AdS Boundary Conditions and the Black-Hole Region

An asymptotically anti-de Sitter spacetime has a timelike conformal boundary, so two logically separate choices must be made. The metric and its conformal completion determine which bulk events can send causal signals to that boundary; this defines the black-hole region. Boundary conditions on scalar, electromagnetic, or gravitational perturbations determine whether initial data have a unique evolution and how fields return from, or exchange flux through, the boundary. Changing field boundary conditions on a fixed metric does not change the metric causal past and therefore does not move its event horizon.

Helpful background. Exact Solutions of Einstein’s Equations supplies conformal completions and asymptotic metrics; Causality and Global Structure supplies causal futures, pasts, and domains of dependence; The Einstein Equation supplies the background and linearized field equations; The Gibbons-Hawking-York Boundary Term supplies the role of prescribed boundary data in a variational problem; and Black-Hole Geometry and Mechanics supplies the event-horizon definition generalized here to a timelike infinity.

The causal definition at a timelike boundary

Let $(\widetilde{\mathcal M},\widetilde g)$ be a conformal completion of the physical spacetime $(\mathcal M,g)$, with

$$\widetilde g=\Omega^2 g, \qquad \mathscr I=\partial\widetilde{\mathcal M}, \qquad \Omega\big\rvert_{\mathscr I}=0, \qquad d\Omega\big\rvert_{\mathscr I}\ne0.$$

Assume $\mathscr I$ is timelike and choose a connected component $\mathscr I_0$ representing the exterior under study. The boundary-accessible region and black-hole region relative to that component are

$$\mathcal E_0=J^-_{\widetilde{\mathcal M}}(\mathscr I_0)\cap\mathcal M, \qquad \mathcal B_0=\mathcal M\setminus\mathcal E_0,$$

and the future event horizon is

$$\mathcal H_0^+=\partial\mathcal E_0\cap\mathcal M.$$

If several boundary components exist, the subscript matters: a point can communicate with one asymptotic region and not another. With chronological sets, the associated domain of outer communications is $I^+(\mathscr I_0)\cap I^-(\mathscr I_0)$. This conformal-boundary formulation and the component-dependent exterior are set out in Galloway, Schleich, Witt, and Woolgar 1999, §II, equation (1), Open PDF, pp. 4–8.

The sets $J^-(\mathscr I_0)$ and $\mathcal H_0^+$ are computed from the null cones of $\widetilde g$, equivalently those of $g$ in the interior. A scalar Robin coefficient, a reflecting condition, or a choice of perturbation falloff is not an input to this causal calculation. Boundary conditions can affect the metric only in a different problem in which backreaction is solved and a new spacetime results.

Why initial data are not sufficient

Timelike $\mathscr I$ can both receive and emit causal curves. A spacelike bulk slice without its boundary is therefore not a Cauchy surface: data can enter later from infinity. Pure AdS already displays this failure of global hyperbolicity, although it has no black-hole region. Ishibashi and Wald state this obstruction and the need for additional data for deterministic linear evolution in Ishibashi and Wald 2004, §1, Open PDF, pp. 2–4.

The precise static theorem has narrower hypotheses than the slogan “AdS needs boundary conditions.” Assume:

• a static, stably causal background with a complete hypersurface-orthogonal timelike Killing flow;
• a linear field equation reducible on a static slice $\Sigma$ to $\partial_t^2\phi=-A\phi$;
• a positive, symmetric, elliptic operator $A$ on the appropriate $L^2$ space; and
• dynamics that agrees locally with the field equation, has conserved positive energy, respects time translation and reflection, and satisfies the stated convergence condition.

Under these assumptions, admissible dynamics are in one-to-one correspondence with positive self-adjoint extensions $A_E$ of $A$. For initial data $(\phi_0,\dot\phi_0)$ the evolution is

$$\phi(t)=\cos\!\left(A_E^{1/2}t\right)\phi_0 +A_E^{-1/2}\sin\!\left(A_E^{1/2}t\right)\dot\phi_0.$$

Different extensions agree with the locally determined solution inside the initial domain of dependence. They can differ only after causal contact with the boundary. The theorem and this evolution formula appear in Ishibashi and Wald 2004, §1, equations (1)–(4), Open PDF, pp. 2–4.

