Chaos at the rim of black hole and fuzzball shadows

We study the scattering of massless probes in the vicinity of the {\it 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 $\lambda$ governing the exponential growth in $d$ dimensions and show that it is bounded from above by $\lambda_b = \sqrt{d{-}3}/2b_{\rm min}$ where $b_{\rm min}$ 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 $\lambda$ is typically below the advocated bound on chaos $\lambda_H=2\pi \kappa_B T/\hbar$, 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 micro-structures at horizon scales via the detection of ring-down modes with time scales $\lambda^{-1}$ longer than those expected for a BH of the given mass and spin.


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 the BH gravitational potential.
The limiting 'circular' photon orbits form the BH photon-sphere that discriminates between the scattering and absorption regimes. It appears as the rim of the black hole shadow projected by the BH on a distant observer. The size and shape of the photonsphere depends on the mass, charge and angular momentum of the BH and can help to uniquely identify it. For instance, for a Schwarzchild BH the photon-sphere is a sphere of radius 3/2 times that of the BH horizon 2 . Photon orbits are also useful to explain the optical appearance of stars undergoing gravitational collapse and the response of the BH to perturbations, the so called quasi-normal modes, that can be interpreted as null particles trapped at the unstable orbit and slowly leaking out [2].
Instability of the photon orbits signals the onset of chaos on the geodetic motion near the photon-sphere [2]. A measure of chaos or, better, of the high sensitivity to the initial conditions is given by the Lyapunov exponent λ, that can be defined via the Poissòn bracket describing the exponential growth with time of the deflection along an angular direction φ, for small variations of the corresponding conjugate momentum P φ . In the case of spherically symmetric BH's, the Lyapunov exponent λ can be related to the time decay of the basic QNM [2]. In an asymptotically AdS space, the chaotic behaviour manifests as a random angular dispersion of nearby trajectories re-emerging at the boundary after a long time t, much longer than the time it takes to a free massless particle to travel from/to the boundary and back. Our work is inspired by the results in [3], where holography was used to diagnose chaos in quantum mechanical systems at finite temperature. The amount of chaos in the quantum theory was characterised by the exponential growth of out-of-time-ordered correlators, involving commutators that replace the Poissòn brackets in (1.1) and encode the perturbation induced by an operator on a later measurement. The long time behaviour of the field theory correlator was related to the exponential growth e λ H t of the center mass energy of the high energy scattering process in the dual gravity theory [4,5]. The exponent λ H = 2πκ B T / , with T the temperature of the BH, is known to be determined by the local Rindler structure of the horizon [6] and it was proposed as a universal bound on the chaos that a quantum thermal system can develop [3].
It this paper we compute the Lyapunov exponent for scattering near the photonsphere of asymptotically AdS charged rotating BH's in d = 4 and, later on, generalise our analysis to diverse dimensions d and to horizon-free micro-state geometries a.k.a. fuzzballs [7][8][9][10]. We relate the Lyapunov exponent λ to the vanishing rate of the radial velocitẏ r ≈ −2 λ(r − r c ) near the photon-sphere, and derive analytic formulae for λ as a function of the radius r c of the limiting photon orbit.
We find that λ is typically smaller than λ H characterising the vanishing rate of the radial velocity at the horizon 3 , but the bound is violated inside a very narrow window near extremality when the innermost region of the BH photon-sphere coalesces with the BH horizon 4 .
We find 5 instead that with d the dimension and b min the minimal impact parameter under which a particle moving in the background of the BH gets trapped in the gravitational potential. The bound is saturated by non-rotating, uncharged BH's. We remark that the existence of circular photon orbits and an instability timescale ∆t ≈ 1/λ is a property common to many solutions in gravity, including BH-looking horizon-less geometries known as fuzzballs or BH microstate geometries with either flat or AdS asymptotics. A large class of horizon-free microstate geometries is known [13][14][15] (and references therein) including their stringy origin [16][17][18][19]. Geodesic motion in these geometries has been investigated in [20][21][22][23][24]. Here we focus on critical scattering for a class of three-charge BH microstate geometries with asymptotically flat or AdS 3 × S 3 geometries. We find that only geodesics scattered in asymptotically flat micro-state geometries can reach the fuzzball photon-sphere, exhibiting random angular dispersion at infinity. More interestingly, we find that the Lyapunov exponent λ for asymptotically flat micro-state geometries is smaller than λ b , which in turn is typically smaller than λ H . Since λ is related to the time decay of QNM's dominating at late times the response of the geometry to perturbations, our results suggest that ring-down signals can be used to discriminate between BH's and geometries with non trivial micro-structures at the horizon scale 6 .
The plan of the paper is as follows. In Section 2 we discuss geodesics in asymptotically AdS Kerr-Newmann space-times in d = 4. We derive the Hamiltonian for massless probes and discuss radial and angular motion by stressing the role of the impact parameter(s). We then consider geodesics ending inside the horizon. In section 3 we study critical and nearly critical geodesics, we introduce the concepts of critical impact parameter b c and Lyapunov exponent λ and propose a bound on the latter. In section 4 we discuss some BH examples in flat and AdS space-times. We also extend our analysis to arbitrary dimension d. In section 5 we discuss three-charge fuzzball micro-state geometries in the near horizon limit or, more interestingly, in flat space-time. Section 6 contains a discussion of our results and possible outlooks. In appendix we discuss integration of the angular motion in d = 4. 4 The role of the ratio of λ, related to the minimum of the imaginary part of the quasi-normal mode frequency, and λH , related to the surface gravity of the inner horizon, based on [11] has been recently stressed in [12] in connection with the strong cosmic censorship. We thank J. Maldacena for pointing this out to us as well as the presence in this context of a sliver near extremality, similar to the one we find. 5 We have an analytical proof of this bound only for spherically symmetric BH's and numerical evidences for rotating black holes in four dimensions. 6 Similar proposals were recently put forward in different contexts for the modifications of gravity at the horizon scale [25,26].
2 Geodesics on AdS Kerr-Newman space-times

