Numerical Holography and the DeTurck Method

Why numerical holography is part of the dictionary

Most analytic calculations in AdS/CFT exploit a large amount of symmetry. Homogeneous black branes reduce Einstein’s equations to ordinary differential equations. Linear response reduces transport to fluctuation equations. RT surfaces in vacuum AdS often reduce to geodesics or simple minimal surfaces.

Research problems are rarely so polite.

A boundary theory may live on a curved spacetime, contain a spatially modulated source, form localized deconfined plasma droplets, develop striped order, or have a stationary heat current between reservoirs. The dual bulk geometry is then a nonlinear solution of Einstein’s equations depending on two or more coordinates. In such cases the AdS/CFT dictionary is still conceptually simple:

$$\text{boundary sources and state} \quad\longleftrightarrow\quad \text{bulk boundary conditions and regularity data},$$

but the hard work is solving the bulk boundary value problem.

Numerical holography is the discipline of doing this reliably. Its output is not just a metric. Its output is a controlled approximation to a CFT state or saddle, together with enough error checks that the extracted stress tensor, currents, entropy, transport data, or entanglement observables can be trusted.

The most widely used method for static and stationary gravitational boundary value problems is the Einstein-DeTurck method. It is a gauge-fixed formulation of Einstein’s equations designed to turn the nonlinear gravitational problem into an elliptic problem, much like fixing a gauge in electromagnetism turns a degenerate operator into an invertible one.

A typical Einstein-DeTurck workflow. The reference metric $\bar g$ fixes the generalized harmonic gauge, the DeTurck vector $\xi^a$ must vanish in the final solution, and holographic observables are extracted only after convergence, residual, and Ward-identity checks.

This page focuses on stationary or static boundary value problems. Fully time-dependent numerical holography is a different, equally important subject: one usually solves an initial-boundary value problem using characteristic, generalized harmonic, ADM/BSSN-like, or other evolution schemes. DeTurck methods are most natural when the physical problem is elliptic after symmetry reduction.

The basic problem

Consider a bulk theory in $D=d+1$ dimensions,

$$S = \frac{1}{16\pi G_{d+1}} \int d^{d+1}x\sqrt{-g} \left( R+\frac{d(d-1)}{L^2} \right) + S_{\mathrm{matter}} + S_{\mathrm{GHY}} + S_{\mathrm{ct}}.$$

The classical holographic problem is to find a solution satisfying:

1. asymptotically AdS boundary conditions that encode the CFT sources,
2. regularity at horizons, axes, caps, or other interior boundaries,
3. fixed conserved charges or potentials appropriate to the ensemble,
4. any imposed discrete or continuous symmetries.

For pure gravity, the equations are

$$R_{ab}+\frac{d}{L^2}g_{ab}=0.$$

With matter, it is often convenient to use the trace-reversed form

$$R_{ab}+\frac{d}{L^2}g_{ab} = 8\pi G_{d+1} \left( T_{ab}-\frac{1}{d-1}Tg_{ab} \right),$$

where $T=g^{ab}T_{ab}$. The problem is nonlinear and diffeomorphism invariant. That last phrase is not a philosophical ornament; it is a numerical headache.

If $g_{ab}$ is a solution, then $\varphi^*g_{ab}$ is also a solution for any diffeomorphism $\varphi$ that preserves the boundary conditions. Therefore the principal symbol of the Einstein operator is degenerate: the equations do not determine pure gauge components of the metric. Before solving the equations numerically, one must fix this degeneracy.

In perturbation theory we might choose de Donder gauge. In numerical boundary value problems, the corresponding robust nonlinear choice is the DeTurck gauge.

The DeTurck vector

Choose a smooth reference metric $\bar g_{ab}$ on the same manifold as the desired solution. It must have the same causal and boundary structure as the desired metric: the same conformal boundary, the same horizons or axes, and the same coordinate domain. It does not need to solve Einstein’s equations.

