# Planck length The Planck length, $l_p$ is given by $l_p^2 = \frac{\hbar G}{c^3}.$In [[0001 AdS-CFT|holography]], $\frac{l_\text{AdS}^{d-1}}{G_N} \sim \left(\frac{l_\text{AdS}}{l_p}\right)^{d-1} \sim N^2.$ For example, in [[2013#Sarkar, Wall]], they expand the metric perturbation in half powers of $\hbar$, which should be thought of as expansion of some length scale ($l_p$) while keeping G = 1. ## Relation to coupling constant - in gravity, $V=\frac{G_N m^2}{r}$. Coupling $\lambda\equiv \frac{V(r_c)}{mc^2}$ = $m^2/m_P^2$ ($r_c$ is Compton wavelength) - => when $m\sim m_P$, coupling is order 1, i.e. at Planck mass scale, gravitational coupling becomes important. ## Relation to minimal probing length - low energy: higher energy -> probe smaller lengths - very high energy: higher energy -> larger BH forms -> probe larger lengths - Planck length: the minimal length one can probe - **alt.**: use uncertainty relation & that probing distance must be bigger than Schwarzschild radius => $\Delta x>l_P$ ## Classical v.s. quantum - Schwarzschild radius $r_S$ is the shortest scale one can probe an object - Compton length $r_P$ is the scale at which quantum effects are important - find $r_S/r_P=m^2/m_P^2$, so when mass large, expect no quantum effects - but not true for black holes: BH can have *macroscopic* quantum effects ## Refs - some of the discussions above are taken from [[Rsc0010 Hong Liu Lectures on holography]]