Black Holes

Introduction

Black holes are regions of spacetime where gravity is so strong that nothing, not even light, can escape. They are exact solutions to Einstein's field equations and represent some of the most extreme environments in the universe.

📺 Video Lectures

World-class lectures on black hole physics, Hawking radiation, and conformal cyclic cosmology from Nobel laureates.

Professor Sir Roger Penrose: Hawking Points, Conformal Cyclic Cosmology & Black Hole Entropy

Nobel laureate Sir Roger Penrose explores Hawking points in the cosmic microwave background radiation, using conformal mapping to examine spacetime geometry. The presentation delves into the second law of thermodynamics and black hole entropy, challenging conventional cosmological models.

Key Topics:

  • Hawking points in the CMB as evidence for pre-Big Bang eons
  • Conformal Cyclic Cosmology (CCC) model
  • Black hole evaporation via Hawking radiation
  • Information paradox and entropy considerations
  • The universe's remote future and cyclic time

The 30th Hintze Lecture – Prof Sir Roger Penrose and Prof Janna Levin: 'A Universe of Black Holes'

A conversation between Nobel laureate Sir Roger Penrose and astrophysicist Prof Janna Levin exploring the profound implications of black holes throughout the universe. They discuss the nature of singularities, event horizons, gravitational waves from black hole mergers, and how black holes shape our understanding of spacetime and cosmology.

Discussion Topics:

  • Formation and evolution of black holes across cosmic time
  • LIGO/Virgo detections of gravitational waves from mergers
  • The nature of singularities and what lies beyond
  • Supermassive black holes at galactic centers
  • The role of black holes in galaxy formation and evolution

1. Schwarzschild Black Hole

The simplest black hole solution: spherically symmetric, static, and uncharged.

Schwarzschild Metric:

$$ds^2 = -\left(1-\frac{r_s}{r}\right)c^2dt^2 + \left(1-\frac{r_s}{r}\right)^{-1}dr^2 + r^2d\Omega^2$$

Schwarzschild Radius:

$$r_s = \frac{2GM}{c^2}$$

For the Sun: $r_s \approx 3$ km. For Earth: $r_s \approx 9$ mm.

Event Horizon:

The surface at $r = r_s$ is the event horizon - a one-way boundary where the escape velocity equals the speed of light.

2. Kruskal-Szekeres Coordinates

To remove the coordinate singularity at $r = r_s$, use Kruskal-Szekeres coordinates:

$$U = \left(\frac{r}{r_s}-1\right)^{1/2}e^{r/2r_s}\sinh\left(\frac{ct}{2r_s}\right)$$ $$V = \left(\frac{r}{r_s}-1\right)^{1/2}e^{r/2r_s}\cosh\left(\frac{ct}{2r_s}\right)$$

The metric becomes:

$$ds^2 = \frac{4r_s^3}{r}e^{-r/r_s}(-dV^2 + dU^2) + r^2d\Omega^2$$

This reveals the maximal analytic extension including white holes and parallel universes.

3. Reissner-Nordström Black Hole (Charged)

A spherically symmetric black hole with electric charge $Q$:

$$ds^2 = -\left(1-\frac{2GM}{c^2r}+\frac{GQ^2}{c^4r^2}\right)c^2dt^2 + \left(1-\frac{2GM}{c^2r}+\frac{GQ^2}{c^4r^2}\right)^{-1}dr^2 + r^2d\Omega^2$$

Two horizons at:

$$r_\pm = \frac{GM}{c^2} \pm \sqrt{\frac{G^2M^2}{c^4} - \frac{GQ^2}{c^4}}$$

Outer horizon $r_+$ is the event horizon, inner horizon $r_-$ is the Cauchy horizon.

4. Kerr Black Hole (Rotating)

Most astrophysical black holes rotate. The Kerr metric in Boyer-Lindquist coordinates:

$$ds^2 = -\left(1-\frac{r_sr}{\Sigma}\right)c^2dt^2 - \frac{r_sr\sin^2\theta}{\Sigma}(c\,dt)(a\,d\phi)$$ $$+ \frac{\Sigma}{\Delta}dr^2 + \Sigma d\theta^2 + \left(r^2+a^2+\frac{r_sra^2\sin^2\theta}{\Sigma}\right)\sin^2\theta d\phi^2$$

where:

$$\Sigma = r^2 + a^2\cos^2\theta, \quad \Delta = r^2 - r_sr + a^2$$ $$a = \frac{J}{Mc}, \quad r_s = \frac{2GM}{c^2}$$

$J$ is the angular momentum and $a$ is the spin parameter.

