 = z directions. Further from the symmetry considerations of the present formulation I set F 1= 0 and treat it as the Case I. Also for the case Ia, I adopt that u = -v and for the case Ib, I adopt that v=0.
Thus for the CASE Ia, v=0
I get h2=-2 F22
e -2u and
for the CASE Ib, u= -v
h2 = - 2 F22e -4u.
The Electrostatic Energy Momentum Tensor, if v=0
E 22 = 1/ (8π (x2) 2)
And, if u = -v then
E 22 = -ε2/ 2 f4 (x2) 4 /(x4± x 1) 4-------------- (36)
Here f is an integral constant and x 4, x 1 are the coordinates (t, x). Obviously, one would take ε, as the measure of the electrostatic bare (mass-less) charge. The two universes of the Taub metric, in my formalism, obtained with u= -v and v=0 exponential parameters of the metric, may be noted to be entirely distinct.
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:
As an example the Einstein Tensor
for the case Ia u= -v is given by
G 22 = e -2u(- u 12 +u 42) + e 2u u22 =-KT22+ κμ h2-8πE22---------------------(37)
where I have retained the magnetic permeability κ and the dielectric capacity μ, which may be scalar functions of position or constants. K has been previously defined.
The assumption made here, that any pressure term would be balancing the magnetic Energy-Momentum tensor term and would be succinctly equal to it. It is a physically reasonable and plausible criterion for finding the set of new class of solutions for the Einstein-Dirac-like equations enunciated above.
CASE Ia. u=- v
For the Taub Universe, it is observed that component G 3 3 of Einstein’s Tensor , the Einstein Equations need to have, an additional quantity -4 f2/ (x 4 ± x 1) 2 on the right hand side of the expression for G 3 3 in addition to the -8πE3 3term. We assert, that in our formulation, the magnetic Energy Momentum Tensor contribution to the G 33 and G 22 vanish in toto. The component G 44 gives the energy matter density expression as
ρκ=4/ f2 (x 2 ) 2 + f2 /(x 4 ± x 1) 2 + 4 πε2 f4 (x2 ) 4/(x4 ± x 1)4------------(38)
In order that G 2 2 = G 3 3 we need to adopt an additional contribution of -4/(x 2 ) 2 for the G 3 3 component of Einstein Tensor . With T 2 2=T 3 3=0 we may be able to retain the identity of
E2 2=E 3 3.
The G 2 2 + G 1 1 = e-2u ( -2 u44 + 6 u42) + κμ h2-8π(E2 2+E1 1) relation simplifies to
4/ f2 (x 2 ) 2 = κμ h2 - 8 π (E 2 2 + E1 1).
But we note that Eq.43 also involves the quantity
4/ f2 (x 2) 2 .These results are surmised to imply that we have to consider the explicit possibility of the individual components of the Magnetic Energy Momentum Tensor contributions.
CASE Ib. v=0.
But we note that Eq.43 also involves the quantity
4/ f2 (x 2) 2 .These results are surmised to imply that we have to consider the explicit possibility of the individual components of the Magnetic Energy Momentum Tensor contributions.
CASE Ib. v=0.
G 2 2= e -2u (- u 44 - u11 )+ (-u2 2 ) = - K T22- 8πE 2 2----------------(39)
Since T22=0 and our set of solutions shows that u 44 = u 11, we get for the possible strength of the electric charge ε,
ε2 = - 1/ 4 π (x2) 2-------------------------- (40)
in the form of a Green’s function propagator. Both E 2 2 and E 3 3 are equal to -ε2/ 2.
We note that G 11= 0 specifies the relation between both h 2 and E11.
Simple arithmetic shows
that the charge ε = 2 F2 (x 4 ± x 1) 2 / f2 (x 2) 2----- (41)
This is a significant result of the present formulation which yields a dimensionless relation of the important constants of the formulation i.e. between ε, F2 and f2.
Note that we have G 22 - G 11 = + κμ h2 because we have adopted that pressure term balances the magnetic term.
