AdS Coordinate Systems

This appendix is a coordinate atlas for the AdS geometries used throughout the course. It is deliberately practical. The same spacetime can look like a cylinder, a half-space, a black hole exterior, or a radial RG flow depending on the coordinates. The physics is invariant; the coordinate system chooses the questions that are easy to ask.

Two warnings prevent most mistakes.

First, the AdS boundary is conformal data, not an ordinary finite-distance wall. A coordinate system usually chooses a representative of the boundary conformal class.

Second, not every coordinate chart covers all of AdS. Poincare coordinates are indispensable for flat-space CFT correlators, but they cover only a patch of global AdS. Global coordinates are indispensable for Hilbert-space questions on the cylinder.

Unless stated otherwise, the bulk dimension is $d+1$, the boundary dimension is $d$, and the AdS radius is $L$.

The most common coordinate systems for $\mathrm{AdS}_{d+1}$. Global coordinates see the boundary cylinder, Poincare coordinates see flat boundary space, Fefferman–Graham coordinates organize near-boundary data, black-brane and Eddington–Finkelstein coordinates describe thermal states, and hyperbolic/Rindler coordinates are natural for ball-shaped regions and modular flow.

Quick coordinate dictionary

Coordinates	Typical metric form	Boundary frame	Best used for
embedding	$-X_{-1}^2-X_0^2+X_1^2+\cdots+X_d^2=-L^2$	none chosen	symmetries, global definitions
global	$-(1+r^2/L^2)dt^2+dr^2/(1+r^2/L^2)+r^2d\Omega_{d-1}^2$	$\mathbb R_t\times S^{d-1}$	states, normal modes, global black holes
compact global	$L^2\cos^{-2}\chi(-d\tau^2+d\chi^2+\sin^2\chi d\Omega_{d-1}^2)$	$\mathbb R_\tau\times S^{d-1}$	causal diagrams, boundary at $\chi=\pi/2$
Poincare	$L^2z^{-2}(dz^2-dt^2+d\vec x^{\,2})$	$\mathbb R^{1,d-1}$	vacuum correlators, RG intuition
Euclidean Poincare	$L^2z^{-2}(dz^2+d\vec x^{\,2})$	$\mathbb R^d$	Euclidean correlators, $H^{d+1}$
Fefferman–Graham	$L^2z^{-2}(dz^2+g_{ij}(z,x)dx^idx^j)$	arbitrary $g_{(0)ij}$	holographic renormalization
domain wall	$dr^2+e^{2A(r)}ds_d^2$	usually flat or curved QFT metric	RG flows, relevant deformations
planar black brane	$L^2z^{-2}[-f(z)dt^2+d\vec x^{\,2}+dz^2/f(z)]$	thermal flat-space CFT	thermodynamics, transport
ingoing EF	$L^2z^{-2}[-f(z)dv^2-2dvdz+d\vec x^{\,2}]$	thermal flat-space CFT	horizons, real-time correlators
hyperbolic/Rindler	$-(r^2/L^2-1)d\tau^2+dr^2/(r^2/L^2-1)+r^2dH_{d-1}^2$	$\mathbb R_\tau\times H^{d-1}$	ball entanglement, modular flow

The table hides many normalization choices. The sections below spell out the versions used in this course.

Embedding-space definition

Lorentzian $\mathrm{AdS}_{d+1}$ can be defined as the hyperboloid

$$-X_{-1}^2-X_0^2+X_1^2+\cdots+X_d^2=-L^2$$

inside $\mathbb R^{2,d}$ with metric

$$ds^2_{\mathbb R^{2,d}} = -dX_{-1}^2-dX_0^2+dX_1^2+\cdots+dX_d^2.$$

The induced metric has constant negative curvature,

$$R_{abcd} = -\frac{1}{L^2}\left(g_{ac}g_{bd}-g_{ad}g_{bc}\right), \qquad R_{ab}=-\frac d{L^2}g_{ab}, \qquad R=-\frac{d(d+1)}{L^2}.$$

The manifest symmetry group of the hyperboloid is $SO(2,d)$. This is the global conformal group of a $d$-dimensional Lorentzian CFT, up to global/discrete subtleties.

