# A Brane World in an Arbitrary Number of Dimensions without Z 2 Symmetry - General Relativity and Quantum Cosmology

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Abstract: We consider a brane world in an arbitrary number of dimensions without Z 2symmetry and derive the effective Einstein equation on the brane, where itsright-hand side is given by the matter on the brane and the curvature in thebulk. This is achieved by first deriving the junction conditions for a non-Z 2symmetric brane and second solving the Gauss equation, which relates the meanextrinsic curvature of the brane to the curvature in the bulk, with respect tothe mean extrinsic curvature. The latter corresponds to formulating an explicitjunction condition on the mean of the extrinsic curvature, analogue to theIsrael junction condition for the jump of the extrinsic curvature. We find thatthere appears a new type of an effective anisotropic fluid on the right-handside of the effective Einstein equation due to the fact that there is no Z 2symmetry. The derived equation is a basic equation for the study ofKaluza-Klein brane worlds in which some dimensions on the brane arecompactified or for a regularization scheme for a higher codimension braneworld, where the Kaluza-Klein compactification on the brane is regarded as ameans to regularize the uncontrollable spacetime singularity caused by thehigher codimension brane.

Author: ** Daisuke Yamauchi, Misao Sasaki**

Source: https://arxiv.org/