Coordinate Systems and the Boundary

The main idea

Anti-de Sitter space is one spacetime, but AdS/CFT constantly uses several coordinate systems because different coordinates make different parts of the dictionary transparent.

Global coordinates make the Hilbert-space interpretation manifest: the boundary is the cylinder $\mathbb R_\tau \times S^{d-1}$, and global time evolution is generated by the CFT dilatation operator after radial quantization. Poincaré coordinates make the flat-space CFT vacuum and ordinary momentum-space correlators manifest: the boundary looks like $\mathbb R^{1,d-1}$. Fefferman—Graham coordinates make the near-boundary dictionary manifest: sources, expectation values, counterterms, and Weyl transformations are organized in a radial expansion.

The crucial point is this:

The AdS boundary is not a finite-distance wall. It is a conformal boundary. A choice of coordinates near infinity is simultaneously a choice of conformal frame and a choice of UV regulator for the boundary theory.

This page develops the coordinate systems that will be used throughout the course. Later pages will use them almost mechanically: global coordinates for spectra and states, Poincaré coordinates for correlators and black branes, Fefferman—Graham coordinates for holographic renormalization, and horizon-regular coordinates for real-time response.

Three complementary ways to describe $\mathrm{AdS}_{d+1}$. Global coordinates display the cylinder $\mathbb R_\tau\times S^{d-1}$; Poincaré coordinates display a flat boundary patch with radial UV/IR relation; Fefferman—Graham coordinates organize the near-boundary data $g_{(0)}$, $g_{(2)}$, $\ldots$ used in holographic renormalization.

Global coordinates: AdS as a spacetime with a cylinder at infinity

Start from the embedding of $\mathrm{AdS}_{d+1}$ in $\mathbb R^{2,d}$,

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

A convenient global parametrization is

$$\begin{aligned} X_{-1} &= L\cosh\rho\,\cos\tau,\\ X_0 &= L\cosh\rho\,\sin\tau,\\ X_i &= L\sinh\rho\, n_i,\qquad i=1,\ldots,d, \end{aligned}$$

where $n_i n_i=1$ parametrizes $S^{d-1}$. The induced metric is

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

Here $\rho\in[0,\infty)$ and $\tau$ is global time. Strictly, the hyperboloid has closed timelike curves because $\tau$ is periodic. The physical Lorentzian AdS spacetime used in holography is the universal cover, where $\tau\in\mathbb R$.

It is also useful to define

$$r=L\sinh\rho, \qquad t=L\tau.$$

Then

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

This form makes the large-$r$ asymptotics especially visible. Near infinity,

$$ds^2\sim -\frac{r^2}{L^2}dt^2+\frac{L^2}{r^2}dr^2+r^2d\Omega_{d-1}^2.$$

Multiplying the metric by $L^2/r^2$ and taking $r\to\infty$ gives the boundary conformal metric

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

Equivalently, using dimensionless time $\tau=t/L$,

$$ds^2_{\partial}=L^2\left(-d\tau^2+d\Omega_{d-1}^2\right),$$

and the overall factor $L^2$ is part of the chosen conformal frame.

Compact radial coordinate

A particularly clean conformal compactification follows from

$$\sinh\rho=\tan\theta, \qquad 0\leq \theta<\frac{\pi}{2}.$$

Since $\cosh\rho=\sec\theta$ and $d\rho=\sec\theta\,d\theta$, the global metric becomes

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

The conformal boundary is at $\theta=\pi/2$. Removing the divergent conformal factor gives

$$ds^2_{\partial}\sim -d\tau^2+d\Omega_{d-1}^2.$$

This is why one often says that global AdS has boundary $\mathbb R\times S^{d-1}$. More precisely, global AdS has a conformal boundary whose natural global conformal frame is the Lorentzian cylinder.

Poincaré coordinates: the flat boundary patch

For most practical calculations in AdS/CFT, especially local correlation functions, one uses Poincaré coordinates:

$$ds^2=\frac{L^2}{z^2} \left( dz^2+\eta_{\mu\nu}dx^\mu dx^\nu \right) =\frac{L^2}{z^2} \left( dz^2-dt^2+d\vec x^{\,2} \right),$$

where $z>0$ and $\mu=0,\ldots,d-1$. The conformal boundary is at $z=0$. Multiplying by $z^2/L^2$ gives the boundary metric

$$ds^2_{\partial}=\eta_{\mu\nu}dx^\mu dx^\nu.$$

Thus Poincaré coordinates naturally describe a CFT on flat Minkowski space.