The literal hyperboloid contains closed timelike curves because the time coordinate is periodic. In AdS/CFT one normally means the universal cover, where the global time coordinate is unwrapped.

Global coordinates

A standard global parametrization is

$$X_{-1}=\sqrt{L^2+r^2}\cos\frac{t}{L}, \qquad X_0=\sqrt{L^2+r^2}\sin\frac{t}{L}, \qquad X_a=r\,\Omega_a,$$

where $a=1,\ldots,d$, $\sum_a\Omega_a^2=1$, and $\Omega_a$ coordinates a unit $S^{d-1}$. The induced metric is

$$ds^2 = -\left(1+\frac{r^2}{L^2}\right)dt^2 + \frac{dr^2}{1+r^2/L^2} + r^2d\Omega_{d-1}^2.$$

The boundary lies at $r\to\infty$. Multiplying by $L^2/r^2$ and taking the limit gives the boundary conformal representative

$$ds^2_{\partial} = -dt^2+L^2d\Omega_{d-1}^2.$$

Thus global AdS is adapted to a CFT on

$$\mathbb R_t\times S^{d-1}.$$

This is the natural coordinate system for state-operator correspondence, normal modes, finite-volume spectra, and global AdS black holes.

Compact global coordinates

Define

$$r=L\tan\chi, \qquad 0\le \chi<\frac{\pi}{2}, \qquad \tau=\frac{t}{L}.$$

Then global AdS becomes

$$ds^2 = \frac{L^2}{\cos^2\chi} \left( -d\tau^2+d\chi^2+\sin^2\chi\,d\Omega_{d-1}^2 \right).$$

This is often the cleanest form for causal diagrams. The conformally related metric

$$d\tilde s^2 = -d\tau^2+d\chi^2+\sin^2\chi\,d\Omega_{d-1}^2$$

is a finite cylinder with boundary at $\chi=\pi/2$. Null rays reach the boundary in finite global time. This is why asymptotically AdS spacetimes require boundary conditions: the boundary is timelike.

Poincare coordinates

Poincare coordinates are adapted to the flat-space CFT vacuum. In Lorentzian signature,

$$ds^2 = \frac{L^2}{z^2} \left( dz^2-dt^2+d\vec x^{\,2} \right), \qquad z>0.$$

The boundary is at $z=0$, and the Poincare horizon is at $z\to\infty$. The boundary conformal representative is Minkowski space,

$$ds^2_{\partial}=-dt^2+d\vec x^{\,2}.$$

A useful embedding map is

$$X_{-1}=\frac{L^2+z^2+x^\mu x_\mu}{2z}, \qquad X_d=\frac{L^2-z^2-x^\mu x_\mu}{2z}, \qquad X_\mu=\frac{Lx_\mu}{z},$$

where $x^\mu x_\mu=-t^2+d\vec x^{\,2}$ and $\mu=0,\ldots,d-1$. This makes clear that Poincare coordinates cover only the region where

$$X_{-1}+X_d>0.$$

Poincare AdS is often also written with

$$r=\frac{L^2}{z}.$$

Then

$$ds^2 = \frac{r^2}{L^2}\left(-dt^2+d\vec x^{\,2}\right) + \frac{L^2}{r^2}dr^2.$$

In these coordinates the boundary is at $r\to\infty$, while the Poincare horizon is at $r\to0$.

Euclidean AdS

Euclidean AdS is hyperbolic space $H^{d+1}$. The Poincare half-space metric is

$$ds^2 = \frac{L^2}{z^2}\left(dz^2+d\vec x^{\,2}\right), \qquad z>0.$$

This coordinate system is ideal for Euclidean CFT correlators on $\mathbb R^d$. The regularity condition in the interior is usually simple: the Euclidean solution should be smooth and should not grow pathologically as $z\to\infty$.

