Asymptotically AdS Boundary Data

For a four-dimensional vacuum spacetime with negative cosmological constant, the conformal boundary is three-dimensional and timelike. In Fefferman–Graham gauge the leading boundary metric determines the local $z^2$ coefficient, while a symmetric transverse-traceless $z^3$ coefficient remains undetermined by the radial recursion. These coefficients are not two arbitrary metrics: the radial Einstein equations impose trace and divergence constraints, a defining function fixes a conformal representative, and an evolution problem still requires boundary and corner conditions.

Required background. The Gibbons–Hawking–York Boundary Term supplies the regulated boundary variation, while PDEs and Green Functions for Gravitation supplies asymptotic expansions and timelike-boundary evolution.

Helpful background. The Einstein Equation supplies radial constraints; Hypersurface Normals and Tangential Projectors supplies conformal-boundary geometry; Formulations of General Relativity orients the volume; and Causality and Global Structure explains why timelike infinity needs boundary data.

Four-dimensional asymptotically locally AdS form

Fix bulk dimension four and solve

$$R_{\mu\nu}=-\frac{3}{\ell^2}G_{\mu\nu}, \qquad \Lambda=-\frac{3}{\ell^2}.$$

Near a smooth conformal boundary choose a geodesic defining function $z$ and write

$$G=\frac{\ell^2}{z^2} \left(dz^2+g_{ij}(x,z)dx^idx^j\right),$$ $$g(x,z) =g_{(0)}+z^2g_{(2)}+z^3g_{(3)}+O(z^4).$$

The unphysical metric $\bar G=z^2G/\ell^2$ extends to $z=0$. A different defining function $z'=e^{\omega(x)}z+O(z^2)$ changes $g_{(0)}$ by a Weyl rescaling. Thus the intrinsic datum before choosing $z$ is the conformal class $[g_{(0)}]$, not a preferred metric.

De Haro, Skenderis, and Solodukhin state the general $(d+1)$-dimensional expansion and the radial Einstein equations in de Haro, Skenderis, and Solodukhin 2001, §2, eqs. (2.3)–(2.5), pp. 5–7, PDF. Their radial coordinate is $\rho=z^2$, and their Ricci convention is opposite to the one inherited here. In particular, the positive Schouten expression in their equation (A.1) becomes the negative Schouten expression below. The explicit AdS calculation later on independently fixes the sign and length factors.

Coefficients fixed by the radial equations

For boundary dimension $d=3$, the first determined coefficient is the negative Schouten tensor in the standard Lorentzian Einstein convention:

$$g_{(2)ij} =-\left( R_{ij}[g_{(0)}] -\frac14R[g_{(0)}]g_{(0)ij} \right).$$

The $z^3$ coefficient obeys

$$g_{(0)}^{ij}g_{(3)ij}=0, \qquad \nabla_{(0)}^i g_{(3)ij}=0.$$

Its transverse-traceless part is not fixed locally by $g_{(0)}$. The exact dimension-specific trace and divergence statements are collected in de Haro, Skenderis, and Solodukhin 2001, Appendix A, eqs. (A.1)–(A.4), pp. 21–22, PDF.

Because the boundary dimension is odd, no $z^3\log z$ obstruction term appears in this pure-gravity expansion. Even boundary dimensions generally acquire a logarithmic coefficient, and matter with resonant falloff can introduce logs even when the vacuum metric series would not. The formula above must therefore not be copied unchanged to another dimension or matter content.

What is free and what is selected

Locally in the radial expansion:

• $[g_{(0)}]$ specifies the boundary conformal geometry;
• choosing $z$ selects a representative $g_{(0)}$;
• $g_{(2)}$ is a local curvature functional of that representative;
• $g_{(3)}$ contains undetermined transverse-traceless data;
• higher coefficients are recursively determined by these data where the formal expansion remains regular.

