Timelike-Boundary Causality and Boundary-Value Problems

A radial light ray reaches the conformal boundary of global anti-de Sitter space in finite global time. Initial data on a spacelike slice therefore do not determine what happens after a signal can reach that boundary: one must also say what the boundary does. This page develops that initial–boundary value problem for a free real scalar on fixed AdS, solves the standard reflective problem explicitly, and then separates five questions that are often blurred together—well-posedness, symplectic closure, self-adjointness, stability, and causality.

Required background. Anti-de Sitter geometry and its conformal boundary supplies the compactified global geometry and the finite boundary travel time. Timelike boundaries and AdS boundary conditions supplies the general operator-domain analysis. Helpful background. Covariant symplectic structure and conserved inner products supplies the Klein–Gordon pairing used below.

The AdS initial–boundary value problem

Take a real scalar on Lorentzian AdS$_{d+1}$, with $d\geq2$, signature $(+,-,\ldots,-)$, and bulk action

$$S_{\mathrm{bulk}} =\frac12\int_M\mathrm d^{d+1}x\sqrt{\lvert g\rvert} \left(g^{MN}\partial_M\phi\,\partial_N\phi-m^2\phi^2\right).$$

The equation of motion is $(\Box+m^2)\phi=0$. The spacetime used below is the universal cover $\widetilde{\mathrm{AdS}}_{d+1}$, with $\tau\in\mathbb R$. In compactified global coordinates,

$$\mathrm ds^2=\frac{L^2}{\cos^2\chi} \left( \mathrm d\tau^2-\mathrm d\chi^2 -\sin^2\chi\,\mathrm d\Omega_{d-1}^2 \right), \qquad 0\leq\chi<\frac{\pi}{2},$$

where $\tau$ is dimensionless. Radial null curves obey $\mathrm d\tau=\pm\mathrm d\chi$, so the boundary at $\chi=\pi/2$ is reached from the center after $\Delta\tau=\pi/2$. A constant-$\tau$ slice is consequently not a Cauchy surface for AdS without a boundary prescription.

The global-AdS causal strip draws this finite outward trip and shows why a return ray is conditional on the boundary law.

An initial–boundary value problem supplies

$$\phi\big\rvert_{\Sigma_{\tau_0}}=\phi_0, \qquad n^M\nabla_M\phi\big\rvert_{\Sigma_{\tau_0}}=\pi_0, \qquad \mathcal B[\phi]\big\rvert_{\mathscr I}=J,$$

where $n^M$ is the future-directed unit normal to the initial slice, $\mathcal B$ is the boundary operator, $J$ is prescribed boundary data, and $\mathscr I$ is the timelike conformal boundary. Smooth initial and boundary data are not independent: at the corner $\overline{\Sigma}_{\tau_0}\cap\mathscr I$, they must obey the boundary law and its time-differentiated compatibility conditions to the regularity order being claimed. For Dirichlet data, for example, the trace of $\phi_0$ must equal $J(\tau_0)$.

For a static closed system, the wave equation can be written schematically as

$$\partial_\tau^2\phi=-A\phi.$$

Under the assumptions of time-translation and time-reflection symmetry, local agreement with the field equation, a suitable conserved positive energy, and the convergence and continuity condition in the theorem, admissible dynamics are generated by a positive self-adjoint extension of the spatial operator $A$ Ishibashi and Wald 2003, §2 and theorem 3.1. This is a classification of conservative static dynamics under stated hypotheses, not a theorem that every well-posed boundary problem must be self-adjoint.

Dissipative AdS problems can also be well posed; their bulk region is an open system unless the degrees of freedom receiving the flux are included Holzegel, Luk, Smulevici, and Warnick 2020, §4.

Asymptotic data and renormalized symplectic flux

Near the boundary, use Fefferman–Graham coordinate $z$ with bulk region $z>0$ and

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

Define

$$\Delta_\pm=\frac d2\pm\nu, \qquad \nu=\sqrt{\frac{d^2}{4}+m^2L^2}.$$

The Breitenlohner–Freedman bound is $m^2L^2\geq-d^2/4$, for which $\nu$ is real. The generic $\nu>0$ power-law case carries the main argument; the logarithmic formulas below keep the endpoints from being treated by an invalid substitution. At a generic nonresonant mass with $\nu>0$, a solution begins as

$$\phi(z,x)=z^{\Delta_-}\alpha(x)+z^{\Delta_+}\beta(x)+\cdots.$$

At the BF bound the roots coalesce, and with the convention used here the expansion instead begins

$$\phi(z,x)=z^{d/2} \left[\alpha(x)\log\!\left(\frac z{\ell_\mu}\right)+\beta(x)+\cdots\right],$$

where $\ell_\mu$ is a reference length.

