Consistency between Bulk and Boundary Causalities in Asymptotically Anti-de Sitter Spacetimes

TL;DR

This work addresses the problem of consistency between bulk and boundary causality in static, spherically symmetric, asymptotically AdS spacetimes by deriving a general, perturbative criterion for time advance based on an effective bulk metric. The authors express the time-advance conditions through turning-point analyses of null geodesics, capturing corrections via coefficients C_n and beta_n from asymptotic expansions, and provide explicit formulas for Δφ and Δt in terms of the turning point r_m. Applying the framework to RNAdS shows no time-advance geodesics, while Einstein–Euler–Heisenberg theory admits time advance for certain positive couplings below an AdS-scale bound, with geodesics passing through regions of negative quasi-local energy and violating the effective-null convergence. The results yield concrete parameter constraints to avoid superluminal bulk propagation, contribute to understanding AdS/CFT causality, and offer a general method applicable to higher-curvature corrections and more general spacetimes.

Abstract

We investigate the consistency between bulk and boundary causalities in static, spherically symmetric, asymptotically anti-de Sitter (AdS) spacetimes. We derive a general formula that provides sufficient conditions for time advance, where bulk propagation arrives earlier than any boundary propagation. As an application, we show that in Reissner--Nordström--anti de Sitter spacetime, no geodesic satisfies the sufficient conditions for time advance even in the super-extremal case. Furthermore, we demonstrate that the Einstein--Euler--Heisenberg theory exhibits time advance when one or a linear combination of the coupling constants is positive and below an upper bound determined by the AdS length scale.

Figures (2)

• Figure 1: The left figure depicts the asymptotically AdS spacetime, while the right figure presents an unfolded view of the left diagram. The red and blue curves represent null geodesics along the boundary, whereas the purple and green curves represent null geodesics within the bulk. In the right diagram, the gray region is causally related to $p$ from the perspective of boundary causality, while the white region is spacelike-separated. Consequently, the purple curve ends at $q'$ within the gray region, indicating that no time advance occurs. In contrast, the green curve terminates at $\tilde{q}$ in the white region, leading to a time advance.
• Figure 2: Plots of $J(n,|w|)$ as a function of $|w|$ with fixed $n$. The first, second, and third rows correspond to the case with $n=0.01$, $1$, $100$, respectively. The left and right columns show the plots for $0<|w|<100$ and $100<|w|<1000$, respectively.