The Inverse Gravity Inflationary Theory of Cosmology

Journal of Modern Physics, 2016, 7, 1762-1776 http://www.scirp.org/journal/jmp ISSN Online: 2153-120X ISSN Print: 2153-1196

Edward A. Walker

Mathematics Department, Florida Memorial University, Miami Gardens, FL, USA

How to cite this paper: Walker, E.A. Abstract (2016) The Inverse Gravity Inflationary Theory of Cosmology. Journal of Modern Cosmological expansion or inflation is mathematically described by the theoretical Physics, 7, 1762-1776. notion of inverse gravity whose variations are parameterized by a factor that is a http://dx.doi.org/10.4236/jmp.2016.713158 function of the distance to which cosmological expansion takes prominence over

gravity. This assertion is referred to as the inverse gravity inflationary assertion. Received: August 12, 2016 Accepted: September 25, 2016 Thus, a correction to Newtonian gravitational force is introduced where a paramete- Published: September 28, 2016 rized inverse gravity force term is incorporated into the classical Newtonian gravitational force equation where the inverse force term is negligible for distances less than Copyright © 2016 by author and the distance to which cosmological expansion takes prominence over gravity. Con- Scientific Research Publishing Inc. This work is licensed under the Creative versely, at distances greater than the distance to which cosmological expansion takes Commons Attribution International prominence over gravity. The inverse gravity term is shown to be dominant generat- License (CC BY 4.0). ing universal inflation. Gravitational potential energy is thence defined by the http://creativecommons.org/licenses/by/4.0/ integral of the difference (or subtraction) between the conventional Newtonian gra- Open Access vitational force term and the inverse gravity term with respect to radius (r) which al- lows the formulation, incorporation, and mathematical description to and of gravitational redshift, the Walker-Robertson scale factor, the Robinson-Walker metric, the Klein-Gordon lagrangian, and dark energy and its relationship to the energy of the big bang in terms of the Inverse gravity inflationary assertion. Moreover, the dynamic pressure of the expansion of a cosmological fluid in a homogeneous isotropic universe is mathematically described in terms of the inverse gravity inflationary assertion using the stress-energy tensor for a perfect fluid. Lastly, Einstein’s field equations for the description of an isotropic and homogeneous universe are derived incorporating the mathematics of the inverse gravity inflationary assertion to fully show that the theoretical concept is potentially interwoven into the cosmological structure of the universe.

Keywords Isotropic and Homogeneous Universe, Inverse Gravity, Cosmological Inflation,

DOI: 10.4236/jmp.2016.713158 September 28, 2016 E. A. Walker

Gravitational Redshift, Robertson-Walker Scale Factor, Klein-Gordon Lagrangian, Dark Energy, Stress-Energy Tensor, Friedman-Walker-Robertson Metric, Photon

1. Introduction

The theoretical notion that cosmological expansion or universal inflation occurs due to inverse variations in gravitational force whose rate of change is regulated by a limiting factor or parameter is introduced. Thus, it is asserted that cosmological expansion or inflation is an inherent property of nature mathematically described by the difference between conventional Newtonian gravitational force and its inverse term which is multiplied by an inflationary parameter which regulates its rate of change. The inflationary parameter multiplied by the inverse term of Newtonian gravitational force is determined by (and is a function of) an astronomical distance to which cosmological expansion over takes gravitational force on a cosmic scale. The establishment of the core concept of the inverse gravity inflationary assertion aforementioned is the foundation to describing the universe in terms of the new assertion. Thus, the aim of this paper is the incorporation of the inverse gravity inflationary assertion (IGIA) into proven and established mathematics describing cosmological inflation. A more detailed introduction is that this paper formulates the correction to the Newtonian gravitational force equation incorporating an inverse gravity term and its conditions. This permits the derivation of gravitational potential energy in terms of the IGIA. Resultantly, the relationship between gravitational potential energy, dark energy, gravitational redshift, the Klein-Gordon Lagrangian, the energy of the big bang, the Robertson-Walker scale factor, and the Friedman-Walker-Robertson metric in terms of the inverse gravity inflationary assertion (IGIA) is formulated and defined. The IGIA is then applied to the stress-energy tensor for describing the dynamic pressure of an expanding cosmological fluid in a homogeneous and isotropic universe. Lastly, the IGIA is applied to Einstein’s field equations for the description of a spherically homogeneous isotropic universe which establishes the inverse gravity inflationary assertion. This will elucidate how the IGIA is interwoven into the cosmological structure of the universe.