Euclidean global AdS can be written as

$$ds^2 = \left(1+\frac{r^2}{L^2}\right)d\tau_E^2 + \frac{dr^2}{1+r^2/L^2} + r^2d\Omega_{d-1}^2.$$

If $\tau_E$ is periodically identified, the boundary is $S^1_\beta\times S^{d-1}$, which prepares a thermal state of the CFT on the sphere.

Fefferman–Graham gauge

Fefferman–Graham coordinates are not primarily a global coordinate system. They are a near-boundary gauge. In one common convention,

$$ds^2 = \frac{L^2}{z^2} \left( dz^2+g_{ij}(z,x)dx^idx^j \right), \qquad z\to0.$$

The expansion has the schematic form

$$g_{ij}(z,x) = g_{(0)ij}(x) +z^2g_{(2)ij}(x) +\cdots +z^d g_{(d)ij}(x) +z^d\log z^2\,h_{(d)ij}(x) +\cdots.$$

The coefficient $g_{(0)ij}$ is the boundary metric source. The coefficient $g_{(d)ij}$ contains state-dependent information and is related to the renormalized stress tensor. The logarithmic term appears for even boundary dimension $d$ and is tied to the Weyl anomaly.

A second common convention uses $\rho=z^2$:

$$ds^2 = \frac{L^2}{4\rho^2}d\rho^2 + \frac{L^2}{\rho}g_{ij}(\rho,x)dx^idx^j.$$

Fefferman–Graham gauge fixes

$$g_{zz}=\frac{L^2}{z^2}, \qquad g_{zi}=0,$$

but it does not fix all diffeomorphisms. The residual transformations include boundary diffeomorphisms and Weyl transformations of $g_{(0)ij}$.

Domain-wall coordinates

For holographic RG flows it is often useful to write

$$ds^2 = dr^2+e^{2A(r)}ds_d^2,$$

where $ds_d^2$ is usually a flat or curved boundary metric. Pure Poincare AdS is recovered from

$$A(r)=\frac rL, \qquad z=L e^{-r/L},$$

up to a constant rescaling of boundary coordinates. The boundary is $r\to\infty$, and moving inward toward smaller $r$ corresponds roughly to flowing toward the infrared.

For Einstein-scalar flows, scalar profiles $\phi^I(r)$ are interpreted as running couplings or expectation values. Domain-wall coordinates make this interpretation transparent but are often less convenient for extracting precise counterterms than Fefferman–Graham coordinates.

Planar black brane coordinates

The planar AdS-Schwarzschild black brane is

$$ds^2 = \frac{L^2}{z^2} \left[ -f(z)dt^2+d\vec x^{\,2}+\frac{dz^2}{f(z)} \right], \qquad f(z)=1-\left(\frac{z}{z_h}\right)^d.$$

The boundary is at $z=0$, and the horizon is at $z=z_h$. Smoothness of the Euclidean geometry gives

$$T=\frac{d}{4\pi z_h}.$$

The same geometry can be written in $r=L^2/z$ coordinates as

$$ds^2 = -\frac{r^2}{L^2}f(r)dt^2 + \frac{L^2}{r^2f(r)}dr^2 + \frac{r^2}{L^2}d\vec x^{\,2}, \qquad f(r)=1-\left(\frac{r_h}{r}\right)^d.$$

Then

$$T=\frac{d\,r_h}{4\pi L^2}.$$

Black-brane coordinates are adapted to a thermal CFT on $\mathbb R^{d-1}$. They are the default setup for transport, real-time correlators, and finite-density generalizations.

Ingoing Eddington–Finkelstein coordinates

Schwarzschild-like black-brane coordinates are singular at the future horizon. For real-time problems one often uses ingoing Eddington–Finkelstein coordinates. Define

$$v=t-z_*(z), \qquad \frac{dz_*}{dz}=\frac{1}{f(z)}.$$

Then

$$dt=dv+\frac{dz}{f(z)},$$

and the black-brane metric becomes

$$ds^2 = \frac{L^2}{z^2} \left[ -f(z)dv^2-2dvdz+d\vec x^{\,2} \right].$$

This form is regular at the future horizon. Infalling fields are smooth functions of $v$ and $z$ near $z=z_h$. That is the geometric origin of the infalling prescription for retarded Green functions.