The metric
We consider geodetic motion in asymptotically AdS 4 Kerr-Newmann space-time 7 with metric [27] with M the 'mass', a = J/M the angular momentum 'parameter', the AdS radius and Q the charge. A BH exists for any choice of a < 8 , Q and M such that ∆ r has at least a real positive root, and the horizon radius r H is then the largest root. A peculiar feature of the above metric is a non-zero angular velocity ω = g tφ /g tt at infinity Ω ∞ = ω| ∞ = −a/ 2 . The angular velocity that is relevant for the thermodynamics is the difference Ω = Ω H − Ω ∞ with Ω H = ω| r H = aα /(r 2 H + a 2 ) the angular velocity at the horizon [27].
The Hawking temperature of the BH is given by If ∆ r (r H ) = 0, the BH is said to be extremal and the temperature vanishes. The explicit expression of r H in terms of the BH parameters (M , a, Q and the AdS 'radius' ) can be found using Cardano-Tartaglia formulae for the zeroes of a quartic polynomial. We will refrain from exposing it since it is quite cumbersome and not crucial for our analysis.

The Hamiltonian and momenta
In the Hamilton-Jacobi formulation of the geodesics equation, the Hamiltonian reads For a massless spin-less particle in the AdS Kerr-Newmann metric H can be written in the 'separate' form More precisely, the metric is not asymptotically AdS4 but rather tends to a rotating Einstein universe with Ω∞ = −a/ 2 . with E = −P t and P φ conserved quantities. Following Carter, Teukolsky and Chandrasekhar [28], the null condition H = 0 is solved by introducing a 'separation' constant K, representing the total angular momentum, and setting Defining the impact parameters as equations (2.5) can be written in the form with R and Θ quartic polynomials of r and cos θ, respectively. While b (as well as a and E) can be taken to be non-negative without loss of generality, ζ can be negative or positive depending on P φ . One can distinguish three classes of geodesics according to the sign of We notice that at infinity so positivity of P 2 r requires 9 b < (2.10) In fact we will later see that b is also bounded from below for the motion to take place at all. Using the monotonically decreasing variable r as time variable in the branch where the geodesics evolves towards the BH, so thatṙ, P r < 0, the equations of motionẋ m = ∂H/∂P m can be written in the form