Positive integer $\nu$ can likewise introduce logarithmic and source-local terms. In those resonant cases, the appropriate canonical response datum in a stated renormalization scheme is the renormalized radial momentum $\pi_{\mathrm{ren}}$, not an unqualified coefficient called $\beta$ de Haro, Skenderis, and Solodukhin 2001, §5.1, especially eqs. (5.2)–(5.11). Alternate or mixed conditions also require the counterterms and finite boundary terms appropriate to the renormalized action

$$S_{\mathrm{ren}}=S_{\mathrm{bulk}}+S_{\mathrm{ct}}+S_{\mathrm{bdy}}.$$

Here $S_{\mathrm{ct}}$ removes cutoff divergences, while a finite $S_{\mathrm{bdy}}$ encodes mixed or dynamical boundary data when such a choice is intended Witten 2001, §3, eqs. (3.1)–(3.3), PDF. The corresponding mixed-boundary integrability condition is given in Amsel and Marolf 2006, §§II–III, eqs. (2.17) and (3.10), PDF.

Let the outward normal point toward decreasing $z$. For nonresonant $\nu>0$, the renormalized symplectic flux of two tangent variations through a boundary segment is

$$\begin{aligned} \Omega_{\mathscr I}^{\mathrm{out}}(\delta_1\phi,\delta_2\phi) =-2\nu L^{d-1}\int_{\mathscr I}\mathrm d^d x \sqrt{\lvert g_{(0)}\rvert}\, \bigl( \delta_1\alpha\,\delta_2\beta -\delta_2\alpha\,\delta_1\beta \bigr). \end{aligned}$$

Reversing the orientation reverses the overall sign, not the zero-flux criterion. At the BF point the preceding expression must not be used with $\nu=0$. Direct substitution of the logarithmic pair into the radial symplectic current cancels the terms proportional to $d/2$ and $\log(z/\ell_\mu)$; for the displayed $+\alpha\log(z/\ell_\mu)$ convention, the finite term is

$$\Omega_{\mathscr I}^{\mathrm{out}} =L^{d-1}\int_{\mathscr I}\mathrm d^d x \sqrt{\lvert g_{(0)}\rvert}\, \bigl( \delta_1\alpha\,\delta_2\beta -\delta_2\alpha\,\delta_1\beta \bigr).$$

More generally, the flux is the antisymmetrized pairing of the source datum with $\pi_{\mathrm{ren}}$. The non-BF phase-space flux and its mixed-boundary integrability condition are developed in Amsel and Marolf 2006, §§II–III, especially eqs. (2.9)–(2.17) and (3.10), PDF.

The tangent-space wording matters. The following familiar conditions make the appropriate flux vanish:

• Fixed standard source: $\delta\alpha=0$. For an autonomous source-free problem one sets $\alpha=0$; calling this merely “the normalizable condition” is misleading because both branches can be normalizable in the BF window.
• Fixed alternate source: $\delta\beta=0$, in the ordinary alternate-quantization window $0<\nu<1$. The endpoints $\nu=0$ and $\nu=1$ require separate logarithmic or null-state analysis.
• Real mixed condition: in a convention where $\beta=W'(\alpha)$, variations about one background obey $\delta_i\beta=W''(\alpha)\delta_i\alpha$, and hence

$$\delta_1\alpha\,\delta_2\beta -\delta_2\alpha\,\delta_1\beta=0.$$

For a functional $W[\alpha]$, the same statement requires its Hessian to be symmetric in the integrated boundary pairing. Two unrelated finite nonlinear solutions do not obey this cancellation simply because each satisfies $\beta=W'(\alpha)$; symplectic flux is a statement about tangent variations around a common solution.

The scalar regimes are therefore:

• $\nu$ imaginary: the BF bound is violated, and the spatial operator is unbounded below.
• $\nu=0$: the two powers merge into a logarithmic pair; flux and positivity require the endpoint analysis.
• $0<\nu<1$: in each fixed spherical-harmonic sector, the radial endpoint is limit-circle and admits a one-parameter family of self-adjoint extensions. A local rotation-invariant Robin law selects a restricted common family for the full field; unrestricted extensions can mix angular modes and need not be local. Only a subset of the sectorwise extensions is positive.
• $\nu\geq1$: in the Ishibashi–Wald positive-energy $L^2$ framework, every radial sector is limit-point at infinity and the extension is unique, corresponding to standard falloff. The ordinary alternate window is open and excludes $\nu=1$.