2. The Correction to the Newtonian Gravitational Force Equation and IGIA Inflationary Parameter

To begin the heuristic derivation, mass values m1 and m2 are the combined masses of cosmological bodies (such as galaxies) evenly dispersed over an isotropic and homogeneous universe and G is the gravitational constant. The terms of gravitational force which are a function of radius r are given such that [1]:

Gm m r 2 12 ′ Frg ( ) = 2 , Frg ( ) = B0  (1.0) r Gm12 m where Frg ( ) is the classical expression of Newtonian gravitational force and Frg′ ( )

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is the inverse term of Newtonian gravitational force, constant B0 is the inflationary

factor or parameter. Inflationary factor B0 is stated such that: 1 B0=2 ;1 >> Br 00 0; > 1 (1.01) r0

The constant r0 is the astronomical distance to which cosmological expansion takes

prominence over gravity. The inverse term of gravity Frg′ ( ) can be re-expressed in

terms of distance r0 such that: 2 ′ 1 r Frg ( ) = 2  (1.02) r0 Gm12 m

The total gravitational force FrT ( ) or the Newtonian correction is stated as the dif-

ference between force values Frg′ ( ) and Frg ( ) such that:

FrTg( ) = Frg′ ( ) − Fr( ) (1.03)

Therefore,

2 1 r Gm12 m FrT ( ) = 22− (1.04) rr0 Gm12 m

The direction (+ or −) of the value of total force FrT ( ) has relationships defined by

the inequalities of radius r and distance r0 given by the conditions below.

For r>+ r0 ; FrT ( ) (1.05)

For r<− r0 ; FrT ( ) (1.06)

Condition (1.05) describes cosmological expansion +FrT ( ) away from the gravita-

tional force center or gravitational source for distances rr> 0 .Conversely, for condi-

tion (1.06), inverse gravity term Frg′ ( ) in total force FrT ( ) is negligible at distance

rr< 0 causing force direction −FrT ( ) toward the center of gravitational force.

The value of the cosmological parameter of distance r0 is determined where total

force value FrT ( ) equals zero and radius r equals cosmological parameter r0 ( rr= 0

and FrT ( 0 ) = 0 ). Furthermore, this implies that for the condition of FrT ( 0 ) = 0 , the

force terms Frg′ ( 0 ) and Frg ( 0 ) in total force FrT ( 0 ) are equal ( Frg′ ( 00) = Frg ( ) ).

Therefore, total force FrT ( 0 ) can be presented such that: 2 1 (r0 ) Gm12 m FrT ( 0 ) = −=0 (1.07) r 22Gm m 0 12 (r0 )

This reduces to:

1 Gm12 m −=2 0 (1.08) Gm12 m (r0 )

Solving Equation (1.08) for the cosmological parameter of distance r0 gives a value such that:

r0= Gm 12 m (1.09)

Conclusively Equation (1.09) above, gives the value of distance r0 to which cosmo-

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logical expansion takes prominence over gravity. In describing an isotropic and Ho- mogeneous spherical universe, mass values m1 and m2 will be evenly (and uniformly) distributed about the spherical volume. Mass value mu denotes the total mass of the universe, therefore the dispersion of cosmological mass mu will be described via the function gm( ,,θφ) at mass variable m and the spherical coordinates at θ and φ

[2]. Resultantly, mass mц′ (corresponding to each mass value of m1 and m2 ) which represents a portion of cosmological mass dispersed about the sphere is set equal to function gm( ,,θφ) as shown below.

mц′ = gm( ,,θφ) (1.10)

Function gm( ,,θφ) takes on a value of [2]:

12 θφ= θφ2 ++ θφ 22 φ gm( , ,) ( m cos sin) (m sin sin) (m cos ) (1.11)

The gravitational interaction of symmetric portions of cosmological mass mц′ and

mц′ separated by distance r0 is stated as the product between the two mass values such that:

2 mmц′′ц = ( gm( ,,θφ))( gm( ,, θφ)) = ( gm( ,, θφ)) (1.12)

Thus, the continuous sums (or integration) of gravitational mass interaction mmц′′ц to whom are located on opposite sides of distance r0 is taken to a value π and to the value of the mass of the universe mu . Thus, the gravitational interaction of mass values mm12 in Equation (1.09) equals the triple integral shown below [2]. m π π mm= u m m′′ mddd m θφ (1.13) 12 ∫0 ∫∫ 00 ( ц ц )

As the continuous sums of the integrals in Equation (1.13) progress in concert with angles θ and φ and sum up to a value π , the interacting mass values of m1 and

m2 on opposite sides of distance r0 sum up in concert with variable mass m to en- compass both halves of the spherical volume, giving the gravitational interaction of the entire spherical volume. Thus, the integration of (1.13) gives the gravitational mass interaction of the entire spherical volume. Therefore substituting the value of Equation

(1.13) into (1.09) gives the proper mathematical description of distance r0 shown below.

mu π π r= G m( mm′′)ddd m θφ (1.14) 0 ∫0 ∫∫ 00 ц ц

The aim and scope of this paper is to introduce the notion and mathematics of the Inverse gravity inflationary assertion, thus we leave the calculation and value of Equa- tion (1.14) as an exercise to the scientific community based on data obtained (The value of universal mass mu ) by astronomical observations.

