# Gravitational constant in string theory

On page 5 of the notes by Veronika Hubeny on The AdS/CFT correspondence, we find the following:

Given that any string theory is a quantum theory which necessarily includes gravity, it naturally invites the examination of its consequences and implications for quantum gravity. Since spacetime plays a central role in general relativity, the obvious question to ask is how does string theory give rise to the curved dynamical spacetime of general relativity, and what happens to the 'stringy geometry' when the classical description breaks down. In the perturbative formulation valid at small string coupling $g_s$, the spacetime coordinates specifying the position of the string appear as scalar fields in the $2$-dimensional sigma model describing the dynamics of the string worldsheet, and the spacetime metric then enters as a coupling constant. But because the strength of gravitational interactions is governed by the string coupling – the $d$-dimensional Newton’s constant is $G_{N} = g^{2}_{s}\ell^{d−2}_{s}$ in units of the string length $\ell_s$ – it is difficult to directly access the most interesting, strongly gravitational, regime.

Consider the final sentence of the above paragraph:

But because the strength of gravitational interactions is governed by the string coupling – the $d$-dimensional Newton’s constant is $G_{N} = g^{2}_{s}\ell^{d−2}_{s}$ in units of the string length $\ell_s$ – it is difficult to directly access the most interesting, strongly gravitational, regime.

1. Why is the $d$-dimensional Newton’s constant given by $G_{N} = g^{2}_{s}\ell^{d−2}_{s}$ in units of the string length $\ell_s$?
2. Why does this mean that it is difficult to directly access the strongly gravitational regime?

1. In natural units $c=\hbar=1$, so velocity is dimensionless, length and time each have mass dimension $-1$ and acceleration has dimension $+1$, so force has dimension $2$. In $d$-dimensional spacetime, there are $d-1$ space dimensions and Gauss's law for gravity implies the force between masses $m,\,M$ is $F_G=G_NmMr^{2-d}$. Setting $G_N=g_s^2\ell_s^{d-2}$ gives $F_G=mMg_s^2\left(\frac{\ell_s}{r}\right)^{d-2}$, which has the same dimension as $mM$ as expected.
2. The value of $g_s$ satisfying this formula for $G_N$ is very small. It is thus straightforward to access the weak-coupling regime, but not so straightforward to access the strong-coupling regime. In natural units $G_N=\ell_\text{Pl}^{d-2}$ with $\ell_{\text{Pl}}$ the Planck length, so $g_s=\left(\frac{\ell_\text{Pl}}{\ell_s}\right)^{d/2-1}\ll 1$ since $\ell_\text{Pl}\ll\ell_s$ and $d>2$. (If $d=2$ Planck units can't be defined, and if $d<2$ we don't have both space and time dimensions.)