Measuring the shape of the extra dimension

We study the possibility of extracting geometric information on the shape of the extra dimension from four-dimensional data such as the mass of the Kaluza-Klein (KK) mode. Assuming one compact extra dimension whose geometry can be considered as perturbations in the flat background, we show that if there is a Z₂ symmetry in the extra dimension, for example the KK parity in models with Universal Extra Dimensions, then the warp factor in the metric is completely determined by the KK mass alone. Without KK parity, additional information depending on the boundary conditions is needed to fully reconstruct the metric, even though such information may be experimentally challenging to obtain. The case in a general background geometry is also considered.