Chaos at the rim of black hole and fuzzball shadows

Abstract

We study the scattering of massless probes in the vicinity of the photon-sphere of asymptotically AdS black holes and horizon-free microstate geometries (fuzzballs). We find that these exhibit a chaotic behaviour characterised by exponentially large deviations of nearby trajectories. We compute the Lyapunov exponent λ governing the exponential growth in d dimensions and show that it is bounded from above by λ b = √ d−3/2b min where b min is the minimal impact parameter under which a massless particle is swallowed by the black hole or gets trapped in the fuzzball for a very long time. Moreover we observe that λ is typically below the advocated bound on chaos λ H = 2πκ B T / , that in turn characterises the radial fall into the horizon, but the bound is violated in a narrow window near extremality, where the photon-sphere coalesces with the horizon. Finally, we find that fuzzballs are characterised by Lyapunov exponents smaller than those of the corresponding BH's suggesting the possibility of discriminating the existence of microstructures at horizon scales via the detection of ring-down modes with time scales λ −1 longer than those expected for a BH of the given mass and spin.

The Regge limit of AdS3 holographic correlators

“…Despite similarities between the conical defect and the effective 3D geometries describing the heavy pure states, the resulting eikonals are different already at this order. We show that results obtained from CFT correlators are in perfect agreement with the eikonal derived by studying geodesics in the dual microstate geometry -the properties of geodesics in microstate geometries have been studied from various perspectives also in [34][35][36][37][38][39]. By following [12,15] we study the relevant bootstrap relation and show that it is satisfied by a different set of CFT data than in the conical defect case [14].…”

The multipolar structure of fuzzballs

“…Moreover these invariants grow always monotonically with the size (average distance between the centers) of the microstate. These properties are analogous to the fact that the quasi-normal mode exponential decay rate (the Lyapunov exponent of unstable null geodesics near the photon sphere) is maximum for the BH solution [43] and provide a portal to test the fuzzball proposal phenomenologically.…”

“…It is also intriguing to note that the Lyapunov exponent of unstable null geodesics near the photon sphere was found to be maximum for the BH solution relative to the microstate geometries [43]. This suggests that the BH metric is an extremum in the parameter space of the solutions of the theory for several (apparently disconnected) quantities.…”

“…From this vantage point, BH microstates are associated to smooth horizonless geometries with the same asymptotics (mass, charges, and angular momenta). Classical properties of BHs emerge as a result of a coarse-graining averaging procedure or as a 'collective behavior' of fuzzballs [39][40][41][42][43]. Unfortunately, so far, finding a statistically significant fraction for five-dimensional (3charge) and for four-dimensional (4-charge) BPS BHs have proven to be too challenging JHEP01(2021)003 of a task.…”

Slow scrambling in extremal BTZ and microstate geometries

“…Scrambling in BTZ has also been studied from the perspective of mutual information in [26]. In the context of microstate geometries resembling maximally rotating BTZ black holes, note that an interesting recent paper [27] has described a different kind of Lyapunov behaviour associated to geodesic instability near photon spheres. This latter Lyapunov exponent is related to quasi-normal decay [28], while the focus of our work is on the Lyapunov growth displayed by OTOCs, which is of a different nature.…”