Four-Dimensional Planck Scale is Not Universal in Fifth Dimension in Randall-Sundrum Scenario

It has recently been proposed that the hierarchy problem can be solved by considering the warped fifth dimension compactified on $S^{1}/Z_{2}$. Many studies in the context have assumed a particular choice for an integration constant $\sigma_{0}$ that appears when one solves the five-dimensional Einstein equation. Since $\sigma_{0}$ is not determined by the boundary condition of the five-dimensional theory, $\sigma_{0}$ may be regarded as a gauge degree of freedom in a sense. To this time, all indications are that the four-dimensional Planck mass depends on $\sigma_{0}$. In this paper, we carefully investigate the properties of the geometry in the Randall-Sundrum model, and consider in which location $y$ the four-dimensional Planck mass is measured. As a result, we find a $\sigma_{0}$-independent relation between the four-dimensional Planck mass $M_{\rm Pl}$ and five- dimensional fundamental mass scale $M$, and remarkably enough, we can take $M$ to TeV region when we consider models with the Standard Model confined on a distant brane. We also confirm that the physical masses on the distant brane do not depend on $\sigma_{0}$ by considering a bulk scalar field as an illustrative example. The resulting mass scale of the Kaluza-Klein modes is on the order of $M$.