Cosmological Meaning of Geometric Curvatures

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arXiv:1911.05518v6 [math.DG] 16 Feb 2020 Introduction 1 o n hsclapiaino hyaeecsie ewl p will paper. We this excessive. in are question they this or to application physical any metric e for of the part one, (real) Einstein’s symmetric commonl the the of in of dimension unlike one additional model, is this 8] [7, In cosmology. model cosmological Kaluza-Klein The by determined are works Moreov Einstein’s electromagnetism. in the spaces to of suits part imaginer the but . hsclmtvto o ieeta emty aiso basics geometry: differential for motivation Physical 1.1 anti-symme or complex and [5] space Riemannian generalized ehooia eeomn,GatN.174012 No. Grant Development, Technological ute eerhsaottesae ihtrinae[15,19 are torsion with wo spaces Einstein’s the these about where researches papers further other Some explained. well vnv Zlatanović [6] Ivanov, ∗ OMLGCLMAIGO GEOMETRIC OF MEANING COSMOLOGICAL eba cdm fSine n rs ahmtclInstitu Mathematical Arts, and Sciences of Academy Serbian h isensTer fGnrlRltvt sacosmologic a is Relativity General of Theory Einstein’s The h usinta rssi hte h nismercpart anti-symmetric the whether is arises that question The h anproeo hsppri ofidapyia enn fs of meaning physical a find to is paper this of purpose main The isenivle h ocp facmlxmti hs real whose metric complex a of concept the involved Einstein aygoercppr tr ihtemtvto rmGenera from motivation the with start papers geometric Many e words: p Key this of s end the the and At energy-density pressure, spaces. tensor, Riemannian energy-momentum generalized of concepts state-parameters ent Afte and energy-densities tensors. pressures, energy-momentum tensors, pressure corresponding for the formulae Madsen’s to the respect developed is It space. mannian 2010 density nti ae,w nlzdtepyia enn fsaa curvatur scalar of meaning physical the analyzed we paper, this In ah uj Classification: Subj. Math. uvtr,lgaga,mte,eeg-oetmtno,pre tensor, energy-momentum matter, lagrangian, curvature, h hsclmtvto ihrsett h isenswor Einstein’s the to respect with motivation physical the , CURVATURES ea .Vesić O. Nenad 35,4B5 34,53B05 53A45, 47B15, 53B50, Abstract 1 h isenMtiiyCondition Metricity Einstein the e eba iityo dcto,Sineand Science Education, of Ministry Serbian te; ∗ r nlzdwt epc odiffer- to respect with analyzed are 2]admn others. many and –23] s. r h ffiecneto coefficients connection affine the er, yial n emtial answer geometrically and hysically k r ie stemtvtosfor motivations the as cited are rks etoants scvrdb the by covered is lectromagnetism aeprmtraeexamined. are tate-parameter rcmtisa well. as metrics tric ht h energy-momentum the that, r fmti esr r important are tensors metric of s sdmdl nteter of theory the in models used y lmdlwihwsdeveloped. was which model al atcrepnst h gravity the to corresponds part n nrydniiswith energy-densities and s sfragnrlzdRie- generalized a for es eaiiy ntepaper the In Relativity. l pr ierte fthe of linearities aper, aa uvtrsfra for curvatures calar cosmology f sr,energy- ssure, . s[–]is [2–4] ks The Kaluza-Klein model will not be studied here. The computational methodology applied in the book [1] but also in the article [17] combined with torsion tensors will be used in this study.

1.2 Geometrical motivation: Generalized Riemannian space

An N-dimensional manifold MN , equipped with a non-symmetric metric tensorg ˆ of the type (0, 2) whose components are gij is the generalized Riemannian space GRN in the sense of Eisenhart [5] . S. M. Minˇcić[13–15], M. S. Stanković[19,20,22,23], Lj. S. Velimirović[15,19,22], M. Lj. Zlatanović[6, 22, 23] and many others have continued the research about these spaces, the mappings between them and their generalizations. 1 The symmetric and anti-symmetric part of the tensorg ˆ are the tensorsg ˆ = (ˆg +g ˆT ) and 2 1 T gˆ = 2 (ˆg − gˆ ), respectively. Their components are ∨

1 1 g = (g + g ) and g = (g + g ) (1.1) ij 2 ij ji ij 2 ij ji

For our research, the matrix gij N×N should be non-singular. The metric determinant for the space R4 is g = det[gij] and it is different of 0 because of the non-singularity of the matrix ij gij . The components g for the contravariant symmetric part of the metric tensorg ˆ are the −1 corresponding elements of the inverse matrix gij N×N . The components of the affine connection coefficients for the space GRN are the components of the generalized Christoffel symbols of the second kind

i 1 iα Γjk = g gjα,k − gjk,α + gαk,j , (1.2) 2 for partial derivative ∂/∂xi denoted by comma. The components of the symmetric and anti-symmetric parts of the generalized Christoﬀel symbol of the second kind are

i 1 i i i 1 i i Γjk = Γjk +Γkj and Γjk = Γjk − Γkj . (1.3) 2 ∨ 2 After some computing, one gets

