Asymptotic Expansions near the AdS Boundary

Near a conformally compact anti-de Sitter boundary, the Einstein equation organizes the metric into a radial expansion. The leading coefficient is a representative of the boundary conformal class; lower subleading coefficients are fixed locally by its curvature; the coefficient at boundary order $d$ contains global state data subject to trace and divergence constraints. In even boundary dimension, a logarithmic term records the conformal obstruction.

Helpful background. Lorentzian Metrics and Causal Cones supplies conformal metrics; Geodesics and Jacobi Fields supplies geodesic defining functions; Causality and Global Structure supplies the timelike boundary; Exact Solutions of Einstein’s Equations supplies exact AdS; Formulations of General Relativity supplies radial constraints; Gravitational Actions and Boundary Terms supplies boundary variations.

Defining functions and normal form

Let the bulk have dimension $d+1$ and solve

$$\widetilde R_{\mu\nu}+\frac d{\ell^2}\widetilde g_{\mu\nu}=0$$

near a timelike conformal boundary. A defining function $z$ satisfies $z=0$ and $dz\neq0$ at the boundary, with $z\gt 0$ in the bulk. Choosing $z$ to be geodesic for the unphysical metric puts the physical metric into Fefferman–Graham normal form,

$$d\widetilde s^2 =\frac{\ell^2}{z^2} \left(dz^2+g_{ij}(z,x)dx^i dx^j\right).$$

Because the normal is spacelike, $g_{ij}(z,x)$ is Lorentzian on constant-$z$ slices. Its leading value

$$g_{(0)ij}(x)=g_{ij}(0,x)$$

is a boundary metric representative. Changing defining function $z\mapsto e^{\sigma(x)}z+\cdots$ induces a Weyl transformation of $g_{(0)}$. Thus the conformal class $[g_{(0)}]$, not a unique metric, is part of the intrinsic boundary structure.

The normal form is local near the boundary. Caustics of the normal geodesics, multiple boundary components, and global topology can obstruct one FG chart from covering the entire bulk.

Recursive Einstein expansion

For smooth data away from exceptional dimensions, expand

$$g(z,x) =g_{(0)}+z^2g_{(2)}+z^4g_{(4)}+\cdots +z^dg_{(d)}+\cdots.$$

Odd powers below order $d$ vanish for pure Einstein gravity in the standard smooth expansion. Substituting into the radial, mixed, and tangential Einstein equations determines $g_{(2)},g_{(4)},\ldots,g_{(d-2)}$ locally from $g_{(0)}$. The first coefficient is the negative Schouten tensor of the boundary metric:

$$g_{(2)ij} =-\frac{1}{d-2} \left(R_{ij}[g_{(0)}] -\frac{R[g_{(0)}]}{2(d-1)}g_{(0)ij}\right), \qquad d\gt 2.$$

For a flat boundary representative this coefficient vanishes. For a curved Einstein-cylinder representative it does not, even for exact AdS. Subleading coefficients therefore do not all represent excitations; many are fixed kinematically by the chosen boundary geometry.

At order $d$, the radial recursion no longer fixes the entire tensor. The constraints determine its divergence and trace—possibly with local anomaly terms—while its transverse-traceless part is free state data. This is the asymptotic imprint of global bulk information. It cannot be computed from $g_{(0)}$ alone without interior conditions.

The standard mathematical expansion and its renormalized interpretation are developed in de Haro, Skenderis, and Solodukhin 2001, §§2–4. Their curvature convention has the opposite Ricci sign to this site’s convention (their footnote 6); translating it produces the minus sign in the Schouten formula above.

Logarithms and dimension dependence

When the boundary dimension $d$ is even, the recursion generally produces

$$g(z,x) =\cdots+z^d \left(g_{(d)}+\log(z^2/z_0^2)\,h_{(d)}\right)+\cdots.$$

Here $z_0$ is a fixed reference scale with the same units as $z$. The logarithmic coefficient $h_{(d)}$ is a local tensor built from $g_{(0)}$ and is related to the conformal obstruction and anomaly. It vanishes for special boundary metrics, including conformally flat metrics in even boundary dimensions greater than two, but cannot be discarded generally. Its presence limits the smoothness of the compactified metric in a pure power expansion. A change of $z_0$ shifts the coefficient called $g_{(d)}$ by a fixed multiple of $h_{(d)}$ without changing the full metric.

For odd $d$, there is no local conformal anomaly and no generic logarithm at this order; $g_{(d)}$ is trace free and divergence free for pure gravity. For even $d$, the trace and divergence of $g_{(d)}$ are specified local functionals. At fixed defining function and logarithmic scale these geometric constraints do not depend on the counterterm scheme; finite counterterms instead shift local pieces of the renormalized response derived from the same metric.