Motion in the radial direction
We can view the motion in the radial direction as that of a particle in a one-dimensional potential. We write R(r) Starting from infinity, a massless particle with large enough impact parameter b moves towards the BH till it reaches a turning point r * , where V + (r * ) = 1 and the radial velocity vanishes, see figure 1. After this point, the particle bounces back to infinity along a symmetrically reversed trajectory obtained by flipping the sign of the radial velocity. Critical geodesics are obtained by tuning the impact parameters so that the value of V + (r) at the maximum is one, i.e.
V + (r c ) = 0 and V + (r c ) = 1 (2.14) for some r c > r H . We will see that for b smaller than a critical value b c (ζ c ), the potential satisfies V + (r) < 1 for all r > r H and the geodesics finds no turning point. So, we can divide the impact parameter plane (b, ζ) into three regions We will mainly focus on the second class of geodesics. 9 One can also consider geodesics that never reach infinity and allow for b > . We will comment on this case later on.

Motion in the θ direction
Motion along the θ direction takes place inside intervals bounded by the zeros of the quartic polynomial Θ(cos θ). Denoting χ = cos θ, the polynomial Θ in (2.7) can be conveniently written in the form Geodetic motion is allowed for choices of the parameters (b, ζ) such that P 2 θ > 0 for some χ ∈ [−1, 1]. To determine this region it is convenient to write P θ in the following form with the angular effective potential The effective potential is even in χ and bounded from below inside the interval [−1, 1]. For ζ = −a it has a maximum at χ * = 0, and minima at χ * = ±1. For ζ = −a it reaches infinity at χ = ±1. The extrema of the potential (solutions of V eff (χ * ) = 0) are located at χ = ±χ * (if real and with χ 2 * ≤ 1) with We are interested on geodesics starting from infinity, so we will take b 2 < 2 . In addition b 2 should be bigger than the absolute minimum of Depending on the value of ζ, the profile of V eff (χ) changes and the global minimum can be located at any of the three extrema in (2.20). We have two different allowed regions: 10 The zeros of the polynomial are located at A) The geodesics bounces inside the interval For ζ * < ζ < 0, the effective potential has two minima and a maximum in χ = 0, while outside this interval the potential has a unique minimum in χ = 0.

B) The geodesics bounces inside the interval
The effective potential has two minima and a maximum in χ = 0.
We display in colors the allowed region in the impact parameter plane (b, ζ) in figure 2. Geodesics belonging to the classes A and B correspond to the regions above and below b = |ζ| (dashed line in the graph) respectively. We distinguish with two colors, red and green, the absorption and scattering phases respectively (see below for details). The light red region collects all geodesics allowed by the kinematics that cannot reach the AdS boundary at infinity. Geodesics in this class always fall into the horizon.
In figure 3, we display the form of V eff (χ) for various choices of ζ setting a = 1, = 2, Q = 0.3. Geodetic motion is allowed when V eff (χ) is below the b 2 -dashed line and it bounces inside the intervals limited by the intersection points of the two graphs. Whenever b 2 coincides with an extremum of V eff (χ) one has either a shear-free (un)stable geodesics at fixed θ = θ 0 or a critical geodesics at the equator (see next section). Geodesics at fixed θ-angle are stable around the minima of V eff (χ), and unstable around the maximum.
We will see in the next section that critical geodesics always correspond to the case b > |ζ|, so critical motion will always bounce inside the interval χ ∈ [−χ + , χ + ] with χ + = 1 for ζ = −a. Figure 3: Effective potential V eff (χ) for a BH with angular parameter a = 0.7 , = 3: (a) Co-rotating and non-rotating geodesics (b) Counter-rotating geodesics.

Geodesics inside the absorption region
For b < b c (ζ c ), the particle falls inside the BH horizon. It is important to observe that near the horizon both radial and angular velocities along the φ-direction are independent of the impact parameters (b, ζ). Indeed expanding (2.11) We notice that near the horizon both the distance from the horizon and the radial velocity vanish but their ratio is finite and proportional to the BH temperature. We will show later that λ H is typically bigger than the ratio λ describing the vanishing rate of the radial velocity at the photonsphere. This inequality is however violated for extremal and near-extremal black holes.

