Mathematical formalism for Black Hole Physics and Quantum Chaos, compiled from peer-reviewed literature for
educational purposes.
I. Kerr Metric & Boyer-Lindquist Coordinates
Full Kerr Metric in Boyer-Lindquist Coordinates
\[
ds^2 = -\left(1 - \frac{2Mr}{\Sigma}\right)dt^2 - \frac{4Mar\sin^2\theta}{\Sigma}dtd\phi +
\frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2 + \left(r^2 + a^2 +
\frac{2Ma^2r\sin^2\theta}{\Sigma}\right)\sin^2\theta \, d\phi^2
\]
Where: \(\Sigma = r^2 + a^2\cos^2\theta\), \(\Delta = r^2 - 2Mr + a^2\), \(a = J/M\) (spin
parameter)
Event Horizons (Inner & Outer)
\[
r_{\pm} = M \pm \sqrt{M^2 - a^2}
\]
Horizons occur where \(\Delta = 0\): \(r^2 - 2Mr + a^2 = 0\)
Extremal limit: \(a = M\) → horizons merge at \(r = M\). Naked singularity if \(a > M\) (cosmic
censorship conjecture).
Ergosphere Boundary
\[
r_{\text{ergo}}(\theta) = M + \sqrt{M^2 - a^2\cos^2\theta}
\]
Static limit surface where \(g_{tt} = 0\). Observer cannot remain stationary relative to infinity.
ISCO for Kerr Black Hole
\[
r_{\text{ISCO}} = M\left[3 + Z_2 \mp \sqrt{(3-Z_1)(3+Z_1+2Z_2)}\right]
\]
\(Z_1 = 1 + (1-a_*^2)^{1/3}\left[(1+a_*)^{1/3} + (1-a_*)^{1/3}\right]\)
\(Z_2 = \sqrt{3a_*^2 + Z_1^2}\), where \(a_* = a/M\)
Surface Gravity (Kerr)
\[
\kappa = \frac{r_+ - r_-}{2(r_+^2 + a^2)} = \frac{\sqrt{M^2 - a^2}}{2M(M + \sqrt{M^2 - a^2})}
\]
III. Quasinormal Modes & Linear Perturbation
∿ Teukolsky Master Equation
Master equation for perturbations on a Kerr background (spin-s perturbations):
\[
\left[\frac{(r^2+a^2)^2}{\Delta} - a^2\sin^2\theta\right]\frac{\partial^2\Psi}{\partial t^2} +
\frac{4Mar}{\Delta}\frac{\partial^2\Psi}{\partial t\partial\phi} + \left[\frac{a^2}{\Delta} -
\frac{1}{\sin^2\theta}\right]\frac{\partial^2\Psi}{\partial\phi^2}
\]
\[
- \Delta^{-s}\frac{\partial}{\partial r}\left(\Delta^{s+1}\frac{\partial\Psi}{\partial r}\right) -
\frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial\Psi}{\partial\theta}\right)
- 2s\left[\frac{a(r-M)}{\Delta} +
\frac{i\cos\theta}{\sin^2\theta}\right]\frac{\partial\Psi}{\partial\phi}
\]
\[
- 2s\left[\frac{M(r^2-a^2)}{\Delta} - r - ia\cos\theta\right]\frac{\partial\Psi}{\partial t} +
(s^2\cot^2\theta - s)\Psi = 4\pi\Sigma T
\]
QNM Frequencies (WKB Approximation)
\[
\omega_n = \Omega_c \ell - i\left(n + \frac{1}{2}\right)|\lambda_L|
\]
For highly damped modes (large n):
\[
\omega_n \approx \frac{\ln 3}{8\pi M} + i\frac{2\pi T_H}{\hbar}\left(n + \frac{1}{2}\right)
\]
QNM poles in retarded Green's function → connection to quantum chaos via pole-skipping.
IV. Out-of-Time-Order Correlator (OTOC)
OTOC Definition
\[
F(t) = \langle W^\dagger(t) V^\dagger(0) W(t) V(0) \rangle_\beta
\]
Regularized version (finite temperature):
\[
F(t) = \text{Tr}\left[\sqrt{\rho} W^\dagger(t) \sqrt{\rho} V^\dagger \sqrt{\rho} W(t) \sqrt{\rho}
V\right]
\]
where \(\rho = e^{-\beta H}/Z\)
Exponential Growth
\[
F(t) \approx 1 - \frac{\epsilon}{N} e^{\lambda_L t} + \mathcal{O}(\epsilon^2)
\]
Commutator squared:
\[
C(t) = -\langle [W(t), V]^2 \rangle_\beta \sim \frac{\epsilon}{N} e^{\lambda_L t}
\]
\(\lambda_L\) = Lyapunov exponent. Scrambling time: \(t_* \sim \frac{1}{\lambda_L}\ln N\)
MSS Chaos Bound (Maldacena-Shenker-Stanford 2016)
\[
\boxed{\lambda_L \leq \frac{2\pi k_B T}{\hbar}}
\]
Derivation outline:
1. Analytic continuation to imaginary time: \(F(t) \to F(t + i\beta/4)\)
2. Bound from positivity of spectral function
3. Uses KMS condition for thermal correlators
Black holes SATURATE this bound! This makes them the fastest scramblers in nature.
