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Lagrangian Densities, Gravitational Field Equations and Conservation

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Lagrangian Densities, Gravitational Field Equations and Conservation Fundamental Properties of Lagrangian Density for A Gravitational System and Their Derivations Fang-Pei Chen Department of Physics, Dalian University of Technology, Dalian 116024, China. E-mail: [email protected] Abstract Through the discussion of the fundamental properties of Lagrangian density for a gravitational system, the theoretical foundations of the modified Einstein’s field equations µν 1 µν µν µν µν − g − λ g − = −8πG and the Lorentz and Levi-Civita’s R 2 R D T (M ) ∂ conservation laws ( − g T µν + − g T µν ) = 0 are systematically studied. Our ∂ xµ (M ) (G) study confirms the view that they could be used as the premises to establish a new cosmology. Keywords: New theory of cosmology ; Lagrangian density ; energy-momentum tensor density ; gravitational field equations; conservation laws ; dark energy; dark matter 1. Introduction In the reference [1] a new theory of cosmology has been established which leads to the following distinct properties of cosmos: the energy of matter field might originate from the gravitational field; the big bang might not have occurred; the fields of the dark energy and some parts of the dark matter would not be matter fields but might be gravitational fields. These distinct properties are deduced from the following premises: 1), the Lagrangian density of pure gravitational field is supposed to be − g −gL (x) = [ R(x) + 2λ + 2D(t)] ， from which the modified Einstein’s field equations G 16πG 1 µν µν µν − g − λ g − µν = −8πG µν (1) R 2 R D T (M ) δ ( − g ) def LG Can be derived; 2), we follow Lorentz and Levi-Civita to use − g µν 2 as the T (G) = δ gµν energy-momentum tensor density for the gravitational field, therefore the conservation laws of energy-momentum tensor density for a gravitational system including matter fields and gravitational fields are the Lorentz and Levi-Civita’s conservation laws: ∂ ( − g T µν + − g T µν ) = 0 (2) ∂ xµ (M ) (G) 1 µν µν and + = (3) T(M) T(G) 0 but not the Einstein’s conservation laws : ∂ µν µν = 0 . ( −gT (M )+ −g t (G)) ∂ xµ Some people doubt that Ref.[1] contains fundamental mistakes and do not believe the above conclusions of Ref.[1]. The aim of this paper is to clear up this doubt. For this purpose we shall study firstly the fundamental properties of Lagrangian density for a gravitational system in the more general case and then explain the rationality for the Lagrangian density of a pure gravitational field − g −gL (x) = [ R(x) + 2λ + 2D(t)] and the modified Einstein’s field equations G 16πG 1 µν µν µν − g − λ g − µν = −8πG µν R 2 R D T (M ) At the same time we shall expound the correctness of the Lorentz and Levi-Civita’s conservation laws and compare the virtues and the defects between the new theory of cosmology and the prevalent big bang cosmology. These studies might help clearing up the doubt about the premises of the new theory of cosmology and the conclusions of Ref.[1]. 