Global AdS black holes and topological black holes

A useful family of static asymptotically AdS metrics is

$$ds^2 = -f_k(r)dt^2+ \frac{dr^2}{f_k(r)} +r^2d\Sigma_{k,d-1}^2,$$

with

$$f_k(r)=k+\frac{r^2}{L^2}-\frac{\mu}{r^{d-2}}.$$

Here $d\Sigma_{k,d-1}^2$ is a unit metric of constant curvature

$$k=1,0,-1.$$

The cases are:

$k$	Horizon geometry	Boundary spatial geometry	Common use
$1$	sphere $S^{d-1}$	$S^{d-1}$	Hawking–Page transition
$0$	plane $\mathbb R^{d-1}$	$\mathbb R^{d-1}$	thermal plasma, transport
$-1$	hyperbolic space $H^{d-1}$ or quotient	$H^{d-1}$ or quotient	ball entanglement, hyperbolic black holes

The planar case is obtained from the large-black-hole or infinite-volume limit of the spherical case.

Hyperbolic/Rindler AdS coordinates

A particularly important hyperbolic slicing is

$$ds^2 = -\left(\frac{r^2}{L^2}-1\right)d\tau^2 + \frac{dr^2}{r^2/L^2-1} +r^2dH_{d-1}^2.$$

This is pure AdS written as a hyperbolic black hole with horizon at $r=L$. It is not a new state of the full theory; it is a coordinate patch adapted to an accelerated observer or, on the boundary, to the domain of dependence of a ball-shaped region.

This coordinate system is central in the relation between vacuum entanglement across a sphere and thermal physics on $\mathbb R\times H^{d-1}$.

Coordinate transformations worth memorizing

The most useful transformations are:

$$r=L\tan\chi \qquad \text{global radial coordinate to compact global coordinate},$$ $$r=\frac{L^2}{z} \qquad \text{Poincare } r\text{ coordinate to Poincare }z\text{ coordinate},$$ $$z=L e^{-r/L} \qquad \text{domain-wall radial coordinate to Poincare }z,$$

and

$$v=t-z_*(z), \qquad \frac{dz_*}{dz}=\frac{1}{f(z)} \qquad \text{Schwarzschild time to ingoing EF time}.$$

You do not need to memorize the full global-to-Poincare transformation. What matters more is the conceptual fact: Poincare coordinates cover only a wedge of global AdS, and the Poincare boundary is conformal to Minkowski space plus a point.

Which coordinates should I use?

Use global coordinates when the boundary theory lives on $\mathbb R\times S^{d-1}$, when you care about the spectrum, or when the state-operator correspondence is central.

Use Poincare coordinates when the boundary theory lives on flat space and translation invariance is present.

Use Fefferman–Graham coordinates when you need near-boundary expansions, sources, counterterms, and one-point functions.

Use domain-wall coordinates when you want physical intuition for RG flows.

Use black-brane coordinates when you want thermodynamics.

Use ingoing Eddington–Finkelstein coordinates when you want retarded real-time correlators or regularity at a future horizon.

Use hyperbolic coordinates when the boundary problem involves ball-shaped regions, modular flow, or hyperbolic thermal states.

Common confusions

“The coordinate $z$ is the energy scale.”

More carefully, small $z$ corresponds to the UV and large $z$ corresponds to the IR in many Poincare-domain-wall setups. But $z$ is a bulk coordinate, not literally an energy. The energy-scale interpretation is a powerful organizing principle, not a replacement for a holographic calculation.

“The Poincare horizon is a black-hole horizon.”

In pure AdS, the Poincare horizon at $z\to\infty$ is a coordinate horizon, not a thermal black-hole horizon. It becomes physically analogous to a horizon in certain wedges, but it does not by itself imply a nonzero CFT temperature.

“Fefferman–Graham coordinates always reach the horizon.”

No. Fefferman–Graham coordinates are guaranteed only near the conformal boundary under suitable asymptotic assumptions. They often break down before reaching a black-hole horizon.

