Chaos at the rim of black hole and fuzzball shadows
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Springer
Received: February 25, 2020 Accepted: April 30, 2020 Published: May 18, 2020
Chaos at the rim of black hole and fuzzball shadows
a
Dipartimento di Fisica, Universit` a di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy b I.N.F.N. Sezione di Roma “Tor Vergata”, Via della Ricerca Scientifica, 00133 Roma, Italy
E-mail: [email protected], [email protected], [email protected] 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 λb = d−3/2bmin where bmin 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 λH = 2πκB 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. Keywords: Black Holes, Black Holes in String Theory, AdS-CFT Correspondence, Gaugegravity correspondence ArXiv ePrint: 2002.05574
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP05(2020)078
JHEP05(2020)078
M. Bianchi,a,b A. Grilloa,b and J.F. Moralesb
Contents 1
2 Geodesics on AdS Kerr-Newman space-times 2.1 The metric 2.2 The Hamiltonian and momenta 2.3 Motion in the radial direction 2.4 Motion in the θ direction 2.5 Geodesics inside the absorption region
4 4 4 6 7 9
3 Critical and nearly critical geodesics 3.1 Critical geodesics 3.2 The Lyapunov exponent
10 10 11
4 Analytic results for specific BH’s 4.1 Non rotating BH’s in AdS 4.2 Rotating BH’s 4.3 Spherical symmetric BH’s in higher dimensions
13 13 15 15
5 Fuzzball geometries 5.1 The BH geometry 5.2 The asymptotically flat fuzzball 5.3 Asymptotically AdS geometry
17 18 18 19
6 Discussion and outlook
21
A Integrating the angular motion
23
1
Introduction
A light ray travelling near a massive object gets deflected and delayed with respect to its propagation in flat space-time. If the massive object is a black hole (BH), the time delay and angular deflection, that are small at large impact parameter, can get arbitrarily large at some critical value. For critical impact parameters even light gets trapped in (unstable) ‘circular’1 orbits around the BH. Below this critical threshold no signal can escape from
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