On a time-dependent extra spatial dimension

In the usual brane-world scenario matter fields are confined to the four-dimensional spacetime, called a 3-brane, embedded in a higher-dimensional space, usually referred to as the bulk spacetime. In this paper we assume that the 3-brane is a de Sitter space; there is only one extra spatial dimension, assumed to be time dependent. By using the form of the brane-world energy-momentum tensor suggested by Shiromizu et al. in the five-dimensional Einstein equations, it is proposed that the cosmological expansion of the 3-brane may provide a possible explanation for the collapse of the extra dimension, as well as for the energy stored in the resulting curled-up dimension. More precisely, whenever the bulk cosmological constant \Lambda is negative, the extra spatial dimension rapidly shrinks during the inflation of the brane. When \Lambda is positive, on the other hand, the extra spatial dimension either completely follows the cosmological expansion of the brane or completely ignores it, thereby shrinking relative to the expanding space. This behavior resembles the all-or-nothing behavior of ordinary systems in an expanding universe, as recently demonstrated by R.H. Price.