Relativistic Astrophysics & Cosmology

Black holes, compact objects, gravitational waves and the expanding universe

About this course

General relativity applied to black holes, compact stars, accretion and gravitational waves, followed by the geometry, dynamics and history of the universe, cosmological data, and the early and perturbed universe.

What you will be able to do

  • Compute particle orbits and energies in the Schwarzschild metric and describe rotating black holes
  • Explain how compact objects form and how accretion powers active galactic nuclei
  • Describe gravitational waves from binary systems and the physics of gravitational lensing
  • Use the Friedmann equations to relate the geometry, dynamics and contents of the universe
  • Interpret modern cosmological data on the primordial and perturbed universe

What this course covers

  • Stellar evolution life cycle and hydrostatic support
  • Nuclear stellar forge and the binding energy curve
  • The Eddington luminosity limit
  • Virial theorem and the γ = 4/3 collapse threshold
  • Degeneracy pressure: non-relativistic and relativistic regimes
  • Chandrasekhar mass and the white-dwarf mass–radius relation
  • Supernovae, the r-process, and the origin of heavy elements
  • Binary star systems: orbital mechanics and Lagrange points
  • Roche lobe overflow and close binary classification
  • Orbital response to conservative mass transfer
  • The binary pulsar as a test of general relativity
  • Linearized gravity and the weak-field wave equation
  • Gauge freedom in linearized gravity and the Lorenz gauge
  • Transverse-traceless gauge and the two physical polarizations
  • Geodesic equations and conserved quantities
  • Conjugate momentum and the true particle energy
  • Master orbit-shape equation and relativistic correction
  • Energy of a circular orbit in Schwarzschild geometry
  • Circular orbits in a general spherically symmetric (AB) metric
  • Innermost stable circular orbit (ISCO) and accretion efficiency
  • Photon orbits and the unstable photon sphere
  • Course scope and observational grounding
  • Gravitational redshift from energy conservation
  • Pound–Rebka experimental confirmation
  • Incompatibility of gravitational redshift with flat spacetime
  • Equivalence of gravitational and inertial mass
  • Weak and strong equivalence principles
  • Gravity as spacetime curvature
  • The Cosmological Principle
  • Deriving the Three Geometries of Space
  • The Hubble–Lemaître Law and Cosmic Expansion
  • The Hubble Time as the Age of the Universe
  • The Scale Factor and Co-moving Observers
  • The FLRW Metric and Cosmic Time
  • The Cosmological Redshift Formula
  • Linearized gravity with a source: retarded Green's function solution
  • Quadrupole formula for the far-field metric perturbation
  • Gravitational wave energy flux and the quadrupole luminosity formula
  • Binary inspiral: chirp mass and orbital evolution
  • Gravitational wave strain and interferometric detection
  • First detection GW150914 and the chirp waveform
  • LIGO as a gravitational-wave telescope: sky localisation, the neutron-star merger, standard sirens, and the gravitational-wave spectrum
  • FLRW Metric and Kinematics Review
  • Spacetime Curvature of the FLRW Metric
  • Gravitational Field Equations and Trace Reversal
  • Connection Coefficients of the FLRW Metric
  • Ricci Tensor and Perfect-Fluid Source
  • The Friedmann–Lemaître Cosmological Field Equations
  • Equation-of-State Parameter and Fluid Densities
  • Dimensionless Cosmological Parameters
  • Special Relativity Foundations and Notation
  • Equivalence Principle and the Einstein Tower
  • Coordinates, Index Notation, and the Metric
  • Raising and Lowering Indices with the Metric
  • Gravity as Curvature: Local Flatness and Tidal Effects
  • Geodesics: Straightest Path and Shortest Path
  • Euler-Lagrange Equations and the Geodesic Equation
  • Conserved Quantities from Metric Symmetry
  • Why rotating black holes are physically necessary
  • History and uniqueness of the Kerr metric
  • Four definitions and the Kerr line element
  • Boyer-Lindquist coordinates and their non-spherical geometry
  • Charged black holes and the four-solution table
  • Horizons and ring singularity of the Kerr metric
  • Stationary limit surface, frame dragging, and the ergosphere
  • Photon energy, frequency, and wavelength relations
  • Atmospheric opacity and the accessible windows
  • Band-by-band survey of the electromagnetic spectrum
  • Multi-messenger astronomy: gravitational waves and neutrinos
  • Accelerated charge as the unifying principle of radiation
  • Line radiation, bremsstrahlung, and the Larmor formula
  • Synchrotron radiation and Compton/inverse-Compton scattering
  • Photon angular velocity near a spinning black hole
  • Frame dragging and the stationary limit surface
  • The Penrose process
  • Geodesic equations in the Kerr equatorial plane
  • Innermost stable circular orbit (ISCO) in Kerr
  • Accretion efficiency and black hole spin
  • Hawking temperature and black hole entropy
  • Black holes as the universe's entropy reservoir
  • FLRW metric and cosmological field equations recap
  • Master equation for a multi-fluid universe
  • Einstein–de Sitter power-law solutions
  • Geometry and fate in Friedmann models
  • Conformal time and Friedmann model solutions
  • Einstein static universe and its instability
  • Flat lambda universe and the physical case-2 solution
  • De Sitter universe and the maximal Copernican principle
  • Gravitational lensing as a cosmic telescope
  • Newtonian deflection of light
  • Stationary weak-field metric derivation
  • Factor-of-two enhancement: Einstein vs Newton
  • Einstein ring radius and cosmological lensing scale
  • Microlensing and weak lensing
  • Event Horizon Telescope and black hole shadow
  • Bondi accretion and the sonic radius
  • Accretion disk structure and viscosity cancellation
  • Disk temperature profile and multi-colour spectrum
  • Accretion efficiency compared to nuclear burning
  • Effective potential and the innermost stable circular orbit
  • Newton–Schwarzschild–Kerr potential hierarchy
  • Spin-dependent efficiency and the Thorne limit
  • Riemann Curvature Tensor and Derived Scalars
  • Intrinsic vs. Embedded Curvature
  • Gauss's Theorema Egregium
  • The Weak-Field Metric and Newtonian Limit
  • Gravitational Time Dilation from the Metric
  • Metrics of Constant Curvature and the General Radial Metric
  • The Schwarzschild Metric and Black Holes
  • Newtonian hydrostatic equilibrium and central pressure
  • The stress-energy tensor: from dust to perfect fluid
  • Pressure-to-energy-density ratio and radiation
  • Reformulating Einstein's equations via trace reversal
  • Derivation of the metric coefficient B and gravitational binding energy
  • The Oppenheimer–Volkoff equation
  • Schwarzschild interior solution and Buchdahl's theorem
  • Cosmological observables and redshift primacy
  • Proper distance and Hubble's law derivation
  • Linking χ, z, t and H(z)
  • Einstein–de Sitter specialisation and comoving horizon
  • Luminosity distance and angular diameter distance
  • Angular diameter distance minimum in Einstein–de Sitter
  • Age of the universe: models, globular clusters, and radioactive dating
  • Radiative efficiency and the spin table
  • Black hole mass measurement methods
  • AGN taxonomy and the unified model
  • Iron line profile and black hole spin measurement
  • Eddington e-folding time and growth challenge
  • Sagittarius A* mass and the M–σ correlation
  • Superluminal jet motion and Doppler boosting
  • Gamma-ray bursts and the compactness argument
  • Neutron star formation and radius estimates
  • Spin-up from collapse and the breakup density limit
  • Pulsars: emission mechanism and the voltage chain
  • Pulsar applications: clocks, timing arrays, and glitches
  • Orbital precession from the GR correction term
  • Lecture synthesis and the tilde convention
  • Inventory of the Universe's Constituents
  • Photon-to-Baryon Ratio and Baryogenesis
  • Thermal History and the T(t) Relation
  • Nucleosynthesis Epoch
  • Recombination and Why the CMB Remains a Black Body
  • Epoch of Equality and Reionization
  • Evidence for Dark Matter
  • The omega-M / omega-lambda parameter plane
  • Power spectra as language for random fields
  • Spherical harmonics and the CMB power spectrum C_l
  • Cosmic variance
  • CMB acoustic peaks as cosmological barcode
  • Type Ia supernovae as standard candles
  • Baryon acoustic oscillations as a standard ruler
  • Joint cosmological constraints and cosmic tensions
  • Three Cosmological Horizons
  • Particle Horizon as Conformal Time
  • Particle Horizon Dependence on Equation of State
  • Horizon Problem and Flatness Problem
  • Acceleration Condition w < −1/3 as the Unified Fix
  • Scalar Field as a Perfect Fluid (Inflation Mechanism)
  • Slow-Roll Inflation and Its Predictions
  • Scalar Fields, the Cosmological Constant, and Lorentz Invariance
  • Jeans instability in a static fluid
  • Algebraic vs exponential growth in an expanding universe
  • Spherical collapse and the EDS perturbation equation
  • Virialization and the factor-of-200 overdensity
  • The dark matter argument from CMB and growth rate
  • Perturbed FRW metric and gauge freedom
  • Cosmological perturbation theory: scope and summary
  • Scalar perturbations in the Newtonian gauge
  • Scalar field perturbations and the Mukhanov-Sasaki equation
  • Gauge invariance and the comoving curvature perturbation R
  • Freezing of modes during inflation
  • Radiation-era evolution and CMB acoustic peaks
  • Bunch-Davies vacuum and quantum initial conditions
  • Primordial power spectrum, spectral tilt, and observational tests

