Gravity localization in a noncompact warped dimension ($M_4^2=M_5^3{\int }_{-\infty }^{\infty }{e}^{-2k|y|}dy=M_5^3/k$)

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 the original single-brane warped-gravity paper.

An inertial observer of a fixed-$y$ flat brane in metric $d{y}^2+e^{-2y/L}{\eta }_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$ is accelerated relative to the higher-dimensional bulk. Normalization and ${\Gamma }_{\mu \nu }^y=e^{-2y/L}{\eta }_{\mu \nu }/L$ give $a^y=-1/L$. This is its invariant bulk proper acceleration, not a nonzero intrinsic four-dimensional acceleration on the flat induced brane metric.