General Relativity

Tensors and curvature through gravitational radiation, cosmology and black holes.

What you will be able to do

  • Work with the geometric viewpoint on gravity: manifolds, tensors and the metric
  • Derive and apply the Einstein field equations
  • Analyze Schwarzschild spacetime, orbits, and black holes
  • Understand gravitational radiation and cosmological solutions

What this course covers

  • Spacetime as a manifold with metric
  • Special relativity as gravity-free limit
  • Inertial reference frame
  • Einstein clock synchronization procedure
  • Natural units with speed of light equal to one
  • Spacetime displacement vector
  • Lorentz transformation of vector components
  • Einstein summation convention
  • Greek vs. Latin index conventions
  • Definition of a spacetime vector
  • Spacetime as a manifold of events
  • Geometric objects vs. coordinate representations
  • Four-vectors and Lorentz transformation
  • Basis vectors in spacetime
  • Transformation law for basis vectors
  • Index notation and dummy indices
  • Spacetime interval and Lorentz invariance
  • Inner product of four-vectors
  • Timelike, spacelike, and null vectors
  • The metric tensor
  • Coordinate basis vs. orthonormal basis
  • Inner product and metric tensor
  • Tensors as geometric, frame-independent objects
  • Extracting tensor components via basis vectors
  • Tensor component transformation law
  • One-forms (dual vectors)
  • Contraction of a one-form with a vector
  • Basis one-forms and the gradient as a one-form
  • Tensors as geometric objects
  • Four-velocity and its timelike norm
  • Four-momentum and massless particles
  • Dust as a continuum matter model
  • Number density and rest-frame description
  • Lorentz contraction of number density
  • Number-flux four-vector
  • Flux across a surface via one-forms
  • Number four-vector and conservation law
  • Energy density transforms with gamma squared
  • Stress-energy tensor as a tensor product
  • Physical interpretation of T^{alpha beta} components
  • Stress-energy tensor for dust
  • Symmetry of the stress-energy tensor
  • Perfect fluid stress-energy tensor
  • Covariant form of perfect fluid T^{alpha beta}
  • Stress-energy tensor as source of gravity
  • Curvilinear coordinate bases in spacetime
  • Jacobian transformation matrix between coordinate systems
  • Metric tensor in curvilinear coordinates
  • Basis one-forms in curvilinear coordinates
  • Derivatives of curvilinear basis vectors
  • Christoffel symbols
  • Covariant derivative of a vector
  • Covariant derivative of one-forms and higher-rank tensors
  • Tensor equations are representation-independent
  • Christoffel Symbols and Covariant Derivative
  • Gravitational Redshift
  • Impossibility of a Global Lorentz Frame with Gravity
  • Local Lorentz (Freely Falling) Frames
  • Equivalence of Gravitational and Inertial Mass
  • Tidal Forces and Limits of Local Frames
  • Geodesics and Spacetime Curvature
  • Weak equivalence principle
  • Einstein equivalence principle
  • Local flatness of spacetime metric
  • Partial derivatives of tensors fail on curved manifolds
  • Tangent spaces on curved manifolds
  • The connection and covariant derivative
  • Metric compatibility condition
  • Parallel transport
  • Geodesics as straightest paths in curved spacetime
  • Lie derivative
  • Covariant derivative and parallel transport
  • Lie derivative
  • Killing vectors and symmetries
  • Tensor densities
  • Determinant of the metric as a tensor density
  • Converting tensor densities to tensors
  • Covariant volume element
  • Contracted Christoffel symbol identity (party trick)
  • Covariant divergence of a vector field
  • Divergence of tensors and stress-energy conservation
  • Geodesic motion of a test body
  • Geodesic equation and free-fall trajectories
  • Affine parameterization of geodesics
  • Momentum form of the geodesic equation
  • Geodesic equation for massless (null) trajectories
  • Lowering indices with the metric in the geodesic equation
  • Constants of motion from metric symmetry
  • Killing vectors and conserved quantities
  • Riemann Curvature Tensor
  • Symmetries of the Riemann Tensor
  • Local Lorentz Frame Simplification
  • Riemann Normal Coordinates
  • Ricci Curvature Tensor
  • Ricci Scalar (Curvature Scalar)
  • Weyl Curvature Tensor
  • Riemann curvature tensor symmetries
  • Ricci tensor and Ricci scalar
  • Geodesic deviation and tidal gravity
  • Bianchi identity
  • Contracting the Bianchi identity
  • Einstein tensor
  • Minimal coupling principle
  • Geodesic equation via minimal coupling
  • Local conservation of energy-momentum in curved spacetime
  • Einstein field equation and Newtonian limit
  • Tensorial formulation of physics laws
  • Einstein field equations from physical reasoning
  • Einstein-Hilbert action and its motivation
  • Action principle and Lagrange density in field theory
  • Euler-Lagrange equations for scalar fields
  • Massive Klein-Gordon equation
  • Choosing the gravitational Lagrangian