1 1 Γi = giα g − g + g and Γi = giα g − g + g . (1.4) jk jα,k jk,α αk,j jk jα,k jk,α αk∨ ,j 2 ∨ 2 ∨ ∨ i The doubled components of the anti-symmetric part Γjk are the components of the torsion ∨ ˆ GR i i tensor T for the space N . The components of the torsion tensor are Tjk = 2Γjk. The ∨ α components of the covariant torsion tensor are Tijk = giαTjk. The manifold MN equipped with the tensorg ˆ is the associated space RN of the space GRN . The components of the symmetric part (1.3) of the generalized Christoffel symbols are the Christoffel symbols of the second kind. Hence, they are the affine connection coefficients of the associated space RN .

2 The associated space RN is the Riemannian space in the sense of Eisenhart’s definition [5] . N. S. Sinyukov [18], Josef Mikeˇswith his research group [10–12] and many other authors have developed the theory of Riemannian spaces. With respect to the affine connection of the associated space RN and a tensora ˆ of the type (1, 1), it is defined one kind of covariant derivative [10–12, 18]

i i i α α i aj|k = aj,k +Γαkaj − Γjkaα. (1.5) Based on this covariant derivative, it is founded one identity of the Ricci Type. From this identity, it is obtained one curvature tensor Rˆ of the space RN see [10–12,18] . The components of this tensor are

i i i α i α i Rjmn =Γjm,n − Γjn,m +ΓjmΓαn − ΓjnΓαm. (1.6) α The components of the corresponding tensor of the Ricci curvature are Rij = Rijα. The αβ scalar curvature of the associated space RN is R = g Rαβ. A. Einstein studied the spaces whose aﬃne connection coeﬃcients are not functions of the metric tensor [2–4]. With respect to the Einstein Metricity Condition

α α gij,k − Γikgαj − Γkjgiα = 0, (1.7) as the system of differential equations which generate the affine connection coefficients for the affine connection space, two kinds of covariant derivatives are defined

i i i α α i i i i α α i aj |k = aj,k +Γαkaj − Γjkaα and aj |k = aj,k +Γkαaj − Γkjaα. (1.8) 1 2

M. Prvanović[16] obtained the fourth curvature tensor for a non-symmetric affine connection space. S. M. Minˇcić[13,14] defined four kinds of covariant derivatives. These four kinds are the covariant derivatives (1.8) and two novel ones

i i i α α i i i i α α i aj |k = aj,k +Γαkaj − Γkjaα and aj |k = aj,k +Γkαaj − Γjkaα. (1.9) 3 4 With respect to these four kinds of covariant derivatives, S. M. Minˇci´cobtained four curvature tensors, eight derived curvature tensors and ﬁfteen curvature pseudotensors of the space GRN . The components of curvature tensors for the space GRN are elements of the family

i i i ′ i α i ′ α i α i Kjmn = Rjmn + uT jm|n + u T jn|m + vT jmT αn + v T jnT αm + wT mnT αj, (1.10) for the corresponding coeﬃcients u, u′, v, v′, w. Six of them are linearly independent.

3 The components of Ricci-curvatures for the space GRN are

α ′ α β Kij = Rij + uT ij|α − (v + w)T iβT jα. (1.11) Three of these tensors are linearly independent. γδ The family of scalar curvatures K = g Kγδ for the space GRN is

′ γδ αǫ βζ K = R − (v + w)g g g T αγβT ǫδζ. (1.12) Two of these curvatures are linearly independent. In this paper, we will stay focused on the space-time GR4 equipped with a non-symmetric metric tensorg ˆ whose symmetric part is diagonal.

1.3 Motivation In the Theory of General Relativity, the Einstein-Hilbert action is the action that yields the Einstein ﬁeld equations through the principle of least action. Let the full action of the theory be

4 S = d x |g| (R + LM ), (1.13) Z p for the scalar curvature R of the associated space R4. The term LM in the last equation is describing matter ﬁelds. The Ricci tensor Rij and the scalar curvature R for the space GR4 satisfy the Einstein’s equations of motion 1 R − Rg = T , (1.14) ij 2 ij ij where Tij are the components of the energy-momentum tensor Tˆ. The last equations are generalized by the cosmological constant Λ as [1,17] 1 R − Rg + Λg = T . (1.15) ij 2 ij ij ij Remark 1.1. The equation (1.14) is obtained from the Einstein-Hilbert action 4 S = d x |g| R + LM but the equation (1.15) holds from the Einstein-Hilbert action 4 SΛ = Rd x p|g| R − 2Λ +LM , S = S0. InR general,p the components Tij of the corresponding energy-momentum tensor Tˆ at the right sides of the equations (1.14, 1.15) are multiplied by the constant κ, κ = 8πGc−4 for the speed of light c and Newton’s gravitational constant G but we will chose such coordinates to be κ = 1 in further research.