Define the DeTurck vector

$$\xi^a = g^{bc} \left( \Gamma^a_{bc}[g] - \bar\Gamma^a_{bc}[\bar g] \right).$$

The condition

$$\xi^a=0$$

is a generalized harmonic-coordinate condition relative to $\bar g$. The Einstein-DeTurck equations are obtained by adding a gauge-fixing term to Einstein’s equations. For pure Einstein-AdS gravity, use

$$R_{ab} + \frac{d}{L^2}g_{ab} - \nabla_{(a}\xi_{b)} = 0.$$

With matter, use

$$R_{ab} + \frac{d}{L^2}g_{ab} - 8\pi G_{d+1} \left( T_{ab}-\frac{1}{d-1}Tg_{ab} \right) - \nabla_{(a}\xi_{b)} = 0,$$

supplemented by the matter equations and appropriate gauge choices for matter fields.

The key point is simple:

$$\xi^a=0 \quad\Longrightarrow\quad \text{Einstein-DeTurck solution is an Einstein solution}.$$

A solution of the Einstein-DeTurck equations with $\xi^a\ne0$ is not the desired physical solution. It is a Ricci soliton or, with matter, a generalized soliton. Therefore every numerical DeTurck calculation must check that

$$\|\xi\|^2 = \xi^a\xi_a$$

converges to zero in the continuum limit. In practice one often monitors a dimensionless maximum such as

$$\max_{\mathcal M} L^2 |\xi^a\xi_a|.$$

For certain static Riemannian problems with suitable boundary conditions, maximum-principle arguments can rule out nontrivial solitons. In research practice, one still checks $\xi^2$ numerically. Trust, but verify; gravity has never been shy about ambushing innocent gauge choices.

Why this helps: ellipticity

The DeTurck term changes the principal part of the Einstein equations. Schematically, for a Riemannian or static problem after appropriate continuation, the gauge-fixed equations have leading behavior

$$\Delta g_{ab} + \text{lower-derivative terms}=0,$$

rather than a degenerate diffeomorphism-invariant operator. This is why the method is useful: it converts the gravitational problem into a well-posed elliptic boundary value problem for the metric functions, provided the ansatz and coordinates are chosen sensibly.

Elliptic equations have the right flavor for equilibrium holography. Boundary data specified on the conformal boundary, horizon, axes, and caps determine the solution globally. This is analogous to electrostatics, not wave propagation.

The contrast is important:

Physical problem	Bulk mathematical problem	Common numerical approach
Static black holes	elliptic boundary value problem	Einstein-DeTurck, Newton iteration, relaxation
Stationary rigid black holes	often elliptic after symmetry reduction	harmonic/DeTurck-type formulations
Spatial holographic lattices in equilibrium	elliptic PDEs	DeTurck plus spectral or finite-difference methods
Real-time thermalization	initial-boundary value problem	characteristic or generalized-harmonic evolution
Quasinormal modes on a known background	eigenvalue problem	shooting, pseudospectral, determinant methods

The DeTurck method is not magic powder sprinkled on any Einstein problem. It is a powerful tool when the physical problem has enough stationarity and regularity that the reduced equations are elliptic.

A minimal example of the logic

Suppose we want a static asymptotically AdS$_{d+1}$ black hole whose boundary metric or sources break translation invariance along one spatial direction $x$. We might choose coordinates $(t,z,x,y_i)$ and an ansatz of the schematic form

$$ds^2 = \frac{L^2}{z^2} \left[ - f(z) Q_{tt}(z,x) dt^2 + \frac{Q_{zz}(z,x)}{f(z)} dz^2 +Q_{xx}(z,x)\bigl(dx+z^2 Q_{zx}(z,x)dz\bigr)^2 +Q_{yy}(z,x)d\vec y^{\,2} \right].$$

Here $z=0$ is the conformal boundary and $z=z_h$ is the horizon. The unknown functions are

$$Q_{tt},\quad Q_{zz},\quad Q_{xx},\quad Q_{zx},\quad Q_{yy},$$

possibly supplemented by matter functions such as $A_t(z,x)$ or scalar profiles.

A natural reference metric might be obtained by setting

$$Q_{tt}=Q_{zz}=Q_{xx}=Q_{yy}=1, \qquad Q_{zx}=0,$$

while keeping the same function $f(z)$ and the same coordinate locations of the boundary and horizon. The reference metric has the same skeleton as the desired solution. It is not asked to know the solution in advance.