  • Variation of the Einstein-Hilbert action
  • Palatini variation as a rigorous alternative
  • Vacuum and sourced Einstein equations from the action
  • Einstein Field Equations as Metric Differential Equations
  • Three Solution Strategies for GR
  • Linearized General Relativity (Weak Gravity)
  • Gauge Transformations in Linearized GR
  • H as a Tensor Field in Flat Spacetime
  • Raising and Lowering Indices with the Flat Metric
  • Inverse Metric to Linear Order
  • Linearized gravity and the wave equation
  • Static-source limit and Newtonian gravity
  • Green's function technique overview
  • Field point vs. source point notation
  • Proof that Green's function gives the solution
  • Retarded Green's function for the wave operator
  • Retarded time and causal signal propagation
  • Retarded solution for linearized gravity
  • Apparent radiative character and gauge subtleties
  • Linearized gravity and gauge freedom
  • SVT decomposition of metric perturbation
  • Transverse-traceless (TT) gravitational wave
  • TT metric perturbation matrix form
  • Geodesic equation for freely falling bodies in GW spacetime
  • Coordinate dependence and the geodesic calculation pitfall
  • Geodesic deviation equation
  • Riemann tensor components for TT wave
  • Physical effect of GW on a ring of test bodies
  • Light travel time between separated bodies as GW observable
  • LIGO as a gravitational wave detector
  • Gravitational waves in linearized theory
  • Limitation of flat-background assumption
  • Gravitational waves on a curved background
  • Separation of length and time scales
  • Water-wave analogy for scale separation
  • Averaging procedure to isolate background and wave
  • Brill-Hartle averaging of tensors in curved spacetime
  • Linearization around a curved background
  • Symmetry-based approach to exact solutions
  • Maximally symmetric spaces
  • Homogeneity and isotropy
  • Riemann tensor form in maximally symmetric spaces
  • Spatial homogeneity and isotropy of the universe
  • The FLRW metric and scale factor
  • Co-moving coordinates
  • Spatial curvature parameter k
  • Robertson-Walker Metric Recap
  • Friedmann Equations
  • Critical Density and Omega Parameters
  • Recipe for Building a Cosmological Model
  • Scale Factor as Observational Target
  • Density Evolution with Scale Factor
  • Killing Tensors and Conserved Quantities
  • Cosmological Redshift via Killing Tensor
  • Compact spherical body spacetime setup
  • General static spherically symmetric metric
  • Areal radius coordinate
  • Schwarzschild coordinates vs. isotropic coordinates
  • Perfect fluid stress-energy tensor for a static star
  • Exterior vacuum solution and Schwarzschild metric
  • Birkhoff's theorem
  • Mass function M(R) for the stellar interior
  • Schwarzschild interior line element
  • Tolman-Oppenheimer-Volkoff (TOV) equations
  • Numerical integration of stellar structure
  • Constant-density (incompressible) star model
  • Analytic pressure profile for constant-density star
  • Maximum compactness and pressure divergence
  • Buchdahl's theorem
  • Cold equations of state for dense matter
  • Tabulated equations of state from dense-matter physics
  • Schwarzschild vacuum solution recap
  • Proper time vs coordinate time for infalling bodies
  • Gravitational redshift near R=2GM
  • Infinite time dilation of pulse intervals
  • R=2GM as the infinite-redshift surface
  • Need for better time coordinates
  • Radial null geodesics in Schwarzschild coordinates
  • Geodesic equation approach to black hole orbits
  • Exploiting spacetime symmetries and Killing vectors
  • Spherical symmetry and orbital plane reduction
  • Conserved energy and angular momentum in Schwarzschild
  • Four-momentum constraint and effective potential
  • Effective potential analysis of orbital motion
  • Circular orbits and stability in Schwarzschild
  • Innermost stable circular orbit (ISCO)
  • Kepler's third law analog in Schwarzschild
  • Perihelion precession of bound eccentric orbits
  • Astrophysical implications: accretion disks and ISCO

Syllabus

  1. Lectures

    1. 1. Introduction and the geometric viewpoint on physics.
    2. 2. Introduction to tensors.
    3. 3. Tensors continued.
    4. 4. Volumes and volume elements; conservation laws.
    5. 5. The stress energy tensor and the Christoffel symbol.
    6. 6. The principle of equivalence.
    7. 7. Principle of equivalence continued; parallel transport.
    8. 8. Lie transport, Killing vectors, tensor densities.
    9. 9. Geodesics.
    10. 10. Spacetime curvature.
    11. 11. More on spacetime curvature.
    12. 12. The Einstein field equation.
    13. 13. The Einstein field equation (variant derivation).
    14. 14. Linearized gravity I: Principles and static limit.
    15. 15. Linearized gravity II: Dynamic sources
    16. 16. Gravitational radiation I
    17. 17. Gravitational radiation II
    18. 18. Cosmology I
    19. 19. Cosmology II
    20. 20. Spherical compact sources I
    21. 21. Spherical compact sources II
    22. 22. Black holes I
    23. 23. Black holes II