The exact positivity restrictions, including the convention-dependent Robin threshold, are given by Ishibashi and Wald 2004, §3.3, theorem 3.2, especially eqs. (169)–(172), PDF. The theorem’s inequality is sectorwise: its parameter $\sigma=\ell+(d-2)/2$ depends on angular momentum, so a full-field Robin law is positive only if the bound holds in every sector. The two possible CFT dimensions in the open window and the scalar unitarity-bound endpoint are explained in Klebanov and Witten 1999, §2.1.

Flux, self-adjointness, stability, and causality

These are separate tests:

1. Vanishing boundary form makes $A$ symmetric on the proposed domain.
2. Self-adjointness additionally requires a maximal domain: $\operatorname{Dom}(A)=\operatorname{Dom}(A^\dagger)$.
3. Positivity, $A\geq0$, excludes negative-spectrum modes that grow exponentially in global time. Uniform boundedness additionally requires zero modes to be absent or controlled, because $A\psi=0$ permits $(c_1+c_2\tau)\psi$.
4. Boundary locality and retarded support control causal propagation.

Thus zero flux does not by itself prove self-adjointness, self-adjointness does not imply positivity, and neither operator property proves that a spatially or temporally nonlocal boundary law respects a causal cone. The distinctions are easiest to keep straight in one comparison. The last three rows preview the open-system constructions developed below.

What different boundary laws do—and do not—establish

Boundary law	Flux and closure	Causal question	Strongest justified claim
Fixed local α; α = 0 when source-free	tangent symplectic flux vanishes	local in boundary spacetime	for a source-free time-independent law: candidate conservative evolution; maximality, positivity, and zero modes still require checks
Fixed local β in 0 < ν < 1	tangent symplectic flux vanishes	local in boundary spacetime	alternate closed evolution after mass-window, maximality, and positivity checks
Real local β = W′(α)	the tangent Hessian cancels the flux	causal if it is local in boundary spacetime and belongs to a well-posed hyperbolic problem	a candidate closed nonlinear phase space; its linearized Hessian gives a candidate symmetric domain, while maximality and positivity remain independent
Real symmetric but spatially nonlocal kernel	integrated flux can vanish	instantaneous coupling can join boundary-spacelike points	flux conservation without a local causal domain of dependence
Action-derived local boundary field with conservative flux matching	bulk flux is balanced by boundary-field flux	causal only if the coupled problem is well posed and its characteristic cone lies within the declared boundary causal cone	a candidate closed bulk-plus-boundary system
Retarded dissipative kernel	bulk flux is generally nonzero	causal support must be checked	a causal open bulk sector, or a closed system only after its bath is supplied
Arbitrary f(Ω, k)	no automatic flux or norm statement	reality, analyticity, growth, and spacetime support all require tests	only a formal boundary relation until those tests pass

Fixed boundary data can also inject energy explicitly even when their variations obey the variational boundary condition; a prescribed time-dependent $J$ is external driving. Autonomous unitary evolution requires a time-independent operator domain, or explicit dynamical boundary variables whose energy and symplectic form are included.

A solved reflective scalar problem in global AdS

Now impose the standard source-free condition $\alpha=0$ and choose the positive Friedrichs extension for $m^2L^2\geq-d^2/4$. Separate a complexified solution as

$$\phi=e^{-i\omega\tau}R_{\omega\ell}(\chi)Y_{\ell\sigma}(\widehat{\mathbf n}), \qquad -\nabla_{S^{d-1}}^2Y_{\ell\sigma}=\ell(\ell+d-2)Y_{\ell\sigma}.$$

The radial equation is

$$R''+\frac{d-1}{\sin\chi\cos\chi}R' +\left[ \omega^2-\frac{\ell(\ell+d-2)}{\sin^2\chi} -\frac{m^2L^2}{\cos^2\chi} \right]R=0.$$

