Numerical metric extraction in AdS/CFT

TL;DR

The paper develops an iterative, geometry-driven method to extract bulk metrics of asymptotically AdS spacetimes from boundary data, leveraging the RT relation between entanglement entropy and geodesic length in AdS3. A central insight is that the derivative of geodesic length with respect to angular separation, dL/dφ, directly yields the geodesic's angular momentum J and turning point r_min, enabling stepwise recovery of f(r). It then compares this spacelike-geodesic approach with Hammer's null-geodesic method, highlights their complementary strengths, and demonstrates how combining them extends metric recovery to general static, spherically symmetric spacetimes, including a toy AdS3 radiation star. The work shows robustness to moderate approximations and emphasizes practical applicability for probing bulk properties from boundary data, with clear paths for extension to less symmetric and higher-dimensional settings.

Abstract

An iterative method for recovering the bulk information in asymptotically AdS spacetimes is presented. We consider zero energy spacelike geodesics and their relation to the entanglement entropy in three dimensions to determine the metric in certain symmetric cases. A number of comparisons are made with an alternative extraction method presented in arXiv:hep-th/0609202, and the two methods are then combined to allow metric recovery in the most general type of static, spherically symmetric setups. We conclude by extracting the mass and density profiles for a toy model example of a gas of radiation in (2+1)-dimensional AdS.

Figures (16)

• Figure 1: A sample of geodesic paths in $AdS_{3}$ (with $R = 1$), all beginning at the same point on the boundary, with varying $J$ and $E$. The null geodesics (left plot) all terminate at the same (antipodal) point, whereas this is not the case for spacelike geodesics (right plot).
• Figure 2: A static spacelike geodesic in $AdS_{3}$ (left plot), with the regions A and B highlighted (right plot).
• Figure 3: A plot of the proper length, $\mathcal{L}$, vs the angular separation of the endpoints, $\phi$, for static spacelike geodesics in an asymptotically AdS spacetime (red, lower curve), and in pure AdS (black, upper curve). The gradient, $d \mathcal{L}/d \phi$ at each point provides the angular momentum, $J$, for the corresponding geodesic. When the angular separation is small, the geodesics remain far from the centre, away from the deformation, and hence both curves coincide.
• Figure 4: The data points for the largest two step size estimates for $f_{1}(r)$, compared with the actual curve (in blue). Whilst both give good estimates to the curve, the step size of $0.1$ (left) deviates at a higher $r$ than when using a step size of $0.05$ (right).
• Figure 5: The data points for the next-to-smallest step size estimate for $f_{1}(r)$, compared with the actual curve (in blue). The fit here appears very good even close to $r = 0$, however, Table \ref{['table11']} shows that we still need to go to a smaller step size in order to accurately extract values for $\alpha$, $\beta$ and $\gamma$.