The Friedman-Lemaitre-Robertson-Walker (FLRW) and the Bianchi Type-I cosmological models are solutions of the Einstein’s equations of motion.

4 This article is consisted of ﬁve sections plus conclusion. The purposes of this paper are:

1. To recall and develop the Madsen’s formulae [9] for pressure, energy-density and state parameter,

2. To correlate the space-time model caused from the article [6] with the Einstein’s equations of motion,

3. To analyze the linearity of energy-momentum tensors, pressures, energy-densities and state-parameters under summings of ﬁelds.

2 Pressure, density, state parameter and Madsen’s formulae

2 4 1 MP 2 1 α Based on the action I = d x |g| 8π − ξφ R + 2 ∂αφ∂ φ − V [φ] , the energy-mo- p 2 mentum tensor is see [9] R 2 −1 2 2 Tij = 1 − ξφ Sij + ξ gij (φ ) − (φ )|i|j , (2.1) h i for the constant ξ, the time-like scalar field φ which has unit magnitude, the operator defined i αβ i 1 αβ as aj = g aj|α|β and the tensor Sij = φ,iφ,j − 2 g φ,αφ,β − V [φ] gij. Madsen also chosen ~ 2 the units such that = c = 1 and MP = 8π for the Planck mass MP . In the second section of the paper [9], Madsen deals with the problem of finding a unique vectoru ˆ related to the scalar field φ, appears in the energy-momentum tensor (2.1). The components for this vector are ui and they satisfy the equation

α u uα = 1. (2.2) The components ui are

i α −1/2 i u = ∂ φ∂αφ ∂ φ. (2.3) For the symmetric tensor hˆ of the type (0, 2) whose components are [9]

hij = gij − uiuj, (2.4) α the energy-density ρ, the pressure p, the vector-ﬁeldq ˆ such that uαq = 0 and the tensorπ ˆ of α α the type (0, 2) whose components satisfy the equalities πiαu = 0 and πα = 0, the components of the energy-momentum tensor Tˆ of the type (0, 2) are [9]

Tij = ρuiuj + qiuj + qjui − phij + πij . (2.5) 1 α α β α β It holds see [9] p = 3 Πα and ρ = Tαβu u , for Πij ≡ phij + πij = −Tαβhi hj . After jk k k k composing the equation (2.4) by g , one obtains hi = δi − uiu . If compose the equation (2.5) ij α by the tensor g , use the symmetry Tij = Tji, the relation u uα = 1 and the previously founded i components hj, we will acquire the following expressions

α α α β α α α β Πij = −Tij + Tiαu uj + Tjαu ui − Tαβu u uiuj and Πα = −Tα + Tαβu u . (2.6)

5 Hence, the energy-momentum tensor Tˆ, the pressure p, the energy-density ρ and the state parameter ω satisfy the next equalities

1 α 1 α β α β 1 α β γ −1 1 p = − T α + T αβu u , ρ = T αβu u , ω = − T α T βγu u + . (2.7) 3 3 3 3 i i In the comoving reference frame, u = δ0, the equalities (2.7) reduce to

1 α 1 1 α −1 1 p0 = − T α + T 00, ρ0 = T 00, ω0 = − T α T 00 + . (2.8) 3 3 3 3 After composing the equation (1.15) by uiuj and gij but using the equation (2.2) as well, we get 1 T uαuβ = R uαuβ − R +Λ and T α = −R + 4Λ. (2.9) αβ αβ 2 α In the case of i = j = 0, one obtains

(1.15) 1 T = R − Rg + Λg . (2.10) 00 00 2 00 00 After substituting the equations (2.9, 2.10) into the expressions (2.7, 2.8), we will ﬁnd

In a reference system ui In the comoving reference system ui = δi0 1 α β 1 1 1 p = 3 Rαβu u + 6 R − Λ p0 = 3 R00 + 6 R − Λ α β 1 1 ρ = Rαβu u − 2 R + Λ ρ0 = R00 − 2 R + Λ −1 −1 ω = pρ ω0 = p0ρ0 Table 1: Pressures, energy-densities and the state parameters

3 Generalized Einstein’s equations of motion I

Let us consider the Einstein-Hilbert action

S = d4x |g| K − 2Λ , (3.1) Z p with respect to the Shapiro’s cosmological model [17]. Based on the equation (1.12), we transform the Einstein-Hilbert action (3.1) to

S = d4x |g| R − 2Λ − (v′ + w)gγδ T α T β . (3.2) Z γβ αδ p After lowering the contravariant indices into the last equation, one obtains