In this case G 11 = - (u22 + u2 2) = 0 and as well
G 44 = (u 22 + u2 2) = 0= -8 πE44+ κμ/2 h2 ----------------------- (42)
But the later gives the expression for the material density
Κρ = 4 πε 2-½κμ h2---------------------------------------- (43).
This emphasizes the result that a balance of electric and magnetic field energies only makes the material density to vanish.
The solution further asserts that all the non diagonal Einstein Tensor components such as T νμ (with ν ≠ μ) vanish, since we have assumed that the exponential parameters u, and v of the metric are not functions of x 3= z. This finding is found to be of paramount importance to readily deduce the Energy Momentum Tensor for the Dirac-like equations. In the next section, I present the details of this discovery.
This is a significant result of the present formulation which yields a dimensionless relation of the important constants of the formulation i.e. between ε, F2 and f2.
Note that we have G 22 - G 11 = + κμ h2 because we have adopted that pressure term balances the magnetic term.
In this case G 11 = - (u22 + u2 2) = 0 and as well
G 44 = (u 22 + u2 2) = 0= -8 πE44+ κμ/2 h2 ----------------------- (42)
But the later gives the expression for the material density
Κρ = 4 πε 2-½κμ h2---------------------------------------- (43).
This emphasizes the result that a balance of electric and magnetic field energies only makes the material density to vanish.
The solution further asserts that all the non diagonal Einstein Tensor components such as T νμ (with ν ≠ μ) vanish, since we have assumed that the exponential parameters u, and v of the metric are not functions of x 3= z. This finding is found to be of paramount importance to readily deduce the Energy Momentum Tensor for the Dirac-like equations. In the next section, I present the details of this discovery.
CASE IIa:
With v=0 and with F 1, F 2, F 3 individual magnetic field strengths. The individual magnitudes of magnetic field components of the flow vector hα= (h1, h2, h3, 0) are specified by,
h 12= F1 2 e-4v-u,
h22 = F 2 2e-4u-2v, and
h32 = F 32 e-4u-2v------------------- (44)
The component G11=0
led to the expression
κμ F 12 = 4 πε2 f2(x 2) 2/ (x 4± x 1) 2 ------------------------- (44a)
The component G 44 =0 yielded the simple result that
- 4πε2 + κρ=0. -------------------------------------------------------- (45)
It says the only electrical energy density gives the matter density of the cosmic entity with a constant magnetic permeability. The matter density happens to be the ratio of electrical density and the magnetic permeability.
The components G 22, G 33 give the expression ε2 = - 1/ 4 π (x2) 2.
CASE IIb:
if u = -v
then G 11= e – 2u( u 12+ 3 u 42 )- u 22 = 4 π ε2 e 4u +κμ F 12 e 2u yields the solution
κμ F 12 = (x 4 ± x 1) 2 / f4 (x 2 ) 4- 1/ (x 2 ) 2 - 4 πε2 f2 (x 2 ) 2/ (x 4±x 1) 2 -------(46)
While the difference of G 22 - G 33yields the solution,
2 κμ ( F 32 - F 22 )=- 8 f4(x 2 ) 2 / (x 4 ± x 1 ) 4--------(47)
This shows that F 2 ≠ F 3.
SECTION IV: THE KILLING VECTORS AND THE MAGNETIC STRENGTHS:
The Taub metric with exponential parameters u, v as functions of x 1, x2, t could reveal the space-time properties admitting group of conformal motions. I could obtain explicit expressions for the Killing vector components and found out how they depend on the magnitudes of the magnetic strengths. I next, assume that the space-time admits a group of conformal motions
i.e. Lξ = 2 ψ’g i k (A, B, C, D) ------------------------- (48)
where Lξ signifying the Lie derivative along ξ i and ψ’(x i) is the conformal factor and I have introduced the constants A, B, C, D and respectively for the expressions involving the diagonal metric tensor components. I expect that these killing vectors would generate a set of exact solutions of the field equations. Among other significant features of the conformal killing vectors, I concentrate on their relation with magnitude of magnetic strengths, as proposed in the present formalism, for the two distinct space-time universes of the Taub metric.
