Why Black Hole is maximally chaotic? I understand intuitively that black holes are chaotic. However, people say black holes are not just chaotic, they are "maximally chaotic". What is the quantitative definition of "maximally chaotic," and how can we know that black holes are maximally chaotic?
 A: A measure of chaos in a system is given by the so-called Lyapunov exponent $\lambda$. In a classical chaotic system, nearby trajectories typically diverge exponentially fast in time:
$$
\{q(t),p(0)\}=e^{\lambda t}
$$
One can show that the correct way to implement this in a quantum system is by considering the behaviour of thermal (i.e. at finite temperature $T=1/\beta$) 4-point functions of two operators, say $W$ and $V$. What one wants to compute is the object:
$$
\langle V(0)W(t)V(0)W(t) \rangle_{\beta}
$$
Notice that they have alternating time $t$. Such correlators are called out-of-time-ordered. How do we compute them? A way to do this is provided by the AdS/CFT correspondence.
The near-horizon region of near-extremal black holes is given by the AdS$_2$ geometry. In this particular spacetime, one can define a particular theory of gravity by coupling the metric to a field called dilaton (gravity alone is non-dynamical in 2 dimensions). In addition to that, one can add some matter fields. In the AdS$_{d+1}$/CFT$_d$ correspondence, such matter fields are supposed to be dual to conformal primary operators living on the boundary of AdS$_2$. More technically, the partition function of the gravity theory is identified with the generating functional of the CFT. If $\phi$ is a matter field with boundary value $\phi_0$ dual to an operator $\mathcal O$ with conformal dimension $\Delta$, then we have
$$
Z_{\text{AdS}}[\phi_0]=\int\limits_{\lim_{z\to0} \phi z^{\Delta-d}=\phi_0} D\phi\, e^{-I[\phi]}\equiv Z_{CFT}[\phi_0]= \bigg\langle \exp \left(\int_{\partial \Omega}d^dx\, \mathcal{O}\phi_0\right)\bigg\rangle 
$$
By applying the above equation, we are able to compute the four-point functions of the conformal operators.
In the simplest case without supersymmetry, one finds $[1]$
$$ 
\langle V(0)W(t)V(0)W(t) \rangle_{\beta} \sim \beta e^{\frac{2\pi}{\beta}}
$$
meaning that the Lyapunov exponents is $\lambda=\frac{2\pi}{\beta}$.
Maldacena showed that this is the maximum value allowed for Lyapunov exponents $[2]$, so these non-supersymmetric black holes are maximally chaotic.
In supersymmetric black holes there are matter fields which are actually spinors, and for them it turns out that the Lyapunov exponent is less than the maximal value $[3]$.
References
$[1]$  Maldacena J. Conformal symmetry and its breaking in two dimensional Nearly Anti-de-Sitter space. PTEP 2016, 12C104 (2016).
$[2]$  Maldacena, J. A bound on chaos. J. High Energy Phys. 08, 106 (2016).
$[3]$ Campos Delgado, R. and Foerste, S. Lyapunov exponents in N=2 supersymmetric Jackiw-Teitelboim gravity. Phys.Lett.B 835, 137550 (2022).