Here $\widehat{\mathbf n}\in S^{d-1}$, while $\sigma$ labels the angular degeneracy and is unrelated to the scalar mass $m$. Set $a_\ell=\ell+d/2-1$ and normalize the spherical harmonics by $\int_{S^{d-1}}\mathrm d\Omega_{d-1}\,\lvert Y_{\ell\sigma}\rvert^2=1$. The positive-frequency modes are

$$\begin{aligned} u_{n\ell\sigma} ={}&N_{n\ell}e^{-i\omega_{n\ell}\tau} (\sin\chi)^\ell(\cos\chi)^{\Delta_+}\\ &\times P_n^{(a_\ell,\nu)} \!\left(1-2\sin^2\chi\right)Y_{\ell\sigma}(\widehat{\mathbf n}),\\ \omega_{n\ell} ={}&\Delta_++2n+\ell, \qquad n,\ell\in\mathbb N_0. \end{aligned}$$

Every factor has a job. Near the center, $(\sin\chi)^\ell$ selects the regular solution. Near the boundary, $(\cos\chi)^{\Delta_+}$ sets the slow coefficient $\alpha$ to zero. The Jacobi polynomial arises when the regular hypergeometric series terminates; that termination quantizes the dimensionless global frequency. The physical frequency is $\omega_{n\ell}/L$ Aharony, Gubser, Maldacena, Ooguri, and Oz 2000, §2.2.2, eqs. (2.33)–(2.42), PDF.

On a constant-$\tau$ slice, the Klein–Gordon pairing becomes

$$\begin{aligned} (u,v)_{\mathrm{KG}} =iL^{d-1}\int_0^{\pi/2}\!\mathrm d\chi\, \tan^{d-1}\chi\int_{S^{d-1}}\!\mathrm d\Omega_{d-1}\, \left(u^*\partial_\tau v-v\partial_\tau u^*\right). \end{aligned}$$

With the action normalization above, unit positive norm is obtained from

$$\lvert N_{n\ell}\rvert^2 =\frac{n!\,\Gamma(n+\ell+\Delta_+)} {L^{d-1}\Gamma(n+\ell+d/2)\Gamma(n+\nu+1)}.$$

The Jacobi orthogonality relation then gives

$$(u_{n\ell\sigma},u_{n'\ell'\sigma'})_{\mathrm{KG}} =\delta_{nn'}\delta_{\ell\ell'}\delta_{\sigma\sigma'}.$$

Smooth finite-energy initial data satisfying the boundary condition and corner compatibility can be expanded as

$$\phi=\sum_{n,\ell,\sigma} \left(a_{n\ell\sigma}u_{n\ell\sigma} +a_{n\ell\sigma}^*u_{n\ell\sigma}^*\right), \qquad a_{n\ell\sigma}=(u_{n\ell\sigma},\phi)_{\mathrm{KG}}.$$

Together with completeness from the self-adjoint spectral theorem, this mode construction solves the scalar initial–boundary value problem: regularity, the boundary domain, and initial data fix a unique linear evolution within the declared finite-energy class.

The conservation check is independent of the mode sum. Let $\mathscr I_{12}$ be the boundary segment between two global slices. Stokes’ theorem gives, with the outward convention above,

$$(u,v)_{\Sigma_2}-(u,v)_{\Sigma_1} =-i\,\Omega_{\mathscr I_{12}}^{\mathrm{out}}(u^*,v).$$

Both modes have $\alpha=0$, so the boundary term vanishes and the pairing is slice independent. The Stokes calculation independently proves conservation of the Klein–Gordon pairing for this domain. It does not by itself prove spectral completeness, construct an interacting QFT, establish nonlinear stability, or establish a holographic duality.

Domains of dependence and boundary-returned signals

The compactified metric makes the first causal test almost visual:

$$\Delta\tau_{\mathrm{center}\to\mathscr I} =\int_0^{\pi/2}\mathrm d\chi=\frac\pi2, \qquad \Delta\tau_{\mathrm{round\ trip}}=\pi.$$

The causal strip linked above shows the same result: the outward ray is fixed by the bulk metric, whereas the dashed return exists only after a reflective boundary law has been chosen.

For a local causal boundary condition, the field at a bulk point $p$ depends on initial data in $J^-(p)\cap\Sigma$ and on boundary data in $J^-(p)\cap\mathscr I$, where $J^-(p)$ is the causal past of $p$. Before that causal past reaches $\mathscr I$, local hyperbolicity makes the result independent of the boundary condition. After first contact, reflective, absorbing, or transmitting-to-a-declared-bath laws give different contributions. Energy estimates establish this finite-speed initial–boundary value formulation for Dirichlet, Neumann, and Robin classes under suitable asymptotically AdS hypotheses Warnick 2013, §§4.1, 4.4, and 6.2.

