# Fefferman-Graham expansion The Fefferman-Graham (FG) expansion is an expansion of the metric in an asymptotically AdS spacetime near the boundary. It states that, for vacuum [[0554 Einstein gravity|Einstein gravity]], the on-shell metric can be put into the form$d s^2=\frac{\ell^2}{z^2}\left(d z^2+\gamma_{i j}(x, z) d x^i d x^j\right),$where $\ell$ is the AdS length, and $z=0$ at the asymptotic boundary. As we can see, there are no $g_{zi}$ components by a choice of gauge. The essence of the FG expansion is to then expand$\gamma_{i j}(x, z)=\gamma_{i j}^{(0)}(x)+z \gamma_{i j}^{(1)}(x)+\cdots$and study the dependence of these coefficients among themselves. This power series expansion will always work for even bulk dimensions; in the odd case, we actually need to allow logarithmic terms. The more general ansatz turns out to look like:$\gamma_{i j}(x, z)=\gamma_{i j}^{(0)}+z^2 \gamma_{i j}^{(2)}+\cdots+z^d \gamma_{i j}^{(d)}+z^d \bar{\gamma}_{i j}^{(d)} \log z^2+\cdots.$Here, $d$ is the total spacetime dimension of the boundary. It might be useful to point out that the log term starts to appear at order $\mathcal{O}(z^d)$ relative to the leading order, which is the same order that contributes to the [[0592 Gravitational energy|mass]] of the spacetime. The most important application of the FG expansion is perhaps to relate dual CFT quantities to these coefficients. It is also useful for analysing the [[0060 Asymptotic symmetry|asymptotic symmetry]] of AdS, including the special case of [[0085 Asymptotic symmetry of AdS3|AdS3]], and the [[0209 Holographic renormalisation|holographic anomaly]]. See [[0001 AdS-CFT|the AdS/CFT dictionary]] for more details. ## Free parameters - odd boundary dimensions ($d$) (no conformal anomaly) - no log terms - free parameters are $\gamma^{(0)}$ and $\gamma^{(d)}$ - even boundary dimensions: - $\gamma_{i j}(x, z)=\gamma_{i j}^{(0)}+z^{2} \gamma_{i j}^{(2)}+\cdots+z^{d} \gamma_{i j}^{(d)}+z^{d} \bar{\gamma}_{i j}^{(d)} \log z^{2}+\cdots$ - $\bar{\gamma}_{i j}^{(d)}$ totally determined by $\gamma^{(0)}$, but $\gamma^{(d)}$ still free ## 3D gravity is special Unlike in higher dimension, the series ==terminates== in 3D. ## Global issues - FG expansion is fixed by the boundary metric and the boundary stress tensor - need to impose regularity in the interior by hand: - this is a global issue that FG does not know about - convergence - [[2022#Serantes, Withers]]: complexified radial coordinate - [[HolzegelShao2022]][](https://arxiv.org/pdf/2207.14217.pdf): if a domain $D$ on the boundary satisfies generalised null convexity condition (GNCC), the bulk metric near the boundary is determined from the conformal metric and the stress tensor ## Refs - reviews - [[Rsc0020 Chapter 19 Conserved charges in AdS by FischettiKellyMarolf]] - [[Rsc0004 CompereFiorucci Ch2 3d gravity]] - [[0006 Higher-derivative gravity|HDG]] extensions - quadratic gravity [[AksteinerKorovin2015]][](https://arxiv.org/pdf/1511.08747.pdf) - modified FG expansion (3D) - [[2016#Grumiller, Riegler]] - Weyl-covariant approach (3D) - [[2024#Arenas-Henriquez, Diaz, Rivera-Betancour]]