One explicit relation to the embedding coordinates is useful for orientation. With $x^2=\eta_{\mu\nu}x^\mu x^\nu$,

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

These equations obey

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

where $X_\mu$ includes the Lorentzian boundary time direction. They also show immediately that the Poincaré patch is the region with $X_{-1}+X_d>0$.

The Poincaré patch does not cover all of global AdS. The surface $z\to\infty$ is the Poincaré horizon of pure AdS: it is a coordinate horizon, not a curvature singularity. This distinction matters. A calculation in the Poincaré vacuum is often simpler than a global calculation, but it is not automatically a statement about every global state.

The scaling isometry and the radial direction

The Poincaré metric is invariant under

$$x^\mu\to \lambda x^\mu, \qquad z\to \lambda z.$$

On the boundary this becomes the CFT dilatation $x^\mu\to\lambda x^\mu$. This is the first precise hint of the radial/energy-scale relation:

$$z \text{ small} \quad \leftrightarrow \quad \text{boundary UV}, \qquad z \text{ large} \quad \leftrightarrow \quad \text{boundary IR}.$$

The statement is not that $z$ is a gauge-invariant energy scale by itself. Radial position depends on coordinates and gauge choices. The invariant statement is that radial diffeomorphisms act near the boundary as Weyl transformations and RG transformations of the boundary data.

Cutoff surfaces

A surface $z=\epsilon$ has induced metric

$$\gamma_{\mu\nu}(\epsilon)=\frac{L^2}{\epsilon^2}\eta_{\mu\nu}, \qquad \sqrt{-\gamma}=\left(\frac{L}{\epsilon}\right)^d.$$

In holographic calculations one usually does not evaluate the action directly at $z=0$. Instead one regulates at $z=\epsilon$, computes the on-shell action, adds local counterterms on the cutoff surface, and then sends $\epsilon\to0$:

$$S_{\mathrm{ren}} = \lim_{\epsilon\to0} \left( S_{\mathrm{bulk}}^{z\geq\epsilon} +S_{\mathrm{GHY}}^{z=\epsilon} +S_{\mathrm{ct}}^{z=\epsilon} \right).$$

This is the gravitational version of UV renormalization. The divergence near $z=0$ is an infinite-volume divergence in the bulk, but it maps to a UV divergence in the boundary CFT.

A useful diagnostic is the proper radial distance between $z_1$ and $z_2$ at fixed $x^\mu$:

$$\ell=L\int_{z_1}^{z_2}\frac{dz}{z} =L\log\frac{z_2}{z_1}.$$

The boundary $z=0$ is infinitely far away in proper distance. This is why the boundary is not a literal wall placed at finite distance.

Euclidean AdS and hyperbolic space

For Euclidean correlators one Wick-rotates $t=-i t_E$. The Poincaré metric becomes

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

This is hyperbolic space $H_{d+1}$ in the upper-half-space model. The boundary is $\mathbb R^d$ plus a point at infinity. Euclidean AdS is especially useful for deriving conformal two- and three-point functions because the bulk-to-boundary propagator takes a simple form,

$$K_\Delta(z,x;x') \propto \left(\frac{z}{z^2+|x-x'|^2}\right)^\Delta.$$

The Lorentzian continuation is not a formality: real-time correlators require choices of contour and boundary condition. Infalling conditions at horizons, Schwinger-Keldysh contours, and $i\epsilon$ prescriptions will appear later.

The boundary is a conformal class

The clean mathematical definition of the AdS boundary uses a defining function. A spacetime $(M,g)$ is asymptotically locally AdS if one can find a function $\Omega$ such that

$$\Omega>0 \text{ in the bulk}, \qquad \Omega=0 \text{ at } \partial M, \qquad d\Omega\neq0 \text{ at } \partial M,$$

