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Geodesics are introduced and the geodesic equation analysed for the geometries introduced in chapter 2, using variation principles of classical mechanics. Geodesic motino on a sphere is described as well as the Coriolis effect and the Sagnac effect. Newtonian gravity is derived as the non-relativistic limit of geodesic motion in space-time. Geodesics in an expanding universe and heat death is described. Geodesics in Schwarzschild space-time are treated in detail: the precession of the perihelion of Mercury; the bending of light by the Sun; Shapiro time delay; black holes and the event horizon. Gravitational waves and gravitational lensing are also covered.
The simplest curved spacetimes of general relativity are the ones with the most symmetry, and the most useful of these is the geometry of empty space outside a spherically symmetric source of curvature – for example, a spherical star. This is called the Schwarzschild geometry. To an excellent approximation, this is the curved spacetime outside the Sun and therefore leads to the predictions of Einstein’s theory most accessible to experimental test. In this chapter, we explore the geometry of Schwarzschild’s solution, assuming it’s given. We will concentrate on predicting the orbits of test particles and light rays in the curved spacetime of a spherical star that exhibit some of the famous effects of general relativity – the gravitational redshift, the precession of the perihelion of a planet, the gravitational bending of light, and the time delay of light.
We derive the equations of motion for a variety of physical systems in the PPN formalism, including photons, fluid systems, and N-body systems consisting of well-separated self-gravitating objects. We also specialize to two-body systems and describe the framework for calculating perturbations of Keplerian orbit elements induced by post-Newtonian corrections in the equations of motion. For a class of theories based on an invariant action, we obtain the Lagrangian that describes the dynamics of an N-body system. We derive the locally-measured, or effective gravitational constant, as measured by a Cavendish experiment, within the PPN formalism. For spinning bodies, we obtain the equations of motion and the equations of spin precession.
We describe three central tests of general relativity, sometimes call the "classical" tests: the deflection of light, the Shapiro time delay and the perihelion advance of Mercury. After deriving each effect in detail within the PPN formalism, we describe the various measurements that lead to tight bounds on the relevant PPN parameters.
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