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# A quantum theory of spacetime in spinor formalism and the physical reality of cross-ratio representation: the equation of density parameters of dark energy, matter, and ordinary matter is derived: Ω M2 = 4 Ω b Ω Λ

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### ABSTRACT

By theorizing the physical reality through the deformation of an arbitrary cross-ratio, we leverage Galois differential theory to describe the dynamics of isomonodromic integratable system. We found a new description of curvature of spacetime by the equivalency of isomonodromic integratable system and Penrose’s spinor formalism of general relativity. Using such description, we hypothetically quantize the curvature of spacetime (gravity) and apply to the problem of the evolution of the universe. The Friedmann equation is recovered and compared so that the mathematical relationship among dark energy, matter (dark matter + ordinary matter), and ordinary matter, $ΩM2≃4ΩbΩΛ$, is derived; the actual observed results are compared to this equation (calculated ΩM = 0.33 vs. observed ΩM = 0.31); the model might explain the origin of dark energy and dark matter of the evolution of the universe.

### Main article text

#### INTRODUCTION

We looked for the simplest mathematical object to identify the underlying reality of nature, and we found it to be cross-ratio. By defining cross-ratio over Riemann sphere, infinity is just another usual point; similarly, there shouldn’t be any point in the universe more special than others. However, the variety of nature must be realized as a condition for such an underlying object.

In “Cross-ratio arbitrariness and the constraint to the parameter space of projective space basis” section, the article explains how potential physical varieties come from different representations of the same thing – cross-ratio deformation. So, the cross-ratio consists of both simplicity and variety. A successful example is like Einstein’s masterpiece – general theory of relativity. Although Einstein’s field equation is simple, many interesting solutions emerged.

In “Cassidy’s work on isomonodromic system” section, we introduce Galois differential theory and related Cassidy’s work. It is a mathematical machinery to manifest the deformation of cross-ratio. Cassidy’s work consists of introducing a 2 by 2 matrix differential equation and related isomonodromic integratable system, so it can describe the deformation. By such machinery, we formulate an alternative theory of the dynamics of curvature of spacetime to recover the spinor general relativity equivalent counterpart (for which a brief introduction is given in “Brief overview of spinor formulation of general relativity” section). By such connection, we hypothetically claim the origin of spacetime is from the isomonodromic integratable system, and spacetime is more fundamentally described by the curvature rather than metric or coordinated mathematical framework, that is, spinor formulation of general relativity might be more fundamental than classical general relativity; a similar argument was postulated by Penrose (1960).

In “As an application to the problem of modeling the universe evolution” section, we apply the calculation of the deformation of the isomonodromic integratable system with certain simplified conditions so a solution is found. The solution is used to recover Friedmann equation and related density parameters such that observed and calculated results are compared. This proposes an explanation of the origin of dark matter and dark energy without new kind of matter or energy, as they are new kind of gravitational field of spacetime’s curvature.

#### BRIEF OVERVIEW OF SPINOR FORMULATION OF GENERAL RELATIVITY

##### Penrose’s spinor approach to general relativity

Spinor formulaism of general relativity (i.e., Spinor GR) (Penrose, 1960) adopted a coordinate-free approach. The correspondence between tensors and spinors is obtained by the use of a hybrid spinor ΣΝCD′ (2 × 2 Hermitian matrix per CD index, denoted as ΣΝ in short) (Einstein’s summation notation is used in this article):

$ΣΜAC′ΣΝBC′+ΣΝAC′ΣΜBC′=gΜΝεAB,forA,B,C′=0,1;Μ,Ν=0,1,2,3,$  (1)

where $εBC$ represents 2 × 2 skew-symmetric “metric” spinor (components’ values are 0 or ± 1). To raise or lower spinor index, either one of , $εB′C′$, and $εB′C′$ is used:

$ΞA=εABΞB,ΞB=ΞAεAB$  (2)

where Ξ denotes spinor. Primed indexes refer to the complex conjugate spin space. Please note that

(3)
$ΣΜAB′=defgΜΝΣΝCD′εCAεD′B′$  (4)

is Hermitian because gΜΝ (g is the metric) is real.

Spinor GR requires the covariant derivatives of $ΣΝCD′$ , gΜΝ, and $(εCAεD′B′)$ to be zero.

The spinor equivalent of tensor GΜΝ is stated as

$GAC′BD′=ΣΜAC′ΣΝBD′GΜΝ,GΜΝ=ΣΜAC′ΣΝBD′GAC′BD′$  (5)

##### Curvature tensors, symmetry and Einstein tensors in spinor general relativity

By $RAE′BF′CG′DH′$ (spinor version of Riemann-Christoffel tensor RΜΝΡΣ), the curvature in spinor GR is defined by curvature spinors - ΧABCD and $ΦABG′H′$ (Penrose, 1960):

$RAE′BF′CG′DH′=12(ΧABCDεE′F′εG′H′+εCDΦABG′H′εE′F′+εABΦ⋆E′F′CDεG′H′+εABεCDΧ⋆E′F′G′H′)$

($⋆$ indicates complex conjugate), which satisfy the following equations (Penrose, 1960):

$(∇AH′∇BH′+∇BH′∇AH′)ΞF=ΧABFEΞE(∇AD′∇AC′+∇AC′∇AD′)ΞF=ΦFED′C′ΞE$  (6)

where $∇AG′$ and $∇BG′$ are spinor covariant derivatives and raised spinor derivatives, respectively. They can be modified to

$εGF(∇AH′∇BH′+∇BH′∇AH′)ΞF=εGFΧABFEΞE→(∇AH′∇BH′+∇BH′∇AH′)ΞG=ΧABGEΞEεGF(∇AD′∇AC′+∇AC′∇AD′)ΞF=εGFΦFED′C′ΞE=εGF(Φ⋆D′C′FE)ΞE→(∇AD′∇AC′+∇AC′∇AD′)ΞG=Φ⋆D′C′GEΞE$  (7)

for which we use the symmetry property of ΧAB(FE) and $ΦD′C′(FE)$, $ΦD′C′FE=Φ⋆FED′C′$ (Penrose,1960).