3. The IGIA Mathematical Integration into Established Fundamental Concepts in Cosmology

This section applies the mathematical concept of the inverse gravity assertion to gravitational potential energy, gravitational Redshift, the Robertson-Walker scale factor,

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Friedman-Walker-Robertson metric, the Klein-Gordon lagrangian, dark energy, and

the energy of the big bang. Gravitational potential energy UrT ( ) describing the energy of inflation in terms of the IGIA is equal to the conventional integral of total force

FrT ( ) (Equation (1.04)) with respect to radius r (for the condition of rr> 0 ) as shown below [1]. = UTT( r) ∫ Fr( )d r (2.0)

Thus after evaluating the integral above, one obtains a value of potential energy

UrT ( ) such that:

3 1 r Gm12 m UrT ( ) = 2 + (2.01) r0 3Gm12 m r

As a photon propagates across the expanding cosmological expanse, its energy is af-

fected by the gravitational potential energy UrT ( ) . Thus, photonic energy E is set

equal to potential energy UrT ( ) in terms of the IGIA; this equivalence is displayed below [1]. hc E= pc = = UT ( r) (2.02) λg

where λg is the photon’s wavelength influenced by potential energy UrT ( ) , the

photonic energy affected by the potential energy field of UrT ( ) can be expressed such that [1]: 3 hc1  r Gm12 m = 2 + (2.03) λg r0 3Gm12 m r

Resultantly, wavelength λg affected by the potential energy field of UrT ( ) of the IGIA has a value expressed as [1]: −1 3 1 r Gm12 m λg = hc 2 + (2.04) r0 3Gm12 m r

Energy E0 is the initial energy value ( E00= ( hc λ ) ) of the emitted photon prior to it traversing through a region of space-time under the influence of gravitational poten-

tial energy UrT ( ) in terms of the IGIA, thus redshift z is given such that [3]:

hc − E λ 0 UrT ( ) − E0 g λ z = = ≡−0 1 (2.05) EE00λg

This reduces to [3]: 1 hc λ z = −≡110 − (2.06) λλ E0 gg

Red shift value z can then be expressed in terms of the IGIA such that: 3 11r Gm12 m z =2  +−1 (2.07) E0r0 3 Gm12 m r

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Thus, the photonic energy value hc λg is substituted by the value of potential energy UrT ( ) in Equation (2.06) giving the value of redshift z in Equation (2.07).

Observe the expression below where a0 is the scale factor of the universe as it is pre- sently and at( em ) is the scale factor at the emission time tem of the photon (or a scale factor of the universe as it was in the past as some authors state it) [4]. a 1+=z 0 (2.08) at( em )

Substituting the value of redshift z of Equation (2.07) into (2.08) above gives [4]:

11r3 Gm m a 11++12 −= 0 (2.09) 2  E0r0 3 Gm12 m r a( tem )

This reduces to:

11r3 Gm m a +=12 0 2  (2.10) E0r0 3 Gm12 m r a( tem )

The value of the scale factor at the time of the emitted photon at( em ) is given such that [4]: −1 3 11r Gm12 m at( em ) = a0 2 + (2.11) E0r0 3 Gm12 m r  where Equation (2.11) is of the form at( ) =11( + z) [4] which implies that a0 equals

1 ( a0 = 1). Equation (2.11) adequately shows the relationship of the Robertson-Walker scale factors and gravitational redshift to the IGIA. At this juncture, the IGIA connec- tion to Friedman-Lemaitre-Walker-Robertson metric dΣ2 is shown below [3].

