Chapter 12 - Curvature, Strong Gravity, and Gravitational Waves

Abstract

Classical differential geometry is introduced. Gaussian curvature is shown to be an intrinsic property of a surface. The Gauss–Bonnet theorem is discussed. The geodesic equation of curved four-dimensional space–time is derived. Parallel transport is introduced and illustrated. The covariant derivative is introduced and Christoffel symbols are discussed from several perspectives. Riemannian geometry, which only deals with intrinsic properties of space–time, is introduced and the Riemann and Einstein tensors are defined, illustrated, and discussed. Formulas for the Riemann tensors in terms of the metric of space–time are derived. The Einstein field equation (“curvature=energy–momentum density”) is introduced and is shown to reduce to Newtonian gravity in the weak field, nonrelativistic case. General relativity is shown to be an intrinsically nonlinear field theory for the metric of space–time. The Schwarzschild metric is derived, and the Schwarzschild black hole is obtained and analyzed. The relativistic generalization of tidal forces is introduced, and the importance of quadrupole moments of mass distributions is emphasized. Gravitational waves, linearized gravity, and the Laser Interferometer Gravitational Wave Observatory (LIGO) experiment to observe such waves are presented. General relativity is contrasted with special relativity, and photons and gravitons are discussed. The similarities of general relativity to modern gauge theories of elementary particles are drawn. Dark energy is discussed, and challenges to the field are presented.