On Torsion-free Vacuum Solutions of the Model of de Sitter Gauge Theory of Gravity
aa r X i v : . [ g r- q c ] A p r On Torsion-free Vacuum Solutions of the Model of de SitterGauge Theory of Gravity
Chao-Guang Huang a,e ∗ , Yu Tian b,e † , Xiaoning Wu c ‡ Han-Ying Guo d,e § a Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100049, China b Department of Physics, Beijing Institute of Technology, Beijing 100081, China c Institute of Mathematics, Academy of Mathematics and System Science,Chinese Academy of Sciences, Beijing 100080, China d Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing 100080, China and e Kavli Institute for Theoretical Physics China at the Chinese Academy of Sciences, Beijing, 100080, China (Dated: November 2007)It is shown that all vacuum solutions of Einstein field equation with a positivecosmological constant are the solutions of a model of dS gauge theory of gravity.Therefore, the model is expected to pass the observational tests on the scale ofsolar system and explain the indirect evidence of gravitational wave from the binarypulsars PSR1913+16.
PACS numbers: 04.50.+h, 04.20.Jb
The astronomical observations show that our universe is probably an asymptotically deSitter(dS) one [1, 2]. It arises the interests on dS gauge theories of gravity. There is a modelof the dS gauge theory of gravity, which was first proposed in the 1970’s [3, 4]. The modelcan be stimulated from dS invariant special relativity [5–7] and the principle of localization[8], just like that the Poincar´e gauge theory of gravity may be stimulated from the Einsteinspecial relativity and the localization of Poincar´e symmetry [9]. The principle of localizationis that the full symmetry of the special relativity as well as the laws of dynamics are bothlocalized. The gravitational action of the model takes the Yang-Mills form of [3, 4, 8] S GYM = 14 g Z M d xe Tr dS ( F µν F µν ) , (1)where e = det( e aµ ) is the determinant of the tetrad e aµ , g is a dimensionless coupling constantintroduced as usual in the gauge theory to describe the self-interaction of the gauge field, F µν = (cid:0) F ABµν (cid:1) = (cid:18) F abµν + R − e abµν R − T aµν − R − T bµν (cid:19) (2)is the curvature of dS connection B µ = (cid:0) B ABµ (cid:1) = (cid:18) B abµ R − e aµ − R − e bµ (cid:19) ∈ so (1 , , (3) ∗ Email: [email protected] † Email: [email protected] ‡ Email: [email protected] § Email: [email protected] The same dS-connection with different dynamics has also been explored in Ref. [10]. and Tr dS is the trace for the so (1 ,
4) indices
A, B . In Eq.(2), F abµν and T aµν are the curvatureand torsion tensors of the Lorentz connection B abµ ∈ so (1 , R is the dS radius,and e µνab = e µa e νb − e νa e µb . In terms of F abµν and T aµν , the gravitational action can be rewrittenas S GYM = − Z M d xe (cid:20) g F abµν F µνab − χ ( F − − χ T aµν T µνa (cid:21) , (4)where F = F abµν e µνab the scalar curvature, the same as the action in the Einstein-Cartantheory, χ = 1 / (16 πG ) is a dimensional coupling constant, Λ = 3 /R = 3 χg is the cosmo-logical constant.The gravitational field equations, obtained by the variation of the total action S T = S GYM + S M (5)with respect to e aµ , B abµ , are T µνa || ν − F µa + 12 F e µa − Λ e µa = 8 πG ( T µ M a + T µ G a ) , (6) F µνab || ν = R − (16 πGS µ M ab + S µ G ab ) . (7)Here, S M is the action of the matter source with minimum coupling, || represents the covari-ant derivative using both Christoffel symbol { µνκ } and connection B abµ , F µa = − F µνab e bν , T µ M a := − e δS M δe aµ (8) T µ G a := g − T µ F a + 2 χT µ T a (9)are the tetrad form of the stress-energy tensor for matter and gravity, respectively, where T µ F a := − e δδe aµ Z d xe Tr( F νκ F νκ )= e κa Tr( F µλ F κλ ) − e µa Tr( F λσ F λσ ) (10)is the tetrad form of the stress-energy tensor for curvature and T µ T a := − e δδe aµ Z d xeT bνκ T νκb + T µνa || ν = e κa T µλb T bκλ − e µa T λσb T bλσ (11)the tetrad form of the stress-energy tensor for torsion, and S µ M ab = 12 √− g δS M δB abµ (12)and S µ G ab are spin currents for matter and gravity, respectively. Especially, the spin currentfor gravity can be divided into two parts, S µ G ab = S µ F ab + 2 S µ T ab , (13)where S µ F ab := 12 √− g δδB abµ Z d x √− gF = − e µνab || ν = Y µλν e λνab + Y νλν e µλab (14) S µ T ab := 12 √− g δδB abµ Z d x √− gT cνλ T νλc = T µλ [ a e b ] λ (15)are the spin current for curvature F µνab and torsion T µνa , respectively.In Ref.[12], it is shown that all vacuum solutions of Einstein field equation without cosmo-logical constant are the solutions of Eq.(6) and Eq.