2 2 2 2 dr 22 2 2 2 dΣ=− dt + at( ) 2 + rdθ + r sin θφ d (2.12) 1− kr

The scale factor at( ) in Equation (2.12) above is set equal to the scale factor at( em ) of Equation (2.11) ( at( ) = at( em ) ) giving Equation (2.12) such that (note: for our purposes tt= em ) [3]: 2 2 2 2 dr 22 2 2 2 dΣ=− dt + at( em ) 2 + rdθ + r sin θφ d (2.13) 1− kr

Substituting the value of scale factor at( em ) Equation (2.11) into Equation (2.13) above gives the Friedman-Walker-Robertson metric in terms of the IGIA such that: ddΣ=−22t

−1 2 32   (2.14) 11 rrGm12 m d 22 2 2 2 +a0 + ++rrdθ sin θφ d  Er 223 Gm m r 1− kr 00 12  

This establishes the IGIA relationship to the Friedman-Walker-Robertson metric, where constant k in Equation (2.14) is equal to +1 ( k = +1) and rr> 0 for positive spherical curvature describing the expansion of the cosmological fluid [3].

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The inverse gravity inflationary assertion (IGIA) can be defined in terms of field theory via its relationship to the Klein-Gordon lagrangian. In formulating the expres-

sions describing this relationship, IGIA potential energy UrT ( ) is set equal relativistic

energy denoted Erel as shown below [1]. 12 =+≡24 24 UT ( r)  pc mc Erel (2.15)

This can be expressed such that:

3 1 r Gm m 12 = +=+12 24 24 Erel 2  pc mc (2.16) r0 3Gm12 m r

where φ ( xц ) is a scalar field function at Minkowski coordinates xц , the momenta p

is expressed as a tangent vector on the Minkowski coordinates xц which is a function

of (or parameterized by) time t ( xц = xtц ( ) ) as shown below [3].

11∂∂φφ( xxц ) ( ц ) ∂xц 1 px= = = ∇цφ ( ц ) (2.17) c∂ t cx ∂∂ц tc

Expressing Equation (2.16) in terms of field φ ( xц ) and substituting the value of momentum p (of Equation (2.17)) into Equation (2.16) gives [5]:

3 12  22 1 r Gm12 m 24 Exrel ( ц ) ≡ +=∇+цφφ( xц ) m( xц ) c (2.18) 2 ( ) ( ( ) ) r0 3Gm12 m r

where the relativistic energy Erel is expressed as a function of Minkowski coordinates

xц ( Exrel ( ц ) ) which gives a form of the Klein-Gordon equation. The speed of light c is

set equal to unity ( c = 1 ). A priori is that the differential momentum value ∇цφ ( xц ) relates to the energy value of the IGIA such that (or solving Equation (2.18) for the dif-

ferential term ∇цφ ( xц ) ): 12 222 ∇=φφ( x) Ex( ) − m( x) (2.19) ц ц ( rel ц ) ( ( ц ) )

Equation (2.19) can be alternatively expressed in terms of the IGIA such that:

2 12 3  2 1 r Gm12 m 2 ∇=цφφ( xц ) + −mx( ( ц )) (2.20) r 2 3Gm m r ( ) 0 12

where r is a function of the Minkowski is coordinates ( xц = { it,,, x123 x x } ), distance r can be expressed such that [3]: 12 3 2 r=−+ t2 x = rx (2.21) ∑1 ( ц ) ( ц )

Equation (2.19) can then be expressed in terms of radius rx( ц ) ( r= rx( ц ) ) as shown below.

2 12 3 rx( ) 2 1 ( ц ) Gm12 m 2 ∇=цφφ( xц )  + −mx( ( ц )) (2.22) r2 3Gm m rx 0 12 ( ц ) 

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ц Therefore, we introduce momentum p′ expressed as the differential term ∇ φ ( xц ) (observe the superscript ц ) as shown below [3].

∂φ ( xц ) px′ =∇=∇=цφηφη( ) цσσ( x) ц (2.23) ц ц ц ∂t (Note: Recall thatthe speed of light c is set equal to unity ( c = 1 )) Where the Min- σ σ kowski metric η ц is expressed such that η ц =diag [ −1,1,1,1] [3], relativistic energy Exrel′ ( ц ) corresponds to momentum p′ and the IGIA such that:

3  2 12 1 r Gm12 m ц 24 E′ x ≡ +=∇+φ x mc (2.24) rel ( ц ) 2  ( ( ц )) ( ) r0 3Gm12 m r

Equation (2.25) below is the Klein-Gordon equation expressed such that [3]:

22 ц 2 (EKGφ( xц )) =∇∇−ц φφ( xц ) ( xmц ) ( φ( xц )) (2.25)