It is necessary to adopt that all the quantities depending on the coordinate x 3=z vanishes.
If A = D then ξ 4, 4+ ξ 1, 1= 2 u1ξ 1+2 u 4 ξ4 and, for the case v=0 we find that B= 0. I have due to symmetry adopted C=0. Simplification arises for both u=-v and v=0 cases with u 1 =- u 4 and ξ 1 = ξ 4. I have derived the following set of each six relations involving non-diagonal terms, for the two cases of u = -v and v=0 respectively, illustrated below side by side.
For u= -v for v=0
ξ 1,2+ ξ 2,1= 2 u2 ξ 1 – 2 u 1 ξ 2 ξ 1,2 + ξ 2,1= 2 u2 ξ 1
ξ1,3+ ξ 3,1= 2 u3 ξ 1 – 2 u 1 ξ 3 ξ1,3 + ξ 3,1= 2 u3 ξ 1
ξ 1,4 + ξ 4,1= 2 u4 ξ 1+ 2 u 1 ξ 4 ξ 1,4 + ξ 4,1= 2 u4ξ 1 + 2 u 1 ξ 4
ξ 2, 3 + ξ 3, 2= -2 u3 ξ 2 – 2 u 2 ξ 3 ξ 2, 3 + ξ 3, 2=0
ξ 2,4 + ξ 4,2= -2 u4 ξ 2 + 2 u 2 ξ 4 ξ 2,4 + ξ 4,2= 2 u 2 ξ 4
ξ 3,4 + ξ 4,3= -2 u4 ξ 3 + 2 u 3 ξ 4 ξ 3,4 + ξ 4,3= 2 u 3 ξ 4-------------------- (49)
The exact solutions obtained by the use of killing vectors, differ slightly from that of the Einstein-Dirac-like equations of the General Theory of Relativity. Instead of (x 4 ± x1) only the x 4 - x 1 seems to get involved in the result of e u = f (x 2) / (x 4 - x 1). But of more interest, are the expressions for the killing vectors obtained involving the magnetic field magnitude h 2.
For the case v=0 the obtained expressions are when the total magnitude of magnetic field
h2= -2 F 2 2 e – 4u with F2=F 3, F 1=0 and
ξα = (-2 F 2 2 /h2f2, 0, 0,-2F22 /h2f2)--------------(50)
And when F2, F 3, and F 1 are distinct and non-vanishing (Eq.44)
ξα = (-F1 2 /h12 f 2, 0, 0, 0) ----- (51)
For the case u=-v the two corresponding expressions are respectively,
ξα = (-2 F 2 2 /h2f 2, f2 h2 x2 2 / -2 F 22 , 0, -2 F 2 2 /h2 f 2) -------------(52)
When the total magnitude of magnetic field is
h2 = -2 F 2 2 e – 2u with F2=F 3, F 1=0
And when F2, F3 and F1 are distinct and non-vanishing (Eq.44)
ξα = (- F 1 2 / h12f2 , - f2 x22 F2 2 / h2 2 , 0 , 0)---------------------------------(53)
The results found are ξ2 = ψ’ B(x 4 – x1) 2 /2 f 2 x2 2 + 2(x4 – x 1) ---- (54)
and with A=D ξ 4 = ξ1 = ψ’ A e2u/4u1= x22/(x4–x1)2--------------(55)
Since the metric tensor given by Eq.12 becomes for u=-v;
ds2=f2 (x2)2/ (x4±x1) 2(dx2-dt2)+ (x4–x1)2/2 f2x2 2(dy2+dz2) ----------(56)
We get, as an example,
ρκ = ψ’ A/ u 1 x 2 + 4 u 1/ ψ’ A(x4 ± x1) 2 + 64 π ε2u 12( x 2 ) 4 / (x 4 ± x1) 4 ψ’2 A2--------(57)
Involving both the ψ’ and the constant A = D.