Syllabus

  1. General Relativity

    1. L1 · Introduction
    2. L2 · General Relativity Recap
    3. L3 · Curvature
    4. L4 · General Relativity and Matter

  2. Black Holes

    1. L5 · Particle Motion and Energy in the Schwarzschild Metric
    2. L6 · Rotating Black Holes
    3. L7 · Rotating Black Holes II

  3. Compact Objects and High-Energy Astrophysics

    1. L8 · Compact Objects and the Life Cycles of Stars
    2. L9 · Astrophysics Across the Electromagnetic Spectrum (and Beyond)
    3. L10 · Accretion Disks
    4. L11 · Active Galactic Nuclei
    5. L12 · Neutron Stars and Pulsars

  4. Gravitational Waves and Lensing

    1. L13 · Binary Systems and Gravitational Waves
    2. L14 · Gravitational Waves
    3. L15 · Gravitational Lensing

  5. Cosmology

    1. L16 · The Geometry of the Universe
    2. L17 · The Dynamics of the Universe
    3. L18 · The Evolution of the Universe
    4. L19 · Measuring the Universe
    5. L20 · Constituents of the Universe
    6. L21 · Modern Cosmological Data
    7. L22 · The Primordial Universe
    8. L23 · The Perturbed Universe
    9. L24 · The Quantum Universe