Gravitational plane waves and their detection start the chapter off in parallel with electromagnetic plane waves and their detection. Geodesic deviation is reviewed and allows for a brief introduction to LIGO. The stress tensor source of radiation (and therefore, at least local, plane waves) and its conservation of energy and momentum are tied to the radiation solution of the linearized form of Einstein’s equation. A highlight is the role of gauge freedom in making gravitational plane waves physically relevant (by gauge fixing to reveal the underlying plus and cross polarizations). Power loss to gravitational radiation is discussed in the linearized limit by comparison with electric quadrupole radiation.

This chapter discusses the problem of de-quantization. Starting with some given quantum space defined through a matrix configuration, a general procedure is developed which allows to associate a manifold and further geometrical structure underlying the given quantum space.

Spherical symmetry for a metric is defined and used to build a two-function ansatz. The Schwarzschild spacetime emerges as the solution to Einstein’s equation with, and we see how the Newtonian potential is related to the linearized Schwarzschild metric. The lightlike and spacelike geodesics of the metric are explored using exact, approximate, and numerical approaches. Many of the usual experimental tests are covered in detail: perihelion precession, bending of light, and time dilation, for example. The structure of the singularities in the Schwarzschild spacetime is studied using Eddington–Finkelstein and Kruskal–Szekeres coordinates. At the end of the chapter, the Kerr spacetime is introduced, and students are invited to explore its geodesics.

This chapter explains the generation of fuzzy extra dimensions in the context of ordinary Yang–Mills gauge theory and Yang–Mills matrix models. This will play an important role in the context of gravity.

Newtonian gravity is reviewed and an attempt is made to combine it with special relativity, first by expanding the sources from mass to more general mass-energy, and then by considering relativistic force predictions. The gravito-electro-magnetic field equations are developed by analogy with Maxwell’s equations, and using dynamical source configurations familiar from the study of E&M. In addition to the fields, there are predicted particle interactions, like the bending of light, that go beyong Newtonain gravitational forces. Finally, it is clear that this attempt to combine gravity and special relativity lacks the necessary self-coupling of the gravitational field, which carries energy and therefore acts as its own source.

An introduction to field Lagrangians for scalars, vectors, and the Einstein–Hilbert Lagrangian for gravity provides a venue to think about coupling together different field theories. The natural expression of that coupling comes from an action, and we show how the “Euler–Lagrange” field equations enforce the universal coupling of all physical theories to gravity. As an example, the combined field equations of electricity & magnetism and gravity are solved in the spherically symmetric case to give the Reissner–Nordstrøm spacetime associated with the exterior of charged, massive, spherically symmetric central bodies.

Starting from the definition of tensorial objects by their response to coordinate transformation, this chapter builds the flat space vector calculus machinery needed to understand the role of the metric and its associated geodesic curves in general. The emphasis here is on using tensors to build equations that are “generally covariant,” meaning that their content is independent of the coordinate system used to express them. Motivated by the transformation of gravitational energy sources, the gravitational field should be a second-rank tensor, and given the way in which that tensor must show up in a particle motion Lagrangian, it is natural to interpret that tensor as a metric.