We may introduce different expressions for ψ and, obtain different Lie elements to accommodate the so-called Bag Models of confined quarks, or strange fields[13] or even the super symmetry fields and as well could realize the mass-less Wess Ghosts (W ± , W0) and Zumino Z0(Zumino Ghost.)
SECTION V: A. THE COLLAPSING COSMIC ENTITY GRAVITATIONAL SUPERCONDUCTING CURRENTS:
In my formulation one should note that the Dirac –like wave function is a bilinear entity of the neutrino and the bare electrostatic charge. The physical entities, earlier in this article, referred as Χ 11, Χ10, Χ 1 -1, and Χ00, , involve Χ 10, and Χ 00, that eventually correspond to, in an analogy, with W 0 and Z 0 and an angle of their mixing could be specified designated as θcsc where csc stands for collapsing superconductivity currents, as enunciated in the present formulation. But our physical entities are the Ghosts. Also the two 2 x 2 matrices for the Χ 10 and Χ 00 combination of the neutrino and the bare electrostatic charge involved, may be specified by the 2 x 2 matrices as:
J= {0-1; 1 0} and Γ={1 0; 0 -1}
designated by J and Γ respectively. They yield the pseudo-scalar and the symmetric scalar, bilinear combinations. [11]
The matrix element for the elastic neutrino-electrostatic charge scattering due to the Lagrangean, given below,
£=(Gc/2)Ψ†e’γ k (GV – GA γ 5 ) Ψ ν’e Ψ† e γ k ( G V – G A γ 5 ) Ψ νe -------(58)
yields a cross-section proportional to
s= (p νe + p e) 2
where p νe, pe are respectively the momentums of neutrino and the bare electrostatic charge. Gc should be taken to be the current coupling constant of the collapsing cosmic entity. The G V =1 and G A which is negligibly small, are the vector and the axial vector coupling constants. A similar type of matrix elements for the current-current interactions of the anti-neutrino (Ghost) with the bare electrostatic charge may be written down. This way of analysis is convenient, for comparison with the high energy studies of emergence of new particles in high energy collisions, but may not be substantially valid in the spirit of the present formulation.
This expression, in physical content, differs from the previously considered several Lagrangeans of this form enumerated and discussed by J. Leite Lopes in his book on Gauge Field theories. [ 12]. In the Gauge field approach several authors, discussed the origin and acquisition of mass by the elementary or the fundamental particles, by physical mechanisms and phenomenon that occur, at very high energies of the order above TeV in LHC, SSC, TRISTAN, TEVATRON, RHIC, LEP Colliders etc., machines. The only solace in my approach to study the elementary particle physics is to accept the existence of universally valid coupling or the interaction constants such as, the Fermi coupling constant, V-A coupling constants etc. and most probably they are universal. This would be a digression and wouldn’t be taken up for further elaboration.
SECTION VI: HOW THE COLLAPSING COSMIC ENTITY GRAVITATIONAL SUPERCONDUCTIVITY CURRENTS ORIGINATE?
It is possible to conceive it since we are dealing with a novel phase of the Ghost neutrino and the bare electrostatic charge, which display the unusual property of the state of a collapsing cosmic entity of infinitely high conductivity. An understanding of this new phase is by recognizing the fact that the equilibrium state of the cosmic entity becomes unstable, by the interplay of the weak attractive interactions between each pair of the bare electrostatic charges. Such an interaction would evidently be initiated by the ghost neutrinos. Space-time deformation of the cosmic entity, results in excess of positive charge, due to the displacement of negative charges, and a passing by second bare electrostatic charge would be attracted to it.
The space-time deformation, of course, in terms of the world lines geometry, produces a weak attractive interaction between the pair of negative charges. The condition of a vibrating string of the space time would become characterized by a vibration period and offers a time duration in which the bare electrons get correlated via the neutrino bilinear association and dissociations. Since spatial extension of the ghost neutrinos is negligibly small, one could take the extremely long interaction range of the bare electrostatic charges, to be about a few hundred times the size of the bare charges in order, to account for the considerable screening distance of the repulsive coulomb interaction. Continuous association and dissociation of the bare charges with neutrinos might make them as well virtual.