Shear free geodesics
A particular simple class of geodesics in the absorption phase are the so called shear-free geodesics (radial falling) corresponding to trajectories with zero total angular momentum. For b = 0, non-negativity of P 2 θ requires that the polynomial Θ(cos θ) in (2.7) exactly vanishes. This happens when the impact parameter ζ and the initial angle θ 0 are related by ζ = −a cos 2 θ 0 (2.27) For this choiceθ = 0, i.e. θ = θ 0 along the whole trajectory. Plugging these values into the polynomial R(r) one finds that is strictly positive, so no turning points are found before the geodesics reaches the BH horizon. The rates of change of t, φ and θ with r take the simple form that reduce to (2.25) near the horizon.

Critical and nearly critical geodesics
In this section we study critical and nearly critical geodesics in asymptotically AdS Kerr-Newman BH's.

Critical geodesics
A critical geodesics is obtained when the two largest roots of the polynomial coincide. More precisely, we say that a geodesics is critical, if the largest zero r c of R(r) is a double zero, or in other words These equations can be easily solved for ζ and b as a function of the critical radius Plugging these formulae into Θ(χ), one finds that the polynomial is positive for r c belonging to a finite interval with r ± saturating the inequality. We remark that for non-extremal BH's , r − is always outside the horizon since equations (3.2) do not admit any solution with r c = r H unless On the other hand, critical geodesics with r c inside the interval [r − , r + ] display a nontrivial motion in the θ direction. In figure 4 we display the dependence of (ζ c , b c ) and P θ on the critical radius r c . We notice that in the region where P 2 θ is positive and therefore motion is allowed b c > |ζ c |, so critical geodesics fall always into the first of the two classes identified in Section 2.3. The motion in the χ-plane is confined inside the interval [−χ + , χ + ].
The collection of all points leading to a critical geodesics delimits the edge of the BH shadow as seen by an observer at infinity. For an observer on a plane at fixed θ, the rim of the shadow is defined by the parametric curve [28] with r c running inside the sub-interval of (3.4) allowed by the chosen value of χ = cos θ, see Fig. 4. In figure 5 we depict the shadow of a Kerr-Newmann AdS BH for particular values of mass M , charge Q and angular momentum parameter a at different values of θ = 0, π/6, π/2. Note the asymmetric shape, due to the rotation, as one approaches the equatorial plane θ = π/2.

The Lyapunov exponent
In this section we consider nearly critical geodesics along the equatorial plane (θ = π/2, χ = 0) and compute the Lyapunov exponent determining the exponential growth with time of geodetic deviation.
Approaching the critical radius r ≈ r c with χ =χ = 0, the radial velocity (2.11) vanishes as dr dt and We will now show that λ, governing the vanishing rate ofṙ, coincides with the Lyapunov exponent of nearly critical geodesics scattered around r ≈ r c . A nearly critical geodesics can be obtained by varying the impact parameters (b, ζ) slightly away from the critical values (b c , ζ c ). There are two distinct cases depending on whether the variation lifts the double zero or splits it into two. In the former case, the turning point disappears and the (massless) particle falls into the BH. We are interested in the latter case whereby the (massless) particle reaches the largest of the two nearby zeroes and bounces back to infinity. Near the critical point the polynomial R(r) can be approximated by where parametrises the distance between the critical radius and the turning point of the nearly critical geodesics located at r * = r c + . The time delay to reach r * and get back becomes ∆t = 2 r * dṙ where (3.9) has been used. Similarly for the deflection angle one finds with ω c =φ(r c ) = α ζ c ∆ r (r c ) + a r 2 c − a ζ c aζ c ∆ r (r c ) + (r 2 c + a 2 ) (r 2 c − a ζ c ) (3.15) the angular velocity at the critical point. Taking the variation with respect to the deviation in the incoming momenta one finds We conclude that the Lyapunov exponent λ is determined by the vanishing rate (3.9) of the radial velocity of an infalling neutral massless particle at the photon-sphere. In figures 6 we plot the functions (3.10) λ ± = λ(r ± ), the minimum and maximum values of λ, λ H and as functions of the angular momentum and charge, where b min = b c (r − ) is the minimal choice of the impact parameter that an impinging massless particle can have without falling into the BH's. We notice that λ ≤ λ b with the bound saturated for non-rotating and uncharged BH's.