Thus, as presented by Wald [3], the Klein-Gordon lagrangian is of the form shown below. 1122ц 2 LKG =( EKGφ( xц )) =−∇ц φφ( xц ) ∇( xmц ) + ( φ( xц )) (2.26) 22 Expressing the Klein-Gordon lagrangian (Equation (2.26)) above in terms of the IGIA, the value of Equation (2.23) is substituted into Equation (2.26) (where ц цσ ∇=∇φ( xxц ) ηφц ( ц ) ) giving: 2 3 rx 22 11цσ ( ( ц )) Gm12 m 22 L=−η+−mxφφ + mx (2.27) KG 2  ( ( ц )) ( ( ц )) 23r0 Gm12 m rx( ц )  

ц This implies that the product of differential terms ∇∇цφφ( xxц ) ( ц ) takes a value incorporating the IGIA such that:

2 3 rx( ) 2 ц цσ 1 ( ц ) Gm12 m 2 ∇∇цφ( xц ) φη( xц ) = + −mx(φ( ц )) (2.28) r2 3Gm m rx 0 12 ( ц ) 

Solutions to Equations (2.27) and (2.28) pertain to mathematical methods of solving differential equations. Conclusively, Equation (2.28) is the IGIA correlation to various areas of field theory especially quantum energy fields describing vacuum energy (and the stress-energy tensor described in the next section). Lastly, dark energy Edark in terms of the IGIA is given by the conditions shown below.

1 r3 = Edark 2  (2.29) r0 3Gm12 m

Thus, dark energy is interpreted according to the IGIA as the inverse term of potential energy UrT ( ) (of Equation (2.01)). An important consideration is that the energy of expansion or energy Edark is different from the energy of the big bang which will be denoted energy EBB for the purposes of this explanation. Thus energy EBB is far

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greater in magnitude than the energy of gravity at the big bang denoted UBB = ( GMm ri ) (along with other elementary forces such as electromagnetism, the strong nuclear force, and the weak nuclear force opposing universal expansion) at the big bang where radius

ri is the infinitesimally small distance of the big bang ( EUBB> BB ). Where big bang

energy EBB is composed of kinetic energy and electromagnetic energy of the universe,

this implies that energy EBB is also sufficient for accelerating the total cosmological

mass beyond the astronomical distance or radius r0 to which cosmological expansion

takes prominence over gravity and where energy Edark can generate cosmological expansion.

4. The Dynamic Pressure of an Expanding Cosmological Fluid in Terms of the IGIT

This section mathematically defines the dynamic pressure of a cosmological fluid in a homogeneous isotropic universe in terms of the IGIA. The stress-energy tensor for a perfect fluid used to describe the expansion of the cosmological fluid is given such that [3]:

Tab=∇∇+ aφφ bgL ab KG (3.0) where φ is the field function of the space-time manifold [5] that the stress-energy

tensor is defined on, gab is the metric tensor, and LKG is the Klein-Gordon lagrangian such that [3]: 1 L =−∇∇cφφφ +2mc 22 (3.01) KG 2 c

The 4 space tangent vectors ∇∇abφφ in Equation (3.0) obey the geodesic rule such that [3]:

c ∇∇∇c( abφφ) =∇∇∇ c( ab φφ) +Γ abab( ∇∇ φφ) ≡0 (3.02)

c Thus showing the appropriate use of the Christoffle symbol Γab [3]. The partial

derivative ∇c in Equation (3.02) is of the form ∇cc =( ∂∂x ) . The tangent vector of

the form ∇µφ is defined by the chain rule such that [2]:

∂φ ∂xц ∇=µφ (3.03) ∂∂xtц

where the time coordinate has a value ct and the speed of light c is set to unity ( c = 1 ) [5], 4 the spatial coordinates xц in R are the Minkowski coordinates ( xц = { it,,, x123 x x }

and xц = xtц ( ) ) [5]. Thus the tangent vector ∇µφ at time t is the 4-velocity of the 4 cosmological fluid denoted uц ( uRц ∈ ) as shown below [3].

∂φ ∂xц uц =∇=µφ (3.04) ∂∂xtц

Therefore the product of tangent vectors ∇aφ and ∇bφ ( ∇∇abφφ) are given such that [3]: ∇∇φφ = ⋅ = 2 a buu ab∑ ц ( uц ) (3.05)

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It must be noted that tangent vectors ∇aφ and ∇bφ are symmetric ( ∇=∇abφφ) which implies the fluid velocities ua and ub are also symmetric ( uuuab= = ц ), ⋅ = 2 therefore uuab∑ ц ( uц ) . Total dynamic pressure Pab (where the subscripts ab pertain to a 4 by 4 matrix) is given in terms of fluid 4-velocity uц such that [1] [3]: 2 ρ∑ (uц ) P = ц (3.06) ab 2 This implies that [1]: 2 ρ∑ (uц ) ρ (∇∇φφ) P = ц = ab (3.07) ab 22 In the task of defining the expansion of the cosmological fluid in terms of the IGIT, 4 4 4 consider the unit vector uˆ in R ( uˆ ∈ R and xRµ ∈ ) shown below [2].

