Energy, momentum and angular momentum conservations in de Sitter gravity

Abstract

In de Sitter (dS) gravity, where gravity is a gauge field introduced to realize the local dS invariance of the matter field, two kinds of conservation laws are derived. The first kind is a differential equation for a dS-covariant current, which unites the canonical energy-momentum (EM) and angular momentum (AM) tensors. The second kind presents a dS-invariant current which is conserved in the sense that its torsion-free divergence vanishes. The dS-invariant current unites the total (matter plus gravity) EM and AM currents. It is well known that the AM current contains an inherent part, called the spin current. Here it is shown that the EM tensor also contains an inherent part, which might be observed by its contribution to the deviation of the dust particle's world line from a geodesic. All the results are compared to the ordinary Lorentz gravity. divergence vanishes [7,8]. The Lorentz-invariant currents can be used to define the total EMA currents in Lorentz gravity.Note that in Lorentz gravity, gravity is not a gauge field, in the sense that it is not an Ehresmann connection of some principal fiber bundle. On the other hand, there exists the de Sitter (dS) gravity [9-13], where gravity is described by an Ehresmann connection which is introduced to realize the local dS invariance of the matter field. The dS gravity is well motivated for some cosmological reasons. Firstly, the observed cosmological constant may be related to that of the internal dS space, which is a characteristic structure in dS gravity. Moreover, an interesting investigation shows that [13,14], the dS symmetry together with a Kaluza-Klein-type ansatz can pick out the only one model that is free of the big-bang singularity in the Robertson-Walker universe filled with a spin fluid [15], among the R + βS abc S abc models of gravity [16], where R is the scalar curvature, β is a parameter, and S abc denotes the torsion tensor.In this paper, the EMA conservation laws are generalized to dS gravity. The result consists of two kinds of conservation laws. Firstly, it is shown that the diffeomoriphism and dS symmetries lead to a differential equation for a dS-covariant current, which unites the EM and spin tensors. Secondly, provided the gravitational field equation is satisfied, each one-parameter group of dS rotations result in a dS-invariant current, which is conserved in the sense that its torsion-free divergence vanishes. The dS-invariant current can be used to define the total (matter plus gravity) EMA currents in dS gravity. In the analysis of the first kind conservation law, it is found that the EM tensor contains an inherent part, just like the fact that the angular momentum (AM) current contains an inherent part (the spin current). Also, the dust particle's world line is studied, which deviates from a geodesic for two reasons. The first is the existence of the spin tensor, while the second is the existence of the inherent EM tensor discovered here.The paper is arranged as follows. In section 2, the dS gravity is briefly introduced. In sections 3...

de Sitter-covariant Hamiltonian formalism of Einstein--Cartan gravity

“…Also, the conservation law with respect to the local dS symmetry and diffeomorphism symmetry is discussed in Ref. [18]. With the help of this, we have…”

“…By definition, H[q, p] = −L[q, 0]. With the help of the Noether identity (∂L/∂D µ ξ A )D ν ξ A + 2(∂L/∂F AB µσ )F AB νσ = L δ µ ν with respect to the diffeomorphism invariance [18], it follows that L[q, 0] = Σ d 3 x δL/δΩ AB t • Ω AB t . Moreover, in virtue of C AB = δL/δΩ AB t , we have…”

“…To find out these constraints, it is convenient to use the ADM decomposition [20] of the time direction basis vector: (∂ t ) µ = Nn µ + N µ , where N is named the lapse function, N µ is named the shift vector, which is tangent to Σ, and n µ is normal to Σ, with n µ n µ = −1. Accordingly, D t ξ A can be decomposed as the dS transformation δ A , while putting A A B = −Ω A Bµ v µ yields the dS-invariant diffeomorphism δ v [18]. These transformations can be generated by the first-class constraints (20)- (22) in the following way.…”

“…Besides, the linear realization is able to include those matter fields transforming under the full representations of the P/dS/AdS group [15,16]. Such new matter fields might be responsible for new physics [17,18].…”

“…Firstly, the dS/AdS group provides an elegant unification of the Lorentz transformations and translation, resulting in a 5d covariant theory. For example, the energy-momentum conservation and angular momentum conservation can be united into a 5d covariant conservation in the dS/AdS gravity [18]. Secondly, the dS group is consistent with the asymptotic symmetry of the expanding universe.…”