V. SYK Model - Exact Solvability
SYK Hamiltonian
\[
H = \sum_{1 \leq i < j < k < l \leq N} J_{ijkl} \, \psi_i \psi_j \psi_k \psi_l \]
Majorana fermions: \(\{\psi_i, \psi_j\} = \delta_{ij}\)
Disorder average: \(\overline{J_{ijkl}^2} = \frac{3! J^2}{N^3}\)
Schwinger-Dyson Equations
\[
G(\tau_1 - \tau_2) = -\langle T \psi_i(\tau_1)\psi_i(\tau_2)\rangle
\]
\[
\Sigma(\tau) = J^2 G(\tau)^{q-1}
\]
\[
G(i\omega_n)^{-1} = -i\omega_n - \Sigma(i\omega_n)
\]
In the IR conformal limit \((\beta J \gg 1)\):
\[
G(\tau) \propto \frac{\text{sgn}(\tau)}{|\tau|^{2\Delta}}, \quad \Delta = \frac{1}{q}
\]
Four-Point Function & Chaos
\[
\frac{\mathcal{F}(t)}{\mathcal{F}(0)} = 1 - \frac{6}{\beta J} e^{\frac{2\pi}{\beta}t} + \ldots
\]
Lyapunov exponent: \(\lambda_L = \frac{2\pi}{\beta} = 2\pi T\) - saturates the chaos bound!
Schwarzian Action (Low Energy)
\[
S[\tau(u)] = -C \int_0^\beta du \, \{f(u), u\}
\]
Schwarzian derivative:
\[
\{f, u\} = \frac{f'''}{f'} - \frac{3}{2}\left(\frac{f''}{f'}\right)^2
\]
This action governs the dynamics of nearly-AdS₂/nearly-CFT₁ and JT gravity.
VII. Page Curve & Unitarity
Page Curve Evolution
\[
S_{\text{rad}}(t) = \begin{cases}
S_{\text{thermal}}(t) & t < t_{\text{Page}} \\ S_{\text{BH}}(t) & t> t_{\text{Page}}
\end{cases}
\]
Page time: \(t_{\text{Page}} \sim \frac{t_{\text{evap}}}{2}\)
At Page time, \(S_{\text{rad}} = S_{\text{BH}}\)
Page's Result for Random States
\[
\langle S_A \rangle = \ln(d_A) - \frac{d_A}{2d_B} + \mathcal{O}(1/d_B^2)
\]
Where \(d_A, d_B\) are the Hilbert space dimensions of subsystems A and B, with \(d_A \leq d_B\).
VIII. Island Formula & QES
Generalized Entropy with Islands
\[
\boxed{S_{\text{gen}}(R) = \min_{\mathcal{I}} \text{ext}_{\mathcal{I}}
\left[\frac{\text{Area}(\partial\mathcal{I})}{4G_N} + S_{\text{matter}}(R \cup \mathcal{I})\right]}
\]
Key insight: Island \(\mathcal{I}\) is a region inside the horizon that "contributes"
to the entropy of radiation \(R\).
Quantum Extremal Surface (QES)
\[
\frac{\delta}{\delta X^\mu}\left[\frac{A[\gamma]}{4G_N} + S_{\text{bulk}}[\Sigma_\gamma]\right] = 0
\]
QES generalizes RT/HRT surfaces to include quantum corrections. Crucial for resolving information
paradox.
Replica Trick for Islands
\[
S = \lim_{n \to 1} \frac{1}{1-n} \log \text{Tr}(\rho_R^n)
\]
Path integral with replica saddles:
\[
\text{Tr}(\rho_R^n) = \sum_{\text{topologies}} e^{-I[\text{topology}]}
\]
IX. Quantum Complexity & Holography
Complexity = Volume (CV)
\[
\mathcal{C}_V = \frac{V(\Sigma)}{G_N \ell}
\]
\(\Sigma\) = maximal codimension-1 surface anchored at boundary time slice, \(\ell\) = AdS scale
Complexity = Action (CA)
\[
\mathcal{C}_A = \frac{I_{\text{WDW}}}{\pi\hbar}
\]
Wheeler-DeWitt patch action:
\[
I_{\text{WDW}} = \frac{1}{16\pi G_N}\int_{\mathcal{M}} d^{d+1}x\sqrt{-g}(R - 2\Lambda) + \text{boundary
terms}
\]
Lloyd's Bound
\[
\frac{d\mathcal{C}}{dt} \leq \frac{2M}{\pi\hbar}
\]
For eternal AdS black hole at late times:
\[
\frac{d\mathcal{C}}{dt} = \frac{2M}{\pi\hbar} \quad \text{(saturated!)}
\]
Black holes are not only the fastest scramblers but also the fastest computers!
Second Law of Complexity
\[
\frac{d\mathcal{C}}{dt} \geq 0 \quad \text{until} \quad t \sim e^S
\]
Complexity grows linearly for exponentially long time \(\sim e^{S_{\text{BH}}}\), then saturates and
fluctuates (Poincaré recurrences).