and the rescaled metric

$$\widetilde g_{ab}=\Omega^2 g_{ab}$$

extends smoothly to $\partial M$. The boundary metric is then

$$g_{(0)\mu\nu}=\left.\widetilde g_{\mu\nu}\right|_{\partial M}.$$

But $\Omega$ is not unique. If $\Omega$ is a defining function, then so is

$$\Omega'=e^{\sigma(x)}\Omega$$

near the boundary. This changes the boundary representative by a Weyl rescaling,

$$g_{(0)\mu\nu}\to e^{2\sigma(x)}g_{(0)\mu\nu}.$$

Therefore the natural boundary datum of pure AdS geometry is not one metric $g_{(0)}$, but the conformal class

$$[g_{(0)}].$$

When the boundary CFT is coupled to a background metric, choosing a representative $g_{(0)}$ is choosing a conformal frame. In even boundary dimension there can be a Weyl anomaly, so the quantum generating functional is not strictly invariant under all Weyl rescalings; the anomaly is local and is encoded holographically by logarithmic terms in the near-boundary expansion.

Fefferman—Graham coordinates: the near-boundary expansion

Fefferman—Graham coordinates put any asymptotically locally AdS metric into the near-boundary form

$$ds^2=\frac{L^2}{z^2} \left( dz^2+g_{\mu\nu}(z,x)dx^\mu dx^\nu \right),$$

at least in a collar neighborhood of the boundary. The coordinate $z$ is a defining function, and $g_{\mu\nu}(z,x)$ has an expansion of the form

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

The logarithmic term is present in even boundary dimension $d$ for generic sources and boundary curvature. For pure Einstein gravity, the lower coefficients $g_{(2)},g_{(4)},\ldots$ up to order $z^{d-2}$ are locally determined by the boundary metric and matter sources. The coefficient $g_{(d)}$ contains state-dependent information; in particular, it is related to the CFT stress tensor.

For a flat boundary and no anomaly terms, the relation has the schematic form

$$\langle T_{\mu\nu}\rangle \sim \frac{L^{d-1}}{G_{d+1}}\,g_{(d)\mu\nu}.$$

With conventional Einstein normalization, and for simple cases where local curvature/anomaly contributions vanish,

$$\langle T_{\mu\nu}\rangle = \frac{dL^{d-1}}{16\pi G_{d+1}}g_{(d)\mu\nu}.$$

The omitted local terms are not optional decoration. They are essential on curved boundaries, in even $d$, and when one keeps track of renormalization schemes.

What Fefferman—Graham gauge is good for

Fefferman—Graham coordinates are excellent for questions that live near the boundary:

Task	Why Fefferman—Graham helps
Identify sources	$g_{(0)}$, scalar leading terms, and boundary gauge fields appear directly.
Compute one-point functions	canonical momenta and subleading coefficients give vevs after counterterms.
Derive Ward identities	radial constraints become CFT conservation, trace, and anomaly equations.
Renormalize the action	divergences are local functionals of cutoff-surface data.
Track Weyl transformations	radial diffeomorphisms implement boundary Weyl transformations.

They are not ideal for everything. Fefferman—Graham coordinates often break down at horizons or caustics of the geodesics normal to the boundary. For black holes and real-time correlators, one usually switches to coordinates that are regular at the future horizon.

Global cylinder versus Minkowski boundary

The global boundary cylinder and the Poincaré boundary $\mathbb R^{1,d-1}$ are conformally related. In unit-radius notation, write the cylinder metric as

$$ds^2_{\mathrm{cyl}} =-d\tau^2+d\theta^2+ \sin^2\theta\,d\Omega_{d-2}^2.$$

Minkowski polar coordinates $(t,r,\Omega_{d-2})$ are related to cylinder coordinates by

$$t=\frac{\sin\tau}{\cos\tau+\cos\theta}, \qquad r=\frac{\sin\theta}{\cos\tau+\cos\theta}.$$

Then

$$-dt^2+dr^2+r^2d\Omega_{d-2}^2 = \frac{-d\tau^2+d\theta^2+ \sin^2\theta\,d\Omega_{d-2}^2} {(\cos\tau+\cos\theta)^2}.$$

So flat Minkowski space is a conformal patch of the cylinder. This is the boundary reason why global AdS and Poincaré AdS are both natural. A CFT can be placed on either background; the two descriptions are related by conformal transformations, with the usual caveats about operator insertions, states, anomalies, and global domains.

The bulk version is subtler. Poincaré coordinates cover only a portion of global AdS, and the Poincaré vacuum is adapted to the time translation $\partial_t$, not to the global Hamiltonian $\partial_\tau$. For local vacuum correlators this distinction is often harmless; for horizons, thermodynamics, and global questions it is not.

