## A Brane World in an Arbitrary Number of Dimensions without Z_2 Symmetry

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Yamauchi, Daisuke

Sasaki, Misao

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We consider a brane world in an arbitrary number of dimensions without Z_2
symmetry and derive the effective Einstein equation on the brane, where its
right-hand side is given by the matter on the brane and the curvature in the
bulk. This is achieved by first deriving the junction conditions for a non-Z_2
symmetric brane and second solving the Gauss equation, which relates the mean
extrinsic curvature of the brane to the curvature in the bulk, with respect to
the mean extrinsic curvature. The latter corresponds to formulating an explicit
junction condition on the mean of the extrinsic curvature, analogue to the
Israel junction condition for the jump of the extrinsic curvature. We find that
there appears a new type of an effective anisotropic fluid on the right-hand
side of the effective Einstein equation due to the fact that there is no Z_2
symmetry. The derived equation is a basic equation for the study of
Kaluza-Klein brane worlds in which some dimensions on the brane are
compactified or for a regularization scheme for a higher codimension brane
world, where the Kaluza-Klein compactification on the brane is regarded as a
means to regularize the uncontrollable spacetime singularity caused by the
higher codimension brane.

Comment: 12 pages, no figure, PTPtex. Typos corrected and references updated. The title has been slightly modified. Final version to be published in Prog. Theor. Phys

Comment: 12 pages, no figure, PTPtex. Typos corrected and references updated. The title has been slightly modified. Final version to be published in Prog. Theor. Phys

##### Keywords

General Relativity and Quantum Cosmology, Astrophysics, High Energy Physics - Theory