By Bianchi identity of Riemann curvature,

$∂ΤRΜΝΡΣ+∂ΡRΜΝΣΤ+∂ΣRΜΝΤΡ=0$

it implies the relationship of curvature spinors, which is called Bianchi identity of spinor GR (Penrose, 1960):

$∇BD′ΧFEAB=∇AC′ΦFED′C′$  (8)

One can further modify it by the symmetry of (Penrose, 1960)

$ΧFEAB=ΧABFE,ΦFED′C′=Φ⋆D′C′FE$

to

$→∇BD′ΧABFE=∇AC′Φ⋆D′C′FE→∇BD′ΧABFEεGF=∇AC′Φ⋆D′C′FEεGF→∇BD′ΧABGE=∇AC′Φ⋆D′C′GE$

Note that we can express Bianchi identity as

$∇BD′ΧABGE=∇AC′Φ⋆D′C′GE$  (9)

where $ΧABGE$ and $Φ⋆D′C′GE$ are 2 × 2 traceless matrices if considering G and E as the matrix indexes because of the symmetry property of ΧAB(FE) and $Φ⋆D′C′(FE)$ (Penrose, 1960) (trivially proved by contracting $ΧABFEεGF$ and $Φ⋆D′C′FEεGF$). The author claims this form (or in matrix equivalent), and Equation (7) is more fundamental to the original version of Equations (6) and (8).

By the symmetry of Riemann curvature RΜΝΡΣ, the curvature spinor ΧFEAB needs to obey (Penrose, 1960)

(10)

Penrose (1960) mentioned that the geometry of a Riemannian space (with signature +,−,−,−) can be described by spinor terms with the role of curvature tensor being taken over spinors ΧFEAB and $ΦFED′C′$ satisfying their symmetry property, Bianchi identity (Equation 8), and Equation (6).

Finally, Ricci spinor is equivalent to Ricci tensor (Penrose, 1960):

$RAC′BD′=ΛεABεCD−ΦABC′D′$  (11)

Ricci scalar is (Penrose, 1960)

$R=4Λ$  (12)

Einstein spinor is equivalent to Einstein tensor (Penrose, 1960):

$GAC′BD′=−ΛεABεCD−ΦABC′D′$  (13)

#### APPLICATION OF GALOIS DIFFERENTIAL THEORY TO SPINOR FORMULATION OF GENERAL RELATIVITY

##### Cassidy’s work on isomonodromic system

For a commuting set of derivations $∂∂zi$ (i=0,...,m), there exists an integratable (parameterized second-order linear differential) system [Cassidy & Singer, 2005: Prop6.3(3)]

$∂∂ziY=BziY$  (14)

with integratability condition: $[Bzi,Bzj]=∂∂ziBzj−∂∂zjBzi(i,j=0,...,m)$, where Y is in GL2(f(zi)), $Bz0=sl2(f(zi))$, $Bzk=gl2(f(zi))$ (k=1,…,m), f(zi) is a set of functions which $∂∂zi$ acts on. The variable set {zk}, for k=1,…,m, is called parameters-set. This system is called isomonodromic system. Furthermore, the related Gal-group is a proper Zariski dense subgroup of SL2(cz0), where Gal-group is the parameterized Picard–Vessiot group (PPV-group) and cz0 is the differential constant of $∂∂z0$ (but function of parameters-set) or called $∂∂z0$-invariant. Y is called the fundamental solution matrix.

The PPV-group is a -automorphism group such that Gal ∈ PPV-group if Gal:Y → Y and Gal = Gal $∀$. Therefore, Gal(Y) is another fundamental solution matrix to the system.

Let us define the Lie algebra corresponding to one-parameter subgroup of Gal by G through exponential map:

By expanding the exponential map with parameter Κ → 0 and commuting property of Gal (i.e., Gal = Gal), the commuting property of Lie algebra G and exists:

$G∂=∂G,∀∂$  (15)

It is obvious that G ∈ sl2(cz0).

There is a property of integratability condition that change of variables from parameter-set zi to $zΜ′$ preserves this property (by change of variables) if there exists smooth maps $∂zi∂zΜ′$ for $zi(zΜ′)$. The transformed new integratable system (together with $∂∂z0Y=Bz0Y$) is

$∂∂zΜ′Y=B′zΜY,whereB′zΜ=∑i∂zi∂z′ΜBziforΜ=1,...,n;i=1,...,m$  (16)

note that n is not necessarily equal to m.

##### Cross-ratio arbitrariness and the constraint to the parameter space of projective space basis

A cross-ratio (https://en.wikipedia.org/wiki/Cross-ratio) can be defined by a function of four points on Riemann sphere:

$Cross-Ratio≡(z0−z2)(z1−z3)(z1−z2)(z0−z3)$

If we allow any arbitrary cross-ratio fixed value to be assigned, those four points are then constrained (only three points are free to move). If we further assume any arbitrary reference fixed point to be assigned, say for z0, then those other points are also constrained by four real parameters (x0, x1, x2, x3), that is,

$Cross−Ratio=(z0−z2(x0,x1,x2,x3))(z1(x0,x1,x2,x3)−z3(x0,x1,x2,x3))(z1(x0,x1,x2,x3)−z2(x0,x1,x2,x3))(z0−z3(x0,x1,x2,x3))$

So, the arbitrariness of description of a cross-ratio allows four real parameters. Physically, we mean the principle of cross-ratio arbitrariness is natural because no particular point over Riemann sphere is more special and no reference point is more special than others.

By Yoshida (Chapters I and IV), the realization of cross-ratio has direct relationship with hypergeometric function (three regular singular points):

$z(1−z)d2ydz2+(c−(a+b+1)y)dydz−aby=0,where a,b,c are complex numbers$

Since hypergeometric function is a solution of related hypergeometric differential equation, so if we deform the hypergeometric differential equation, we also deform the related hypergeometric function and its underlying cross-ratio realization.