Another method of finding the extent of long range interaction between the two electrostatic ghost charges via the ghost neutrino participation is to consider a possible collapsing superconductivity current surface.* Around this surface (analogous to the Fermi surface in conventional physics of superconductivity) there exist the bilinear physical entities. This is the basic precept of the present formulation for generation of novel bilinear physical entities. Band gaps are the result of the bilinear entities energy states. The corresponding momentum distribution over these band gaps, determines the range of the spatial extension of the csc interaction. Also instead of just the exchange of ghost neutrinos mechanism, it is conceivable to have the virtual bilinear physical entities in their excitation near the postulated current surface to contribute to collapsing superconductivity current.
Speaking in terms of frequency of such excitation mechanism, say ω ex and ω eν the charge-neutrino frequency of the current surface, an inverse dielectric function would be,
1/εex= ωeν/ [ω 2 – (ω ex - i δ) 2] together with some screening contribution due to deformations. Here
εex and ω ex are dependent on the wave vectors of excited bilinear physical entity spread over the current surface. [12a].
I refer these excited bilinear physical entities as “quasi-ghost entities”. Tomographic bosonization, renormalization procedures and broken symmetry aspects of conventional theories of the superconductivity in metals and materials, adds only to the richness of my formulation, in view of the role played by the quasi-ghosts in the considerations of the collapsing gravitational superconductivity currents of a cosmic entity. [12b.]
To understand how a magnetic field affect the color superconducting currents formed by the pairing of the Dirac-like wave function bilinear physical entities, it is necessary to use a linear combination of bare electrostatic charge and the neutrino that are both mass-less. In this paper I have detailed how to formulate Einstein-Dirac-like field equations of the General Theory of Relativity, specifically with the bilinear physical entities (subject to strange fields, or magnetic fields, or fluid pressures and the electrostatic fields etc. or some combinations of these [13, 14]). This approach led to the determination of current densities of pairing of electrostatic charges via the mass-less neutrinos (Ghost neutrinos).
The band gaps of the superconducting currents arise due to the different energy states of the bilinear physical entities split as symmetric and pseudo-symmetric bi-products. They have been clubbed into a single Dirac-like wave function constituting its four components. The Dirac –like wave function is a bilinear entity of the neutrino and the bare electrostatic charge spins. The physical entities, earlier in this article, referred as Χ 11, Χ 10, Χ 1-1 and Χ 00 involve Χ 10 and Χ 00 that eventually correspond to, in an analogy, with W0 and Z 0 and an angle of their mixing could be specified designated as θcsc where csc stands for collapsing superconducting currents, as enunciated in the present formulation. The obvious direct product SU (2) x U (1) type of gauge quantities become realizable with the said mixing of the appropriate bilinear physical entities. An angle of mixing of the symmetric and pseudo-symmetric bi-products of wave functions of neutrino and the electrostatic charge spin states may be obtained as the collapsing superconducting currents (csc) angle. It has remarkable analogy with that of Weinberg and Salam theory in high energy physics. Except that the particles that we surmise are mass-less. For a comparison with the Wess and Zumino particles we may take in an abbreviated notation that
Χ+=Χ11,
Χ0=Χ10, Χ-=Χ1-1 and ζ0=Χ00 so that,Wess(W±, W0) and Zumino Z0correspond to (Χ ±, Χ 0) and ζ0 bilinear entities respectively.
These bilinear entities are the associations of the neutrino and electrostatic charge Ghost fields. Introduction of color and flavor characterization adds to the enchanting variety of collapsing superconducting current densities.
SECTION VII: PARTICLES VERSUS VECTOR FIELDS:
The Salam-Weinberg models Spinors in 6D-manifold are described by an 8-component wave function. These are of course split into 2 4-component spinors which correspond to electron and electron-neutrino. The idea that rest masses arise due to interactions with scalar fields, from a conformal factor, via a mechanism that of Higgs seem to ascertain that particles are originally mass-less. But I have adopted that the fields are ghosts and no particle concept has been utilized.
