Black Holes

The following sections are included:

• Preface

• Contents

The following sections are included:

• What is a black hole?

• Why study black holes?

• Elements of general relativity
  • The principle of equivalence
  • The Newtonian affine connection
  • Newtonian gravity
  • Metrics in relativity
  • The velocity and momentum 4-vector
  • General vectors and tensors
  • Locally measured physical quantities
  • Derivatives in relativity
  • Acceleration 4-vector
  • Paths of light
  • Einstein's field equations
  • Symmetry and Killing's equation

The following sections are included:

• The Schwarzschild metric
  • Coordinates
  • Proper distance
  • Proper time
  • Redshift
  • Interpretation of M and geometric units
  • The Schwarzschild radius
  • The event horizon
  • Birkoff's theorem
  • Israel's theorem

• Orbits in Newtonian gravity
  • Newtonian Energy
  • Angular momentum
  • The Newtonian effective potential
  • Classification of Newtonian orbits

• Particle orbits in the Schwarzschild metric
  • Constants of the motion
  • Conserved Energy
  • Angular momentum
  • The effective potential
  • Newtonian approximation to the metric
  • Classification of orbits
  • Radial infall
  • The locally measured energy of a particle
  • Circular orbits
  • Comparison with Newtonian orbits
  • Orbital velocity in the frame of a hovering observer
  • Energy in the last stable orbit

• Orbits of light rays
  • Radial propagation of light
  • Capture cross-section for light
  • The view of the sky for a stationary observer

• Classical tests

• Falling into a black hole
  • Free-fall time for a distant observer
  • Light-travel time
  • What the external observer sees
  • An infalling observer's time
  • What the infalling observer feels

• Capture by a black hole
  • Case I: Capture of high angular momentum particles
  • Case II: Capture of low energy particles

• Surface gravity of a black hole
  • The proper acceleration of a hovering observer
  • Surface gravity
  • Rindler coordinates

• Other coordinates
  • Null coordinates
  • Eddington–Finkelstein coordinates

• Inside the black hole
  • The infalling observer

• White holes

• Kruskal coordinates
  • The singularities at r = 0 and cosmic censorship
  • The spacetime of a collapsing star

• Embedding diagrams

• Asymptotic flatness
  • The Penrose–Carter diagram for the Schwarzschild metric
  • The Penrose–Carter diagram for the Newtonian metric

• Non-isolated black holes
  • The infinite redshift surface
  • Trapped surfaces
  • Apparent horizon

• The membrane paradigm

The following sections are included:

• The Kerr metric

• The event horizon
  • The circumference of the event horizon
  • The area of the event horizon

• Properties of the Kerr metric coefficients
  • Identities
  • Contravariant components

• Interpretation of m, a and geometric units

• Extreme Kerr black hole

• Robinson's theorem

• Particle orbits in the Kerr geometry
  • Constants of the motion
  • Energy
  • Angular momentum
  • The Carter integral
  • The radial equation
  • The effective potential

• Frame-dragging
  • Free fall with zero angular momentum
  • Orbits with non-zero angular momentum

• Zero angular momentum observers (ZAMOs)
  • Some applications of ZAMOs

• Photon orbits
  • The photon effective potential
  • Azimuthal motion
  • Photon capture cross-section

• The static limit surface

• The infinite redshift surface

• Circular orbits in the equatorial plane
  • Innermost (marginally) stable circular orbit
  • Period of a circular orbit
  • Energy of the innermost stable orbit
  • Angular momentum of the innermost stable orbit
  • Marginally bound orbits
  • Unbound orbits

• Polar orbits
  • Orbital period

• The ergosphere
  • Negative energy orbits
  • Energy and angular momentum
  • The Penrose process
  • Realising the Penrose process

• Spinning up a black hole
  • From Schwarzschild to extreme Kerr black hole

• Other coordinates

• Penrose–Carter diagram
  • Interior solutions and collapsing stars

• Closed timelike lines

• Charged black holes

The following sections are included:

• Black hole mechanics
  • Surface gravity
  • Redshift
  • Conservation of energy

• The area of a Kerr black hole horizon cannot decrease
  • Area change by accretion
  • Area change produced by the Penrose process
  • The area theorem
  • Irreducible mass
  • Maximum energy extraction
  • Naked singularities

• Scattering of waves
  • Superradiance

• Thermodynamics
  • Horizon temperature
  • The four laws of black hole thermodynamics

• Hawking radiation
  • Introduction
  • Casimir effect
  • Thermal vacua in accelerated frames
  • Hawking radiation

• Properties of radiating black holes
  • Entropy and temperature
  • Radiating black holes
  • Black hole in a box

• Entropy and microstates

The following sections are included:

• Introduction

• Wormholes

• Traversible wormholes

• Creating a wormhole

• Weak energy condition

• Exotic matter

• Time machines

• Chronology protection

The following sections are included:

• Introduction

• Stellar mass black holes
  • Formation
  • Finding stellar mass black holes
  • The black hole at the centre of the Galaxy

• Supermassive black holes in other galaxies
  • Intermediate mass black holes
  • Mini black holes

• Further evidence for black hole spin

• Conclusions

The following sections are included:

• Solutions to Problems

• References

• Bibliography

• Index