Flux cancellation alone does not guarantee this causal support. For example,

$$\beta(t,\mathbf x)=\int_{\partial\Sigma_t}\mathrm d\Sigma_y\, K(\mathbf x,\mathbf y)\alpha(t,\mathbf y), \qquad K(\mathbf x,\mathbf y)=K(\mathbf y,\mathbf x)^*,$$

where $\mathrm d\Sigma_y=\mathrm d^{d-1}y\sqrt{\sigma(\mathbf y)}$ is the spatial boundary measure. For a real scalar kernel, the adjointness condition reduces to real symmetry. It makes the integrated tangent flux vanish, but a kernel joining boundary-spacelike points makes the boundary condition respond instantaneously across space. The completed initial–boundary value problem—not the bulk differential operator alone—determines the actual domain of dependence.

Open boundaries and retarded response

Frequency dependence is not itself pathological. Let $t$ be a physical boundary time—on the global cylinder, $t=L\tau$—and let $\Omega$ be its conjugate frequency. Fix the Fourier convention

$$\alpha(t,\mathbf x)=\int_{-\infty}^{\infty} \frac{\mathrm d\Omega}{2\pi}\, e^{-i\Omega t}\alpha(\Omega,\mathbf x).$$

A linear response law is retarded only if it can be written as

$$\beta(t,\mathbf x)=\int\mathrm dt'\int_{\partial\Sigma_{t'}}\mathrm d\Sigma_y\, K_R(t,\mathbf x;t',\mathbf y)\alpha(t',\mathbf y),$$

with $K_R=0$ unless the source point lies in the appropriate past. If the boundary theory is claimed to be local, mere support at $t'\leq t$ is not enough: $K_R$ must also vanish outside the boundary causal cone.

For a stationary translation-invariant relation $\beta=f(\Omega,\mathbf k)\alpha$, reality requires

$$f(-\Omega^*, -\mathbf k)=f(\Omega,\mathbf k)^*,$$

For a stable tempered retarded kernel, $f$ must be analytic in $\operatorname{Im}\Omega>0$ and obey suitable growth or subtraction bounds; Son and Starinets 2002, §2, eq. (2.10) illustrates the upper-half-plane condition. With the outward-flux convention used here, the scalar flux formula below makes passivity of positive-frequency modes require $\Omega\,\operatorname{Im}f(\Omega,\mathbf k)\geq0$. Spatial locality still requires support inside the boundary causal cone. None of these tests by itself establishes all the others, positivity, or completeness of the boundary state space. The site’s retarded-response conventions give the general time-domain interpretation. The holographic radial-flux prescription appears in Son and Starinets 2002, §3.2, eq. (3.14).

A clean closed completion can contain a genuine boundary field $\psi_{\partial}$. Choose a local boundary action whose Euler–Lagrange equation is hyperbolic and whose matching condition cancels the bulk flux. The bulk flux then need not vanish separately, while

$$\Omega_{\mathrm{total}}=\Omega_{\mathrm{bulk}}+\Omega_{\partial}$$

is conserved. Generalized Wentzell conditions give an explicit bulk-plus-boundary realization of this mechanism Dappiaggi, Ferreira, and Juárez-Aubry 2018, §§III.A and IV. This proves cancellation of symplectic flux in the combined system; it does not by itself prove nondegeneracy, positivity, self-adjoint Hamiltonian evolution, or unitarity. Absence of bound states, well-posedness of the coupled problem, and compatibility of the boundary characteristic cone with the declared causal cone remain separate checks.

Failure tests: dissipation, instability, and acausality

Three failures that look similar in a mode equation have different meanings.