xµ uuˆ = such that ˆ ⋅uˆ = 1 (3.08) xµ

Multiplying unit vector uˆ to the IGIT force value FrT ( ) of Equation (1.03) gives 4 ∈ 4 vector valued force (FrT ( ))µ in R ( (FrT ( ))µ R) such that [2]: = ˆ (FrTT( ))µ u (F ( r)) (3.09)

This can be expressed such that: x =µ ′ − (FrTg( ))µ  Frg ( ) Fr( ) (3.10) xµ ′ 2 where A′ ( Ar= 4π [3]) is a spherically symmetric area, the sum or superposition of 4 pressure components is as expressed below [3]. [Where pressure = force ÷ area ( P= FA)][1] Fr  ( T ( ))µµx F =  Fr′ ( ) − Fr( ) → (3.11) ∑∑µµ′ ′ g g ∑ AAAxµ

(Note: that the pressure component P00 of the 4 by 4 matrix of the stress-energy tensor has a value of energy density ρE ( P00 = ρE ) [3]) This is set equal to total dynamic pressure Pab of Equation (3.07) as shown below [1]. ρ 2 Fr( ) ∑ ц (uц ) ( T )µ Pab = = ∑ (3.12) 2 µ A′ This implies that [1]: ρ 2 Fr( ) ∑ ц (uц ) ( T )µ ρ (∇∇abφφ) = ∑ = ≡ Pab (3.13) 22µ A′ which also implies the equivalence of: Fr 22( T ( ))µ ∇∇aφφ b = ∑ ≡ Pab (3.14) ρρµ A′

Therefore this can be expressed such that:

1771 E. A. Walker

 22xµ ∇φφ ∇ =  Fr′ ( ) − Fr( ) ≡ P (3.15) ab ρρ∑ µ ′ g g ab Axµ

Now substituting the value of force term Frg′ ( ) − Frg ( ) into Equation (3.15) gives: 2 21xµ r Gm12 m 2 ∇aφφ ∇ b =  − ≡ Pab (3.16) ρρ∑ µ ′ 22 Axµ rr0 Gm12 m

Thence, substituting the value of ∇∇abφφ (shown above) for the value of Equation

(3.15) into the stress-energy tensor Tab , the stress-energy tensor can be expressed such that:  2 xµ T=  Fr′ ( ) − Fr( ) − gL (3.17) ab ρ ∑ µ ′ g g ab KG Axµ

which is equivalent to the stress-energy tensor such that: 2 21xµ r Gm12 m Tab =  − −gLab KG (3.18) ρ ∑ µ ′ 22 Axµ rr0 Gm12 m

Equation (3.18) can be expressed in matrix form such that:

ρE TTT01 02 03  2 T10 P 11− gL 11 KG T12 T13 ρ  Tab = 2 TT20 21 P22− gL 22 KG T23 ρ  2 TT30 31 T32 P33− gL 33 KG ρ

The correlation of the stress-energy tensor Tab formulated in terms of the IGIA to Einstein’s field equations describing an isotropic and homogeneous universe is given in the conclusion. Thus, we conclude with the formulation and incorporation of the IGIA in Einstein’s field equation in its entirety.

5. Conclusions: Einstein’s Field Equations Describing an Expanding Homogeneous and Isotropic Universe in Terms of the IGIA

In describing an expanding homogeneous isotropic universe in terms of the IGIA, it is of great importance that the IGIA is fully incorporated to Einstein’s field equations as a whole. Thus, we began the heuristic derivation according to Wald [3] with expressions of Einstein’s Field equations such that:

GTtt=8π tt = 8πρ (4.0)

GTP**=8π ** = 8π (4.1)

The expressions of the stress-energy tensor in Equations (4.0) and (4.10 (Ttt and T** ) are related the average value of cosmological mass density ρ and to pressure value P

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[3]. The two expressions of Einstein’s tensor Gtt and G** are given such that [3]:

ab Gtt= G ab uu (4.2)

ab G** = Gab ss (4.3) where uц and sц are contra variant unit tangent vectors of homogenous surfaces of the isotropic and expanding universe [3]. Thus as depicted by Wald, the Robertson- Walker metric for describing a flat spatial geometry is expressed such that [3]: ds2=−+ d t 22 at( )( ddd x 2 + y 2 + z 2) (4.4)