The introduction of each new dimension of space time by one adds a new vector field, just as in the case of a 5D-theory of space-time by Kaluza-Klein. Four vector fields became necessary for Weinberg-Salam 8D-models. The interaction between particles and the fields has been described by the hyper-charge and the projection of iso-spin to account for the charge as Q=Y/2+ T 3 in a six-dimensional model.
The supergravity models adopt usual 4D space-time but additional anti-commuting quantities Grassmanian variables. Minimum gauge group needed is SU (3)xSU(2)xU(1) that goes with a minimum 7 extra dimensions and with 4D of space-time, hence total dimensions would be just 11=7+4.
It was noticed some years ago that the for mass-less fields in a flat 4D space time equations are invariant with a 15-parameter group of conformal transformations, of which 6 correspond to Lorentz group( a sub group). The whole group of conformal transformations is non-linear and could be regarded as a group of linear transformations of flat 6D manifold with two time axes and four spaces. The extra time-coordinate is found useful to renormalize the rest masses.
The idea of a complex time variable viz.t+it’ where t’>0 and i is the imaginary number √-1, has been also utilized by many people for path integrals making use of analytical continuation [[15].
SECTION VIII: THE CHARACTERIC SURFACE CONES THAT DEPICT THE WAVE PATTERNS:
Another idea behind the present work is to analyse the 3D wave like patterns of the space-time universes. The case Ib. v=0 solution, for G 22 explicitly given in Eq.39, shows that
(u 44 + u 11)=0----------------------- (59)
Surprisingly, it represents a vibrating string like behavior of the coordinates x 1 and x4 of course, over a membrane defined by (x 1 , u ) in the conceived phase diagram of x 1 , u and x4.
The surface r - x4= constant, form the characteristic solutions of a 2Dphase wave equation. I adopt that
r = {(x 1 - x 1 p ) 2 + (u - u p) 2} 1/2----------------- (60)
Then the characteristic surface r - x4= r 0 intersects the (x 1 , u ) plane in a circle of radius r0 , with the centre at x 1 p , u pon the membrane at time x4 = x 4 p were on the circle of radius r 0 at x4= 0. The idea referred to above is to ascertain that when the membrane is struck by the vibrating string, circular waves go out at a unit velocity and generate the characteristic cone surfaces in the (x1, u, x4) i.e. the 3D space-time. I term these as the collapsing current cones in analogy with the light cones [16].
SECTION X: SUMMARY AND CONCLUSIONS:
At the outset I took that ψ be a solution of Dirac-like equation, for bilinear associations of mass-less neutrino and charges (both ghosts) in a space-time continuum of arbitrary curvature. The infinite conductivity property of a collapsing gravitational cosmic entity has been identified with existence of a possible state of collapsing superconducting currents. These are modelled to be produced by the superconductivity of a long range interaction of pair of bilinear physical entities. Eventually the bare electrostatic ghost charges pair up via the interaction mediated by ghost neutrinos.
An understanding of this new phase is by recognizing the fact that the equilibrium state of the cosmic entity becomes unstable, by the interplay of the weak attractive interactions between each pair of the bare electrostatic charges. Such an interaction would evidently be initiated by the ghost neutrinos.
The formulation is characterized by the possible ground states of 16–band gaps resulting from the pair combinations of a Dirac-like 4-component Spinor function. The Lagrangean formulation of the superconductivity current-current interaction has been given, with the specification of the universal coupling constants. A bilinear physical entity is defined to be an association of ghost neutrino and ghost charge spins. The associations of these ghosts allow the three symmetric and one pseudo-symmetric state that constitute the 4 component Spinor Dirac-like wave field. Further it is found that Χ 0 and ζ0 could generate mixed bilinear physical entities (like γ and Z of Weinberg and Salam gauge theory) with a mixing angle designated as θcsc where csc stands for collapsing superconducting currents.
