The Geometry of Gravity

General Relativity and Its Classical Tests in 18 chapters — from “Minkowski spacetime and relativistic dynamics, completed” to “FRW cosmology as an exact solution”.

Chapters

  1. Minkowski spacetime and relativistic dynamics, completed
    • The Minkowski metric and index position
    • The Lorentz group as the isometries of η
    • Causal structure and signature conversion
    • Four-velocity and four-acceleration
    • The four-force and the covariant equation of motion
    • Problems
  2. Tensors in flat spacetime: index notation and covariant field equations
    • The tensor transformation law
    • Contraction and the quotient rule
    • Manifest covariance and index balance
    • The field-strength tensor and Maxwell's equations
    • The tensor transformation law in curvilinear coordinates
    • Problems
  3. The stress-energy tensor and relativistic continua
    • The stress-energy tensor and its components
    • Dust, then a perfect fluid
    • Local conservation of energy and momentum
    • Trace, radiation and energy conditions
    • Why ρ alone cannot source gravity
    • Problems
  4. The tidal field: what free fall cannot remove
    • The tidal field as a tensor
    • Trace and traceless parts
    • The locality bound on a free-fall frame
    • Non-removability of the tidal field
    • The specification for a geometric replacement
    • Problems
  5. Accelerated observers, Rindler coordinates and the equivalence principle
    • A coordinate system adapted to an observer
    • The Rindler metric
    • Clock rates from g00​
    • The Rindler horizon as a coordinate effect
    • The equivalence principle as a working constraint
    • Problems
  6. Redshift, local inertial frames and the failure of flat spacetime
    • The redshift as a metric component
    • The local inertial frame as a statement about the metric
    • Schild's argument as a statement about geometry
    • Free fall as extremal proper time
    • Clock budgets in orbit
    • Problems
  7. The equivalence principle under test
    • Violation parameters and their bounds
    • Schiff's conjecture
    • The strong equivalence principle and the Nordtvedt effect
    • Metric theories of gravity
    • Problems
  8. Manifolds, vectors, tensor fields and the metric
    • Manifold, chart and atlas
    • Tangent vector as a directional derivative
    • Covectors and the coordinate basis
    • Tensor fields and the transformation law recovered
    • The metric on a manifold
    • Riemann normal coordinates
    • Problems
  9. The covariant derivative, parallel transport and geodesics
    • The connection and the covariant derivative
    • Christoffel symbols from the metric
    • Parallel transport
    • Geodesics
    • The variational route and affine parameters
    • Covariant divergence and the volume element
    • Problems
  10. Curvature: the Riemann tensor, geodesic deviation and the Einstein tensor
    • Parallel transport round a small loop
    • The Riemann tensor and its symmetries
    • The sphere, the Rindler wedge and curvature invariants
    • Geodesic deviation
    • The Newtonian limit of geodesic deviation
    • The Bianchi identity and the Einstein tensor
    • Problems
  11. Symmetry: Lie derivatives, Killing vectors and conserved quantities
    • Geodesics and a cyclic coordinate
    • The Lie derivative along a flow
    • Killing vectors and maximal symmetry
    • Conserved quantities along geodesics
    • Static, stationary and spherically symmetric
    • Problems
  12. The Einstein field equations and the Newtonian limit
    • The field equations
    • The assumptions that fix the left-hand side
    • Fixing the constant by the Newtonian limit
    • Ten equations, four constraints
    • Non-linearity and the absence of a local energy density
    • Problems
  13. Matter in curved spacetime: minimal coupling and relativistic hydrodynamics
    • Minimal coupling
    • Fields and fluids under minimal coupling
    • Projecting the conservation law
    • Inertia of pressure, and why dust follows geodesics
    • Relativistic hydrostatic equilibrium
    • Problems
  14. Linearised gravity: weak fields, gravitomagnetism and the wave equation
    • The weak-field expansion and its coordinate dependence
    • Gauge freedom as residual coordinate freedom
    • The linearised field equations
    • Gravitomagnetism
    • The vacuum wave equation
    • Where the linear theory fails
    • Problems
  15. The Schwarzschild geometry and Birkhoff's theorem
    • The static spherically symmetric ansatz
    • Solving the vacuum equations
    • Birkhoff's theorem
    • Areal radius, proper distance and proper time
    • The exact redshift
    • The Schwarzschild radius read with an invariant
    • Problems
  16. Orbits and light rays in Schwarzschild spacetime
    • Conserved quantities and the radial equation
    • The relativistic effective potential
    • Perihelion advance
    • Light deflection and the factor of two
    • Shapiro delay and gravitational lensing
    • Which tests probe the field equations
    • Problems
  17. Solar-system tests, the PPN framework and alternative theories
    • The PPN expansion
    • Rewriting the tests as bounds
    • Brans-Dicke and alternative theories of gravity
    • What the solar system cannot test
    • Problems
  18. FRW cosmology as an exact solution
    • Homogeneity and isotropy as symmetry
    • The FRW metric
    • The Friedmann equations
    • The cosmological redshift
    • Model universes and the density parameters
    • Problems