Global, Poincaré, and AdS-Rindler Patches

Global coordinates cover the universal cover of AdS, Poincaré coordinates cover one wedge ending on Minkowski space, and AdS-Rindler coordinates cover the causal wedge of a boundary ball or diamond. The latter two have patch horizons but no curvature singularity in pure AdS. Consequently, a frequency decomposition, accessible algebra, or completeness statement is always tied to its chosen Killing time and domain.

Required background. Anti-de Sitter geometry and its conformal boundary supplies the embedding and compactified cylinder. Helpful background. Causal, Killing, trapping, and apparent horizons distinguishes a coordinate Killing horizon from invariant singular or trapped structure.

Coordinate coverage is part of the observable

Work in Lorentzian AdS$_{d+1}$ with signature $(+,-,\ldots,-)$. Global coordinates use

$$\mathrm ds^2=L^2\left(\cosh^2\rho\,\mathrm d\tau^2-\mathrm d\rho^2 -\sinh^2\rho\,\mathrm d\Omega_{d-1}^2\right),$$

and cover the entire universal cover. The boundary is the cylinder $\mathbb R_\tau\times S^{d-1}$.

Poincaré coordinates are defined invariantly from embedding coordinates by

$$z=\frac{L^2}{X_{-1}+X_d}, \qquad x^a=\frac{L X^a}{X_{-1}+X_d}, \qquad z>0.$$

They cover the region $X_{-1}+X_d>0$ and give

$$\mathrm ds^2=\frac{L^2}{z^2} \left(\eta_{ab}\,\mathrm dx^a\mathrm dx^b-\mathrm dz^2\right), \qquad \eta_{ab}=\operatorname{diag}(+,-,\ldots,-).$$

The $z=0$ boundary representative is Minkowski space. It is conformal to the cylinder with a point or null boundary removed; the Poincaré horizon at $z\to\infty$ is the edge of coordinate coverage, not a divergent curvature invariant.

For a spherical boundary diamond, an AdS-Rindler chart may be written schematically as

$$\mathrm ds^2=L^2\left[(r^2-1)\,\mathrm d\eta^2 -\frac{\mathrm dr^2}{r^2-1}-r^2\,\mathrm dH_{d-1}^2\right], \qquad r>1.$$

The horizon $r=1$ is generated by the Killing field $\partial_\eta$. The conformal boundary is $\mathbb R_\eta\times H^{d-1}$, conformally equivalent to the domain of dependence of a ball Casini, Huerta, and Myers 2011, §§2–3. Thus the wedge naturally organizes observables localized to that boundary diamond; it does not supply the complete global boundary algebra.

First application: restricting a global normal mode

A free scalar of standard dimension $\Delta$ has global normal frequencies

$$\omega_{n\ell}=\Delta+2n+\ell, \qquad n=0,1,2,\ldots,$$

with time dependence $e^{-i\omega_{n\ell}\tau}$. Restrict one normalizable global solution to the Poincaré wedge by substituting the embedding-coordinate map above. Near the boundary, its normalizable coefficient transforms as a primary one-point profile under the cylinder-to-Minkowski Weyl map:

$$\langle\mathcal O(x)\rangle_{\mathrm P} =\Omega(x)^{-\Delta}\langle\mathcal O(\tau,\Omega_{S})\rangle_{\mathrm cyl}.$$

The restricted solution is generally a superposition of modes of Poincaré time $t$, because $\partial_t$ is a different $SO(d,2)$ generator from $\partial_\tau$. Restricting further to an AdS-Rindler wedge gives data only in a boundary diamond. This is why local reconstruction in a wedge has a smaller geometric domain than global mode expansion, as made explicit in the AdS-Rindler reconstruction of Hamilton, Kabat, Lifschytz, and Lowe 2006, §§2–3.

The transformation is exact for the classical solution on the overlap. What changes is the basis used to declare positive frequency and the algebra treated as accessible. Any claim about vacua or thermality additionally needs a state and analytic prescription.

Adversarial check: horizons and completeness

Pure AdS has constant curvature everywhere, including $z\to\infty$ and $r=1$. Therefore scalar invariants cannot diagnose either patch horizon as a curvature singularity. The correct tests are:

Claim	Necessary check	Licensed conclusion
“the chart ends”	evaluate an invariant such as $R$	only coordinate coverage ends
“the wedge modes are complete”	state the function space and wedge boundary conditions	completeness holds only for wedge data
“the wedge gives the global state”	compare correlations across the complementary diamond	generally false without extra global data
“the horizon is thermal”	identify the state and the $\eta$-KMS property	a state-dependent statement, not geometry alone

Treating $r=1$ as a physical singularity fails the invariant-curvature check. Promoting a wedge basis to global completeness fails because distinct global solutions can agree on incomplete boundary data outside the relevant domain assumptions. The strongest robust statement is that each chart furnishes a valid coordinate and mode description on its own region.

Controlled limits and handoff

The coordinate transformations are exact on patch overlaps, but mode completeness and vacuum assignments depend on the region, boundary conditions, and analytic continuation. No thermal claim follows from a Killing horizon alone. Timelike-Boundary Causality and Boundary-Value Problems now fixes the boundary data needed for evolution; subregion reconstruction is deferred to the dedicated reconstruction chapter.

Exercise

Why can a Poincaré-boundary source not prepare arbitrary independent data on the entire global cylinder?

Solution

The Poincaré boundary is only a conformal patch of the cylinder. Its complement and the null edges are absent from the source domain. A bulk solution obtained from Poincaré data can be continued when regularity and state conditions supply the missing information, but that continuation is an additional boundary or state prescription. Coordinate transformation alone cannot create independent data on the omitted region.

The chapter overview contains the structure diagram and validity and failure diagram. They are embedded there once so that their shared chapter-level context is not repeated on every article.

For the chapter-wide comparison of assumptions, counterevidence, falsifiers, and claim ceilings, see the claim-domain table.

References

• Casini, Horacio, Marina Huerta, and Robert C. Myers. “Towards a Derivation of Holographic Entanglement Entropy.” Journal of High Energy Physics 2011, 036 (2011). arXiv. DOI.
• Hamilton, Alex, Daniel Kabat, Gilad Lifschytz, and David A. Lowe. “Holographic Representation of Local Bulk Operators.” Physical Review D 74 (2006): 066009. arXiv. DOI.