Falloffs, reflection, and open evolution

For a scalar in $D$ spacetime dimensions, a concrete way to read the endpoint parameter is

$$\nu^2=\frac{(D-1)^2}{4}+m_{\mathrm{eff}}^2L^2, \qquad \Delta_\pm=\frac{D-1}{2}\pm\nu,$$

where $L$ is the AdS radius and $m_{\mathrm{eff}}^2$ includes any curvature coupling. With a defining coordinate $z\sim L/r$ near the boundary, the two scalar branches behave as

$$\phi\sim \alpha z^{\Delta_-}+\beta z^{\Delta_+}.$$

At the Breitenlohner–Freedman endpoint $\nu=0$, the independent second branch is $z^{(D-1)/2}\log z$ rather than another distinct power. For electromagnetic and gravitational master equations, $\nu$ is instead the effective endpoint index obtained after reduction; the scalar mass formula should not be applied to those sectors unchanged.

For the separated AdS problem, the endpoint behavior decides whether $A$ has a unique extension or a family. In the notation of Ishibashi and Wald, $\nu^2\ge1$ gives a unique positive self-adjoint extension. For $0\lt\nu^2\lt1$, there is a one-parameter family characterized by the ratio of the two admissible asymptotic coefficients; only a specified subrange is positive. Generalized Dirichlet and Neumann conditions are distinguished members, and the other real ratios are generalized Robin conditions Ishibashi and Wald 2004, §3.3, equations (158)–(171), Open PDF, pp. 32–35.

At $\nu=0$, a one-parameter extension family still exists, characterized by the ratio of the power-law and logarithmic coefficients, but its boundary-data interpretation differs. Only the theorem-specified subrange is positive; the Friedrichs extension, for which the logarithmic-branch coefficient vanishes, is included in that range. If $\nu^2\lt0$, the spatial operator is unbounded below and has no positive self-adjoint extension Ishibashi and Wald 2004, §3.3, equations (164)–(171) and Theorem 3.2, Open PDF, pp. 32–35. Thus neither the Breitenlohner–Freedman endpoint nor the below-bound regime may be silently included in the positive two-power-law family just described.

These positive self-adjoint choices describe a closed bulk system with conserved positive energy: in that precise sense they are reflecting. A genuinely absorbing or transparent boundary is not another positive self-adjoint extension on the same bulk Hilbert space. It requires an open-system evolution or additional degrees of freedom beyond $\mathscr I$. One must then state the enlarged system and its energy balance. Even so, for a fixed background metric the geometric set $J^-(\mathscr I_0)$ remains unchanged.

Scope and counterexamples

The self-adjoint-extension result is not a nonlinear existence theorem for the Einstein equation, and its static uniqueness assumptions do not automatically extend to Kerr–AdS. Gauge constraints also have to be handled before electromagnetic or gravitational perturbations reduce to the required scalar form. Boundary conditions that violate positivity can admit growing modes; boundary data that change the conformal metric define a different gravitational boundary-value problem.

Two examples prevent common conflations. First, pure AdS needs a field boundary condition but has $\mathcal B_0=\varnothing$: non-global hyperbolicity does not imply a black hole. Second, place the same linear scalar field on one fixed Schwarzschild–AdS metric and choose two allowed Robin conditions. The spectra and reflected signals may differ, yet $f(r_+)=0$ and $\mathcal H_0^+$ are identical because the metric has not changed.

Worked transfer diagnostic

A scalar perturbation on a fixed Schwarzschild–AdS background is evolved once with generalized Dirichlet data and once with an allowed generalized Robin condition. Which of the following can change: the event-horizon radius, the boundary-accessible set, the normal-mode spectrum, and the flux law?

Solution

The horizon radius and boundary-accessible set cannot change. Both are determined by the fixed metric through $f(r_+)=0$ and $J^-(\mathscr I_0)$, respectively. The normal-mode spectrum can change because the domain of $A_E$ has changed. For two positive self-adjoint extensions, the detailed boundary relation changes while the appropriate total bulk energy remains conserved; if one instead imposes an absorbing rule outside the self-adjoint class, the bulk flux law also changes and the system must be treated as open. None of these field choices moves the event horizon unless backreaction produces a different metric.

Next, Extended AdS Black-Hole Mechanics allows the cosmological constant itself to vary while keeping the asymptotic phase space explicit.

References

• Galloway, G. J., Schleich, K., Witt, D. M., and Woolgar, E. “Topological Censorship and Higher Genus Black Holes.” Physical Review D 60 (1999): 104039. doi:10.1103/PhysRevD.60.104039. Open PDF.
• Ishibashi, A., and Wald, R. M. “Dynamics in Non-Globally-Hyperbolic Static Spacetimes III: Anti-de Sitter Spacetime.” Classical and Quantum Gravity 21 (2004): 2981–3013. doi:10.1088/0264-9381/21/12/012. Open PDF.