Horizon-regular coordinates

A common mistake is to use coordinates that are singular exactly where the physical boundary condition is imposed. This appears most often in black-hole backgrounds.

Even in pure Poincaré AdS, one can define an ingoing null coordinate

$$v=t-z.$$

Then

$$dt=dv+dz,$$

and the pure AdS metric becomes

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

For black branes the analogous ingoing Eddington—Finkelstein coordinate is regular at the future horizon and is the natural coordinate for imposing infalling boundary conditions. We will use this in the real-time Green-function and hydrodynamics chapters.

Which coordinates should you use?

Coordinate system	Boundary frame	Best for	Main danger
Global AdS	$\mathbb R\times S^{d-1}$	spectra, states, radial quantization, thermal AdS, Hawking-Page physics	confusing the global cylinder with flat Minkowski space
Poincaré AdS	$\mathbb R^{1,d-1}$	local correlators, momentum space, black branes, RG intuition	forgetting it covers only a patch of global AdS
Euclidean Poincaré AdS	$\mathbb R^d$	Euclidean correlators, Witten diagrams, conformal integrals	losing Lorentzian causal prescriptions
Fefferman—Graham	arbitrary $g_{(0)\mu\nu}$	holographic renormalization, Ward identities, stress tensor	using it through horizons where it may fail
Eddington—Finkelstein	usually flat or thermal	real-time response, infalling conditions, numerical evolution	hiding the simple near-boundary expansion
AdS-Rindler	causal diamond/domain of dependence	subregion physics, entanglement wedges, modular flow	mistaking coordinate horizons for singularities

The best practice is not to love one coordinate system too much. Use the coordinate system that makes the physical question transparent, and translate near the boundary when extracting CFT data.

Common mistakes

Mistake 1: treating the boundary as a finite-radius brane

The boundary is reached only after conformal compactification. A cutoff surface $z=\epsilon$ or $r=R$ is a regulator, not the actual CFT spacetime. The induced metric on the cutoff surface diverges; the finite boundary metric is obtained after a Weyl rescaling.

Mistake 2: confusing a coordinate patch with a state

Poincaré coordinates are a coordinate patch; the Poincaré vacuum is a state. The two are related in practice, but not identical ideas. Global AdS can be described in many coordinates, and the same coordinate system can support different bulk states.

Mistake 3: using the UV/IR slogan too literally

It is useful to remember $z\sim 1/\mu$, but the exact relation between radial position and field-theory scale is scheme- and gauge-dependent. Holographic RG is a statement about how radial evolution reorganizes boundary data, not a local observable called “the energy scale at point $z$.”

Mistake 4: extracting vevs before renormalization

Subleading coefficients in a near-boundary expansion are not automatically renormalized expectation values. One must include the Gibbons—Hawking—York term, counterterms, possible finite scheme-dependent terms, and then vary the renormalized action.

Mistake 5: ignoring the conformal frame

The CFT on $\mathbb R\times S^{d-1}$ and the CFT on $\mathbb R^{1,d-1}$ are conformally related in many situations, but energies, vacua, thermal circles, anomalies, and operator normalizations must be transformed carefully.

Exercises

Exercise 1: Derive the global AdS metric

Using

$$X_{-1}=L\cosh\rho\cos\tau, \qquad X_0=L\cosh\rho\sin\tau, \qquad X_i=L\sinh\rho\,n_i,$$

with $n_i n_i=1$, derive

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

Solution

The ambient metric is

$$ds^2_{\mathbb R^{2,d}} =-dX_{-1}^2-dX_0^2+ \sum_{i=1}^d dX_i^2.$$

For the two timelike embedding coordinates,

$$dX_{-1}^2+dX_0^2 =L^2\left(\sinh^2\rho\,d\rho^2+ \cosh^2\rho\,d\tau^2\right),$$

because the $d\rho\,d\tau$ cross terms cancel. For the spatial coordinates,

$$\sum_i dX_i^2 =L^2\left(\cosh^2\rho\,d\rho^2+ \sinh^2\rho\,dn_i dn_i\right).$$

Since $dn_i dn_i=d\Omega_{d-1}^2$, the induced metric is

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

Using $\cosh^2\rho-\sinh^2\rho=1$ gives the desired result.

