The Gravity Dual of Boundary Causality

TL;DR

This paper reframes boundary causality in AdS/CFT as a geometric condition on the bulk: the averaged light-cone tilt I(γ) = ∫_γ δg_ab k^a k^b dλ along every complete null geodesic γ must satisfy I(γ) ≥ 0 to preserve boundary causality at linear order. The authors prove that this condition is necessary and sufficient for preserving the boundary causal structure under perturbative bulk corrections, and they show it is strictly weaker than the Gao–Wald ANCC. In connecting to established results, they demonstrate that the ANCC implies boundary causality but that boundary causality does not imply the ANCC, and they provide a concrete perturbation that preserves BCC while violating linearized ANCC. Overall, the work identifies a gauge-invariant, background-dependent criterion that governs bulk perturbations consistent with boundary causality and suggests avenues toward a background-independent formulation in perturbative quantum gravity.

Abstract

In gauge/gravity duality, points which are not causally related on the boundary cannot be causally related through the bulk; this is the statement of boundary causality. By the Gao-Wald theorem, the averaged null energy condition in the bulk is sufficient to ensure this property. Here we proceed in the converse direction: we derive a necessary as well as sufficient condition for the preservation of boundary causality under perturbative (quantum or stringy) corrections to the bulk. The condition that we find is a (background-dependent) constraint on the amount by which light cones can "open" over all null bulk geodesics. We show that this constraint is weaker than the averaged null energy condition.

Figures (3)

• Figure 1: \ref{['subfig:GaoWald']}: an illustration of a consequence of the Gao-Wald theorem GaoWal00. In any asymptotically AdS spacetime, lightlike signals fired along the boundary from the point $p_{-}$ reconverge at the point $p_{+}$. When the spacetime obeys the ANCC (and some other technical assumptions), every bulk null geodesic $\gamma$ (red) starting at $p_{-}$ arrives nowhere in the (causal) past of $p_{+}$. \ref{['subfig:AdS']}: in an asymptotically AdS spacetime saturating the BCC (so that the spacetime saturates the ANCC and violates the null generic condition), there exists at least one null bulk geodesic $\gamma$ (red) connecting a pair of boundary points $p_{-}$ and $p_{+}$. Pure AdS saturates the BCC "maximally" in the sense that all null bulk geodesics arrive at the same time as their boundary-contained counterparts.
• Figure 2: \ref{['subfig:Poincare']}: the boundary $\mathcal{H}$ of the future of a point $p_-$ on the boundary of pure AdS; this surface is a Poincaré horizon of AdS. Generators of $\mathcal{H}$ are shown as lines that emanate from $p_-$ and reconverge at the antipodal boundary point $p_+$. Here we also highlight a spatial slice $\Sigma$ (whose geometry is slice-independent), and we illustrate a basis adapted to $\mathcal{H}$. This basis consists of the vector field $k^a$ tangent to the generators of $\mathcal{H}$ as well as coordinate basis vectors $\xi_i^a$. \ref{['subfig:pert']}: boundary causality requires that every null geodesic $\bar{\gamma}$ (red) of the perturbed spacetime which starts at $p_-$ must reach the boundary nowhere to the past of $p_+$. This requires that for all generators $\gamma$ of $\mathcal{H}$, the deviation vector $\eta^a_+(\gamma)$ cannot be past-directed.
• Figure 3: A flowchart showing the logical relationships between assorted conditions often assumed in perturbative and semiclassical gravity. Acronyms are defined in the main text, with the exception of the Einstein field equation (EFE). Ignoring assorted assumptions, the implications are as follows: the QFC implies the GSL when applied to a causal horizon, which in turn was shown in Wal10QST to imply the BCC. The QFC also implies the ANCC in the classical limit ($\hbar \to 0$) when integrated along a complete null geodesic, which in turn likewise implies the BCC by Gao-Wald. In the probe limit ($G_{N}\rightarrow 0$) and under use of the EFE, the QFC implies the QNEC, which in turn implies the ANEC when integrated along complete null geodesics; the ANCC and the ANEC are equivalent when the EFE are invoked. We emphasize that in this paper we are not restricting ourselves to either the $\hbar \to 0$ or $G_N \to 0$ limits. Finally, we note that the QFC also implies the Bousso bound Bou99bBou99cBou02, but it is not clear if and how this bound is related to the BCC.

Theorems & Definitions (5)

• Theorem 1
• Proposition 1
• Theorem 2
• proof
• Proposition 1