Then the problem becomes:

$$\mathcal F[Q_i,A_a,\phi,\ldots]=0,$$

where $\mathcal F=0$ denotes the coupled Einstein-DeTurck and matter equations. The numerical task is to solve this nonlinear system with boundary conditions at $z=0$, $z=z_h$, and perhaps $x=0$, $x=\ell$, or periodic $x$.

Boundary conditions at the conformal boundary

Near the AdS boundary, the metric is usually written in Fefferman-Graham form,

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

with expansion

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

The leading term $g_{(0)\mu\nu}$ is the boundary metric source for $T^{\mu\nu}$. Matter fields have similar source/response expansions. For example, a scalar dual to an operator of dimension $\Delta$ behaves as

$$\phi(z,x) = z^{d-\Delta}\phi_{(0)}(x) + \cdots + z^\Delta \phi_{(2\Delta-d)}(x) + \cdots.$$

At the numerical boundary $z=0$, one normally imposes the source data:

$$g_{(0)\mu\nu}(x)=\text{chosen boundary metric}, \qquad \phi_{(0)}(x)=\text{chosen scalar source}, \qquad A^{(0)}_\mu(x)=\text{chosen gauge source}.$$

But the subleading response data must not be fixed arbitrarily. They are determined by regularity and by the equations. A common beginner mistake is to over-prescribe both source and vev data. That usually kills the solution or hides an inconsistency.

For numerical coordinates that are not exactly Fefferman-Graham, one imposes equivalent asymptotically AdS conditions and later transforms, expands, or evaluates counterterm expressions to extract the renormalized observables.

Boundary conditions at a Euclidean horizon

For a static black hole, it is often useful to work in Euclidean signature near the horizon. Let $\rho=0$ be the horizon. A smooth Euclidean horizon looks locally like

$$ds_E^2 = d\rho^2+\kappa^2\rho^2 d\tau^2 +\gamma_{ij}(\rho,x)dx^i dx^j,$$

with periodic Euclidean time

$$\tau\sim \tau+\frac{2\pi}{\kappa}.$$

This gives the temperature

$$T=\frac{\kappa}{2\pi}.$$

In a numerical ansatz with a horizon at fixed coordinate $z=z_h$, smoothness imposes relations among the metric functions. For instance, if the near-horizon metric contains

$$ds_E^2 \sim \frac{L^2}{z_h^2} \left[ f(z) Q_{\tau\tau} d\tau^2 + \frac{Q_{zz}}{f(z)}dz^2 +\cdots \right],$$

then regularity often requires

$$Q_{\tau\tau}(z_h,x)=Q_{zz}(z_h,x),$$

plus smoothness conditions such as Neumann conditions on other functions. The exact form depends on the ansatz.

For gauge fields, regularity of the one-form at a Euclidean horizon usually imposes

$$A_\tau(z_h,x)=0$$

in a gauge where $A=A_\tau d\tau+\cdots$. In Lorentzian notation this becomes the familiar regularity condition that the electrostatic potential vanish at the future horizon in a regular gauge. The chemical potential is then a gauge-invariant potential difference between boundary and horizon.

Axes, caps, and corners

Many holographic boundary value problems have coordinate axes, compact circles that shrink smoothly, or corners where two boundaries meet. These are often the places where numerics fail first.

Near a smooth axis, the metric must look like polar coordinates:

$$ds^2 = d\rho^2+\rho^2 d\varphi^2+\text{regular transverse directions},$$

with the correct period for $\varphi$. In an ansatz, this typically imposes equality of two metric functions at the axis, plus vanishing normal derivatives for smooth functions.

Near an AdS soliton cap, a spatial circle shrinks rather than the Euclidean time circle. The regularity condition is similar, but the field-theory interpretation is different. The shrinking circle encodes an IR mass gap or confinement scale rather than a temperature.

Corners require compatibility. If the conformal boundary meets a horizon, or a horizon meets an axis, the boundary conditions inherited from the two sides must agree in the overlapping expansion. A surprisingly large fraction of numerical trouble is just corner trouble wearing a fake mustache.

The reference metric is a gauge choice, not a physical ansatz

The reference metric $\bar g$ appears in the equations, so it is tempting to assign it physical meaning. Resist the temptation.