2. The fundamental properties of Lagrangian density for a gravitational system in the more general case In gravitational theories the action integral I = ∫ − g(x) L (x)d 4 x is always used to study the rules of gravitation [2，3，4]. Where g(x) is the determinant of (x) ; − g(x) (x) is the total Lagrangian gµν L density of a gravitational system, it can be split into two parts: − g(x) (x) = − g(x) (x) + − g(x) (x) ; − g(x) (x) is called gravitational Lagrangian L LM LG LG density which is composed of gravitational fields only, − g(x) (x) is called matter Lagrangian density LM which is composed of both matter fields and gravitational fields. Therefore − (g) (x) describe only pure LG gravitational fields; but besides describing matter fields, − g(x) (x) describe also the interactions LM between gravitational field and matter field. In the General Relativity, the gravitational Lagrangian is denoted by 1 (x) = R(x) (4) LG 16πG 1 or (x) = [ R(x) + 2λ ] (5) LG 16πG 2 the matter Lagrangian (x) is denoted by LM (x) = [ (x); (x); i (x)] (6) LM LM ψ ψ |µ h.µ where (x) is the matter field, (x) is the covariant derivative of (x) : ψ ψ |µ ψ 1 ψ (x) =ψ (x) + .λ (x) (x) ij ψ (x) [5] (7) |µ ,µ 2 hi h jλ,µ σ Since the great majority of the fundamental fields for matter field are spinors, it is necessary to use tetrad field i (x) [5]. The metric field (x) is expressed by (x) = i (x) j (x) and we have h.µ gµν gµν h.µ h.ν ηij ∂ h.µ (x) =η g (x) h j (x) ; h (x) = h (x) ; etc. i ij µν .ν iν ,λ ∂ xλ iν In the new theory of cosmology established at Ref.[1], the matter Lagrangian (x) is also denoted by LM Eq.(6): (x) = [ (x); (x); i (x)] LM LM ψ ψ |µ h.µ but the gravitational Lagrangian is denoted by − g −gL (x) = [ R(x) + 2λ + 2D(x)] (8) G 16πG At here we suppose for the moment that D(x) is a scalar function of {x}, from the cosmological principle it can be proved that D(x) =D(t). Owing to Eq.(7) − g(x) (x) can be denoted by the following functional LM form: − g(x) (x) = − g(x) [ (x); (x); i (x) ; i (x)] (9) LM LM ψ ψ ,λ h.µ h.µ,λ λ λ µν ∂ ∂Γ Since = Γ..µν ..µλ σ λ σ λ , R g ( − +Γ..µν Γ..λσ −Γ..µλΓ..νσ ) ∂xλ ∂xν 1 λσ λ = g ( g + g − g ) Γ..µν 2 σµ,ν σν ,µ µν ,σ and i (x) are the dynamical gravitational fields but λ and D(t) are only nondynamical constant or h.µ 3 function , so − g(x) (x) in Eqs.(4,5,8) all can be denoted by the functional form of the dynamical fields LG i (x) and their derivatives: h.µ − g(x) (x) = − g(x) [ i (x); i (x); i (x) ] (10) LG LG h.µ h.µ,λ h.µ,λσ Eq.(9) and Eq.(10) have summarized the general character of the General Relativity and the gravitational theory used at Ref.[1], consequently these two theories must all have the general properties deduced from Eq.(9) and Eq.(10) which we shall talk about in the following. Symmetry exists universally in physical systems, one fundamental symmetry of a gravitational system is that the action integrals = − g(x) (x) 4 = − g(x) (x) 4 and I M ∫ LM d x I G ∫ LG d x = + = − g(x) (x) + (x) 4 I I M IG ∫ (LM LG )d x satisfy = 0 = 0 and = 0 respectively under the following two simultaneous δI M δI G δI transformations [2,6]: (1), the infinitesimal general coordinate transformation µ xµ → x′µ = xµ +ξ (x) (11) (2), the local Lorentz transformation of tetrad frame (x) → ( ) = (x) − mn(x) j (x) (12) ei e′i x′ ei ε δ mηni e j The symmetry (1) is precisely the symmetry of local space-time translations. The sufficient condition of an action integral I = ∫ − g(x) L (x) d 4 x being δI = 0 under above transformations is [2,7]: µ ( − gL) + − gL ≡ 0 (13) δ 0 (ξ ),µ where represent the variation at a fixed value of . Evidently there are also the relations δ 0 x µ µ ( − g ) + − g ≡ 0 and − g + − g ≡ 0 (14) δ 0 LM ( ξ LM ),µ δ 0 ( LG ) (ξ LG ),µ If there exists only the symmetry (2), Eqs.