= Gravity localization in a noncompact warped dimension
{title2=$M_4^2=M_5^3\int_{-\infty}^{\infty}e^{-2k|y|}\,dy=M_5^3/k$}
In a one-brane warped background, the proper extra dimension can be infinite while the four-dimensional <graviton> zero mode has finite norm. The <Einstein-Hilbert action> weighs its four-dimensional curvature term by the warp factor integral shown above, finite for $k>0$. Thus the proper length is not the effective gravitational volume. This differs from an unwarped noncompact direction, whose volume integral diverges. It also differs from the two-brane hierarchy mechanism, where finite brane separation redshifts masses on the distant brane. The noncompact construction is established in https://arxiv.org/abs/hep-th/9906064[the original single-brane warped-gravity paper].
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