The role of $\bar g$ is to define the generalized harmonic gauge. A good reference metric should:

• live on the same coordinate domain as $g$,
• have the same conformal boundary structure,
• have the same horizon, axis, and cap topology,
• obey the same leading boundary and regularity conditions,
• be smooth everywhere in the domain,
• be simple enough to compute with.

It does not need to solve the same equations. It may be pure AdS, an AdS black brane, an AdS soliton, or a simple interpolation between boundary structures.

A bad reference metric can make the Newton iteration struggle, introduce coordinate singularities, or obscure the desired solution branch. But changing $\bar g$ should not change the final Einstein solution once $\xi^a=0$ is achieved.

Discretization choices

Once the ansatz and boundary conditions are fixed, the continuum problem becomes a finite-dimensional nonlinear system. There are two common choices.

Finite differences

Finite differences approximate derivatives on a grid. They are flexible, local, and robust for complicated domains. A derivative such as $\partial_x Q$ becomes a stencil involving nearby grid values. Smooth solutions converge algebraically as the grid is refined:

$$\|Q_N-Q_{\infty}\| \sim N^{-p},$$

where $p$ is determined by the order of the stencil and by the regularity of the solution.

Finite differences are often a good first implementation, especially when the domain has awkward boundaries or when one wants sparse matrices with simple structure.

Spectral and pseudospectral methods

Spectral methods expand functions in global basis functions, for example Chebyshev polynomials in a finite radial interval and Fourier modes in a periodic direction:

$$Q(z,x) \approx \sum_{m=0}^{N_z} \sum_{n=-N_x/2}^{N_x/2} q_{mn} T_m(\tilde z)e^{i n k x}.$$

For smooth solutions, spectral methods often converge exponentially:

$$\|Q_N-Q_{\infty}\| \sim e^{-\alpha N}.$$

This is why they are popular in high-precision stationary holography. Their weakness is that nonsmoothness, corners, bad coordinates, or nonanalytic boundary expansions can destroy the exponential convergence. Then domain decomposition, coordinate changes, or subtraction of known nonanalytic pieces may be necessary.

Newton iteration

The discretized Einstein-DeTurck system is a nonlinear algebraic system,

$$F_I(u_J)=0,$$

where $u_J$ denotes all metric and matter unknowns at all grid points or spectral coefficients. Newton’s method linearizes around the current guess:

$$F_I(u+\delta u) = F_I(u) +J_{IJ}\delta u_J +\cdots,$$

with Jacobian

$$J_{IJ}=\frac{\partial F_I}{\partial u_J}.$$

At each step one solves

$$J_{IJ}\delta u_J=-F_I(u),$$

then updates

$$u\to u+\alpha\delta u,$$

where $0<\alpha\le1$ may be chosen by a line search or damping strategy.

A practical Newton workflow is:

choose initial guess u
repeat:
  assemble residual F[u]
  assemble or approximate Jacobian J[u]
  solve J delta_u = -F
  choose damping alpha
  update u <- u + alpha delta_u
  check residual, DeTurck norm, and boundary conditions
until converged

Newton’s method is fast near a solution and unforgiving far from one. Good initial guesses matter. Common strategies include:

• start from an analytic homogeneous black brane and turn on a source slowly,
• continue in temperature, chemical potential, or lattice amplitude,
• use a coarse grid solution as the initial guess for a finer grid,
• use pseudo-arclength continuation near turning points,
• exploit perturbative small-amplitude solutions as seeds.

Continuation and phase diagrams

Many interesting holographic solutions appear in families. Let $\lambda_{\mathrm{num}}$ be a control parameter: temperature, chemical potential, source amplitude, lattice wave number, boundary curvature, or horizon size. A branch of solutions is a curve

$$F(u,\lambda_{\mathrm{num}})=0.$$

The simplest continuation method increments $\lambda_{\mathrm{num}}$ and uses the previous solution as the next initial guess. This fails at folds, where $d\lambda_{\mathrm{num}}/ds=0$ along the branch parameter $s$.

Near such turning points, pseudo-arclength continuation treats $u$ and $\lambda_{\mathrm{num}}$ together as unknowns and adds a constraint fixing the step along the branch. This is often essential for black-hole phase diagrams.