Analytic results for specific BH's
In this section we compute the Lyapunov exponents and critical impact parameters for some simple instances of BH's in d = 4 and then present some general results for nonrotating BH's in arbitrary dimensions. For simplicity we focus on geodesics withχ = 0, corresponding to one of the choices r c = r ± . In all the examples we compare the results for λ with the one for λ H characterising the vanishing rate of the radial velocity at the horizon and given by

Non rotating BH's in AdS
We consider first the case of a non-rotating charged BH in AdS, i.e. a = 0. The metric reads 3) The peculiar feature of the non-rotating case, is that due to full rotational symmetry, the radial velocity of a probe depends only on the total angular momentum K = bE, so the critical equations determine both b and the critical radius r c , leaving ζ undetermined . The photon-sphere is a round sphere with critical radius given in terms of the BH mass and charge. The critical equations reduce to that can be solved for r c and b c .

Schwarzschild BH in flat space-time
Setting Q = 0 and sending → ∞, one gets a Schwarzchild BH in flat space-time for which We notice that Reissner-Norsdtröm BH in flat space-time and 0 < Q < M . We notice that λ < λ b in the charged case. Comparing with one finds that λ ext > λ H = 0. More generally, one find that the bound is violated inside the very narrow window 0.99M < Q < M near extremality. In particular, in the extremal case Q = M one finds

AdS-Schwarzschild BH
Setting Q = 0 with finite, one finds Solving ∆ r (r H ) = 0 for M in favor of r H , it is easy to check that λ < λ H for any r H and .

AdS-Reissner-Norsdtröm BH
Keeping Q = 0 one finds Note that 2M ≤ r c ≤ 3M , so with the bound saturated for Q = 0.

Rotating BH's
Expressions for λ in the general case of rotating BH's are quite involved so we limit ourselves to special choices of the parameters such that analytic formulae can be found.

Large photon-spheres
In the limit where the size of Anti-deSitter is much smaller than the radius of the photonsphere, i.e. a, << M, Q, r c one finds For the Lyapunov exponents one finds as expected.

Extremal asymptotically flat Kerr-Newman BH
The extremal Kerr-Newman BH in flat space-time ( = ∞) is obtained by taking One finds Here we assume that a < r H /2, in order to ensure that r − is outside the BH horizon. For the Lyapunov exponents one finds

Spherical symmetric BH's in higher dimensions
We now extend our analysis and consider critical null geodesics around spherically symmetric BH's in d-dimensions with AdS or flat asymptotics. The line element of an asymptotically AdS d spherically symmetric BH reads and M , Q, describing the mass, charges and AdS radius of the asymptotic geometry. The Hamiltonian null condition reads with K denoting the total angular momentum. The radial velocity reads with b = K/E the impact parameter and Near the horizon one finds the radial velocity vanishing as given in terms of the BH temperature T . The BH photon-sphere is defined by the critical equations R(r c ) = R (r c ) = 0 leading to with r c solving r c f (r c ) = 2f (r c ) (4.29) Near the critical radius one finds Note that λ is bounded from above with the bound saturated by the uncharged BH's. For Q = 0 one finds Quite remarkably r c does not depend on but b c does.

Fuzzball geometries
In this section we study critical and nearly critical null geodesics in a class of three-charge microstate geometries of type (1, 0, n) first introduced in [29]. We consider both the asymptotically flat and the near-horizon limit exhibiting pretty different behaviour. We focus on scattering along the ϑ = 0 direction, where the geodesic equations have been shown to be integrable in this class of microstate geometries. We will also set Q 1 = Q 5 = L 2 for notational simplicity. The geometries are characterized by an asymptotically AdS 3 × S 3 metric that after a conformal rescaling 11 can be written as with ds 2 4 = ρ 2 +a 2 cos 2 ϑ dρ 2 ρ 2 +a 2 + dϑ 2 + ρ 2 + a 2 sin 2 ϑ dϕ 2 + ρ 2 cos 2 ϑ dψ 2 .
(5.2) the flat space metric on R 4 in oblate spheroidal coordinates. For ϑ = 0, the various functions entering in the metric take the form Motion along the ϑ = 0 direction requires P ϑ = P ϕ = 0, so the Hamiltonian reduces to 11 We recall that conformal rescalings are irrelevant for the study of null geodesics. 12 In the notation of [23] 1 = 2 4 = 2a 2 (ν − 1).

