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.