For holography, each numerical solution should also be assigned thermodynamic data: temperature, entropy, energy, charge, free energy, and sources. Phase dominance is determined by the correct thermodynamic potential, not by which geometry looks more elegant.

Extracting holographic observables

Solving the bulk equations is only half the calculation. The other half is translating the solution into CFT data.

For the stress tensor, one may use holographic renormalization. In Fefferman-Graham coordinates,

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

and the renormalized stress tensor has the schematic form

$$\langle T_{\mu\nu}\rangle = \frac{d L^{d-1}}{16\pi G_{d+1}}g_{(d)\mu\nu} +X_{\mu\nu}[g_{(0)},\text{sources}],$$

where $X_{\mu\nu}$ is a local functional that includes counterterm and anomaly contributions.

For a conserved current dual to a Maxwell field,

$$\langle J^\mu\rangle = \lim_{z\to0} \frac{\delta S_{\mathrm{ren}}}{\delta A^{(0)}_\mu}$$

and in simple coordinates this is proportional to the renormalized radial electric flux.

For entropy,

$$S = \frac{A_{\mathcal H}}{4G_{d+1}}$$

in two-derivative Einstein gravity. For higher-derivative theories one must use the appropriate Wald or generalized entropy functional.

For free energies, one evaluates the renormalized Euclidean on-shell action:

$$\Omega = T I_E^{\mathrm{ren}}$$

in the grand-canonical ensemble, or its Legendre transform in the canonical ensemble.

Every one of these extractions is a potential source of error. A metric that solves the bulk PDE beautifully can still produce a wrong CFT answer if the boundary expansion, counterterms, ensemble, or source normalization are mishandled.

Validation: the part that separates numerics from numerology

A numerical holographic solution should pass several independent tests.

Residuals

The equation residual should converge to zero:

$$\|F[u_N]\|\to0 \qquad \text{as}\qquad N\to\infty.$$

Do not only report the Newton residual on the collocation grid. It is better to also test residuals on a finer or independent grid, because spectral collocation can occasionally flatter you.

DeTurck norm

The DeTurck vector must vanish:

$$\max_{\mathcal M} L^2|\xi^a\xi_a| \to0.$$

If the residual is small but $\xi^2$ is not small, you may have found a DeTurck soliton rather than an Einstein solution.

Convergence

For a smooth solution with spectral discretization, differences between successive resolutions should decrease exponentially:

$$\|u_{N+1}-u_N\| \sim e^{-\alpha N}.$$

For finite differences, the decrease should follow the expected algebraic order. Plotting errors on a log scale is often more informative than quoting one small number.

Constraint equations

Even in a gauge-fixed system, some equations may behave as constraints. Evaluate them independently. In gravity, redundant equations are not useless equations; they are lie detectors.

Ward identities

The extracted stress tensor and currents should obey the correct field-theory Ward identities. For example, with background gauge fields and scalar sources, one expects schematic identities such as

$$\nabla_\mu \langle T^{\mu}{}_{\nu}\rangle = F^{(0)}_{\nu\mu}\langle J^\mu\rangle +\langle \mathcal O\rangle \nabla_\nu \phi_{(0)} +\cdots,$$

and

$$\nabla_\mu\langle J^\mu\rangle=0$$

when the boundary symmetry is not explicitly anomalous or broken.

For a CFT on a curved background, the trace identity has the schematic form

$$\langle T^\mu{}_{\mu}\rangle = \mathcal A[g_{(0)},\text{sources}] + \sum_i (d-\Delta_i)\phi_{(0)}^i\langle\mathcal O_i\rangle,$$

with $\mathcal A$ the conformal anomaly in even $d$.

Thermodynamic identities

For black holes, check the first law and the relevant Smarr-like relations when available:

$$dE=T dS+\mu dQ+\cdots.$$

If the solution branch is computed in the grand-canonical ensemble, check

$$\Omega=E-TS-\mu Q-\cdots.$$

A mismatch may indicate insufficient resolution, an omitted boundary term, a wrong ensemble, or an inconsistent normalization.

Common holographic DeTurck projects

The method appears in many types of holographic calculations.

