Higher order corrections to the Newtonian potential in the Randall-Sundrum model

The general formalism for calculating the Newtonian potential in fine-tuned or critical Randall-Sundrum braneworlds is outlined. It is based on using the full tensor structure of the graviton propagator. This approach avoids the brane-bending effect arising from calculating the potential for a point source. For a single brane, this gives a clear understanding of the disputed overall factor $4/3$ entering the correction. The result can be written on a compact form which is evaluated to high accuracy for both short and large distances.

Figure 1

The figure shows the relative correction $\Delta$ in (83), where $V(r)=V_0(r)(1+\Delta )$, as obtained by a numerical integration using Mathematica. The leading order terms $\Delta \simeq 4/(3\pi \mu r)$ (short distances) and $\Delta \simeq 2/(3{\mu }^2{r}^2)$ (large distances) are clearly seen from the figure, and the transition between the two regions occurs around $\mu r\sim 1$.