Referring to Hypergeometric function (https://en.wikipedia.org/wiki/Hypergeometric_function), any second-order differential equation with three regular singular points can be converted to hypergeometric differential equation by change of variables, so we can describe the hypergeometric differential equation by second-order differential equation. Leveraging Cassidy’s work in previous section, we can deform the hypergeometric differential equation by the deformation of second-order differential equation. Physically, we mean we can deform the realization of cross-ratio by deformation of related isomonodromy integratable system by a deformation space of four real parameters:

(17)

the later is in spinor form, where $BΜ(z0,xΝ)=∑i∂zi(xΜ)∂xΜBzi(z0,zj)$ for Μ, Ν = 0,…,3;i,j = 1,…,3; xΝ ∈ Reals.

Id is 2 × 2 identity matrix; $D^ΜEF≡∂ΜΔEF$; note that BΜ(z0, xΝ) and Bzi(z0,zj) are matrices gl2, and $D^ΜEF,BΜEF$ are hybrid spinors like ΣΜAB′ in spinor GR (Robert, Chapter 13).

##### Operators of integratability condition and quantized Bianchi equation for curvature spinors

Based on the isomonodromy integratable system formulated in previous section, we look for a set of the Lie algebras associated with Gal that can produce the integratability. In order to do so, some fundamental operators, such as derivation operator and integratability operator, are introduced. They are related to mathematical objects in spinor GR so that Bianchi identity is recovered by an operator equation. Finally, the equivalency of spinor GR and this theory is summarized.

Let us define spinor (derivation) operator:

$D^AB′EF≡ΣΜAB′D^ΜEF$  (18)

and $D^ΜEF$ is the differential hybrid spinor operator from previous section (so it is associated with an integratable system), not covariant derivative.

We demand $ΣΜAB′$ (denoted as $ΣΜ$ in short) is the differential constant of $D^Ν$ in the sense that $D^ΜEF−invariant|D^ΜEF(ΣΝAB′X)=ΣΝAB′(D^ΜEFX)$ for any spinor X, so it is consistent with spinor GR requirement (covariant derivative of $ΣΝAB′$ is zero) except $∇→D^$. $ΣΝAB′$ defined here satisfies the same relationship in Equation (4).

Note that commutative Μ implies commutative $D^AB′EF:D^AB′EFD^CD′FG=D^CD′EFD^AB′FG$.

By Equations (16) and (18) as well as $ΣΜAB$ being $D^ΜEF−invariant$, there is a commuting set of spinor operator for an integratable system:

(19)

where

(↔ indicates complex conjugate, so prime and unprime indices are interchanged), and note that the ordering between prime and unprime indices are irrelevant (Robert, Chapter 13).

The integratability condition is used to facilitate a definition on a set of spinor operator.

$J^AB′CD′E F$ is called integratability operator (or, in short, int operator), acting on Y |

and

$J^⋆BA′DC′E′ F′Y⋆F′G′=H⋆ [BA′E′G′H⋆DC′]G′ F′Y⋆F′G′$

[AB’ and CD’] are anti-commutating indices notation as usual in spinor GR. So, by integratability condition (14), this implies

$J^AB′CD′ EFYFG=(D^[AB′EG(HCD′]GF))YFG,J^⋆BA′DC′ E′F′Y⋆F′G′=(D^⋆[BA′E′G′(H⋆DC′]G′F′))Y⋆F′G′$  (20)

There are a total of four sets of spinor operators from the contraction of $J^$ and $J^⋆$, which have a close relationship with int operators and curvature spinors:

$J^ACEF≡J^AB′CB′E F=εB′D′J^AB′CD′E F,J^⋆A′C′E′F′=J^⋆ BA′BC′E′F′=εBDJ^ ⋆BA′DC′E′F′,M^B′D′EF≡J^AB′ AD′EF=εACJ^AB′CD′E F,M^⋆BDE′F′=J^⋆ BA′DA′E′ F′=εA′C′J^⋆ BA′DC′E′F′,$  (20.b)

Specifically defining a bar-spinor: $J¯BDE′F′≡J^ ⋆BA′DA′E′F′=M^⋆BDE′F′$, one can see the $J¯BDE′F′$ has symmetric contraction over prime indices of $J^ACEF$ except over complex space: that is, $Y⋆F′H′$.

Note that $J^(AC)EF$ and $M^(B′D′)EF$ have symmetric property over their first two indices because

$J^ACEF=εB′D′J^ AB′CD′EF=εD′B′J^ CD′AB′EF=J^CAEF$
,

where we use the property of anti-symmetry of $εD′B′$ and $J^ABCD′E F=−J^CD′AB′E F$.

Let us look for an integratable system with the Lie algebra of PPV-group generated by spinor operators $J^ACEF,M^B′D′EF$ on YFG:

$J^ACEFYFG=JACEFYFG,$

where $JACEF$ is a spinor without derivation;

$M^B′D′EFYFG=MB′D′EFYFG$

where $MB′D′EF$ is a spinor without derivation;

such that $JACEF$ and $MB′D′EF$ are matrices (if considering E and F as the matrix indices) of Gal Lie algebra G. By requirements in Equation (15), but with association to $D^C,D′FG$ instead of , commuting property of $G(J^,M^)$ against $D^$ reads:

$(J^ABEFD^CD′FG)YGH=(D^CD′EFJ^ABFG)YGH,(M^B′D′EFD^AC′FG)YGH=(D^AC′EFM^B′D′FG)YGH$

Let’s define an fundamental equation as:

$εBC(J^ABEFD^CD′FG)YGH=εB′C′(M^B′D′EFD^AC′FG)YGH$  (21)

or equivalently:

$(J^ABEFD^BD′FG)YGH=(J¯DBE′F′D^⋆BA′F′G′)⋆YGH$

after applying the commuting property of $(J^,M^)$ against $D^$ above and using symmetry of $M^(B′D′)FG$:

$(D^BD′EFJ^ABFG)YGH=(D^AC′EFM^D′C′FG)YGH$

This is the quantized version (i.e., operator-wise equivalent) of Bianchi identity which describes the differentiation on curvature spinors Φ, Χ (compared to Equation (9) $∇BD′ΧABGE=∇AC′Φ⋆D′C′GE$)), while the spinor covariant derivative is replaced by spinor derivation operator $D^$, and Χ, $Φ⋆$ are replaced by operators $J^$ and $M^$, respectively. The quantization here refers to changing the classical equation of physical observable (spinor derivative of curvature in spinor GR) to operator equation (e.g., operator $D^BD′EFJ^ABFG$ acting on YGH), like Schrodinger’s equation that KE + PE = E →$(p^22m+V)Ψ=EΨ$.