Holographic lattices

A spatially modulated source, such as

$$\mu(x)=\mu_0+A\cos(kx),$$

breaks translation invariance explicitly. The dual charged black hole depends on both $z$ and $x$. The DeTurck method is then used to construct the inhomogeneous black brane. Once the background is known, one can compute thermodynamics, DC conductivities, optical conductivities, and fermion spectral functions.

Plasma balls and localized black holes

In confining holographic models, deconfined plasma can form localized finite-size droplets. The bulk dual is a localized black hole sitting near the IR bottom of the geometry. Such solutions are not captured by homogeneous black branes and typically require nonlinear numerical construction.

CFTs on curved black-hole backgrounds

One can place the boundary CFT on a curved spacetime containing a black hole. The dual bulk may contain funnels or droplets, depending on how heat exchange with the boundary black hole is realized. The problem is naturally a stationary gravitational boundary value problem.

Spontaneous striped phases

If a homogeneous black brane develops a spatially modulated zero mode, the nonlinear endpoint may be a striped black brane. Constructing it requires solving coupled nonlinear PDEs with periodic boundary conditions and comparing free energies to decide phase dominance.

Lumpy black holes and higher-dimensional phases

Outside holography, the same Einstein-DeTurck technology is heavily used to construct higher-dimensional black holes: localized Kaluza-Klein black holes, nonuniform strings, lumpy AdS black holes, and black rings. Holography benefits from this numerical-relativity technology because the underlying boundary value problem is the same kind of animal.

What not to forget about matter fields

The DeTurck term fixes diffeomorphism freedom. It does not automatically fix internal gauge redundancies. If the bulk contains a Maxwell field, there may be a $U(1)$ gauge freedom

$$A_a\to A_a+\partial_a\Lambda.$$

For static electric ansätze, regularity and ansatz choices often effectively fix the gauge. For more general problems, one may impose an additional gauge condition, such as radial gauge or Lorenz gauge, or use gauge-invariant variables.

For charged scalars, a common ansatz uses the gauge freedom to make the scalar real. This is convenient but must be compatible with the horizon and boundary conditions. For phases with currents, vortices, or magnetic fields, gauge choices become more delicate.

The moral is that every continuous redundancy needs either a gauge condition or gauge-invariant variables. DeTurck fixes spacetime diffeomorphisms, not every redundancy in the theory.

A compact research checklist

Before trusting a numerical holography result, ask the following questions.

Step	Question
Physical setup	What boundary sources, ensemble, and symmetries are being imposed?
Bulk theory	What action and truncation are being used? Is it top-down, consistent, or bottom-up?
Ansatz	Does the ansatz include all fields allowed by the symmetries?
Reference metric	Does $\bar g$ have the same boundary, horizon, axis, and cap structure?
PDE type	Is the reduced Einstein-DeTurck system elliptic?
Boundary data	Are sources fixed without over-fixing vevs?
Horizon	Are temperature and gauge regularity imposed correctly?
Numerics	Is convergence demonstrated with resolution studies?
DeTurck check	Does $\xi^2$ vanish in the continuum limit?
Observables	Are holographic renormalization and ensemble choices handled correctly?
Physics	Are Ward identities, thermodynamics, and known limits satisfied?

This checklist is not bureaucratic. It is a survival guide.

Common mistakes

Mistake 1: treating $\xi^2$ as optional

The DeTurck vector is not a diagnostic decoration. A finite $\xi^2$ means the solution is not necessarily a solution of Einstein’s equations. Always report and monitor it.

Mistake 2: using a reference metric with the wrong topology

If the desired solution has a horizon, the reference metric should have a corresponding horizon in the same coordinate domain. If a circle shrinks in the physical geometry, the reference metric should encode the same kind of cap. Otherwise the gauge may fight the solution.

Mistake 3: over-prescribing boundary data

At the AdS boundary, fix sources. Let vevs be determined. At a horizon, impose regularity. Do not impose every coefficient that appears in a local series expansion.

Mistake 4: confusing coordinate singularities with physics

Bad radial coordinates, nonsmooth axes, and inconsistent corner conditions can masquerade as physical phase transitions. Before believing an exotic branch, check the coordinate and boundary regularity.

Mistake 5: comparing free energies in the wrong ensemble

A solution at fixed chemical potential should be compared using the grand potential. A solution at fixed charge should be compared using the canonical free energy. Legendre transforms matter.