S = d4x |g| R − 2Λ − (v′ + w)gγδ gǫαgβζ T T . (3.3) Z ǫγβ αδζ p

6 If compare the variations of the functional (3.3) and the Einstein-Hilbert action 4 S = d x |g| R − 2Λ + LM , we get R p ′ γδ ǫα βζ LM = −(v + w)g g g T ǫγβT αδζ . (3.4) 4 The variation of the functional S1 = S1[ˆg]= d x |g| R − 2Λ is [1] R p 1 δS = d4x |g| R − Rg + Λg δgαβ . (3.5) 1 Z αβ αβ αβ p 2 4 γδ ǫα βζ The variation of the functional S2 = S2[ˆg]= d x |g|g g g T ǫγβT αδζ is R p

4 1 γδ ǫη θζ αβ δS2 = d x |g| 3ταβ + 2Wαβ − g g g TηγθTǫδζ gαβ δg , (3.6) Z p h 2 i γδ δg ǫα βζ γδ ǫα βζ δTαγβ for τij = g g T ǫγβT αδζ and Wij = g g g T ǫδζ . δgij δgij ij Based on the variation rule, the variational derivatives δT α.γβ/δg are the components vαγβij for the tensorv ˆ of the type (0, 5). With respect to the equations (3.5, 3.6), we obtain

4 1 ′ 1 γδ ǫη θζ αβ δS = d x |g| Rαβ − Rgαβ +Λgαβ −(v +w) 3ταβ +2Wαβ − g g g TηγθTǫδζ gαβ g . Z p n 2 h 2 io The right side of the last equation vanishes if and only if

1 ′ 1 γδ ǫα βζ Rij − Rgij + Λgij = (v + w) 3τij + 2Wij − g g g TαγβTǫδζ gij . (3.7) 2 2 The family of Einstein’s equations of motion is presented by the equation (3.7). The next theorem holds. Theorem 3.1. With respect to the equations of motion (3.7), the families of the energy- momentum tensors and its traces are

′ 1 γδ ǫα βζ T ij = (v + w) 3τij + 2Wij − g g g TαγβTǫδζ gij , 2 (3.8) α ′ α α γδ ǫα βζ T α = (v + w) 3τα + 2Wα − 2g g g TαγβTǫδζ , respectively, for the coefficients v′, w and the above defined tensors τˆ and Wˆ in the analyzed cosmological model. With respect to the equations (2.7, 2.8) and the equalities (3.8), the families of the pressures and energy-densities are 3 p = − 1 (v′ + w) 3τ α + 2Wα − gγδgǫαgβζ T T − 3τ + 2W uαuβ ,  3 α α αγβ ǫδζ αβ αβ P ressure : h 2 i  1 ′ α α 3 γδ ǫα βζ p0 = − 3 (v + w) 3τα + 2Wα − g g g TαγβTǫδζ − 3τ00 − 2W00 , 2  (3.9) 3 ρ = (v′ + w) 3τ + 2W uαuβ − gγδgǫαgβζ T T ,  αβ αβ αγβ ǫδζ Energy − density : h 2 i (3.10)  ′ 3 γδ ǫα βζ  ρ0 = (v + w) 3τ00 + 2W00 − g g g TαγβTǫδζ . h 2 i  7 −1 −1 ′ The state-parameters ω = p · ρ and ω0 = p0 · ρ0 do not depend of the coefficients u, u , v, ′ v , w which generate curvature tensors of the generalized Riemannian space GR4. Corollary 3.1. The next equations hold

1 α β 1 1 ′ α α 3 γδ ǫα βζ α β Rαβu u + R−Λ= − (v +w) 3τα +2Wα − g g g TαγβTǫδζ − 3ταβ +2Wαβ u u , 3 6 3 h 2 i (3.11)

α β 1 ′ α β 3 γδ ǫα βζ Rαβu u − R +Λ=(v + w) 3ταβ + 2Wαβ u u − g g g TαγβTǫδζ , (3.12) 2 h 2 i in the reference frame ui and

1 1 1 ′ α α 3 γδ ǫα βζ R00 + R − Λ= − (v + w) 3τα + 2Wα − g g g TαγβTǫδζ − 3τ00 − 2W00 , (3.13) 3 6 3 h 2 i 1 ′ 3 γδ ǫα βζ R00 − R +Λ=(v + w) 3τ00 + 2W00 − g g g TαγβTǫδζ , (3.14) 2 h 2 i in the comoving reference frame.

Proof. After equalizing the expressions of the pressures p, p0 and the energy-densities ρ, ρ0 from the Table 1 and the equations (3.9, 3.10), we complete the proof for this corollary.