5. Ergosphere and Penrose Process

The ergosphere is the region between the event horizon and the static limit:

$$r_{\text{static}} = \frac{r_s}{2} + \sqrt{\frac{r_s^2}{4} - a^2\cos^2\theta}$$

In the ergosphere, rotational energy can be extracted via the Penrose process. Maximum extractable energy:

$$\eta_{\text{max}} = 1 - \sqrt{\frac{8}{9}} \approx 0.29 \text{ (29\%)}$$

for an extremal Kerr black hole ($a = GM/c^2$).

6. Particle Motion and Geodesics

For the Schwarzschild metric, the effective potential for radial motion:

$$V_{\text{eff}}(r) = \left(1-\frac{r_s}{r}\right)\left(1+\frac{L^2}{r^2c^2}\right)$$

Photon Sphere:

$$r_{\text{photon}} = \frac{3r_s}{2} = \frac{3GM}{c^2}$$

Innermost Stable Circular Orbit (ISCO):

$$r_{\text{ISCO}} = 3r_s = \frac{6GM}{c^2}$$

7. Hawking Radiation

Quantum effects near the event horizon cause black holes to radiate as black bodies:

Hawking Temperature:

$$T_H = \frac{\hbar c^3}{8\pi GMk_B} = \frac{\hbar c}{4\pi k_Br_s}$$

For a solar mass black hole: $T_H \approx 60$ nK (extremely cold!).

Evaporation Time:

$$t_{\text{evap}} = \frac{5120\pi G^2M^3}{\hbar c^4} \propto M^3$$

For a solar mass: $t_{\text{evap}} \sim 10^{67}$ years (much longer than the age of the universe).

8. Black Hole Thermodynamics

Bekenstein-Hawking Entropy:

$$S = \frac{k_BAc^3}{4\hbar G} = \frac{k_BA}{4\ell_P^2}$$

where $A = 4\pi r_s^2$ is the area of the event horizon and $\ell_P = \sqrt{\hbar G/c^3}$ is the Planck length.

Four Laws of Black Hole Thermodynamics:

  • 0th Law: Surface gravity $\kappa$ is constant over the event horizon
  • 1st Law: $dM = \frac{\kappa}{8\pi G}dA + \Omega_HdJ + \Phi_HdQ$
  • 2nd Law: $\delta A \geq 0$ (area theorem - horizon area never decreases)
  • 3rd Law: Impossible to reduce $\kappa$ to zero in finite steps

Surface Gravity:

$$\kappa = \frac{c^4}{4GM} = \frac{c^2}{2r_s}$$

Related to temperature by: $T_H = \frac{\hbar\kappa}{2\pi k_Bc}$

9. No-Hair Theorem

Black holes are characterized by only three externally observable parameters:

  • • Mass ($M$)
  • • Angular momentum ($J$)
  • • Electric charge ($Q$)

All other information is lost during gravitational collapse - "black holes have no hair." This leads to the information paradox in quantum mechanics.

10. Accretion Disks

Matter falling into a black hole forms an accretion disk. The efficiency of converting rest mass to radiation:

$$\eta = 1 - \sqrt{1-\frac{2GM}{c^2r_{\text{ISCO}}}}$$

For Schwarzschild: $\eta \approx 0.057$ (5.7%)

For maximal Kerr: $\eta \approx 0.42$ (42%) - more efficient than nuclear fusion!

11. Tidal Forces (Spaghettification)

The tidal force experienced by an object of length $\ell$ at distance $r$:

$$F_{\text{tidal}} \approx \frac{2GM\ell}{r^3}$$

For stellar-mass black holes, tidal forces are enormous at the horizon. For supermassive black holes, they can be survivable at the horizon.

12. Binary Black Hole Mergers

Two black holes in orbit emit gravitational waves, losing energy. The chirp mass:

$$\mathcal{M} = \frac{(M_1M_2)^{3/5}}{(M_1+M_2)^{1/5}}$$

Gravitational wave frequency evolution:

$$f(t) \propto (t_c - t)^{-3/8}$$

where $t_c$ is the coalescence time. LIGO/Virgo detections confirm GR predictions.

13. Penrose-Hawking Singularity Theorems

Under reasonable physical conditions (energy conditions, causality), singularities are inevitable:

  • • Null energy condition: $T_{\mu\nu}k^\mu k^\nu \geq 0$ for all null vectors $k^\mu$
  • • Generic condition: geodesics encounter curvature
  • • Causality: no closed timelike curves

These imply that spacetime is geodesically incomplete - a singularity exists where GR breaks down.