We set
and focus on the case P y = 0. For this choice the Hamiltonian reduces to and the velocities w.r.t. the radial variable are given by and b = P ψ /E. We notice that positivity of R(ρ) at infinity requires The critical equations R(ρ c ) = R (ρ c ) = 0 (5.11) can be solved for b c and ρ c . Near the critical radius one finds

The BH geometry
The BH geometry corresponding to the above micro-states is obtained by sending a → 0, ν → ∞ and n → 0 keeping finite the product L 2 p = n ν a 2 (5.14) leading to The radial function R(ρ) becomes The critical radius and impact parameters read  leading to the Lyapunov exponent We notice that λ BH < 1/( √ 2b c ). The asymptotically AdS solution can be analysed by simply dropping the 1 in Z. It is easy to see that no solutions to the critical equations are found in this case.

The asymptotically flat fuzzball
In the case of the asymptotically flat fuzzball the critical equations can be written in the The second equation is linear in ν and can be solved for ν = ν(ρ c ) that then can be plugged into the first equation to determine b c (ρ c ). For instance for n = 1 one finds Near the critical radius one finds The functions b c (ρ c ), ν(ρ c ), λ(ρ c ) can be used to plot parametrically b c (ν), λ(ν). The plots are displayed in figure 7 for a fuzzball solution with n = 1 and L = 5. The plots show that λ is below the bound (in d = 5) We notice also that λ of the fuzzball is smaller than the Lyapunov exponent λ BH (5.18) associated to the BH of the same charge.

Asymptotically AdS geometry
The asymptotically AdS case is obtained by dropping the 1 term in Z. 13 The positivity of P 2 ρ at infinity requires P ψ = 0 so only geodesics moving entirely inside AdS 3 can arrive from the AdS boundary 14 .
Starting from infinity, geodesics evolve untill they reach the turning point ρ * , i.e. a zero of P ρ (ρ), where the radial velocity vanishes. Zeroes of P ρ (ρ) coincide with the ones of the function where we introduced the impact parameter b = P y E (5.25) The critical equations R(ρ) = R (ρ) = 0 (5.26) can be written in the form The first equation is quadratic in ν, since F n is linear in ν, and we can solve it as a function of ρ c . The second equation determines b c (ρ c ) once ν is replaced by ν(ρ c ) inside F n . One 13 The metric is asymptotically conformally AdS with radius = √ L1L5 = L. 14 Photon orbits with non-trivial angular momentum P ψ exists and were studied in [23]. finds non trivial solutions for n ≥ 2. The explicit form of the solutions are not particularly illuminating, so we will not display it here. In figure 8 we show the curves b c (ρ c ) and ν(ρ c ) for n = 2. First, we notice that there is no solution in the scattering region |b| < 1 where motion is allowed from infinity, so critical geodesics exists but they cannot be reached starting from infinity. Second we observe that for b ≈ ±1, equations (5.27) admit critical solutions for For ν < ν min no critical geodesics exists. We find that geodesics fall into two different categories: scattered geodesics impinging from infinity and finding at most one turning point (green region), and stable orbits confined in the interior of AdS (grey regions) bouncing between two turning points. We conclude that the photon-sphere of fuzzballs in the class (1, 0, n) under study cannot be reached by geodesics scattered from the AdS boundary at infinity.