As a short summary, we look for an integratable system, where G is generated by the int operator $J^$, and G is a solution to the fundamental (equation 21). This system leads to quantized Bianchi identity.

Let us define a set of spinor operators $Δ^AB′CD′E F$ such that

$(Δ^AB′CD′E F−Δ^CD′AB′E F)YFH=(D^[AB′EG(HCD′]GF))YFH$  (22)

By contracting over B’ and D’, we get

$εB′D′(Δ^AB′CD′E F−Δ^CD′AB′E F)YFH=(Δ^AB′CB′E F+Δ^CD′AD′E F)YFH=εB′D′(D^[AB′EG(H*CD′]GF))YFH.$

Using the integratability condition property (Equation 20) for $J^$ operator on the right-hand side, we get

$(Δ^AB′CB′E F+Δ^CD′AD′E F)YFH=εB′D′(J^AB′CD′E F)YFH$

The right-hand side is just $J^ACEF$ (Equation 20b), and the equation becomes

$(Δ^AB′CB′E F+Δ^CD′AD′E F)YFH=J^ACEFYFH$  (23)

Let us do the same for (Equation 22), but contracting over A and C:

$εAC(Δ^AB′CD′E F−Δ^CD′AB′E F)YFH=εAC(D^[AB′EG(HCD′]GF))YFH$

After the same operations, we get

$(Δ^ AB′AD′EF+Δ^ AD′AB′EF)YFH=M^B′D′EFYFH$  (24)

By decomposing YFH as matrix into ΞF and ΗH (both are column vectors of dimension 2) into Equations (23) and (24): Y → ΞΗ, we get: (and same for Η). They are equivalent to the equations of curvature spinors in Equation (7) if

$Δ^AB′CB′E F,Δ^ AB′AD′EF→∇A,B′∇CB′,∇A,B′∇AD′$  (25)

and

$JACEF=ΧACEF,MB′D′EF=Φ⋆B′D′EF(not the operators)$  (26)

In conclusion, spinor GR requires a well-defined spinor covariant derivative ∇∇. ∇∇ gives raise to the relationship as Equation (7), so that curvature spinors Χ and Φ are well established, which obey Bianchi identity (Equation 9). These conditions are said to be equivalent to “generate” spacetime manifold of {+,–,–,–} (Penrose, 1960) (except certain torsionness symmetric properties). In this article, the author claims that the spinor covariant derivative ∇∇ is corresponding to the operator $Δ^$; curvature spinors Φ and Χ are generated by $J^$ and $M^$ operators on Y; Bianchi identity is satisfied by the operators $J^$, $M^$ in such quantized form.

To illustrate the correspondences between spinor GR and this part of the theory (i.e., isomonodromic system):

(27)
$○Curvatures:Φ⋆,Χ↔J^,M^$  (28)
(29)

Note that $J^$ and $M^$ are just linear combinations of int operators, which must exist in an integratable system, while Y is a special solution to quantized Bianchi equation, which is a result of commuting relationship of $J^$, $M^$ and $D^$. The consequence is that, we can only claim that the equivalence of spinor GR to this theory is up to particular set of solution Y and its associated values of $J^$ and $M^$, that is, spinor GR is equivalent to this part of the theory in such quantized sense.

##### The symmetry of curvature spinor

Certain symmetries of curvature spinors are described by Penrose in previous sections, and those symmetries are consequence of classical general relativity. We show that certain symmetries are naturally arisen by the operators introduced in previous section, while some others are torsion related and needed to be met additionally. However, those additional conditions are not considered as mandatary because the key objective is to describe that spacetime’s curvature can be manifested by the deformation of isomonodromy integrable system, not to explain the naturalness of torsion.

From Equation (26) and the traceless property of matrices generated from int operators (Equation 20), it is concluded that $ΧABGE$ and $Φ⋆D′C′GE$ are traceless matrices (if considering G and E as the matrix indices), and

(30)

Because of the symmetry properties of $J^(AC)EF$ and $M^(B′D′)EF$, it implies trivially

(31)

Besides these two symmetries, spinor GR also requires symmetry:

,

and the Λ reality (Equation 10):

(32)

By Gomez-Lobo and Martin-Garcia (2012), GR symmetry (Equation 31) and Λ reality are consequences of the torsion-free property of original GR theory. However, torsion free is known that it is not a mandatory requirement; for example, Einstein–Carton theory removes such constraints.

In conclusion, to recover GR symmetry ΧABFE=ΧFEAB and $Φ⋆D′C′FE=ΦFED′C′$ and Λ reality, there are conditions for spinors generated by operators $J^$ and $M^$ to meet. However, the discussion of sufficient and necessary conditions for $J^$ and $M^$ to meet these symmetries is not covered yet in this paper.

##### Translation to ordinary Ricci, Einstein tensors, and Bianchi equation

The purpose of this section is to show the explicit links between related mathematical objects spinor GR (spinors of Ricci and Einstein tensors) and the isomonodromy integratable counterparts. Because those are key components of classical general relativity, showing explicit relationships helps visualize the equivalency and claim the origin of related tensors.

Recall the spinor equivalences of Ricci tensor (Equation 11), Ricci scalar (Equation 12), and Einstein tensor (Equation 13): $RAC′BD′=ΛεABεC′D′−ΦABC′D′$, R = 4 Λ, and $GAC′BD′=−ΛεABεC′D′−ΦABC′D′$. The following are $J$ and $M$ equivalences:

(33)
$R=2εBDJABAD$  (34)
(35)

Follow the ordinary way of spinor GR to translate from spinor to tensor (Equation 5), Einstein tensor is equivalent to

(36)

If we use the property of $MA′B′EDεEC=MB′A′EDεEC$, and $MA′B′EDεEC=MA′B′ECεED$, for which the symmetries are proved in the previous section, then

(where, in the last right-hand side, we rename the dummy indices). So, it is a clear symmetric for G(ΜΝ) as expected.

