An investigation of equivalence between the bulk-based and the brane-based approaches for anisotropic models

We investigate the relation between the brane-based and the bulk-based approaches for anisotropic case in brane-world models. In the brane-based approach, the brane is chosen to be fixed on a coordinate system, whereas in the bulk-based approach it is no longer static as it moves along the extra dimension. It was shown that these two approaches are equivalent for specific models in Mukohyama et al. (Phys Rev D 62:024028, 2000), Bowcock et al. (Class Quant Gravit 17:4745–4764, 2000). In this paper, it is aimed to get general formalism of the equivalence obtained in Mukohyama et al. (Phys Rev D 62:024028, 2000). We found that calculations driven by a general anisotropic bulk-based metric yield a brane-based metric in Gaussian Normal Coordinates by conserving spatial anisotropy. We also derive solutions for an anisotropic bulk-based model and get the corresponding brane-based metric of the model.