Exercise 2: Compactify global AdS

Let $\sinh\rho=\tan\theta$. Show that the global metric becomes

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

What is the boundary metric in this conformal frame?

Solution

From $\sinh\rho=\tan\theta$ we get

$$\cosh\rho=\sqrt{1+\sinh^2\rho}=\sec\theta.$$

Differentiating $\sinh\rho=\tan\theta$ gives

$$\cosh\rho\,d\rho=\sec^2\theta\,d\theta,$$

so

$$d\rho=\sec\theta\,d\theta.$$

Substituting into the global metric,

$$\begin{aligned} ds^2 &=L^2\left(-\sec^2\theta\,d\tau^2+ \sec^2\theta\,d\theta^2+ \tan^2\theta\,d\Omega_{d-1}^2\right)\\ &=\frac{L^2}{\cos^2\theta} \left(-d\tau^2+d\theta^2+ \sin^2\theta\,d\Omega_{d-1}^2\right). \end{aligned}$$

The boundary is at $\theta=\pi/2$. Multiplying by $\cos^2\theta/L^2$ and restricting to the boundary gives

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

Exercise 3: The Poincaré scaling isometry

Show that

$$ds^2=\frac{L^2}{z^2} \left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right)$$

is invariant under

$$z\to \lambda z, \qquad x^\mu\to \lambda x^\mu.$$

Explain the boundary interpretation.

Solution

Under the transformation,

$$dz\to \lambda dz, \qquad dx^\mu\to \lambda dx^\mu.$$

The numerator transforms as

$$dz^2+\eta_{\mu\nu}dx^\mu dx^\nu \to \lambda^2\left(dz^2+\eta_{\mu\nu}dx^\mu dx^\nu\right),$$

while the denominator transforms as

$$z^2\to \lambda^2 z^2.$$

The factors of $\lambda^2$ cancel, so the bulk metric is invariant. At the boundary, only $x^\mu\to\lambda x^\mu$ remains. This is the CFT dilatation. The fact that $z$ scales together with $x^\mu$ is the geometric origin of the radial/energy-scale relation.

Exercise 4: Proper distance to the Poincaré boundary

At fixed $x^\mu$, compute the proper radial distance between $z=\epsilon$ and $z=z_0$ in Poincaré AdS. What happens as $\epsilon\to0$?

Solution

At fixed $x^\mu$,

$$ds=\frac{L}{z}dz.$$

Therefore

$$\ell(\epsilon,z_0) =L\int_\epsilon^{z_0}\frac{dz}{z} =L\log\frac{z_0}{\epsilon}.$$

As $\epsilon\to0$, the distance diverges logarithmically. Thus the conformal boundary is infinitely far away in the physical bulk metric even though it is placed at a finite coordinate location in the compactified geometry.

Exercise 5: A near-boundary stress tensor check

Suppose an asymptotically AdS$_{d+1}$ metric in Fefferman—Graham form has flat boundary metric and expansion

$$g_{\mu\nu}(z)=\eta_{\mu\nu}+z^d a_{\mu\nu}+\cdots,$$

with no anomaly or source-dependent local terms. What constraints should $a_{\mu\nu}$ satisfy if the dual CFT stress tensor is conserved and traceless?

Solution

In this simple flat-boundary case,

$$\langle T_{\mu\nu}\rangle \propto a_{\mu\nu}.$$

Conservation of the stress tensor gives

$$\partial^\mu a_{\mu\nu}=0.$$

Tracelessness gives

$$\eta^{\mu\nu}a_{\mu\nu}=0.$$

In the bulk these conditions arise from the radial constraint equations. In more general cases, sources, curved boundary metrics, and anomalies modify the trace Ward identity by local terms.

Further reading

• O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, Large N Field Theories, String Theory and Gravity, especially the sections reviewing AdS geometry and the conformal boundary.
• E. Witten, Anti de Sitter Space and Holography, for the original boundary-value viewpoint of holographic observables.
• L. Susskind and E. Witten, The Holographic Bound in Anti-de Sitter Space, for the UV/IR interpretation of radial position.
• K. Skenderis, Lecture Notes on Holographic Renormalization, for Fefferman—Graham expansions, counterterms, Ward identities, and anomalies.