Bianchi identity (Equation 9) in spinor GR describes the dynamics of curvature. Unlike Einstein field equation, which the dynamic of curvature is described by solving (up to) second-order differential equation of metric tensor, spinor GR handles the same problem by first-order differential equation – Bianchi identity (Penrose, 1960). However, spinor GR is coordinate-free yet a classical theory; there is no metric differential analog. Nevertheless, quantized Bianchi equation do provide quantized analog to spinor GR as

$∇BD′ΧABGE=∇AC′Φ⋆D′C′GE↔(D^BD′EFJ^ABFG)YGH=(D^AC′EFM^D′C′FG)YGH.$

Once we solve the operators $Δ^$, $D^$, $J^$, $M^$, and fundamental solution matrix Y, the dynamics of curvature spinors Χ and Φ can be found by equations Equation (26). The ordinary curvature components such as Ricci tensors and Ricci scalar as well as Einstein tensors can be found.

From last three chapters of this section, the overall approach of spacetime dynamics is done by first identifying related classical curvature spinors and the derivatives equations (i.e., Bianchi identity), then manifesting the spacetime by a deeper structure from isomonodromy system; finally, illustrating related tensors such as Ricci and Einstein tensors equivalence by associated spinors by associated objects from isomonodromy system. Physically, the approach claims a more fundamental reality from isomonodromy system over classical spacetime just like Penrose mentioned that spinor GR may be more deep-rooted than tensors (Penrose, 1960).

##### Einstein condition and the connection to quantum operators

It is well known that Einstein’s theory (theory of general relativity) does not assign a definite stress–energy tensor to the gravitational field.

When solving Einstein field equation, we guess the form of metric and energy–momentum–stress tensor TΑΒ, because the physical interpretation to define TΑΒ is not yet known without knowing the metric first. In this article, the author uses the same approach to define TΑΒ but claims that the curvature is caused by $Δ^$ on Y, which constitutes energy–momentum–stress distribution. As required, the energy–momentum conservation is guaranteed in “quantized sense” by quantized Bianchi equation (just as in classical GR, Bianchi identity guarantees conservation). However, it is expected that there must exist a condition that local energy/mass and momentum density do contribute proportional to GΑΒ as Einstein field equation: $GΑΒ=−8ΠGTΑΒ$ ($G$ is gravitational constant).

As GΑΒ is related to $ΣΑEG′ΣΒFH′$ multiplying linear combination of Λ (by constraction of $J$) and enties of $M⋆$ (Equation 36), let’s examinate how to find $J$ and $M$. To find $J$ and $M$, which are contractions of int operators $J^$ on Y (Equation 20), Equation (22) explains that we can get them by $Δ^AB′CD′E F$. Considering the form of the operator $Δ^$:

$Δ^AB′CD′E F=D^AB′EGD^CD′GF+ΣΜAB′ΣΝCD′(ΓΜΝEGD^ΝGF),$  (37)

where $ΓΜΝEG$ is the unknown hybrid-spinor. This implies that int operator $J^$ can be found by

$J^AB′CD′E FYFH=(Δ^[AB′CD′]E F)YFH=ΣΜAB′ΣΝCD′(ΓΜΝEGD^ΝGF−ΓΝΜEGD^ΜGF)YFH$  (38)

for which we use the fact that Μ and Ν are dummy indexes, and $D^ΜEG$ and $D^ΝGF$ commute; this implies $J$ and $M$ are related to terms of $(ΣΜAB′ΣΝCD′(ΓΜΝEGD^ΝGF−ΓΝΜEGD^ΜGF)YFH)(Y−1)HK$, where (Y−1)HK is the inverse of YFH. The introduction of $Δ^$ (Equation 37) is to change from dealing with $HAB′EG$ to$ΓΜΝEG$, while this form $Δ^$ seems not unique.

Let $BΝEF$ is invertible (if considering E and F are considered as the matrix indices), that is, in GL, we may naturally define the hybrid-spinor $ΓΜΝEG$ by

$ΓΜΝEG=(D^ΜEF(BΝFH))(BΝ−1)HG.$

So it implies

$ΣΜAB′ΣΝCD′(ΓΜΝEGD^ΝGF−ΓΝΜEGD^ΜGF)YFH=ΣΜAB′ΣΝCD′(ΓΜΝEGBΝGF−ΓΝΜEGBΜGF)YFH=ΣΜAB′ΣΝCD′(D^[ΜEG(BΝ]GF))YFH=(ΣΜAB′D^[ΜEG(ΣΝCD′BΝ]GF))YFH=(D^[AB′EG(HCD′]GF))YFH$

where we use core equation (Equation 17) and condition, that is, $ΣΜAB′$is $D^$-invariant. So, we get the int operator working as desired.

For scalar case of $D^ΜEF$, we currently consider ($D^ΜEF≡∂ΜΔEF$ as previously stated):

$D^ΜEFYFH=D^ΜYEH=p^ΜiℏYFH$  (39)

where $p^Μ$ is the usual quantum energy, momentum operator, while, for simplicity, pi (i=1,2,3) is negative to usual momentum; we can plug it into the following expression and get the relationship:

note that we choose to ignore the complex conjudge sign $⋆$ of $M$ and detail of the indexes per entity as we just want to illustrate the possible condition to match TΑΒ (as operator-sense) of Einstein field equation.

It motivates the definition of a hybrid-spinor density to match the unit of Einstein field equation:

$ΓΜΝEG≡ihG[Density]$  (40)

such that $Δ^$ with $ΓΜΝEG$ density generates the curvature by the source of local energy–momentum from related ordinary quantum mechanics operators (iħ∂Μ). The detail of $ΣΜAB′,ΣΝCD′$ and $ΓΜΝEG$ results in the freedom that one can assume the structure of TΑΒ as we normally solve problems in classical general relativity (demonstrated in later sections). Nevertheless, one must solve the operators and required equations in section 3.3, in order to find $ΓΜΝEG$ density for a complete solution.