Discussion and outlook
We have studied geodesics for massless neutral probes around the photon-sphere, discriminating between the scattering and absorption phases of asymptotically AdS Kerr-Newman BH and fuzzball geometries. We have found that geodesics near the photon-sphere exhibit always a chaotic behaviour characterised by an exponential growth of the angular dispersion of nearby geodesics. We have computed the critical impact parameters and the Lyapunov exponent λ governing the exponential growth as functions of the radius of the limiting photon orbits. We have related the Lyapunov exponent λ to the ratio between the vanishing radial velocity and the distance from the photon-spherė r ≈ 2λ (r − r c ) as r ≈ r c (6.1) The coefficient λ depends on the parameters characterizing the BH: mass, charges and angular momentum. Geodesics with impact parameters below the critical values fall into the horizon with vanishing radial velocities again proportional to the distance from the horizon, with the ratio λ H given solely in terms of the BH temperature λ H = 2πT . The critical exponent λ H has been recently related to the Lyapunov exponent computing the exponential growth of out-of-time ordered correlators in the quantum mechanic systems describing holographically the near horizon BH geometry [3]. A quantum system dual to a BH is expected to be maximally chaotic, so λ H has been proposed as an upper bound on quantum chaos a thermal theory at temperature T can develop. The BH photon-sphere is far away from the horizon, so chaos in the photon-sphere is not related to that in the quantum mechanics in any obvious way. Still it is interesting to observe that λ is typically below the advocated upper bound λ H , but the bound is badly violated inside a very narrow window near extremality, where the BH temperature T H is very small or vanishing and the photon-sphere coalesces with the BH horizon. Due to the obvious limitations of our classical analysis, we cannot exclude that the violation of the bound could be attributed to quantum effects at low temperatures. Another possibility is that the bound λ < λ H should be revised/reconsidered in near extreme conditions even after including quantum or stringy corrections 15 In arbitrary space-time dimension d, we have found instead that λ is bounded by with b c the minimal impact parameter that a massless particle can travel without falling into the BH. Finally we have considered null geodesics in fuzzball geometries of the so called (1, 0, n) class [29], with asymptotically flat or AdS 3 × S 3 geometries. We have shown that critical photon orbits exists in both cases, but they can be reached from infinity only in the asymptotically flat case. We have also found that the critical exponent λ of the fuzzball is typically smaller than the one of the corresponding BH. This suggests that fuzzball can be distinguished from BH's by the signature of the quasi-normal modes characterising the response of the geometry to small perturbations, that are known to be dominated at late times by a basic mode with time scale given by the inverse of λ [2].
Since the scattering processes under study here can be related to out-of-time-order correlators in the boundary theory (at 'radial' infinity), one expects that the chaotic behaviour of geodesics reflects into an exponential growth of the correlators in the boundary theory (at 'radial' infinity) with Lyapunov exponent λ. High energy scattering in AdS has been also related to correlators involving the insertion of two massless (light-like separated) states (the incoming and outgoing rays) and two heavy states (the BH) in the boundary theory [30]. The time delay and deflection angles experienced by the geodesics measure the derivatives of the (eikonal) phase shift (encoding the relevant part of the correlator) with respect to the energy and angular momentum and determine the location of its poles in coordinate space [31]. One may expect that the results here may provide a detailed prediction about the location of poles in the phase shift of the dual correlators at strong coupling. A complementary picture of the exponential fall off of thermal correlators at large imaginary frequency was obtained in [32] from the study of space-like geodesics probing regions inside the BH horizon. It would be interesting to explore similar connections in the vicinity of the photon-sphere.

A Integrating the angular motion
A different source of chaotic, or more precisely quasi-periodic, behaviour lies in the angular dependence of the metric (2.1). In order to address the existence of such behaviour we investigate the relation between the two angular variables φ and θ.
Being interested in critical geodesics, we know that the angular variable θ is constrained by the values ±χ plus as shown in section (2.5).
For the non rotating BH's (i.e. a = 0) one gets The motion in the φ-direction is given by As one would expect for critical geodesics in static BH's, while cos θ sweeps the interval [−χ plus , χ plus ] back and forth, the angle φ spans its whole 2π domain (an integer number of times) and the particle trajectory is a full circle.
Rotating BH's entail a way more complicated relation between θ and φ ∆φ = i In the case of interest for us m = k 2 < 0. In order to have periodic motion one should require ∆φ = π + 2πn this imposes stringent constraints on r c that are not generically satisfied. As a result, classical motion in the angular variables (for critical radii r c ) is non-periodic and as such chaotic / ergodic in the sense that the orbit covers and entire region of the 'sphere'. At the quantum level, preventing destructive interference is tantamount to imposing the quantisation condition ∆φ = π + 2πn that selects only a finite, yet possibly very large, set of critical radii r c .