The equations (3.11, 3.13) are the equations of equilibrium for metric with respect to the pressure p (pEQM). The equations (3.12, 3.14) are the equations of equilibrium for metric with respect to the energy-density ρ (ρEQM). With respect to the equations (1.4, 3.4) and the meaning of the term LM , the torsion-free aﬃne connection spaces (the Riemannian spaces R4 are the special ones) describe spaces without matter. Hence, the anti-symmetric part of the metric tensor gˆ and the torsion tensor of the space GR4 as well are correlated to a matter.

3.1 Non-symmetric metrics and lagrangian In this part of the paper, we will examine what are components for the anti-symmetric part gˆ of the metric tensorg ˆ which correspond to the summand LM in the Einstein-Hilbert action ∨ 4 d x |g|(R + LM ). R Letp us consider the non-symmetric metricg ˆ whose components are

s0(t) n0(t) n1(t) n2(t)  −n (t) s (t) n (t) n (t)  g = 0 1 1 2 . (3.15)  −n1(t) −n1(t) s2(t) n3(t)     −n2(t) −n2(t) −n3(t) s3(t)  The components of the symmetric and anti-symmetric parts for this metric are

s0(t)0 0 0 0 n0(t) n1(t) n2(t)  0 s (t) 0 0   −n (t) 0 n (t) n (t)  g = 1 and g = 0 3 4 .(3.16) 0 0 s (t) 0 −n (t) −n (t) 0 n (t)  2  ∨  1 3 5   0 0 0 s (t)   −n (t) −n (t) −n (t) 0   3   2 4 5 

8 The components of the corresponding Christoﬀel symbols of the ﬁrst kind are

1 ′ Γ0.00 = 2 s0(t), 1 ′ 1 ′ Γ0.11 = − 2 s1(t), Γ1.01 =Γ1.10 = 2 s1(t), 1 ′ 1 ′ (3.17) Γ0.22 = − 2 s2(t), Γ2.02 =Γ2.20 = 2 s2(t), 1 ′ 1 ′ Γ0.33 = − 2 s3(t), Γ3.03 =Γ3.30 = 2 s3(t), but Γi.jk = 0 in all other cases. The components of the covariant torsion tensor are

′ T012 = −T021 = −T102 = T120 = T201 = −T210 = −n3(t), ′ T013 = −T031 = −T103 = T130 = T301 = −T310 = −n4(t), (3.18) ′ T023 = −T032 = −T203 = T230 = T302 = −T320 = −n5(t), and Tijk = 0 otherwise. As we may see, the components of the torsion tensor Tˆ do not depend of the components n0(t), n1(t), n2(t) of the anti-symmetric part of the metric tensorg ˆ. With respect to the equation (3.3), we obtain that the term LM is

3 ′ −1 ′ 2 ′ 2 ′ 2 LM = − (v + w)˜g s0(t) s3(t) n3(t) + s2(t) n4(t) + s1(t) n5(t) . (3.19) 2 n o After basic computing, one gets

−1 −1 −1 ′ 2 ′ 2 ′ 2 τ00 = 6 s1(t) s2(t) s3(t) n3(t) s3(t)+ n4(t) s2(t)+ n5(t) s1(t) , −1 −1 −1 ′ 2 ′ 2  τ11 = 6s0(t) s2(t) s3(t) n3(t) s3(t)+ n4(t) s2(t) ,   −1 −1 −1 ′ 2 ′ 2  τ22 = 6s0(t) s1(t) s3(t) n3(t) s3(t)+ n5(t) s1(t) ,   −1 −1 −1 ′ 2 ′ 2  τ33 = 6s0(t) s1(t) s2(t) n4(t) s2(t)+ n5(t) s1(t) ,  (3.20)  −1 −1 ′ ′ −1 −1 ′ ′  τ12 = τ21 = 6s0(t) s3(t) n4 (t)n5(t), τ23 = τ32 = 6 s0(t) s1(t) n3(t)n4(t), −1 −1 ′ ′ τ13 = τ31 = −6 s0(t) s2(t) n3(t)n5(t),  − − − − ′  τ α s t 1 s t 1 s t 1 s t 1 n t 2s t s t − s t  α = 0( ) 1( ) 2( ) 3( ) 3( ) 3( ) ( ) 3( )   ′ 2 − ′ 2 n −  + n4(t) s2(t) s(t) s2(t) + n5(t) s1(t) s(t) s1(t) ,    4 4 4 4 4 4 o W (γ) (ǫ) (β) −1 −1 −1 ij = δ(δ) δ(α)δ(ζ) sα(t) sβ(t) sγ (t) T(ǫ).(δ)(ζ)v(α)(γ)(β)ij , (3.21) α=1 β=1 γ=1 δ=1 ǫ=1 ζ=1 X X X X X X for s(t)= s0(t)+ s1(t)+ s2(t)+ s3(t), the above deﬁned tensorv ˆ and τij = 0 in all other cases. The brackets about the indices in the equation (3.21) mean that the Einstein’s Summation Convention should not be applied to them. It holds the next theorem.