The scale factor at time t denoted at( ) in the metric above is equal to the value of scale factor at( em ) of Equation (2.11) ( at( ) = at( em ) ) in terms of the IGIA. This equivalence can be stated such that: −1 3 11r Gm12 m at( ) = at( em ) = a0 2 + (4.5) E0r0 3 Gm12 m r 

Recall that for our purposes tt= em . Substituting the value of Equation (4.5) into Equation (4.4), the flat space Robertson-Walker metric of equation of (4.4) can be stated in terms of the IGIA (similarly to Robertson-Walker metric of Equation (2.14)) such that:

−1 2 3 2 211 r Gm12 m 222 ds=−+ d ta0 + (dddxyz+ + ) (4.6) Er 2 3 Gm m r 00 12 

Thus solving Equation (4.6) for the value of scale factor at( em ) in Equation (4.6) gives the equivalence of values:

1 −1 3 −1  222 2 22 11 r Gm12 m at( ) =d s + dd t x ++ d y d z = a+ (4.7) em ( )( ) 0 2  E0r0 3 Gm12 m r 

The time derivative of scale factor at( em ) is denoted a (and a is simply a= at( em ) ) and can be expressed such that [3]: ∂at( ) a = em (4.8) ∂t

The coordinate xц represent the Minkwoski coordinates ( xц = { it,,, x y z} ) [5]. Where the time coordinate has a value of ct and the speed of light c is set to unity ( c = 1 ) [5]. The question in reference to calculations is “how does the scale factor

at( em ) in terms of the IGIA (Equation (2.11) or (4.5)) relate to the time valued-derivative a of Equation (4.8) shown above?”. The radius r is defined on the Minkowski coordinates ( it,,, x y z ), thus the value (or magnitude) of radius r (where rs22= ∆ ) is expressed such that [5]: 12 22=∆ = ∆2222 +∆ +∆ +∆ rs( t) ( x) ( y) ( z) (4.9) 22 2212 =−+−+−+−it it x x y y z z ( ii) ( ) ( ii) ( )

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Distance r is measured from the center of expansion (or the center of the universe),

thus the initial values of ti, xi, yi, and zi equal zero. Therefore substituting zero for all in-

itial values ti, xi, yi, and zi ( (txyzii, , ii ,) = ( 0,0,0,0) ) and solving for radius r in Equation (4.9) gives (Similarly to Equation (2.20)):

22 2212 r=−+ txyz + + (4.10) Substituting the value of Equation (4.10) into Equation (2.11) (or Equation (4.5)) gives the IGIA scale factor as a function of the Minkowski coordinates ( atxyz( ,,,) ) such that:

at( em ) = atxyz( ,,,) −1 22 2232 −+txyz + + (4.11) 11( ) Gm12 m = a0  + 2 22 2212 E0r0 3Gm12 m −+txyz + +   

Thus pertaining to the time coordinate ( x0 = it ), the scale factor at( em ) in terms of the IGIA is now differentiable to the time valued derivative of Equation (4.8), therefore permitting the continuation of the formulation without ambiguity. Equation (4.11) af- fords the opportunity to briefly present Hubble’s constant in terms of the IGIA such that [3]: a 1 datxyz( ,,,) Ht( ) = =  (4.12) a atxyz( ,,,) d t

In continuing the IGIA’s incorporation to Einstein’s field equation, the scale factors a

(keep in mind that a= at( ) = at( em ) = atxyz( ,,,) ) relate to the symmetric Christoffel symbols such that [3]: ttt Γ=Γ=Γ=xx yy zz aa (4.13)

xxyyzz Γ=Γ=Γ=Γ=Γ=Γ=xt tx ty yt zt tz aa (4.14)

a Thus we acknowledge that the Christoffel symbols Γbc is of the form [3]:

a 1 ad ∂∂∂gggcb ca bc Γ=bc ∑ g + − (4.15) 2 d ∂∂∂xxxbcc

The components of the Ricci tensor are calculated according to the equation of [3]: RR= ab ∑ c acbc (4.16) This can alternatively be expressed such that [3]:

∂∂y c c dc dc Rab =∑ Γab −( ∑∑ Γcb ) +( Γab Γ dc −Γ cb Γ da ) (4.17) c∂∂xxcac dc,

The Ricci tensor is then related to the scale factor a (or at( em ) in terms of the IGIA) by the equations of (where a = d22( atxyz( ,,,)) d t ) [3]:

Rtt = −3 aa (4.18)

2 − a (a) R= aR2 = + 2 (4.19) ** xx a a2

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As stated by Wald, the value of Ricci tensor Rxx in Equation (4.19) above relates to the Christoffel symbol such that [3]:

∂∂y c c dc dc Rxx =∑ Γxx −( ∑∑ Γcx ) +( Γxx Γ dc −Γ cx Γ dx ) (4.20) c∂∂xxcxc dc, Therefore the value of the scalar curvature R is given such that [3]:

RR=−+tt 3 R** (4.21) Substituting the value of Equation (4.18) and (4.19) into Equation (4.21) give a value such that [3]: 2 a (a) RR=−+=36 R  + (4.22) tt ** 2 a a

Conclusively, the values of Einstein tensor values Gtt and G** are given such that [3]: 2 1 3(a) GR=+= R =8πρ (4.23) tt tt 2 a2

2 1 a (a) GR= − R =−−28 =πP (4.24) ** ** 2 a a2 As stated by Wald, using Equation (4.23), Equation (4.24) can be rewritten such that [3]:

2 3(a) =−+4π (ρ 3P) (4.25) a2 Due to the fact that the description of the IGIA is defined in reference to a homogeneous and isotropic universe, the general evolutions for isotropic and homogeneous universe are given such that [3]: 2 3(a) 3k =8πρ − (4.26) aa22 3a =−−4π (ρ +3P) (4.27) a where scale factors a and their corresponding time derivatives ( a and a ) can be defined in terms of the IGIA of Equation (4.10) ( a= atxyz( ,,,) ), constant k is equal to

+1 ( k = +1) and rr> 0 for positive spherical curvature describing the expansion of the cosmological fluid in an homogeneous isotropic universe [3]. At this juncture, the relationship of the scale factor of Equation (4.5) (and Equation (2.11)) in terms of the IGIA to Einstein’s field equation have been formulated.

Equation (4.1) ( GTP**=8π ** = 8π ) shows the relationship between pressure P and the stress-energy tensor Tab of Equation 3.18 [3]. This implies that P can be minimally substituted for the stress-energy tensor ( PT→ ab ). Thus, the pressure term P relates the IGIA stress-energy tensor of Equation (3.18) (of the previous section) such that: 2 21xµ r Gm12 m PT→ab =  − −gLab KG (4.28) ρ ∑ µ ′ 22 Axµ rr0 Gm12 m

1775 E. A. Walker

Resultantly, this can be expressed such that: 2 21xµ r Gm12 m P=  − −gLab KG (4.29) ρ ∑ µ ′ 22 Axµ rr0 Gm12 m

Spherically symmetric area A′ is equal to a value of 8π ( A′ = 8π ), therefore Equa- tion (4.29) can be stated such that: 2 21xµ r Gm12 m P=  − −gLab KG (4.30) ρ ∑ µ 22 8π xµ rr0 Gm12 m

Substituting the value of pressure P presented above into Equation (4.27) gives a value such that:

2 xµ Gm m 3ar  21 12 =−4π ρ +3 ∑  − −gLab KG (4.31) a 8πρ µ x rr22Gm m µ 0 12

Equation (4.31) gives an additional incorporation of the Mathematics of the IGIA showing that the theoretical concept is well ingrained to the cosmological structure of the universe. The incorporation of the IGIA mathematics to Einstein’s field equations gives a complete description to validate the concept and convey a new theoretical pos- sibility to the physics community.

Acknowledgements

I would like to thank Juanita Walker and Mr. Eugene Thompson for their encourage- ment and support to this endeavor.

References [1] Young, H.D., Freedman, R.A. (2004) Sears and Zemansky’s University Physics. 11th Edi- tion, Addison Wesley, San Francisco, Cal., Pearson, 444, 445, 454-457, 517, 529, 530, 1431, 1450. [2] Stewart, J. (2003) Calculus. 5th Edition, Belmont, Cal., Thompson, Brooks/Cole, 877, 968, 1095. [3] Wald, R.M. (1984) General Relativity. Chicago Press, Ltd., Chicago, 17, 23-26, 31-38, 59, 70, 72, 95, 96, 101-104, 120-124. [4] Mukhanov, V.F. (2005) Physical Foundations of Cosmology. Cambridge University Press, Cambridge, New York, Page 58. http://dx.doi.org/10.1017/CBO9780511790553 [5] Penrose, R. (2004) The Road to Reality: Comprehensive Guide to the Laws of the Universe. Alfred A. Knopf Publishing, New York, 222-223, 412, 413.

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