At $d=2$, the displayed Schouten formula is singular and the recursion must be treated separately. Three-dimensional bulk gravity has no local graviton degrees of freedom, yet global identifications and boundary stress data remain nontrivial. Dimension labels are therefore part of every coefficient formula.

Matter fields introduce their own indicial exponents and can source powers or logarithms that mix into the metric. Relevant scalar modes may backreact before order $d$. A vacuum FG expansion cannot simply be reused when matter decay violates its assumptions.

State data and renormalized response

After adding local counterterms, the renormalized boundary stress tensor has the structure

$$\langle T_{ij}\rangle =\frac{d\ell^{d-1}}{16\pi G} \left(g_{(d)ij}+X_{ij}[g_{(0)}]\right),$$

where $X_{ij}$ is a local curvature functional that vanishes in suitable odd-$d$ schemes and carries anomaly terms for even $d$. The normalization assumes the Einstein action and conventional counterterms; higher-curvature gravity changes it.

This formula illustrates the split: the leading boundary metric is source-like boundary data, while the undetermined part of $g_{(d)}$ is response-like global data. In classical gravity the same coefficient determines conserved charges under boundary symmetries. The interpretation as a quantum expectation value belongs to holography and requires the additional duality dictionary; the geometric expansion itself does not prove a dual theory.

The radial constraints imply conservation of the renormalized stress tensor when boundary sources obey the corresponding Ward identity. A conformal anomaly gives a nonzero trace in even $d$. These are boundary consequences of bulk diffeomorphism invariance, with scheme qualifications.

Checks and failure modes

For Poincaré AdS, $g_{(0)}=\eta$ and every displayed correction vanishes: $g_{(d)}=0$ in the vacuum normalization. For a planar AdS black brane, $g_{(0)}$ remains flat but $g_{(d)}$ is nonzero, encoding mass and pressure. This proves that the state coefficient is not locally determined by boundary curvature.

For global AdS, the cylinder boundary has curvature and lower even coefficients are nonzero. Comparing its vacuum energy with the Poincaré normalization can reveal even-dimensional Casimir/anomaly terms; it is not an inconsistency.

An exact check makes this visible without solving a recursion. In global AdS$_4$, let $r$ be the physical areal radius, take dimensionless $t$, and set $r/\ell=z^{-1}-z/4$. Direct substitution gives

$$d\widetilde s^2=\frac{\ell^2}{z^2} \left[dz^2-\left(1+\frac{z^2}{4}\right)^2dt^2 +\left(1-\frac{z^2}{4}\right)^2d\Omega_2^2\right], \qquad 0\lt z\leq2.$$

For $g_{(0)}=-dt^2+d\Omega_2^2$, the boundary Ricci scalar is $2$, with $R_{tt}=0$ and $R_{AB}=q_{AB}$. The $d=3$ Schouten formula gives $g_{(2)tt}=-1/2$ and $g_{(2)AB}=-q_{AB}/2$, exactly matching the expansion above. The free coefficient $g_{(3)}$ vanishes. Nonzero $g_{(2)}$ therefore coexists with the exact AdS vacuum; it records boundary curvature rather than a graviton excitation.

The expansion can fail through insufficient regularity, logarithms outside the assumed index set, matter modes, a degenerate boundary metric, or loss of the FG chart at a radial caustic. It is asymptotic, not generally convergent. Truncating it does not reconstruct the interior or determine which global solution realizes the data.

Exercise

For boundary dimension $d=4$ and flat $g_{(0)}$, suppose a vacuum solution has $g_{(2)}=0$ and constant, transverse, trace-free $g_{(4)}$. Which parts are locally fixed, and what does $g_{(4)}$ represent?

Solution

Flatness makes the boundary Schouten tensor vanish, so the recursion fixes $g_{(2)}=0$. The order-four logarithmic obstruction also vanishes for the flat representative. The constraints on $g_{(4)}$ reduce to

$$\partial^ig_{(4)ij}=0, \qquad \eta^{ij}g_{(4)ij}=0.$$

They do not fix its transverse-traceless value. It is global state data, proportional—after counterterm normalization—to the renormalized boundary stress tensor. Poincaré AdS chooses zero; a black brane supplies a nonzero constant tensor with the same $g_{(0)}$.

Asymptotic Symmetries and Charges in AdS uses the boundary conformal class and $g_{(d)}$ data to compare charge constructions.

References

• de Haro, Sebastián, Kostas Skenderis, and Sergey N. Solodukhin, “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence”, Communications in Mathematical Physics 217, §§2–4, pp. 595–622, 2001.
• Skenderis, Kostas, “Lecture Notes on Holographic Renormalization”, Classical and Quantum Gravity 19, §§2–4, pp. 5849–5876, 2002.