Theorem 3.2. The functions s0(t), s1(t), s2(t), s3(t), n3(t), n4(t), n5(t) are the components of a 4 metric tensor (3.15) which corresponds to the Einstein-Hilbert action S = d x |g| R + LM , for LM given by the equation (3.19) if and only if they satisfy all of theR equationsp of motion (3.7).

In the sense of the research in this subsection, the equations of motion (3.7) are diﬀerential equations by the functions n3(t), n4(t), n5(t) with respect to the known functions s0(t), s1(t),

9 s2(t), s3(t). They express the correlation between the energy-momentum tensor with respect to the symmetric and anti-symmetric parts of metrics. ′ ′ ′ Let us consider a functional proportion n3(t) : n4(t) : n5(t) = α3 : α4 : α5. Hence, it exists ′ a non-trivial function n(t) such that nk(t)= αkn(t), k = 3, 4, 5. With respect to these changes, the equation (3.19) transforms to

3 ′ −1 2 2 2 2 LM = − (v + w)g s0(t) α3s3(t)+ α4s2(t)+ α5s1(t) n(t) . (3.19’) 2 If LM 6= 0, the last equation involving n(t) as the unknown has two solutions if and only if ′ 2 2 2 ′ 2 2 2 (v +w) α3s3(t)+α4s2(t)+α5s1(t) 6= 0. In the case of (v +w) α3s3(t)+α4s2(t)+α5s1(t) LM 6= 0 and with respect to the Existence and Uniqueness Theorem, the diﬀerential equation (3.19) has two solutions by the functions n3(t),n4(t),n5(t) . These solutions are

− − 2 1 2 2 2 1 nk0 (t)= αk − s0(t) α s3(t)+ α s2(t)+ α s1(t) gLM dt, nk1 (t)= −nk0 (t). (3.22) 3(v′ + w) 3 4 5 Z s

In other words, the 3-tuples n30 (t),n40 (t),n50 (t) and n31 (t),n41 (t),n51 (t) are the corresponding components of the anti-symmetric part of the metri c tensorg ˆ. After substituting the expressions (3.20, 3.21) into the equations (3.9, 3.10), one gets the −1 corresponding pressures p and p0 and the densities ρ and ρ0 as well. The proportions p · ρ and −1 p0 · ρ0 are the corresponding state parameters.

4 Generalized Einstein’s equations of motion II

I. Shapiro [17] studied the theory of gravity with torsion. He analyzed the four-dimensional space-time cosmological models. Into the second section of the paper [17], I. Shapiro considered ˜i i the non-symmetric affine connection spaces whose affine connection coefficients are Γjk =Γjk + i i ˆ Kjk, for the Christoffel symbols Γjk (eq. 1.4, left) and the tensor K of the type (1, 2) whose i i ˜ˆ ˜ˆ ˆ components satisfy the equality Kjk = −Kkj. The torsion tensor T is T = 2K in the Shaprio’s article [17]. We will analyze a generalization of this concept below. Fourteen years after Shapiro, S. Ivanov and M. Lj. Zlatanovićpublished the paper [6] where they involved the model of the generalized Riemannian space that generalizes the Eisenhart’s one. We considered above the Einstein’s equations of motion covered by the generalized Rieman- nian space with respect to Eisenhart’s definition [5]. In this section, we will derive the equations of motion with respect to the generalized Riemannian space GR˜ 4 defined by S. Ivanov and M. Zlatanovićin [6] as the manifold M4 equipped with non-symmetric metric tensorg ˆ. The covariant affine connection coefficients Γ˜ijk for the space GR˜ 4 are [6]

˜ 1 ˜ ˜ ˜ 1 Γijk =Γijk + Tjki + Tijk − Tkij − gki|j + gij |k − gkj|i , (4.1) 2h i 2 1 1 1 for the Christoﬀel symbol of the ﬁrst kind Γijk obtained with respect to the symmetric metric i i ˜i α ˜α i tensorg ˆ and the covariant derivative aj |k = aj,k + Γαkaj − Γjkaα. 1

10 With respect to the equation (4.1), we obtain

˜ 1 ˜ ˜ 1 ˜ 1 ˜ Γijk =Γijk − Tjik + Tkij − gki|j + gij |k − gkj |i and Γijk = T ijk. (4.2) 2h i 2 1 1 1 ∨ 2 After rising the index i in the last equation, we get

˜i ˜i ˜i ˜α ˜i ˜α ˜i Rjmn = Γjm,n − Γjn,m + ΓjmΓαn − ΓjnΓαm. (4.5)

These components and the components (1.6) of the curvature tensor Rˆ for the space R4 satisfy the equation

˜i i 1 i 1 i 1 α i i α α i i α Rjmn = Rjmn − ηjm,n + ηjn,m − ηjmΓαn + ηαnΓjm − ηjnΓαm − ηαmΓjn 2 2 2 (4.6) 1 α i α i + ηjmηαn − ηjnηαm , 4 for

i iα ˜ ˜ ηjk = g Tjαk + Tkαj + gkα|j + gαj |k − gkj |α . (4.7) 1 1 1 With respect to the equations (1.5, 4.6), we proved the next proposition.

