Theoretical Physics

Black Holes & Quantum Chaos

A scientific exploration of black hole physics through quantum chaos theory, covering Schwarzschild and Kerr geometries, information scrambling dynamics, the MSS bound, and holographic correspondences.

Event Horizon Hawking Radiation Information Paradox OTOC Lyapunov Exponent AdS/CFT
Fundamentals Advanced Mathematics Simulations
Observational Data

Observed Black Holes

Real black holes from EHT, LIGO/Virgo, and multi-wavelength observations. Select one to see calculated properties.

Black Hole Catalog

Calculated Properties

Geometry

Object -
Schwarzschild radius -
ISCO (Kerr) -
Spin parameter -

Thermodynamics

Hawking temperature -
Bekenstein-Hawking entropy -
Evaporation time -

Quantum Chaos

Lyapunov bound (λL) -
Scrambling time (t*) -

Key Formulas

rs = 2GM/c² TH = ℏc³/(8πGMkB) SBH = kBA/(4lP²) λL ≤ 2πkBT/ℏ
Black Hole Fundamentals

Black Hole Anatomy

Fundamental structure and properties of black holes based on Einstein's general relativity.

Singularity

A point of infinite density at the center of a black hole. All known laws of physics break down here.

ρ → ∞ Singularity density

Event Horizon

The boundary from which nothing, including light, can escape. Radius determined by mass.

rs = 2GM/c² Schwarzschild radius

Photon Sphere

Unstable circular orbit for photons at r = 1.5rs. Light can temporarily orbit.

rph = 3GM/c² Photon sphere radius

Accretion Disk

Matter spinning and falling into the black hole, emitting intense radiation due to frictional heating.

L = ηṀc² Accretion luminosity (η ≈ 0.1)

Ergosphere

Region outside the event horizon of a rotating (Kerr) black hole where spacetime is dragged by rotation.

re = M + √(M² - a²cos²θ) Ergosphere outer boundary

ISCO

Innermost Stable Circular Orbit - the deepest stable circular orbit before matter spirals inward.

rISCO = 6GM/c² (Schwarzschild) 3rs for non-rotating

Black Hole Classification

Schwarzschild

Non-rotating, uncharged. The simplest solution to Einstein's equations.

ds² = -(1-2M/r)dt² + (1-2M/r)⁻¹dr² + r²dΩ²

Kerr

Rotating, uncharged. The most realistic model for astrophysical black holes.

Has spin parameter a = J/Mc

Reissner-Nordström

Non-rotating, charged. Has two horizons (inner & outer).

r± = M ± √(M² - Q²)

Kerr-Newman

Rotating and charged. The most general solution for electrovacuum black holes.

Combination of parameters M, J, Q
Quantum Chaos in Black Holes

Chaos & Scrambling

Black holes as maximally chaotic systems - quantum information scrambling and connection with holography.

λ

Lyapunov Exponent

In classical chaotic systems, distance between trajectories grows exponentially: δx(t) ~ eλt. Black holes have a maximal Lyapunov exponent bounded by quantum mechanics.

λL ≤ 2πkBT/ℏ

Maldacena-Shenker-Stanford bound (2016). Black holes saturate this bound in holographic theories.

For Black Hole: λBH = 2πTH = κ (surface gravity)
C

OTOC

Out-of-Time-Order Correlator - the main diagnostic for quantum chaos.

C(t) = ⟨[W(t), V]†[W(t), V]⟩

Measures how fast perturbations spread in a quantum system.

t*

Scrambling Time

Time required for information to "scramble" evenly throughout the system.

t* ~ (ℏ/2πT) log S

S = Bekenstein-Hawking entropy. For BH: t* ~ rs log(S)

∿

Quasinormal Modes

Black hole oscillation modes that decay exponentially - fingerprint of chaos.

ω = ωR - iωI

Poles in Green's function, related to Lyapunov exponent.

Holographic Duality & SYK Model

AdS/CFT Correspondence

Duality between gravity in Anti-de Sitter (AdS) space and Conformal Field Theory (CFT) on its boundary. Black hole in bulk ↔ thermal state on boundary.

AdS₍ᵈ₊₁₎ Gravity ⟷ CFT₍ᵈ₎

SYK Model

Sachdev-Ye-Kitaev model - system of N fermions with random interactions. Maximally chaotic and has a gravitational dual in nearly-AdS₂.