Mistake 6: solving a too-small ansatz

A symmetric ansatz can artificially forbid the real endpoint of an instability. If a perturbative analysis predicts modes outside your ansatz, the nonlinear solution you find may be only a constrained saddle.

Mistake 7: trusting pretty plots more than convergence

A smooth-looking metric function can be wrong. A jagged residual plot can be honest. In numerical holography, aesthetics are not evidence.

Relation to full time-dependent numerical holography

The Einstein-DeTurck method is mainly an equilibrium and stationary tool. Full time-dependent holography solves different mathematical problems. For example, colliding shocks or gravitational collapse in AdS are initial-boundary value problems. They require specifying initial data and evolving through time.

Characteristic formulations based on infalling null coordinates are especially effective for asymptotically AdS gravitational dynamics. The physics output is different: time-dependent stress tensors, hydrodynamization, entropy production, horizon formation, turbulence, and far-from-equilibrium relaxation.

The shared lesson is that holography is not only a dictionary of exact formulas. It is also a computational framework for strongly coupled quantum dynamics and equilibrium states.

Exercises

Exercise 1: The DeTurck equation implies Einstein when $\xi^a=0$

For pure Einstein-AdS gravity, the Einstein-DeTurck equation is

$$R_{ab}+\frac{d}{L^2}g_{ab}-\nabla_{(a}\xi_{b)}=0.$$

Show that if $\xi^a=0$ everywhere, the metric solves the Einstein equation with cosmological constant

$$\Lambda=-\frac{d(d-1)}{2L^2}.$$

Solution

If $\xi^a=0$, then $\nabla_{(a}\xi_{b)}=0$, so the Einstein-DeTurck equation reduces to

$$R_{ab}+\frac{d}{L^2}g_{ab}=0.$$

Thus

$$R_{ab}=-\frac{d}{L^2}g_{ab}.$$

In $D=d+1$ dimensions, the vacuum Einstein equation is

$$R_{ab}-\frac12 Rg_{ab}+\Lambda g_{ab}=0.$$

Taking the trace of $R_{ab}=-d g_{ab}/L^2$ gives

$$R=-\frac{d(d+1)}{L^2}.$$

Substituting into the Einstein equation gives

$$-\frac{d}{L^2}g_{ab} +\frac12\frac{d(d+1)}{L^2}g_{ab} +\Lambda g_{ab}=0,$$

so

$$\Lambda=-\frac{d(d-1)}{2L^2}.$$

Exercise 2: Horizon smoothness in Euclidean signature

Consider a two-dimensional near-horizon Euclidean metric

$$ds^2 = A(\rho)d\tau^2+d\rho^2, \qquad A(\rho)=\kappa^2\rho^2+O(\rho^4).$$

Show that smoothness at $\rho=0$ requires $\tau$ to have period $2\pi/\kappa$.

Solution

Near $\rho=0$,

$$ds^2=d\rho^2+\kappa^2\rho^2d\tau^2.$$

Define an angular coordinate

$$\theta=\kappa\tau.$$

Then

$$ds^2=d\rho^2+\rho^2d\theta^2,$$

which is the flat metric on the plane in polar coordinates. Smoothness at the origin requires

$$\theta\sim\theta+2\pi.$$

Therefore

$$\tau\sim\tau+\frac{2\pi}{\kappa}.$$

The corresponding temperature is $T=\kappa/(2\pi)$.

Exercise 3: Source versus vev boundary conditions

A scalar field dual to an operator of dimension $\Delta$ has near-boundary behavior

$$\phi(z,x) = z^{d-\Delta}\phi_{(0)}(x) +z^\Delta \phi_{(2\Delta-d)}(x) +\cdots.$$

In standard quantization, which coefficient should be fixed at the boundary, and which should be determined by the solution? What goes wrong if both are imposed arbitrarily?

Solution

In standard quantization, $\phi_{(0)}(x)$ is the source for the boundary operator $\mathcal O$. It is part of the boundary condition and should be fixed by the definition of the field-theory problem.

The coefficient $\phi_{(2\Delta-d)}(x)$ is proportional, after counterterms and possible local terms, to the expectation value $\langle\mathcal O(x)\rangle$. It should be determined dynamically by regularity in the interior and by the bulk equations.

