Locally Localized Gravity: The Inside Story

We derive the exact gravitational field of a relativistic particle localized on an $AdS$ 3-brane, with curvature radius $\ell$, in $AdS_5$ bulk with radius $L$. The solution is a gravitational shock wave. We use it to explore the dynamics of locally localized tensor gravitons over a wide range of scales. At distances below $L$ the shock wave looks exactly like the $5D$ $GR$ solution. Beyond $L$ the solution approximates very closely the shock wave in 4D $AdS$ space all the way out to distances $\ell^3/L^2$ along the brane. At distances between $L$ and $\ell$, the effective 4D graviton is a composite built of the ultralight mode and heavier gravitons, whereas between $\ell$ and $\ell^3/L^2$ it is just the ultralight mode. Finally beyond $\ell^3/L^2$ the shock reveals a glimpse of the fifth dimension, since the ultralight mode wave function decays to zero at the rate inherited from the full $5D$ geometry. We obtain the precise bulk-side formula for the 4D Planck mass, defined as the coupling of the ultralight mode, in terms of the $5D$ Planck mass and the curvature radii. It includes higher-order corrections in $L/\ell$, and reduces to the RS2 formula in the limit $\ell \to \infty$. We discuss $AdS/CFT$ interpretation of these results, and argue that the spatial variation of the effective gravitational coupling read from the shock wave amplitude corresponds to RG running driven by quantum effects in the dual $CFT$.