On the extension of the concept of thin shells to the Einstein-Cartan theory

Abstract

This paper develops a theory of thin shells within the context of the Einstein-Cartan theory by extending the known formalism of general relativity. In order to perform such an extension, we require the general non symmetric stress-energy tensor to be conserved leading, as Cartan pointed out himself, to a strong constraint relating curvature and torsion of spacetime. When we restrict ourselves to the class of space-times satisfying this constraint, we are able to properly describe thin shells and derive the general expression of surface stress-energy tensor both in its four-dimensional and in its three-dimensional intrinsic form. We finally derive a general family of static solutions of the Einstein-Cartan theory exhibiting a natural family of null hypersurfaces and use it to apply our formalism to the construction of a null shell of matter. *

On the junction conditions in $f(R)$ -gravity with torsion

“…In this paper we have derived the junction conditions of two generic spacetimes in the context of f (R)-gravity with torsion. In particular, we have shown that these conditions can be recast into a form that resemble very closely that holding in the ECSK theory [23,28], via the definition of a suitable "effective extrinsic curvature" tensor. Using such a tensor one can both deduce the conditions of junction and determine the stress energy tensor S kj of the shell corresponding to the violation of such conditions.…”

On the stress–energy tensor of a null shell in Einstein–Cartan gravity

“…The requirement (32) on the spin tensor was also obtained in [4] by imposing the condition that the surface stress-energy tensor must be tangential to Σ, while in the present work it is unconditionally obtained. The intrinsic forms of the constraints (30) are [S abc n b ] = [S abc n c ] = 0,…”

“…Since γ cd − ζ cd has 9 independent components, it follows thatγ cd −ζ cd has two independent components contributing to the Weyl tensor on the shell and can be interpreted as representing the two degrees of freedom of polarization of an impulsive gravitational wave traveling along the shell [4].…”

“…Hence, dynamics of the spacetime is determined by torsion being triggered by the intrinsic angular momentum of the matter, and the curvature being stemmed from the mere presence of matter [1,2,3]. The first generalization of the junction conditions for a null hypersurface within the EC gravity was made by Bressange [4](see also [5]). He extended the unified description of shells of any type, including the null case, provided by Barrabes-Israel [6] in the presence of a non-symmetric connection.…”

Some Mathematical Aspects of f(R)-Gravity with Torsion: Cauchy Problem and Junction Conditions

“…On the contrary, at least in the authors' knowledge, very few works deal with junction conditions in ECSK theory: an analysis has been performed by Arkuszewski et al [35], by means of the formalism of tensor-valued differential forms [36][37][38], while the same topic has been indirectly addressed by Bressange [39] following the same approach as in [34]. Concerning f (R)-gravity in purely metric formulation, a discussion of junction conditions has been proposed by Deruelle et al [40] and Senovilla [41].…”