One of the popular ways to regularize a brane is to promote a brane that is a point-like object in two extra-dimensions to a 5D ring . The ring is a codimension one object and it is possible to consider a motion of this ring. However, a problem is that this ring brane is not homogeneous as one of dimensions on the brane is compact and this breaks Birkhoff’s theorem. It has been shown that cosmology obtained by the motion of a brane in a static bulk shows pathological behaviour [351, 326]. This indicates that we need to find fully time-dependent bulk solutions.
At the moment, the only accessible way is to solve the 6D bulk spacetime using the gradient expansion methods assuming physical scales on a brane are much lower than mass scales in the bulk [150, 16, 233]. It is still an open question what are high energy effects in 6D models.
As in 5D models, there are many generalizations such as the inclusion of the induced gravity term on a brane and the Gauss–Bonnet term in the bulk [43, 222, 101, 84, 349, 239, 83, 82].
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