The kinematics of general relativity is described. General relativity is given by intrinsically curved spacetimes, for non-Euclidean geometry, based on two assumptions, leading to two physical principles for the kinematics (plus one equation for the dynamics). The kinematics is based on the same parallel transport as for gauge theories. The motion of free particles is on geodesics in the curved spacetime.

The theory of special relativity incorporates a modification of Newtonian mechanics together with electromagnetism. A natural question to consider is how gravitation fits into the framework of relativity. In this chapter we focus our analysis of this question along two main ideas, that of the equivalence between uniform acceleration and a uniform gravitational field, and that of the gravitational redshift. These will lead us to the Einstein equation, which we then show can be given a variational formulation. We present some solutions of the Einstein equation, with particular attention given to the Schwarzschild spacetime and its Kruskal extension.

Chapter 3 considers a set of arguments, presented by Richard Swinburne and William Lane Craig, claiming that General Relativity can save the A-theory by substantiating objective passage via cosmic time. This argument is assessed and found wanting.

After an introduction to general relativity and supersymmetry, the formalism of supergravity is defined, on-shell, off-shell, and in superspace, using coset theory and local superspace. Higher dimensions, extended susy, and KK reduction are also defined. Then, various applications are described: dualities and solution-generating techniques, solutions and their susy algebra, gravity duals and deformations, supergravity on the string worldsheet and superembeddings, cosmological inflation, no-go theorems and Witten’s positive energy theorem, compactification of low-energy string theory and toward embedding the Standard Model using supergravity, susy breaking and minimal supergravity.

The principle of relativity requires that no interaction can propagate instantly. Gravitational waves (GW) must exist, propagating with the same speed as light. The specific characters of GW are predicted by Einstein’s general relativity (GR). After decades of efforts to develop detectors, on 11 February 2016, the LIGO and Virgo Collaboration published the discovery of a GW.

The elements of GR relevant for GW production, propagation and detection. How the GR field, which is the dimensionless metric tensor, differs from the other fundamental fields, which have physical dimensions. The instruments and the discovery. After the first observation, dozens of gravitational signals have been detected, the vast majority from merging black holes and one, on 17 August 2017, from the merger of neutron stars. In this case, electromagnetic signals are expected, and have been detected, providing unique information to astrophysics and to fundamental physics as well. The measurement of the speed of the GW and the establishment of a bound on the mass of the graviton.

One important area where radio astronomers confirmed theoretical predictions was in tests of General Relativity. Radio interferometer measurements made during the 1970s were able to confirm Einstein’s prediction of the gravitational bending of light to an accuracy better than 1 percent, or an order of magnitude better than the controversial classical optical tests made during the time of a solar eclipse. In 1965, MIT Professor Irwin Shapiro suggested and subsequently confirmed a new fourth test of General Relativity resulting from the excess delay of the reflected radar signal from a planet as the signal passes close to the Sun. Radio observations have also found Einstein’s “gravitational lensing” by which a massive cluster of galaxies can form multiple radio images of a background galaxy or quasar. Observations of small periodic deviations in the time of arrival of pulsar pulses at the Arecibo Observatory led Princeton University graduate student Russell Hulse and his supervisor Joe Taylor to the 1993 Nobel Prize in Physics for the first experimental evidence for the predicted existence of gravitational radiation.

In § 5.1 and § 5.2 we consider the simplest Lorentz metrics: those of constant curvature. The spatially isotropic and homogeneous cosmological models are described in §5.3, and their simplest anisotropic generalizations are discussed in § 5.4. It is shown that all such simple models will have a singular origin provided that A does not take large positive values. The spherically symmetric metrics which describe the field outside a massive charged or neutral body are examined in §5.5, and the axially symmetric metrics describing the field outside a special class of massive rotating bodies are described in §5.6. It is shown that some of the apparent singularities are simply due to a bad choice of coordinates. In §5.7 we describe the Godel universe and in §5.8 the Taub-NUT solutions. These probably do not represent the actual universe but they are of interest because of their pathological global properties. Finally some other exact solutions of interest are mentioned in §5.9.

This chapter has three main aims. First, it gives a pedagogical introduction to Noether’s two theorems and their implications for energy conservation in general relativity, which was a central point of discussion between Hilbert, Klein, Noether, and Einstein. Second, it introduces and compares two proposals for gravitational energy and momentum, one of which is very influential in physics, and neither of which has been discussed in the philosophical literature. Third, it assesses these proposals in connection with recent philosophical discussions of energy and momentum in general relativity. After briefly reviewing the debates about energy conservation between Hilbert, Klein, Noether, and Einstein, the chapter shows that Einstein’s gravitational energy-momentum pseudo-tensor, including its superpotential, is fixed, through Noether’s theorem, by the boundary terms in the action. That is, the freedom to add an arbitrary superpotential to the gravitational pseudo-tensor corresponds to the freedom to add boundary terms to the action without changing the equations of motion. This freedom is fixed in the same way for both problems. The chapter also includes a review of two proposals for energy and momentum in GR: one is a quasi-local alternative to the local expressions, and the other builds on Einstein’s local pseudo-tensor approach.

Famously, Klein and Einstein were embroiled in an epistolary dispute over whether General Relativity has any physically meaningful conserved quantities. This chapter explores the consequences of Noether’s second theorem for this debate and connects it to Einstein’s search for a ‘substantive’ version of general covariance as well as his quest to extend the Principle of Relativity. The chapter’s argument is that Noether’s second theorem provides a clear way to distinguish between theories in which gauge or diffeomorphism symmetry is doing real work in defining charges, as opposed to cases in which this symmetry stems from Kretchmannization. Finally, a comment is made on the relationship between this Noetherian form of substantive general covariance and the notion of ‘background independence’.

Advances in observing techniques, the commissioning of new radio telescopes and the prospect of the Square Kilometre Array are opening new fields of pulsar research. The 55 years since the discovery of pulsars have revealed a rich and evolving population and shown how precise timing can transform our understanding of neutron star structure, binary system dynamics, stellar populations and the interstellar medium, and have opened new prospects in general relativity physics. X-ray and gamma-ray telescopes, and Cerenkov shower arrays, are extending observations over the whole electromagnetic spectrum.

As an appendix, we can look briefly at the central ideas of General Relativity (though we are limited, since much of the maths is beyond our scope). We prepare the ground with a number of thought experiments, and then discuss, in outline, the geometrical ideas we have to use. We can get a sense of what Einstein's equation is doing, and we look at some solutions of Einstein's equation (including the Schwarzschild metric), describing possible spacetimes.