The four killing vector fields’ ξ 1, ξ 2, ξ 3=0, and ξ 4 have been determined, for the two distinct Taub metric universes. Also these are expressed in terms of magnetic field strengths. For the later again two cases have been considered, one with possibility of F 1 =0, F2 = F 3 and the other with F1, F2, F3 to occur individually. The Einstein-Dirac –like field Equations led to determination of a class of exact solutions in terms of the killing vector fields. A conformal group of motions with the constants A, B, C, D and ψ’ has been also presented.
The Dirac-like spin connections, simple currents, and the partial derivative currents (using the condition that Ψ ;j=0 ) have been worked out and listed for the two different conditions of the exponential parameters of the Taub metric tensor, u= -v and v=0 and also for the cases of F 1 =0, F2 = F 3 and the other with F1,F2 , F3 to occur individually.
These have been used to find a class exact solutions of the Einstein-Dirac-like Equations of the General Theory of Relativity, utilizing the explicitly evaluated the Einstein Tensor components. A string and membrane phase situation has been formulated to visualize the csc wave patterns on “Current Cones”, in a (x 1, u, x4)-3D space-time projection.
The Dirac-like field physical entity Energy Momentum tensor vanishes in diagonal terms, but the Dirac-like field and the current densities do not vanish. For u= -v, T 12, T41, and T 42 do not vanish and setting G12, G41, and G42 as zero yields the equations for bilinear physical entities of ghost neutrinos and ghost bare charges. The solutions obtained by solving the magnetic field equations, for the two situations of u=-v and v=0 the partial derivative currents could be solved. Thus certain measurement methods are feasible, in the gravitational fields of Taub universes, to detect the presence of (or generation of) the bilinear physical entities, not only those mentioned herein, but as well other ghost fields. Though they are just associations, it seems that the infinite conductivity of the cosmic entity offers enormous possibilities of existence of novel bilinear physical entities hitherto un-thought of during the phase of collapsing gravitational cosmic entity.
Essentially in my approach I have adopted the philosophy that no allowable system of forces can even produce a difference in energy between the states of Ψ ν and Ψ e, of the bilinear associated system of ghost neutrinos and the ghost charges spins. Such an assumption of irrevocable degeneracy of this kind, of course, persists only under the special conditions of existence of a collapsing superconductivity current surface. One could generate a new solution of Dirac-like equation by the β- rotation Ψ new = e(1/2)(βγ5) with γ5 already defined in terms of the Dirac matrices and as well a new direction of polarization through the angle β.
ACKNOWLEDGEMENT:
I am deeply indebted to (Late) Prof. K. Rangadhama Rao D.Sc. (Madras) D.Sc. (London) for his initiation to do fundamental research work in the subjects of Physical Sciences and wishes to express my gratitude of his help and encouragement.
*The terms of the alkali spectra are closely spaced doublets, extensively researched by Prof. K. Rangadhama Rao as optical doublets, during the decade 1920-1930 in India and abroad , established that for such systems the energy is slightly dependent upon the direction of the "charge" spin. Prof. K. R. Rao has been recognized for his work of an International level of Standards of Wavelengths, I happy to note that his measurements revealed certain fundamental features prompting the present research by the present author.
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13. trusciencetrutechnology@blogspot.com, Sunday, May 3, 2009, “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.
14. a. 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
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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
d. trusciencetrutechnology@blogspot.com, Monday, 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.
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.
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OFF-LINE ADDENDUM
Professor K. Rangadhama Rao with his younger kids of two sons and three
daughters visited the Shanti Ashram, in Visakhapatnam, where the “Omkar
Swami” received him with great reverence. There on the tree hung a board with following sloka of Ishavashya numbered 5 out of 18:
“Thdejati thannaijati thadhure thadvanthike Ha!
Thadhatharashya sarvasya thadu sarvaswyasya bhahyatha Ha !!”
That does moves, doesn’t move, that far away, that nearby that all around inside that all around outside.
The only reason why the Professor took his children to that place, there is situated an excellent beach with shallow waters and children could play in the waters safe, at ease and with comfort and lot’s of play full fun in the waters.
