Extracting the bulk metric from boundary information in asymptotically AdS spacetimes

TL;DR

The paper addresses whether the bulk metric of static, spherically symmetric, asymptotically AdS spacetimes can be reconstructed from boundary boundary-null geodesic endpoint data. It introduces two iterative methods, Method I and Method II, that exploit the relation $y = L/E$ to extract depth-resolved information, with Method II enhanced by a geometric approximation near the geodesic minimum that avoids explicit $f'(r)$ calculation. The main finding is that for spacetimes with a monotonic effective potential, the entire function $f(r)$ can be recovered from endpoint spectra, while non-monotonic potentials or central singularities limit recovery to a finite radius. The work discusses practical tradeoffs, demonstrates substantial gains in accuracy and efficiency with the modified Method II, and outlines limitations and possible extensions such as spacelike geodesics and non-static geometries.

Abstract

We use geodesic probes to recover the entire bulk metric in certain asymptotically AdS spacetimes. Given a spectrum of null geodesic endpoints on the boundary, we describe two remarkably simple methods for recovering the bulk information. After examining the issues which affect their application in practice, we highlight a significant advantage one has over the other from a computational point of view, and give some illustrative examples. We go on to consider spacetimes where the methods cannot be used to recover the complete bulk metric, and demonstrate how much information can be recovered in these cases.

Figures (11)

• Figure 1: Plot of the effective potential in pure AdS (with $R = 1$) for a sample of null geodesics, with $E = 1.0$ and $L = 0.0,0.1,0.2,0.4,0.6$ and $0.8$
• Figure 2: A plot of some null geodesic paths passing through an AdS spacetime, all starting from the same point $t=0$, $\phi = 0$ on the boundary at $r = \infty$ (compactified to lie at $\arctan(\infty) = \frac{\pi}{2}$ in the diagram).
• Figure 3: A plot of the null geodesic paths passing through a modified AdS spacetime, all starting from the arbitrary point $t=0$, $\phi = 0$ on the boundary and with positive angular momentum. The corresponding full spectrum of null geodesic endpoints for this spacetime is shown on the right.
• Figure 4: A plot showing how the integrand of \ref{['eq:trnminusim1']}, is split up into two end curves and a number of trapeziums in order for the integral to be well approximated. The actual curve is shown in black, with the approximations in red.
• Figure 5: A plot showing an example curve (the integrand of \ref{['eq:tphi5']}) to be approximated in method II. The leftmost part of the curve is initially a vertical parabola.