Dissipation without acausality. Let $\nu>0$ and

$$\beta(\Omega)=i\gamma\Omega\alpha(\Omega), \qquad \gamma>0.$$

Because $[\beta/\alpha]=\text{mass}^{2\nu}$, the physical-time coefficient has $[\gamma]=\text{mass}^{2\nu-1}$. With the declared transform, this is the local, reality-compatible condition $\beta(t)=-\gamma\,\dot\alpha(t)$. For a positive-frequency component $\beta=f\alpha$, its outward Hermitian Klein–Gordon flux rate through a spatial boundary section is

$$\begin{aligned} \frac{\mathrm d\Phi_{\mathrm{KG}}^{\mathrm{out}}}{\mathrm dt} &=-i\,2\nu L^{d-1} \int_{\partial\Sigma_t}\mathrm d^{d-1}x\sqrt{\sigma} \left(\alpha^*\beta-\beta^*\alpha\right)\\ &=4\nu L^{d-1} \int_{\partial\Sigma_t}\mathrm d^{d-1}x\sqrt{\sigma}\, \operatorname{Im}f(\Omega)\lvert\alpha_\Omega\rvert^2. \end{aligned}$$

It is therefore proportional to $\gamma\Omega\lvert\alpha_\Omega\rvert^2$, whereas the associated outward Killing-energy flux rate carries one more factor of frequency and is proportional to $\gamma\Omega^2\lvert\alpha_\Omega\rvert^2$. The AdS-region pairing is not conserved, but the boundary relation is not acausal.

The calculation establishes a local, reality-compatible, passive dissipative boundary relation. A causal well-posed bulk evolution still requires an energy estimate or semigroup theorem for this boundary law; a unitary completion additionally requires the bath that receives the flux.

Instability without a causal verdict. Even a self-adjoint domain can have $A\psi=-\lambda^2\psi$, producing $e^{\pm\lambda\tau}\psi$ and violating stability. Likewise, a pole of $f(\Omega)$ in the upper half-plane rules out a stable tempered retarded transfer function defined by the usual real-frequency boundary value.

Such a pole does not, by location alone, prove pre-response: a Bromwich contour above it can instead describe a causal but exponentially growing kernel. The strongest immediate conclusion is loss of the stable retarded prescription; support must be inspected separately.

Acausality by direct support. For $g,\kappa>0$, with $[\kappa]=\text{mass}$ and $[g]=\text{mass}^{2\nu+1}$,

$$K_A(t-t')=g e^{\kappa(t-t')}\theta(t'-t), \qquad f_A(\Omega)=\frac{g}{\kappa+i\Omega},$$

under the declared real-axis inverse transform. The value of $\beta(t)$ depends explicitly on $\alpha(t')$ at later times $t'>t$. This is an advanced boundary law, so a retarded causal claim fails regardless of whether some formal radial solution exists.

Common pitfalls

Treating zero flux as the whole theorem. Zero boundary form establishes symmetry on the proposed domain. Maximality, positivity, and causal support are additional statements with additional tests.

Calling $\alpha=0$ “the normalizable choice.” In the open BF window both branches may have finite renormalized norm. Say “standard source-free boundary condition” and state the mass window.

Testing a nonlinear condition on two finite solutions. Symplectic flux pairs tangent variations about one background. Use the symmetric Hessian of $W$, not $\alpha_1W'(\alpha_2)-W'(\alpha_1)\alpha_2$.

Equating frequency dependence with acausality. Local boundary dynamics, causal memory, dissipation, instability, and advanced support all produce frequency dependence. Diagnose support, analytic growth, flux, and spectrum separately.

Using “transparent” without naming what lies beyond the boundary. The term has been used for inequivalent conditions. State instead whether flux is reflected, absorbed, or transmitted to specified boundary or exterior degrees of freedom.

Exercises

1. Slab balance and conserved pairing

Let $u$ and $v$ solve the Klein–Gordon equation between global slices $\Sigma_1$ and $\Sigma_2$. Derive the balance law for $(u,v)_{\mathrm{KG}}$ and show that the standard source-free condition conserves the pairing.

Solution: slab balance

The current

$$j^M(u,v)=i\left(u^*\nabla^Mv-v\nabla^Mu^*\right)$$

obeys $\nabla_Mj^M=0$. Integrating over the spacetime slab and using the outward orientation on every boundary component gives

$$(u,v)_{\Sigma_2}-(u,v)_{\Sigma_1} =-\Phi_{\mathrm{KG},\mathscr I}^{\mathrm{out}} =-i\Omega_{\mathscr I}^{\mathrm{out}}(u^*,v).$$

For standard source-free data, $\alpha_u=\alpha_v=0$, so the asymptotic flux vanishes. Hence $(u,v)_{\Sigma_2}=(u,v)_{\Sigma_1}$.