“The boundary metric is unique.”

The boundary metric is defined only up to Weyl rescaling. A coordinate system chooses a representative $g_{(0)ij}$ of the boundary conformal class.

Exercises

Exercise 1: Compact global AdS

Starting from

$$ds^2 = -\left(1+\frac{r^2}{L^2}\right)dt^2 + \frac{dr^2}{1+r^2/L^2} +r^2d\Omega_{d-1}^2,$$

set $r=L\tan\chi$ and $t=L\tau$. Derive the compact global metric.

Solution

Since

$$1+\frac{r^2}{L^2}=1+\tan^2\chi=\sec^2\chi, \qquad dr=L\sec^2\chi\,d\chi,$$

we have

$$\frac{dr^2}{1+r^2/L^2} = \frac{L^2\sec^4\chi\,d\chi^2}{\sec^2\chi} = L^2\sec^2\chi\,d\chi^2.$$

Also

$$-\left(1+\frac{r^2}{L^2}\right)dt^2 =-L^2\sec^2\chi\,d\tau^2,$$

and

$$r^2d\Omega_{d-1}^2 =L^2\tan^2\chi\,d\Omega_{d-1}^2 =L^2\sec^2\chi\,\sin^2\chi\,d\Omega_{d-1}^2.$$

Combining terms gives

$$ds^2 = \frac{L^2}{\cos^2\chi} \left( -d\tau^2+d\chi^2+\sin^2\chi\,d\Omega_{d-1}^2 \right).$$

Exercise 2: Black-brane temperature

For

$$ds^2 = \frac{L^2}{z^2} \left[ f(z)d\tau_E^2+d\vec x^{\,2}+\frac{dz^2}{f(z)} \right], \qquad f(z)=1-\left(\frac{z}{z_h}\right)^d,$$

derive $T=d/(4\pi z_h)$ from smoothness at $z=z_h$.

Solution

Near the horizon set $z=z_h-y$, with $y\ll z_h$. Then

$$f(z) = 1-\left(1-\frac{y}{z_h}\right)^d \simeq \frac{d y}{z_h}.$$

The $(\tau_E,z)$ part of the metric is approximately

$$\frac{L^2}{z_h^2} \left( \frac{d y}{z_h}d\tau_E^2 + \frac{dy^2}{d y/z_h} \right).$$

Define

$$\rho^2=\frac{4L^2 y}{d z_h}.$$

Then the metric becomes

$$d\rho^2+ \rho^2\left(\frac{d}{2z_h}d\tau_E\right)^2.$$

Smoothness at the origin requires the angular variable $(d/2z_h)\tau_E$ to have period $2\pi$, so

$$\beta=\frac{4\pi z_h}{d}, \qquad T=\frac{1}{\beta}=\frac{d}{4\pi z_h}.$$

Exercise 3: Why ingoing EF coordinates are regular

Show that the transformation

$$v=t-z_*(z), \qquad \frac{dz_*}{dz}=\frac{1}{f(z)}$$

turns

$$-f(z)dt^2+\frac{dz^2}{f(z)}$$

into

$$-f(z)dv^2-2dvdz.$$

Solution

The definition gives

$$dv=dt-\frac{dz}{f(z)}, \qquad dt=dv+\frac{dz}{f(z)}.$$

Therefore

$$-fdt^2+\frac{dz^2}{f} = -f\left(dv+\frac{dz}{f}\right)^2+\frac{dz^2}{f}.$$

Expanding,

$$-fdv^2-2dvdz-\frac{dz^2}{f}+\frac{dz^2}{f} = -fdv^2-2dvdz.$$

The $1/f$ singularity has disappeared. This is why ingoing EF coordinates are regular at a future horizon.

Further reading

• E. Witten, Anti de Sitter Space and Holography.
• O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large $N$ Field Theories, String Theory and Gravity.
• S. de Haro, S. N. Solodukhin, and K. Skenderis, Holographic Reconstruction of Spacetime and Renormalization in the AdS/CFT Correspondence.
• K. Skenderis, Lecture Notes on Holographic Renormalization.