##### The generalization to gauge operators for spacetime

The gauge theory adopted in this theory is, as usual, to allow certain transformation of mathematical objects including the solution (Y here) such that same set of equations are invariant. So, if we found those gauge transformation, a new dynamic description of the curvature of spacetime exists, which we call it generalization to the gauge operators (or called fields).

The core theory of this article naturally suggests the spinor operation over 2 × 2 matrix (the base mathematical field) instead of scalar because of core equation (Equation 17). So in this section (ONLY), the author illustrates the equations in spinor over 2 × 2 matrix, that is, core equation reads from $DΜEFYFG=BΜEFYFG$ to $DΜY=BΜY$, so it is easy to see the natural simplicity.

Gauge operators (or fields) $=def$ operators used to define the transformation of following objects as well as Y simultaneously while keeping following equations invariant:

Objects to transform by gauge fields: $D^Μ$, $ΣΜAB′$, $HAB′$, BΜ, $J^AB$

Equations to be invariant:

commutativity of $D^$:  $[D^AB′,D^CD′]Y=0$

$D^$ invariant:  $D^ΝΣΜAB′X=ΣΜAB′D^ΝX$ ($ΣΜAB′$ denotes as generalized$ΣΜAB′$, so it is 2 × 2 matrix in general $∀$ indices of A,B’, Μ)

Equations of system (Equation 19):  $D^ABY=HAB$Y, $D^ΜY=BΜ$Y

Integratability: [BΜ,BΝ] = $D^ΜBΝ$$D^ΝBΜ$

Γ operator (if any): $ΓΜΝ=(D^ΜBΝ)BΝ−1$

commutativity of $J^$ and $M^$:   $[J^ABD^CD′]Y=0$, $[M^A′B′D^CD′]Y=0$

Fundamental equation (Equation 21): $εBC(J^ABD^CD′)Y=εB′C′(M^B′D′D^AC′)Y$ or

$(J^ABD^BD′)Y=(J¯DBD^⋆BA′)⋆Y$(original form)

One can easily observe that only certain transformation is allowed because the equations above must be simultaneously satisfied. Physically, it means only certain gauge field related to transformation of curvature of spacetime is allowed. In the next section, we will exercise this generalization in certain way to apply to evolution model of the universe.

A key symmetric structure of fundamental equation in second form $(J^ABD^BD′)Y=(J¯DBD^⋆BA′)⋆Y$ is obvious that the author defines this in the original version. As it has Hermitian conjugate structure if considering the indices of A’D and some $J¯$ and $D^$.

#### AS AN APPLICATION TO THE PROBLEM OF MODELING THE UNIVERSE EVOLUTION

In this section, some forms of operators and solution Y are deployed such that a simple solution of equation (21) is obtained. An equation explaining the evolution model of the universe is derived; usual Friedmann equation is compared as well as density parameters of dark matter, ordinary matter, and dark energy is proposedly originated; furthermore, the prediction of density parameter of ordinary matter is compared to the observation of Planck data (Carroll & Ostlie, 2007; Density data: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html). At last, certain equations of unsolved fields are discussed for the consistency of the theory.

##### Basic cosmological model

There are some assumptions on the form of YGH, $D^ΜEF$, and $ΓΜΝEF$ as well as the new gauge field – hybrid spinor $ΛΜEF$ to the application of a cosmological model we interest:

parameter space are t = x0, x = {x1, x2, x3}; $ΛΝEF(t,x)$ are 2 × 2 commutating matrices (if considering E and F as the matrix indices) such that [ΛΜ, ΛΝ] = 0, and a(t) is a real scalar field for t-dependent;

$ΓΜΝEF→a(t)aX(x)ΓcΜΝEF,$  (41)

where aX(x) is the real scalar field for x-dependent only, $ΓcΜΝEF$ is a constant spinor such that

Finally, we assume $BΜEF$ is an invertible matrix (if considering E and F as the matrix indices).

Let’s examinate $J^EF′GH′I KYKL$ from Equation (38):

$J^EF′GH′I KYKL=ΣΜEF′ΣΝGH′(ΓΜΝIGD^ΝGK−ΓΝΜIGD^ΜGK)YKL=(aaX(ΣΜEF′ΣΝGH′)(a(ΓcΜΝIGΛΝGK−ΓcΝΜIGΛΜGK)+ΓcΜΝIK∂Ν−ΓcΝΜIK∂Μ)YKL$

where we use the commutativity property of $D^$, form of $D^$, form of $ΓΜΝEF$ (Equation 41), and $ΣΜEF′$ is $D^$-invariant. We further apply the form of $ΓcΜΝEF$ (Equation 41) that and YKL is t-dependent only:

$J^EF′GH′I KYKL=a2aX(x)Γc((Σ3i=1ΣiEF′)Σ0GH′−Σ0EF′(Σ3i=1ΣiGH′))Λ0IK(x)YKL(t).$

Let us study the overall t-scaling (i.e., t-dependent) effect of the equation by introducing length scaling factor l from a:

(42)

where a new spinor from combination of $ΣΜAB′:ΣEF′GH′≡(Σ3i=1ΣiEF′)Σ0GH′−Σ0EF′(Σ3i=1ΣiGH′).$

Then we get $J$ and $M$ by $J^,M^$ and the inverse of YGH:

$JACEF=(J^ACEGYGH)(Y−1)HF=(εB′D′J^AB′CD′E GYGH)(Y−1)HF=(εB′D′ΓcΣAB′CD′aX(x)Λ0EG(x)l(t)6YGH)(Y−1)HF=ΣAC(ΓcaX(x)Λ0EF(x)l(t)6),MB′D′EF=(M^B′D′EGYGH)(Y−1)HF=(εACJ^AB′CD′E GYGH)(Y−1)HF=(εACΓcΣAB′CD′aX(x)Λ0EG(x)l(t)6YGH)(Y−1)HF=ΣB′D′(ΓcaX(x)Λ0EF(x)l(t)6),$

where $ΣAC≡εB′D′ΓcΣAB′CD′,ΣB′D′≡εACΓcΣAB′CD′$.