(13,14) reduce to ( − g ) ≡ 0 , ( − g ) ≡ 0 and δ 0 L δ 0 LM ( − g ) ≡ 0 respectively. δ 0 LG 4 From Eq.(9) and Eq.(10) we have ∂( − g L ) ∂( − g L ) ∂( − g L ) ( − g ) = M + M + M i δ 0 LM δ 0ψ δ 0ψ ,λ δ 0 h ∂ψ ∂ψ ∂ i .µ ,λ h.µ (15) ∂( − g L ) + M i δ 0 h ∂ i .µ,λ h.µ,λ ∂( − g L ) ∂( − g L ) ∂( − g L ) ( − g ) = G i + G i + G i (16) δ 0 LG δ 0 h δ 0 h δ 0 h ∂ i .µ ∂ i .µ,λ ∂ i .µ,λσ h.µ h.µ,λ h.µ,λσ As (x) is spinor and i (x) is both tetrad Lorentz vector and coordinate vector, under the ψ h.µ infinitesimal general coordinate transformation and the local Lorentz transformation of tetrad frame, it is not difficult to derive the following induced variations [3]: 1 α (x) = mn (x) (x) − (x) (x) (17) δ 0ψ 2 ε σ mn ψ ξ ψ ,α 1 1 α (x) = mn (x) (x) − mn (x) (x) − (x) (x) δ 0ψ ,λ ε σ mn ψ ,λ ε ,λ σ mn ψ ξ ψ ,αλ 2 2 (18) α − (x) (x) ξ ,λ ψ ,α α α i (x) = mn(x) i j (x) − (x) i (x) − (x) i (x) (19) δ 0 h.µ ε δ mηnj h.µ ξ ,µ h.α ξ h.µ,α α i (x) = mn (x) i j (x) + mn (x) i j (x) − (x) i (x) δ 0 h.µ,λ ε δ mηnj h.µ,λ ε ,λ δ mηnj h.µ ξ ,µ h.α,λ (20) α α α − (x) i (x) − (x) i (x) − (x) i ξ ,µλ h.α ξ h.µ,αλ ξ ,λ h.µ,α 5 i (x) = mn (x) i j (x) + mn i j (x) δ 0 h.µ,λσ ε δ mηnj h.µ,λσ ε ,σ δ mηnj h.µ,λ α + mn i j (x) + mn i j (x) − (x) i (x) ε ,λ δ mηnj h.µ,σ ε ,λσ δ mηnj h.µ ξ ,µ h.α,λσ (21) α α α α − (x) i (x) − (x) i (x) − (x) i (x) − (x) i (x) ξ ,µσ h.α,λ ξ ,µλ h.α,σ ξ ,µλσ h.α ξ h.µ,αλσ α α α − (x) i (x) − (x) i (x) − (x) i (x) ξ ,σ h.µ,αλ ξ ,λ h.µ,ασ ξ ,λσ h.µ,α Putting Eqs.(17-21) into Eq.(15) and Eq.(16); using µ − g + − g ≡ 0 , where or or + ; δ 0 ( Λ ) (ξ Λ ),µ Λ=LM Λ=LG Λ=LM LG owing to the independent arbitrariness of mn(x) , mn (x) , mn (x) , α (x) , α (x) , ε , ε ,λ ε ,λσ ξ ξ ,µ α (x) and α (x) , we obtain the following identities: ξ ,µλ ξ ,µλσ ∂( − g ) ∂( − g ) 1 Λ 1 Λ ∂ ( − g ) + + Λ σ ψ σ ψ hnµ 2 mn 2 ∂ψ mn ,λ ∂ m ∂ψ ,λ h.µ (22) ∂ ( − g Λ ) + = 0 hnµ,λ ∂ m h.µ,λ ∂( − g ) ∂ − g ∂ − g 1 Λ ( Λ ) ( Λ ) ψ + + 2 = 0 (23) 2 σ mn m hnµ Γ.nµ,σ ∂ ∂ h ∂ m ψ ,λ .µ,λ h.µ,λσ ∂ ( − g Λ ) ∂ ( − g Λ ) = (24) hnµ hmµ ∂ m ∂ n h.µ,λσ h.µ,λσ 6 ∂( − g Λ ) ∂( − g Λ ) ∂ ( − g Λ ) + + i ψ ,α ψ ,λα i h.µ,α ∂ ∂ h ∂ψ ψ ,λ .µ (25) ∂ ( − g Λ ) ∂ ( − g Λ ) + hi + hi − ( − g Λ ) = 0 ∂ i .µ,λα ∂ i .µ,λσα ,α h.µ,λ h.µ,λσ ∂( − g Λ ) ∂ ( − g Λ ) ∂ ( − g Λ ) ∂ ( − g Λ ) + i + i + i ψ ,α i hα i h.µ,α i h.α,µ ∂ ∂ h ∂ h ∂ h ψ ,λ .λ .µ,λ .λ,µ (26) ∂ ( − g Λ ) ∂ ( − g Λ ) + i + 2 i − − g λ = 0 h.α,µσ h.µ,σα Λ δ ∂ i ∂ i α h.λ,µσ h.µ,λσ ∂( − g Λ) i ∂( − g Λ) i ∂( − g Λ) i ∂ ∂( − g Λ) i h + h + h − ( ) h ∂ i .α ∂ i .α,σ ∂ i .σ ,α ∂ σ ∂ i .α h.µ,λ h.µ,λσ h.σ ,λµ x h.µ,λσ (27) ∂ ∂( − g Λ) i = − ( h ) ∂ σ ∂ i .α x h.µ,λσ ∂( − g Λ ) ∂( − g Λ ) ∂( − g Λ ) hi + hi + hi = 0 (28) ∂ i .α ∂ i .α ∂ i .α h.µ,λσ h.λσ ,µ h.σ ,µλ From Eq.(28) the another identity: 3 ∂( − g ) ∂ Λ i ( h ) = 0 （29） µ λ σ ∂ i .α ∂ x ∂ x ∂ x h.µ,λσ can be deduced.
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