Proposition 4.1. The components of the curvature tensors R˜ˆ and Rˆ respectively given by the equations (4.5) and (1.6) satisfy the equation

˜i i 1 i 1 i 1 α i α i Rjmn = Rjmn − ηjm|n + ηjn|m + ηjmηαn − ηjnηαm , (4.8) 2 2 4 i for the components ηjk of the tensor ηˆ of the type (1, 2), deﬁned by the equation (4.7).

The space GR˜ 4 is a special aﬃne connection space. For this reason, the components of the curvature tensors K˜ˆ of this space are elements of the family [13,14]

K˜ i = R˜i + uT˜i + u′T˜i + vT˜α T˜i + v′T˜α T˜i + wT˜α T˜i . jmn jmn jm˜|n jn˜|m jm αn jn αm mn αj (4.9)

11 ˜i i 1 i After applying the equations (4.4, 4.8) and the equality Γjk =Γjk − 2 ηjk as well, one proves the next proposition. Proposition 4.2. The family of components of the curvature tensors K˜ˆ for the generalized Riemannian space GR˜ 4 is

1 1 1 K˜ = R − gβγ ηα + gβγ ηα + gγδ ηα ηβ − ηα ηβ − (v′ + w)gγδT˜α T˜β 2 βγ|α 2 αβ|γ 4 γδ αβ βγ αδ γβ δα ′ (4.12) ′ βγ ˜α u αβ δǫ ˜ ˜ ˜γ + u g Tβα|γ + g g T δαǫ + T ǫαδ + gǫα|δ + gαδ |ǫ − gǫδ|α Tβγ . 2 1 1 1 In the equations (4.10, 4.11, 4.12), u, u′, v, v′, w are the corresponding coeﬃcients. As we may see, the part of the scalar curvature K˜ which corresponds to the matter is

12 ′ ′ ′ ′ Let be L˜M = L˜[u , v , w], for the coefficients u , v , w. The meaning of the square brackets in the last equality is that the field L˜ depends of the linear combination of necessary terms with respect to the coefficients u′, v′, w. In this case, the full lagrangian with torsion is L = R − 2Λ + L˜[u′, v′, w] |g|. The correspondingp Einstein-Hilbert action with torsion is

S˜ = d4x |g| R − 2Λ + L˜[u′, v′, w] . (4.14) Z p We need to consider the variations of the functionals

S˜ = d4x |g| R − 2Λ and S˜ = d4x |g|L˜[u′, v′, w]. 1 Z 2 Z p p The variation of the ﬁrst of these functionals is given by the equation (3.5). The variation of the second functional is

δL˜[u′, v, w] 1 δS˜ = d4x |g| − g L˜[u′, v′, w] δgαβ . (4.15) 2 Z αβ αβ p δg 2 With respect to the Quotient Rule, the variational derivatives δL˜[u′, v′, w]/δgij are the components of the tensor V˜ˆ of the type (0, 2), i.e.

δL˜[u′, v′, w] = V˜ij. (4.16) δgij If sum the equations (3.5) and (4.15), we will obtain

1 1 δS˜ = d4x |g| R − Rg + Λg + V˜ − g L˜[u′, v′, w] δgαβ . (4.17) Z αβ αβ αβ αβ αβ p 2 2 The right side of the last equation vanishes if and only if 1 1 R − Rg + Λg = −V˜ + g L˜[u′, v′, w], (4.18) ij 2 ij ij ij 2 ij which are the corresponding Einstein’s equations of motion. It holds the following theorem. Theorem 4.1. With respect to the family of the Einstein’s equations of motion (4.18), the family of energy-momentum tensors and the family of their traces are

1 T˜ = −V˜ + g L˜[u′, v′, w] and T˜α[u′, v′, w]= −V˜α + 2L˜[u′, v′, w]. (4.19) ij ij 2 ij α α The pressure, energy-density and state-parameter may be obtained by the substituting the equation (4.19) into the equations (2.7, 2.8). The pEQM and ρEQM are

1 1 1 1 1 R uαuβ + R − Λ= V˜α − V˜ uαuβ − L˜[u′, v′, w], (4.20) 3 αβ 6 3 α 3 αβ 2 1 1 R uαuβ − R +Λ= −V˜ uαuβ + L˜[u′, v′, w], (4.21) αβ 2 αβ 2

13 in the reference system ui and

1 1 1 1 1 R + R − Λ= V˜α − V˜ − L˜[u′, v′, w], (4.22) 3 00 6 3 α 3 00 2 1 1 R − R +Λ= −V˜ + g L˜[u′, v′, w], (4.23) 00 2 00 2 00 in the comoving reference system.