Let’s examinate the Einstein spinor $GAC′BD′$ (Equation 35) based on these assumptions and the calculation above. $GAC′BD′$ is contributed by Λ and $M⋆ABC′E′$; Λ is contributed by $JABCB$ Equation 32, and Einstein tensor is derived by $GΜΝ=ΣΜEF′ΣΝGH′GEF′GH′$. The t-dependency of $JACEF$ and $MB′D′EF$ is clearly proportional to l−6. Refering to the Equation (1 and 4), ΣΜ and ΣΝ balance each others (because of the metric and its inverse) so we can neglect the overall t-scaling effect of $ΣΜEF′ΣΝFG′×ΣAC$, or $ΣΜEF′ΣΝFG′×ΣB′D′$ from GΜΝ; it makes sense, as in spinor approach of GR, the curvature dynamics is reflected by Χ, Φ curvature spinors, not$ΣΜA,C′$,$ΣΝE,F′$. One can also choose $Σ0GH′$ and $ΣiEF′$ to be the near constants to make computation easy; however, one must check the consistency of the choice with the core equations (refer to section: the generalization to gauge operators for spacetime).

In conclusion, GΜΝ is contributed by the term

$ΓcaX(x)Λ0EF(x)l(t)6,$

and by the following definition (to be consistent with Einstein equation):

$Ordinary matter density tensor=defordinary stress-energy denisity tensor=def−GΜΝ8ΠG$

the t-dependency of ordinary matter density in this case is proportional to $1l(t)6$. Note that, the ordinary matter density defined here is to match Einstein equation for this model only.

##### Calculation of scaling factor and comparison to Friedmann equation

Solving Equation (21, the fundamental equation) is the core task to get the dynamics of the cosmological model, or more generally the physical structure of spacetime. The first step is to calculate the term, $J^CDEFD^AB′FG$, which is left side of the fundamental equation. After using the form of $D^$, $ΓΜΝEF$ (41) and $D^$-invariant, it implies:

$J^CDEFD^AB′FGYGH=εF′H′(ΣΓAB′ΣΜCF′ΣΝDH′)×aaX(a2ΓcΜΝEFΛΓFIΛΝIG−a2ΓcΝΜEFΛΜFIΛΓIG+aΓcΜΝEFΛΓFG∂Ν+aΓcΜΝEFΛΝFG∂Γ+aΓcΜΝEF∂Ν(ΛΓFG)−aΓcΝΜEFΛΓFG∂Μ−aΓcΝΜEFΛΜFG∂Γ−aΓcΝΜEF∂Μ(ΛΓFG)+∂Ν(a)ΓcΜΝEFΛΓFG−∂Μ(a)ΓcΝΜEFΛΓFG+ΓcΜΝEG∂Γ,Ν−ΓcΝΜEG∂Γ,Μ)YGH.$

Applying $ΓcΜΝEF$ for Ν ≠ 0 (condition 41), YGH is t-dependent only, commuting property of $ΛΜEF$ (if considering E and F as the matrix indices), and the following equations from later discussion (Equation 45):

$a′ΛiEF=a∂i(Λ0EF)−a∂0(ΛiEF)aΛiEGΛ0GF=−∂i(Λ0EF)$

for i= 1,2,3 (a′ is t-derivative of a), we get:

$J^CDEFD^AB′FGYGH=aaXΓcεF′H′(ΣCF′DH′)(a2Σ0AB′Λ0EFΛ0FGYGH+a′Σ0AB′Λ0EGYGH+2aΣ0AB′Λ0EG(YGH)′+aΣiAB′ΛiEG(YGH)′+aΣ0AB′∂0(Λ0EG)YGH+Σ0AB′(YEH)′′),$

where Y′,Y′ denote first and second t-derivatives of Y.

By right side of fundamental equation:

$(M^B′D′EFD^CD′FG)YGH=εD′A′(M^B′D′EFD^CA′FG)YGH.$

Combing both sides, we get $εDA$ $(J^CDEFD^AB′FG)YGH=$ $εA′D′(M^B′D′EFD^C,A′FG)YGH$ as

Using the form of $YGH=(eΑ(t)00eΑ(t)),$

differentiating it results in

$(YGH)′→YGHϕ,(YGH)′′→YGH(ϕ2+ϕ′),$

where Α′ ≡ ϕ. After substituting these results to the equation above, all terms are matrices operating on YGH only (if considering G,H as the matrix indices), so we can take it out as YGH is in GL. The equation becomes

$εDAaaXΓcεF′H′(ΣCF′DH′)(Σ0AB′(a2Λ0EFΛ0FG+Λ0EG(a′+2aϕ)+a∂0(Λ0EG)+(ϕ2+ϕ′)ΔEG)+ΣiAB′a(ϕΛiEG))=εD′A′aaXΓcεFH(ΣFB′HD′)(Σ0CA′(a2Λ0EFΛ0FG+Λ0EG(a′+2aϕ)+a∂0(Λ0EG)+(ϕ2+ϕ′)ΔEG)+ΣiCA′a(ϕΛiEG)).$

Finally, we further assume the following conditions in order to look for a simple solution:

${ϕΛiEG→0}(ΛiEGcoupling withϕcontributing very little to the overall dynamics),$

and note that $Λ0FG$ is t-independent. Then, there exists a solution:

$0=a2Λ0EFΛ0FG+(a′+2aϕ)Λ0EG+(ϕ2+ϕ′)ΔEG.$

Let’s pick an abelian matrix structure for $Λ0FG$ (if considering F and G as the matrix indices)below:

$Λ0FG=(s(x)y(x)y(x)s(x)),$

and we have the set of differential equations for the solution:

and we pick a solution that a ≠ 0:

$a′=−2a(as+ϕ),ϕ′=a2(s2−y2)−ϕ2$

To get an equation to compare with usual Friedmann equation, let’s converting the solution to length scaling factor l and getting a scalar $H≡l′l$:

$l′=2(l3ϕ+s)3l2,H=2(l3ϕ+s)3l3$

The equation to be compared with Friedmann equation is

(43)

while Friedmann equation is (spatial curvature k is assumed to be zero, Λcos is usual cosmological constant, Ωn are the usual density parameters for different types of matter):

$(H(l)H0)2=∑nΩnl−n=8ΠG3H02(∑nΡn(l−n))+Λcos3H02$

##### Densities of dark energy, matter, and ordinary matter (ΩΛ, ΩM, Ωb)

As usual in Λ CDM model, we identify the zeroth and negative power terms of scaling factor (l0, l−3 and l−6), in equation $(H(l,ϕ)H0)2$ (Equation 43), so there are three types of matter to be responsible for the evolution of the scaling factor, and their density parameters are namely ΩΛ, ΩM, and Ωb, respectively. The Ωb is proposedly to be responsible for ordinary matter density as the t-dependency of ordinary matter density is proportional to l−6 in preceding discussion:

$Ωb=4s29H02,ΩΛ=4ϕ29H02,ΩM=8sϕ9H02.$

$ΩM2≃4ΩbΩΛ$ (approximation − equal sign because of the approximation of the model)

To compare observation results from observation data (Carroll & Ostlie, 2007; Density data: http://hyperphysics.phy-astr.gsu.edu/hbase/astro/denpar.html), ΩΛ and Ωb data are used:

$ΩΛ→0.69,Ωb→0.04;$

the calculated and observed ΩM are

The observation results of Ωb, ΩΛ, ΩM motivate the speculation for this model to claim the relationship of densities; it is similar that Maxwell speculated that electromagnetic wave is the light itself because the speed of electromagnetic wave from his equations was found accidentally the same as light speed.

The author claims that, more accurate relationship among those sources of evolution of the universe should be manageable after certain assumption is removed such as time-independent of Λ0. Moreover, complete proof of calculation of Ωb requires particle theory, which is also not covered in this article.

##### Spinor gauge field-$ΛΜEF$ dynamics

In preceding discussion of calculation of the solution for fundamental equation (Equation 21), a set of PDE related to $ΛΜEF$ is used. In fact, they are the results of core constraints of the theory.

Let us look at the commutativity of $D^$ after appying gauge field$ΛΜEF$:

$D^A,B′EFD^C,D′FGYGH=D^C,D′EFD^A,B′FGYGH$
,

there exist a solution

$[∂ΝΔEF+aΛΝEF,∂Μ+aΛΜEF]YFH=0$

so$(∂[Ν(aΛΜEF)YFH+a2ΛΝEFΛΜ]FGYGH)=0$;

by $ΛΜEF$ is abelian (if considering E and F as the matrix indices), it implies

$∂[Ν(aΛΜ]EG)YGH=0.$  (44)

Then, let us look at integratability of the system: $B[ΜEFBΝ]FG=D^[ΜEFBΝ]FG$. As $ΓΜΝEF=0$ for Ν ≠ 0 , and it provides $D^ΜEFBΝFG$ by Equation (38) $ΓΜΝEG=(D^ΜEF(BΝFH))(BΝ−1)HG$, so we get

$D^ΜEFBiFG=0,B[iEFBj]FG=0,B[0EFBi]FG=−aaXΓcB0EG(for i,j=1,2,3;Μ=0,1,2,3).$

By the core equation $D^iEFYFG=Bi EFYFG$, $BiEF$ is solved by$(∂iΔEF+aΛiEF)Y(t)FG=Bi EFY(t)FG$; $→$ exist a solution

$BiEF=aΛiEF,$

it is consistent with $B[iEFBj]FG=0$ and $ΛiEF$ abelian property (if considering E and F as the matrix indices). $D^ΜEFBi FG=0,$ equation leads to:

Combining the above equation and Equation (44), the dynamics of gauge field $ΛΜEF$ is got:

(45)

for i,j=1,2,3(a′ is t-derivative of a).

Please note that the calculation of cosmological model in this section does not include solving gauge field-$ΛΜEF$; however, the theory requires that the dynamics of $ΛΜEF$ must exist in order to be consistent.

#### DISCUSSION

The description of quantum mechanical wave function is also not discussed, and this is what this theory should extend to. With such description and spacetime description in this article, a more complete picture of matter and spacetime should be reviewed.

Finally, the author proposes that the physical reality is the general deformation of same cross-ratio.

### References

1. Carroll B. W, Ostlie D. A. 2007. An introduction to modern astrophysics. 2nd ed. Pearson Education. USA:

2. Cassidy P, Singer M. 2005. Galois theory of parameterized differential equations and linear differential algebraic groups. arXiv:math/0502396arxiv.org

3. Gomez-Lobo A. G.-P, Martin-Garcia J. M. 2012. Spinors: A Mathematica package for doing spinor calculus in general relativity. Computer Physics Communications. Vol. 183:2214–2225

4. Penrose R. 1960. Chapters 1–3 of “A spinor approach to general relativity”. Annals of Physics. Vol. 10:171–201

5. Robert M. W. General Relativity. Chicago, IL: University of Chicago Press. 1984

6. Yoshida M. Hyper geometric functions, my love: Modular interpretations of configuration spaces. Springer Fachmedien Wiesbader (1997). Germany:

### Competing interests

The authors declare no competing interests.

### Publishing notes

© 2016 Liu. This work has been published open access under Creative Commons Attribution License CC BY 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com.

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###### Contributors
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SOR-PHYS
ScienceOpen Research
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2199-1006
27 August 2016
25 September 2016
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###### Affiliations
Department of Physics and Astronomy, Louisiana State University, Baton Rouge, LA, USA
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[* ]Corresponding author’s e-mail address: jackieliu0@ 123456gmail.com
###### Article
3802:XE
10.14293/S2199-1006.1.SOR-PHYS.A4HPPH.v1
27ffb33f-f6eb-4638-8316-9f74d68ebae1
© 2016 J. C. H. Liu

This work has been published open access under Creative Commons Attribution License CC BY 4.0 , which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Conditions, terms of use and publishing policy can be found at www.scienceopen.com .

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