Black hole Study Guide

📖 Core Concepts Black hole – an object whose gravity is so strong that nothing, not even light, can escape. Event horizon – the surface beyond which escape is impossible; a one‑way membrane. Singularity – a point (or ring for rotating holes) where spacetime curvature becomes infinite. No‑hair theorem – stationary black holes are completely described by mass (M), spin (J or \(a\)), and electric charge (Q). Hawking radiation – quantum particle emission from the horizon; temperature \(TH \propto 1/M\). Schwarzschild radius – non‑rotating horizon radius \(rs = \dfrac{2GM}{c^2}\). ISCO (Innermost Stable Circular Orbit) – smallest stable orbit; for Schwarzschild \(r{\text{ISCO}} = 3\,rs\). Photon sphere – radius where light can orbit (for Schwarzschild \(1.5\,rs\)). Ergosphere – region outside the horizon of a rotating hole where spacetime is dragged; allows energy extraction. --- 📌 Must Remember Schwarzschild radius: \(rs = \frac{2GM}{c^2}\). Hawking temperature: \(TH = \frac{\hbar c^{3}}{8\pi G M kB}\). Spin parameter: \(a = \frac{cJ}{GM^2}\), 0 ≤ \(a\) ≤ 1. Density scaling: \(\rho \propto M^{-2}\) → supermassive holes are low‑density. Efficiency of accretion: 5.7 %–42 % of rest‑mass energy can be radiated, increasing with spin. M–σ relation: black‑hole mass correlates with galaxy bulge velocity dispersion. Penrose process: extracts rotational energy from the ergosphere. Blandford–Znajek power: \(P{\rm BZ} \propto B^{2} a^{2} M^{2}\). Evaporation time: \(t{\rm evap}\sim \frac{5120\pi G^{2}M^{3}}{\hbar c^{4}}\). --- 🔄 Key Processes Stellar collapse → black hole Iron core > Chandrasekhar/TOV limits → pressure can’t halt gravity → collapse → event horizon forms. Accretion disk heating Viscous torque → angular momentum outward, mass inward → gravitational potential energy → X‑ray emission. Hawking radiation emission Quantum fluctuations near horizon → particle‑antiparticle pair → one escapes, the other falls in → thermal spectrum. Penrose energy extraction Particle splits inside ergosphere → one falls with negative energy → black hole loses spin energy, other escapes with >initial energy. Binary black‑hole merger (GW signal) Inspiral (chirp ↑ frequency) → merger (peak amplitude) → ringdown (quasinormal mode decay). --- 🔍 Key Comparisons Schwarzschild vs. Kerr black hole Spin: none vs. rotating (\(a\neq0\)). Horizon shape: spherical vs. oblate. ISCO: fixed at \(3rs\) vs. moves inward (prograde) / outward (retrograde). Photon sphere vs. Event horizon Photon sphere: radius where light can orbit (1.5 \(rs\) non‑rotating). Event horizon: boundary of no return (1 \(rs\)). Stellar‑mass vs. Supermassive black holes Mass: \(\sim\) few \(M\odot\) – \(10^2 M\odot\) vs. \(\gtrsim 10^6 M\odot\). Density: extremely high vs. low (∝ \(M^{-2}\)). Observational signatures: X‑ray binaries vs. AGN jets, galaxy‑scale effects. --- ⚠️ Common Misunderstandings “Light cannot get close to a black hole.” – Light can orbit at the photon sphere; it is only trapped inside the event horizon. “All black holes evaporate quickly.” – Hawking power ∝ \(1/M^2\); stellar‑mass holes would take ≫ age of Universe to evaporate. “Spin always increases horizon size.” – Adding spin reduces the horizon radius (extremal Kerr → \(rh = rs/2\)). “No‑hair means no structure at all.” – Only external observables are mass, spin, charge; internal physics (jets, disks) still exist. --- 🧠 Mental Models / Intuition Event horizon as a one‑way door – crossing inward feels normal, but nothing can cross outward, just like stepping onto a moving walkway that only goes one direction. Ergosphere as a whirlpool – spacetime is dragged, so any object inside is forced to spin; you can “steal” some spin energy by pushing against the flow (Penrose process). Hawking radiation as a furnace – smaller black holes are hotter because the “surface area” is tiny, analogous to a small hot coal radiating more per unit mass. --- 🚩 Exceptions & Edge Cases Extremal Kerr–Newman: charge or spin at maximal values reduces horizon to half the Schwarzschild radius. Charged astrophysical holes: practically neutral; any net charge is quickly neutralized by surrounding plasma. Primordial black holes < \(10^{12}\) kg: would have evaporated already; not present today. --- 📍 When to Use Which Estimate horizon size → use Schwarzschild formula for non‑spinning; apply Kerr correction if \(a \) known. Predict accretion luminosity → thin‑disk model (\(L = \eta \dot{M} c^2\)) when disk is geometrically thin and optically thick. Identify energy source of jets → prefer Blandford–Znajek if strong magnetic fields and high spin are evident; Penrose process for conceptual explanations. Choose observational probe Stellar‑mass BH: X‑ray spectral fitting, timing, binary dynamics. Supermassive BH: VLBI shadow imaging, stellar orbit tracking, AGN spectra. --- 👀 Patterns to Recognize Spin‑dependent ISCO shift – prograde orbits move inward, raising radiative efficiency; retrograde moves outward, lowering it. Power‑law X‑ray tail + radio flat spectrum → low‑hard state with steady jet. Sudden rise in GW frequency + amplitude → merger phase of binary black holes. Broad, red‑skewed Fe Kα line → relativistic reflection from inner disk, indicating high spin. --- 🗂️ Exam Traps Confusing photon sphere with horizon radius – remember photon sphere is outside the horizon (1.5 \(rs\) vs. 1 \(rs\) for Schwarzschild). Assuming Hawking radiation is observable for astrophysical BHs – the power is negligible for stellar or supermassive masses. Mixing up ISCO with event horizon – ISCO is the inner edge of a stable orbit; matter can still exist between ISCO and horizon (plunging region). Attributing all jet power to accretion alone – the Blandford–Znajek process extracts spin energy, not just accretion energy. Thinking the no‑hair theorem forbids any surrounding matter – it only restricts external gravitational/electromagnetic fields; accretion disks, jets, and surrounding plasma are allowed.