′ ′ The part L˜[u , v , w] which corresponds to the matter with respect to the space GR˜ 4 and the corresponding part LM which corresponds to the matter with respect to the space GR4 satisfy the equality ′ ′ ′ ′ L˜[u , v , w]= LM + L˜[u , v , w] −LM . Indirectly, we will ﬁnd the diﬀerence between the energy-momentum tensors, pressures, energy-densities and state parameters obtained with respect to the families (3.4) and (4.13) in the following section.

5 Linearity

The question which arises is how much would we change the energy-momentum tensor, the pressure, the energy-density and the state-parameter obtained with respect to the part LM if we get the part lLM + fF for some ﬁeld F and the real or complex scalars l and f. We will answer this question more generally below. Let us consider the ﬁeld

⋄ LM = αLM + . . . + αLM , (5.1) 1 1 s s for some s ∈ N, ﬁelds L, ..., L and real or complex coeﬃcients α,...,α. 1 s 1 s The corresponding Einstein-Hilbert action is

⋄ ⋄ S = d4x |g| R − 2Λ + L . (5.2) Z M p The last equation may be equivalently treated as the equation (4.14) after changing the field ⋄ ′ ′ L˜M [u , v , w] by the field LM . Any of the fields L, r = 1,...,s, generates the corresponding r tensor Vˆ analogous to the tensor Vˆ whose components are given by the equation (4.16). For this r ⋄ˆ ⋄ reason, the components of the tensor V obtained with respect to the field LM are

⋄ Vij = αVij + . . . + αVij. (5.3) 1 1 s s Hence, the Einstein’s equations of motion are

s s 1 1 Rij − Rgij + 2Λgij = − αVij + gij αL. (5.4) 2 r r 2 r r Xr=1 Xr=1

14 It is the set of the corresponding Einstein’s equations of motion. The equations of motion (5.4) may be rewritten as

s 1 1 Rij − Rgij + 2Λgij = − α Vij − gijL . (5.5) 2 r r 2 r Xr=1 Based on the equations ( 2.7, 2.8, 5.5), we obtain the following equalities

i i i In reference system u In comoving reference system u = δ0 ⋄ s 1 T ij = − α Vij − gijL r=1 r r 2 r ⋄ P T α = − s α Vα − 2L α r=1 r r α r P ⋄ 1 s α β α ⋄ 1 s α p = − α Vαβu u −V −L p = − α V00 −V −L 3 r=1 r r r α r 3 r=1 r r r α r ⋄ Ps α β 1 ⋄ Ps 1 ρ = − α Vαβu u − L ρ0 = − α V00 − L r=1 r r 2 r r=1 r r 2 r ⋄ ⋄ ⋄P−1 ⋄ ⋄ P⋄−1 ω = pρ ω0 = p0ρ0 Table 2: Linear combinations of energy-momentums, pressures and energy-densities

With respect to the equalities (5.3) and the expressions in the Table 2, we proved the validity of the following theorem.

Theorem 5.1. The energy-momentum tensor and its trace, the pressure and the energy-density are linear by the summands into the lagrangian which corresponds the matter. Their values are equal to the linear combinations of the corresponding values generated by the separate summands of the lagrangian. The corresponding pEQM and ρEQM of the system are the linear combinations of the pEQMs and ρEQMs of the separate subsystems generated by the summands in the lagrangian which corresponds to the whole system. The state-parameter is not linear as the previous magnitudes.

6 Conclusion

In the section 2, we geometrically interpreted the Madsen’s formulae about energy-momentum tensors, pressures and energy-densities. With respect to these interpretations, we computed these magnitudes see the Table 1 . In the section 3, we expressed the energy-momentum tensor with respect to the curvature tensors of a generalized Riemannian space in the sense of Eisenhart’s definition. We also proved that the part LM generates two generalized Riemannian spaces in the sense of Eisenhart’s definition. In this section, it is concluded that the anti-symmetric part of a metric tensor corresponds to a matter. In the section 4, we analyzed the differences and similarities between the results presented in the Shapiro’s paper [17] and the model of the generalized Riemannian space involved by S. Ivanov and M. Lj. Zlatanović[6]. We explicitly obtained the corresponding energy-momentum tensor but the pressure, energy-density and state parameter may be obtained with respect to the corresponding formulae from the section 2.

15 In the Sections 3 and 4, we obtained the systems of equations for the equilibriums between symmetric aﬃne connections and torsions the systems pEQM and ρEQM . In the section 5, we analyzed the linearity of the energy-momentum tensors, energy-densities, pressures and state parameters under summing of matter ﬁelds see the Table 2 . We also analyzed the linearity of pEQM and ρEQM in this section.

Acknowledgements

This paper is ﬁnancially supported by Serbian Ministry of Education, Science and Techno- logical Development, Grant No. 174012.

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