General Relativity — Foundations

From the equivalence principle to black holes, gravitational waves and cosmology.

About this course

Eighteen lectures building general relativity from a single observation: that free fall is indistinguishable from weightlessness. The equivalence principle forces gravity to be geometry, and from there the course develops spacetime and the metric, geodesics as the straightest available paths, proper time and gravitational redshift. The Schwarzschild solution then supplies the classical tests — the effective potential for orbits, Mercury's precession, the bending of light, lensing and the Shapiro delay. A module on black holes covers the horizon, what infall actually feels like, and the Kerr solution behind the Event Horizon Telescope image. The final module builds the curvature machinery properly — geodesic deviation, the Riemann tensor — to arrive at the Einstein field equation, and closes on gravitational waves and cosmology.

What you will be able to do

  • State the equivalence principle and explain why it forces gravity to be described by geometry
  • Describe spacetime with a metric, compute intervals and proper time, and find geodesics
  • Use the Schwarzschild metric to derive the classical tests: orbital precession, light bending and the Shapiro delay
  • Explain the horizon, tidal disruption and the Kerr solution behind the observed black-hole shadow
  • Build curvature from geodesic deviation to the Riemann tensor and interpret the Einstein field equation, gravitational waves and cosmological solutions

What this course covers

  • Geodesic deviation as curvature's definition
  • The geodesic deviation equation and the Riemann tensor
  • The unremovable tide: field vs. curvature
  • Curvature as second derivative of the metric
  • Newtonian tidal acceleration as the limit of geodesic deviation
  • Clause 14 and course synthesis
  • The Equivalence Principle
  • Freely Falling Frames as Inertial Frames
  • Gravity Bends Light
  • Tidal Forces as Geodesic Deviation
  • Gravity Is the Curvature of Space-Time
  • The Schwarzschild Metric as a Gift
  • Reading the Three Terms of the Metric
  • The Schwarzschild Radius rs = 2GM/c²
  • The Factor (1 − rs/r) and the Event Horizon
  • Clause Six of the Course Spine
  • Preview: Orbits and Mercury's Precession
  • Deriving the Schwarzschild radius as the event horizon
  • The outsider's view: freeze, redshift, and fade
  • The faller's own view: smooth finite crossing
  • Reconciling the two views: coordinate time vs. proper time
  • Numerical worked example: rs for the Sun and Earth
  • Clause eleven and course synthesis
  • Orbit as a one-dimensional problem
  • Shape of the effective potential valley
  • Reading orbit type from the energy line
  • Connecting the valley to the real ellipse
  • ISS circular orbit calculation
  • Preview: Schwarzschild metric bends the valley
  • Parallel transport and holonomy
  • The Riemann curvature tensor defined
  • Symmetries of Riemann and the count of twenty
  • Riemann zero as the exact test of flatness
  • Gauss–Bonnet worked example on a sphere
  • Riemann as precursor to Einstein's equation
  • The space-time interval and proper time
  • The metric tensor as a position-varying ruler
  • Gravitational time dilation from the metric
  • Geodesics as paths of maximal proper time
  • The Schwarzschild metric
  • GPS clock correction as empirical proof
  • The metric as the gravitational field: course spine
  • Tidal force as a gravity gradient
  • Spaghettification: radial stretch and tangential squeeze
  • Small vs. supermassive black holes at the horizon
  • Inevitability of the singularity inside the horizon
  • Clause twelve of the course spine
  • Preview: gravitational waves as a travelling tide
  • Orbital precession and the rosette
  • The GR correction term to the effective potential
  • Perihelion advance angle δφ per orbit
  • Precession formula and Mercury's 43 arcsec/century
  • Significance and preview of stronger-field GR
  • Clause 8 of the course spine
  • Contraction ladder: Riemann to Ricci to scalar
  • Einstein tensor and the Bianchi conservation requirement
  • Stress-energy tensor as the source of gravity
  • The Einstein field equation
  • Weak-field limit and the origin of the 8π coefficient
  • Wheeler's synthesis and the self-consistent loop
  • The Kerr Black Hole: Mass and Spin
  • Frame Dragging
  • The Ergosphere and Kerr Structure
  • The Photon Ring and Black Hole Shadow
  • The Event Horizon Telescope Image
  • Clause 13 and Course Synthesis
  • Geodesics: straightest paths on curved surfaces
  • Free fall as the path of maximal proper time
  • Geodesic rule as Newton's first law in curved spacetime
  • Straightest path bends toward a mass — gravity as geometry
  • Recovering Newton's law from the geodesic rule
  • The three-clause spine and lecture synthesis
  • Linearized Gravity and the Wave Equation
  • Physical Degrees of Freedom and Gauge
  • The Two Polarizations of Gravitational Waves
  • Sources of Gravitational Waves: The Quadrupole Rule
  • LIGO and the First Detection
  • Gravitational Waves as a New Observational Window
  • Gravitational light bending: geometry and image shift
  • The factor of two: Newtonian vs. relativistic deflection
  • The 1919 eclipse experiment: design and logic
  • Confirmation and precision of the measurement
  • Clause nine of the course spine
  • Preview: gravitational lensing and black holes
  • Proper time as a path-dependent quantity
  • Gravity slows clocks
  • The gravitational twin
  • GPS as empirical confirmation
  • Clause four of the course spine: proper time in context
  • The Cosmological Principle and FRW Metric
  • The Friedmann Equations and Forbidden Static Universe
  • Cosmological Redshift and Hubble's Law
  • The Big Bang
  • The Cosmic Microwave Background
  • Dark Energy and Accelerating Expansion
  • Gravitational lensing: mass focuses light
  • Einstein ring and image geometry
  • Real gravitational lenses in the sky
  • Shapiro delay: light passing a mass arrives late
  • Numerical estimate of the Shapiro delay
  • Clause ten of the course spine
  • Gravitational redshift: core mechanism
  • Spectral-line barcode as a redshift ruler
  • Redshift ladder: depth determines redness
  • Photon energy vs. speed: resolving Rohan's knot
  • Pound–Rebka experiment: redshift measured
  • Clause five of the course spine and preview of black holes

Syllabus

  1. Foundations — From Equivalence to Geometry

    1. The Equivalence Principle
    2. Spacetime & the Metric
    3. Geodesics — The Straightest Path
    4. Proper Time & the Gravitational Twin
    5. Gravitational Redshift

  2. The Schwarzschild Solution and the Classical Tests

    1. The Schwarzschild Metric
    2. Orbits I — The Effective Potential
    3. Orbits II — Mercury's Precession
    4. Bending of Light
    5. Lensing & the Shapiro Delay

  3. Black Holes

    1. Black Holes — The Horizon
    2. Falling In — Spaghettification
    3. Spin & Shadow — Kerr and the EHT Image

  4. Curvature, the Field Equations and the Universe

    1. Curvature, Concretely: Geodesic Deviation
    2. The Riemann Curvature Tensor
    3. The Einstein Field Equation
    4. Gravitational Waves
    5. Cosmology: General Relativity and the Universe