H = Σijkl Jijkl ψiψjψkψl

Butterfly Effect

In holography, small perturbations at the boundary "fall" into the black hole and spread chaotically.

vB = √(d/2(d-1)) × 2πT

vB = butterfly velocity

Research Applications

01

Information Scrambling Dynamics

Study of how quantum information spreads within the horizon - key to understanding the information paradox.

02

Quantum Circuit Complexity

Connection between volume of the region behind the horizon and computational complexity of the dual quantum state.

03

Eigenstate Thermalization

How individual black hole eigenstates encode thermal physics - ETH in gravitational systems.

04

Random Matrix Theory

Level spacing statistics of quasinormal modes follow random matrix distributions (GUE/GOE).

Black Hole Thermodynamics

Laws of Black Hole Mechanics

Fundamental relationship between gravity, thermodynamics, and quantum mechanics.

0

Zeroth Law

Surface gravity κ is constant on the event horizon of a stationary black hole.

T constant at equilibrium ↔ κ constant on horizon
1

First Law

Mass change is related to area change, angular momentum, and charge.

dM = (κ/8π)dA + ΩdJ + ΦdQ
dE = TdS + work ↔ dM = (κ/8π)dA + ...
2

Second Law

Event horizon area never decreases in any classical process.

δA ≥ 0
Entropy S never decreases ↔ Area A never decreases
3

Third Law

It is impossible to reduce surface gravity κ to zero through any finite physical process.

T = 0 is unattainable ↔ κ = 0 (extremal) is unattainable

Hawking Radiation

Hawking's discovery (1974) that black holes radiate thermally demonstrated that the thermodynamic analogy is remarkably precise - black holes truly have temperature and entropy.

Hawking Temperature

TH = ℏκ/2πkBc = ℏc³/8πGMkB

For M = M☉: T ≈ 60 nanoKelvin

Bekenstein-Hawking Entropy

SBH = kBA/4lP² = kBc³A/4Gℏ

Entropy proportional to AREA, not volume!

Evaporation Time

tevap ≈ 5120πG²M³/ℏc⁴

For M = M☉: t ≈ 10⁶⁷ years

+
−

Virtual pair creation near horizon

Information Paradox

If a black hole fully evaporates via Hawking radiation (which is thermal/random), where does the information about what fell in go? This conflicts with quantum mechanical unitarity!

Page Curve

Radiation entropy initially rises, then falls after "Page time" - information exits gradually.

Island Formula

Generalized entropy formula with "quantum extremal surfaces" reproduces the Page curve.

ER = EPR

Entanglement (EPR) is equivalent to wormholes (Einstein-Rosen bridge) - Maldacena & Susskind.

Firewall?

AMPS paradox: is there a "firewall" at the horizon that destroys infalling observers?

Observational Evidence

Black Holes in Nature

Observational evidence for the existence of black holes from various astronomical methods.

M87 Black Hole
EHT 2019

M87* - First Direct Image

First direct image of a supermassive black hole at the center of Messier 87 galaxy. Mass ~6.5 billion M☉, distance 55 million light years.

6.5 × 10⁹ M☉ Mass
55 Mly Distance
~40 μas Angular Size

Sagittarius A*

Supermassive black hole at the center of the Milky Way. Orbits of S-stars provide undeniable evidence.

4 × 10⁶ M☉ mass 27,000 ly distance

Gravitational Waves

LIGO/Virgo detects binary black hole mergers. GW150914: ~36 + 29 M☉ → 62 M☉

90+ detections ~3 M☉c² radiated

X-ray Binaries

Binary star systems with stellar-mass black holes. Matter accretion emits intense X-rays.

Cygnus X-1 ~21 M☉

Quasars & AGN

Active galactic nuclei powered by supermassive black holes - the most luminous sources in the universe.

L ~ 1047 erg/s z > 7 furthest
Key References

Literature & Resources

Paper

Maldacena, Shenker, Stanford (2016)

"A bound on chaos" - arXiv:1503.01409

Paper

Sachdev & Ye (1993)

SYK model foundations - Phys. Rev. Lett. 70, 3339

Paper

Hawking (1974)

"Black hole explosions?" - Nature 248, 30-31

Paper

Bekenstein (1973)

"Black holes and entropy" - Phys. Rev. D 7, 2333

Review

Harlow (2016)

"Jerusalem Lectures on Black Holes and Quantum Information"

Book

Susskind & Lindesay

"An Introduction to Black Holes, Information and the String Theory Revolution"