Causal Wedges and Subregion Reconstruction

The causal wedge of a boundary region is the set of bulk events that can both receive a signal from and send a signal to its boundary domain of dependence. It is fixed by bulk causal structure, not by an entropy extremization. For free fields with suitable boundary conditions, boundary data on that domain can reconstruct solutions in the causal wedge through causal propagators or equivalent smearing maps. This establishes neither reconstruction outside the wedge nor entanglement-wedge recovery. We use Lorentzian Poincaré AdS of radius $L$, standard reflecting boundary conditions, and retarded/advanced propagators whose support fixes the contour unambiguously.

Required background. The bulk reconstruction problem supplies the region/algebra/state/error specification, and AdS geometry supplies conformal boundary causality.

Helpful background. Hyperbolic causal propagators give the domain-of-dependence theorem; time-band reconstruction distinguishes causal access from analytic continuation; and correlation versus causal exchange prevents interpreting correlations as signaling.

Causal wedge and causal information surface

For a boundary spatial region $A$, let $D[A]$ denote its boundary domain of dependence. The bulk causal wedge is

$$\mathcal C_W[A]=J^-_{\rm bulk}(D[A])\cap J^+_{\rm bulk}(D[A]).$$

Its future and past null boundaries meet at the causal information surface $\Xi_A$. In a smooth spacetime this surface is codimension two, but it is not defined by minimizing or extremizing area. Causal-wedge inclusion is monotone under domain inclusion: if $D[A]\subseteq D[B]$, then $\mathcal C_W[A]\subseteq\mathcal C_W[B]$.

The definition is conformally invariant with respect to the positive Weyl factor relating the AdS metric to its compactification. It remains sensitive to the global state because backreaction changes null geodesics and may create horizons or caustics.

A spherical region in pure AdS

In $\mathrm{AdS}_{d+1}$ Poincaré coordinates,

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

Take the boundary ball $A=\{t=0,\,r<R\}$. Its boundary domain is the diamond

$$D[A]=\{z=0:\ |t|+r<R\},$$

with tips $q_-=(-R,0,0)$ and $q_+=(R,0,0)$. Null curves are straight in the conformally related half-Minkowski metric. A bulk point $(t,z,r)$ lies to the future of $q_-$ and to the past of $q_+$ exactly when

$$(t+R)^2\geq z^2+r^2, \qquad (R-t)^2\geq z^2+r^2.$$

Combining the inequalities gives the explicit wedge

$$\boxed{\mathcal C_W[A]= \left\{(t,z,r): |t|+\sqrt{z^2+r^2}<R\right\}.}$$

At $t=0$, its null boundaries meet on

$$\Xi_A:\qquad z^2+r^2=R^2.$$

For this vacuum ball, $\Xi_A$ happens to coincide with the RT hemisphere. The coincidence follows from the symmetry of this special example and must not be promoted to an equality of definitions. Hubeny and Rangamani introduced the causal information surface precisely as a causal, generally nonminimal object (Hubeny and Rangamani 2012, §§2–3).

Free-field data in the wedge

Let $\phi$ obey $(\Box+m^2)\phi=0$ in the wedge, with $m^2L^2=\Delta(\Delta-d)$ and normalizable branch $\phi\sim z^\Delta\mathcal O$. For a bulk point $X\in\mathcal C_W[A]$, choose a hypersurface made from a portion of the timelike boundary inside $D[A]$ plus null pieces on the wedge boundary. Green’s identity gives schematically

$$\phi(X)=\int_{D[A]}d^dx\,K_A(X|x)\mathcal O(x) +\int_{\mathcal N_A}d\Sigma^\mu \big(G\nabla_\mu\phi-\phi\nabla_\mu G\big).$$

When the characteristic data on $\mathcal N_A$ are fixed by regularity and the chosen state/code sector, the second term is determined and the first becomes a subregion smearing representation. Equivalently, one can prepare boundary sources in the past part of $D[A]$, measure responses in its future part, and solve the hyperbolic inverse problem inside the wedge. This reconstructs the free solution and its correlators only for the specified boundary conditions and spectral domain.

For the vacuum ball, insert $X=(0,z_0,0)$. The point is in the wedge exactly when $z_0<R$. As $z_0\to R$, the available boundary support approaches the diamond tips and the inverse becomes more singular. For $z_0>R$, no retarded/advanced construction using only $D[A]$ has causal support at the target. A formula obtained there by analytic continuation uses additional state information, as on the time-band page.

Why a larger entanglement wedge is not a causal result

In generic geometries, the entropy extremal surface and causal information surface differ. The entanglement wedge may extend strictly beyond $\mathcal C_W[A]$. Choose a point

$$X\in \mathcal E_W[A]\setminus\mathcal C_W[A].$$

By definition, at least one of the causal relations needed for two-way communication with $D[A]$ fails. No argument based only on retarded propagation, causal source-response, or null reachability can reconstruct $X$ from the boundary domain. If a boundary-algebra representation exists, its justification must use the state/code structure, entropy or relative-entropy relations, and quantum error correction. Causal-wedge inclusion inside an entanglement wedge under appropriate hypotheses is a consistency condition, not a proof that the two wedges coincide (Headrick et al. 2014, §§2–4).

This is the adversarial control: in a state with $\mathcal E_W[A]\supsetneq\mathcal C_W[A]$, a claimed causal kernel for every entanglement-wedge point must either acquire noncausal support, depend on extra global/state data, or fail. The strongest surviving claim is free-field causal-wedge reconstruction.

Controlled limits and handoff

The explicit ball calculation assumes pure Poincaré AdS, a spherical region, and negligible backreaction. Caustics can make wedge boundaries nonsmooth; black holes can create holes in causal wedges; interactions require perturbative corrections; gravitational observables require dressing anchored in $D[A]$ or its complement. Reconstruction must therefore record the boundary region, operator algebra, dressing, code sector, perturbative order, and error norm.

Finite-$N$ and horizon limits explain where the semiclassical construction stops. The later entanglement and emergent geometry chapter defines entropy extremal surfaces, while entanglement-wedge reconstruction and QEC are treated separately rather than being smuggled into this causal result.

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

• Headrick, M., Hubeny, V. E., Lawrence, A., and Rangamani, M. (2014). “Causality and holographic entanglement entropy.” Journal of High Energy Physics 2014(12), 162. DOI.
• Hubeny, V. E., and Rangamani, M. (2012). “Causal holographic information.” Journal of High Energy Physics 2012(6), 114. DOI.