If both coefficients are imposed arbitrarily, the boundary value problem is generally overdetermined. One is trying to specify both the source and response of an interacting system, leaving no freedom for the bulk solution to satisfy regularity and the equations of motion.

Exercise 4: What should be reported in a numerical holography paper?

Suppose a numerical solution claims to describe an inhomogeneous holographic black brane. List at least five independent checks that should be reported before one trusts the extracted stress tensor and free energy.

Solution

Good checks include:

1. convergence of the equation residual with increasing resolution,
2. convergence of the DeTurck norm $\max L^2|\xi^2|$ to zero,
3. convergence of physical quantities such as $E$, $S$, $Q$, and $\Omega$,
4. satisfaction of boundary Ward identities for $T_{\mu\nu}$ and $J^\mu$,
5. satisfaction of the first law or relevant thermodynamic identity,
6. independence of results under changes of numerical grid or domain decomposition,
7. agreement with analytic limits, such as the homogeneous black brane when the inhomogeneous source is set to zero,
8. regularity checks at the horizon, axes, and corners.

The important point is that a small Newton residual alone is not enough.

Exercise 5: Design a simple holographic lattice problem

Consider Einstein-Maxwell theory in AdS$_4$ with a boundary chemical potential

$$\mu(x)=\mu_0+A\cos(kx).$$

Describe the minimum ingredients needed to formulate a static DeTurck boundary value problem for the dual black brane.

Solution

A reasonable formulation needs:

1. a metric ansatz depending on $(z,x)$, with enough components to be closed under the equations and the symmetries;
2. a Maxwell field, at least $A_t(z,x)$, and possibly additional components if currents or more general stationary behavior are allowed;
3. a reference metric $\bar g$ with the same AdS boundary, same horizon position, and same coordinate domain;
4. boundary conditions at $z=0$ fixing the boundary metric and

$$A_t(z=0,x)=\mu_0+A\cos(kx);$$

5. regularity conditions at the horizon, including $A_t(z_h,x)=0$ in a regular gauge;
6. periodic boundary conditions in $x$ with period $2\pi/k$;
7. the Einstein-DeTurck equations for the metric and the Maxwell equations for $A_a$;
8. numerical convergence checks, DeTurck norm checks, and extraction of $\langle T_{\mu\nu}\rangle$ and $\langle J^\mu\rangle$ by holographic renormalization.

Further reading

• M. Headrick, S. Kitchen, and T. Wiseman, A New Approach to Static Numerical Relativity, and Its Application to Kaluza-Klein Black Holes. The original modern introduction of the DeTurck method into static numerical relativity.
• T. Wiseman, Numerical Construction of Static and Stationary Black Holes. A concise pedagogical review of the framework and its black-hole applications.
• O. J. C. Dias, J. E. Santos, and B. Way, Numerical Methods for Finding Stationary Gravitational Solutions. The main practical review for stationary gravitational boundary-value problems, including examples in AdS.
• A. Adam, S. Kitchen, and T. Wiseman, A Numerical Approach to Finding General Stationary Vacuum Black Holes. Extends the DeTurck framework to important stationary Lorentzian settings.
• T. Andrade, Holographic Lattices and Numerical Techniques. Holography-oriented notes emphasizing translation-breaking backgrounds.
• G. T. Horowitz, J. E. Santos, and D. Tong, Optical Conductivity with Holographic Lattices. A landmark application to inhomogeneous finite-density transport.
• A. Donos and J. P. Gauntlett, Thermoelectric DC Conductivities from Black Hole Horizons, and E. Banks, A. Donos, and J. P. Gauntlett, Thermoelectric DC Conductivities and Stokes Flows on Black Hole Horizons. Essential complements for extracting DC transport from inhomogeneous horizons.
• P. M. Chesler and L. G. Yaffe, Numerical Solution of Gravitational Dynamics in Asymptotically Anti-de Sitter Spacetimes. A useful bridge to real-time numerical holography, where the mathematical problem is no longer elliptic.

The first four are the most useful starting points for the DeTurck method itself. The later references show how numerical holography is used for lattices, transport, and real-time dynamics.