This is asymptotic constraint accounting, not a standalone Lorentzian existence theorem. A hyperbolic evolution with timelike conformal boundary must still decide which combination of incoming and outgoing gravitational data is allowed. Fixing $g_{(0)}$ as an external source is a common Dirichlet choice and removes renormalized symplectic flux for variations that keep that source fixed. When the source is also time independent, the corresponding energy can be conserved and the usual interpretation is reflective. A prescribed time-dependent boundary metric can instead drive the bulk; fixing it under field-space variation does not make it stationary. Mixed or dissipative conditions require a separate variational and energy analysis; they are not universally admissible merely because the formal radial series can be written.

The finite artificial boundary studied by Friedrich and Nagy is inserted by hand, whereas an AdS conformal boundary is selected by the asymptotic geometry. They emphasize this distinction in Friedrich and Nagy 1999, §1, pp. 619–621. It improves covariance of the boundary description but does not eliminate the need for a boundary condition, because the conformal boundary is timelike.

Independent pure-AdS check

Global AdS$_4$ has boundary representative

$$g_{(0)}=-dt^2+\ell^2d\Omega_2^2.$$

Its Ricci tensor has $R_{tt}=0$, $R_{AB}=\ell^{-2}g_{(0)AB}$, and $R=2/\ell^2$. Hence

$$g_{(2)tt}=-\frac{1}{2\ell^2}, \qquad g_{(2)AB}=-\frac{1}{2\ell^2}g_{(0)AB}.$$

To verify these coefficients from the metric itself, start with global AdS and set

$$r=\frac{\ell^2}{z}-\frac z4, \qquad 0\lt z\lt2\ell.$$

Then the exact Fefferman–Graham tangential metric is

$$g(x,z) =-\left(1+\frac{z^2}{4\ell^2}\right)^2dt^2 +\ell^2\left(1-\frac{z^2}{4\ell^2}\right)^2d\Omega_2^2.$$

Expanding reproduces both $g_{(2)}$ components and gives $g_{(3)}=0$ directly. Time-reflection symmetry alone would not imply that result: a static asymptotically AdS black hole can have nonzero $g_{(3)}$. A proposed nonzero $g_{(3)}$ proportional to $g_{(0)}$ would fail immediately because its trace is nonzero.

Boundary and corner limitations

The boundary metric and $g_{(3)}$ must be compatible with the initial data at the corner where a Cauchy slice reaches conformal infinity. Smoothness fixes a hierarchy of time derivatives there. Global existence, stability, and the formation of horizons or singularities are not decided by the local expansion.

This page also avoids a holographic interpretation. The subleading coefficient can be related to a renormalized boundary stress tensor after counterterms and normalizations are fixed, but that dictionary is not needed to state the gravitational boundary data and radial constraints.

Common pitfalls

The leading and subleading coefficients are not arbitrary independent tensors. Radial equations fix $g_{(2)}$ and constrain $g_{(3)}$.

Fefferman–Graham gauge is not a global coordinate theorem. Geodesic defining functions can fail or charts may need patching away from the boundary.

A formal expansion is not well-posed evolution. Incoming modes, flux, and corner compatibility still require a named Lorentzian IBVP.

Exercise: count the undetermined components

How many local components remain in a symmetric $g_{(3)ij}$ on a three-dimensional boundary after its trace and divergence constraints, before quotienting any further gauge relations?

Solution

A symmetric tensor in three dimensions has six components. One trace equation and three divergence equations remove four local functions in the generic symbol count, leaving two. This matches the two local polarizations of the four-dimensional gravitational field. The count is local: global topology, boundary Killing fields, and regularity can change the solution space of the differential divergence constraint.

References

• de Haro, Sebastian, Kostas Skenderis, and Sergey N. Solodukhin. “Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.” Communications in Mathematical Physics 217 (2001): 595–622. DOI; Open PDF.
• Friedrich, Helmut, and Gabriel Nagy. “The Initial Boundary Value Problem for Einstein’s Vacuum Field Equation.” Communications in Mathematical Physics 201 (1999): 619–655. DOI.