2. Derive the global normal-mode spectrum

Insert $e^{-i\omega\tau}R(\chi)Y_{\ell\sigma}$ into the wave equation. With $y=\sin^2\chi$, first identify the regular exponents at $y=0$ and $y=1$, then factor $R=y^{\ell/2}(1-y)^{\Delta_+/2}F(y)$. Show that regular standard modes have $\omega=\Delta_++\ell+2n$.

Solution: mode quantization

After the stated factorization, $F$ satisfies the hypergeometric equation with

$$a=\frac{\Delta_++\ell-\omega}{2}, \qquad b=\frac{\Delta_++\ell+\omega}{2}, \qquad c=\ell+\frac d2.$$

The solution regular at $y=0$ is ${}_2F_1(a,b;c;y)$. Standard source-free falloff at $y=1$ is obtained for the positive-frequency branch when $a=-n$, $n\in\mathbb N_0$. Therefore

$$\omega=\Delta_++\ell+2n.$$

The terminating hypergeometric polynomial is proportional to $P_n^{(\ell+d/2-1,\nu)}(1-2y)$, reproducing the modes in the text.

3. Mixed flux, nonlocality, and positivity

(a) Show that $\beta=W'(\alpha)$ has zero tangent symplectic flux for a real local $W$. (b) Generalize to a spatial kernel and identify the extra condition needed for integrated flux cancellation. (c) Explain why neither result proves stability.

Solution: mixed-domain tests

(a) About one background, $\delta_i\beta=W''(\alpha)\delta_i\alpha$. Thus

$$\delta_1\alpha\,\delta_2\beta -\delta_2\alpha\,\delta_1\beta=0.$$

(b) If $\delta\beta(\mathbf x)=\int\mathrm d\mathbf y\,H(\mathbf x,\mathbf y)\delta\alpha(\mathbf y)$, the integrated antisymmetric pairing vanishes when $H$ is symmetric with respect to the boundary measure. This symmetry does not prevent $H$ from coupling boundary-spacelike points, so it does not prove finite propagation.

(c) The flux test makes the linearized operator symmetric. A self-adjoint completion can still contain a negative eigenvalue $A\psi=-\kappa^2\psi$, for which $\partial_\tau^2\phi=-A\phi$ has exponentially growing solutions. The invariant stability test is $A\geq0$; a numerical Robin inequality depends on the normalization and signs chosen for $\alpha$ and $\beta$.

4. Retarded, advanced, and dissipative laws

Using the inverse transform $e^{-i\Omega t}$, find the time-domain kernels for

$$f_R(\Omega)=\frac{g}{\kappa-i\Omega}, \qquad f_A(\Omega)=\frac{g}{\kappa+i\Omega}, \qquad f_D(\Omega)=i\gamma\Omega,$$

where $g,\kappa,\gamma>0$, $[\kappa]=\text{mass}$, $[g]=\text{mass}^{2\nu+1}$, and $[\gamma]=\text{mass}^{2\nu-1}$. Classify their support and state whether the AdS-region Klein–Gordon pairing is conserved.

Solution: causal kernels

Contour closure gives the two rational kernels. For the polynomial law, use $\partial_t\delta(t)=\int(\mathrm d\Omega/2\pi)(-i\Omega)e^{-i\Omega t}$. Thus

$$K_R(t)=g e^{-\kappa t}\theta(t), \qquad K_A(t)=g e^{\kappa t}\theta(-t), \qquad K_D(t)=-\gamma\,\partial_t\delta(t).$$

$K_R$ is retarded and decays; $K_A$ is advanced and depends on future data; $K_D$ is local and causal. For real $\Omega>0$,

$$\operatorname{Im}f_R=\frac{g\Omega}{\kappa^2+\Omega^2}, \qquad \operatorname{Im}f_A=-\frac{g\Omega}{\kappa^2+\Omega^2}, \qquad \operatorname{Im}f_D=\gamma\Omega.$$

The retarded and dissipative laws allow outward bulk flux, while the advanced law carries nonzero inward flux under the chosen orientation. The AdS-region pairing is therefore not conserved for any of the three. A closed description requires compensating boundary degrees of freedom; independently, the advanced law fails the causal-support test.

Controlled limits and handoff

The explicit calculation is a free scalar on fixed global AdS with a time-independent positive domain. It does not establish nonlinear gravitational stability, interacting quantum dynamics, or a dual CFT.

Asymptotically locally AdS boundary data replaces the pure-AdS coefficients by the full renormalized source–response data. Boundary conditions, alternate quantization, and deformations develops the BF-window quantizations and their CFT interpretation.

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

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