(7) for the case of sourceless, torsion-free,and vanishing cosmological constant. However, a positive cosmological constant is vitallyimportant for the dS gauge theories of gravity. Without a positive cosmological constant,the gravity should be a Poincar´e or AdS one. Therefore, in order to see whether the modelof dS gauge theory of gravity can pass the observational tests on the scale of solar system,it should be important to explore if the vacuum solutions of Einstein field equation with apositive cosmological constant do satisfy the equations of the model.The purpose of the present Note is to show that it is just the case. Namely, all vacuumsolutions of Einstein field equation with a positive cosmological constant are the solutionsof the torsion-free vacuum equations of the model of dS gauge theory of gravity.For the sourceless case, the torsion-free gravitational field equations of the model reduceto R µa − R e µa + Λ e µa = − πG ( T µ M a + T µ R a ) , (16) R µνab ; ν = 0 , (17)where T µ R a = e νa T µ R ν the tetrad form of the stress-energy tensor of Riemann curvature R µνab ,and a semicolon ; is the covariant derivative using both the Christoffel and Ricci rotationcoefficients. Eq. (16) is the Einstein-like equation, while Eq.(17) is the Yang equation [11].It can be shown [12] that T ν R µ = R abµλ R abνλ − δ νµ ( R abλκ R abλκ )= 12 ( R κσµλ R κσνλ + R ∗ κσµλ R ∗ κσνλ )= 2 C κνλµ R λκ + R R νµ − R δ νµ ) , (18)where R κσµλ is the Riemann curvature tensor, R ∗ κσµλ = R κστρ ǫ τρµλ is the right dual of theRiemann curvature tensor, C λµκν is the Weyl tensor. In the last step in (18), the G´eh´eniau-Debever decomposition for the Riemann curvature, R µνκλ = C µνκλ + E µνκλ + G µνκλ , (19)is used [13], where E µνκλ = 12 ( g µκ S νλ + g νλ S µκ − g µλ S νκ − g νκ S µλ ) , (20) G µνκλ = R
12 ( g µκ g νλ − g µλ g νκ ) , (21) S µν = R µν − R g µν . (22)On the other hand, the vacuum Einstein field equation with a (positive) cosmologicalconstant reads R µν − R δ µν + Λ δ µν = 0 . (23)It results in R = 4Λ , R µν = Λ δ µν , (24)and thus S µν = 0 . (25)Since the Weyl tensor is totally traceless, the stress-energy tenor for Riemann curvaturevanishes, i.e. , T ν R µ = 0 . (26)Therefore, all vacuum solutions of Einstein field equation with a cosmological constant aresolutions of Eq.(16). In addition, the Bianchi identity R µνλσ ; κ + R µνκλ ; σ + R µνσκ ; λ = 0 (27)leads to 0 = R µνλσ ; ν − R µλ ; σ + R µσ ; λ = R µνλσ ; ν . (28)Namely, Yang equation (17) is also satisfied. (The last step of Eq.(28) is valid because ofEq.(24).)Therefore, we come to the conclusion that all vacuum solutions of the Einstein fieldequation with a positive cosmological constant are the torsion-free vacuum solutions of themodel of dS gauge theory of gravity. In particular, the dS, Schwarzschild-dS, and Kerr-de Sitter metrics satisfy the Eqs.(6) and (7). Note that the Birkhoff theorem has beenproved for the gravitational theory (4) without a cosmological constant [14]. So, the modelis expected to pass the observational tests on the scale of solar system. In addition, themodel has the same metric waves as general relativity and thus is expected to explain theindirect evidence of the existence of gravitational wave from the observation data on thebinary pulsar PSR1913+16.One might think that the above results are trivial because the Yang equation does notappear at all if the torsion-free condition is assumed in the action, in which case the tetradand connection are not independent. However, the torsion-free manifold is just the specificsituation of the the model. There is no reason to set the torsion to be zero before thevariation.In fact, it can be shown that all solutions of vacuum Einstein field equation with a positivecosmological constant are also the vacuum, torsion-free solutions of the field equations whenthe terms F µa F aµ , e aν e bµ F µa F νb , e ab λσ e cd µν F µνab F λσcd , e bσ e cµ F µab ν F νσac , e λa e σb T aµλ T bµσ , e σa e µb T aµλ T bλσ are added in the gravitational Lagrangian. Obviously, the last two terms have no con-tribution to the vacuum, torsion-free field equations, while the middle two terms con-tribute the same as the term F µνab F abµν does thus only alter the unimportant coefficients.The first two terms add the term ( R µ [ a e νb ] ) ; ν in Yang equation and the stress-energy tensor R µλ R νλ − δ νµ R σλ R σλ in Einstein equation. Both of them vanish for the solutions of thevacuum Einstein equation with a positive cosmological constant.Obviously, the conclusion is still valid if the integral of the second Chern form of the dSconnection over the manifold is added in the action. Finally, the similar discussions can beapplied to the AdS case as well. Acknowledgments
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