The orbits of test particles and light rays in the Schwarzschild geometry that were worked out in Chapter 9 are not only important for the delicate tests of general relativity in the solar system discussed in Chapter 10. They are also central to a number of astrophysical applications. This chapter introduces three of these applications – gravitational lensing, relativistic frequency shifts from accretion disks, and weighing stars in binary pulsars, which act both as a laboratory for general relativity and and a tool for astronomy. Some tests of Einstein’s theory were the subject of the previous chapter; some of its applications are the subject of this.

The World Trade Organization (WTO) possesses a system for the settlement of disputes between its Members that is, in many respects, unique and, until recently, quite successful. This system is provided for in the WTO Dispute Settlement Understanding (DSU). It creates a single, integrated system for the resolution of disputes arising under any of the WTO ‘covered agreements’.

The stellar endpoint leading to its collapse to a black hole was described in Chapter 12. This chapter explores the other possibility, where a star is supported against gravity by a nonthermal source of pressure. This is realized in nature by white dwarf stars and neutron stars. Unlike black holes, which can be understood entirely in the context of general relativity, an understanding of the stars at the endstate of stellar evolution requires almost all of the rest of physics in some way. We cannot hope to review the range of physics necessary for a complete understanding of the equilibrium endstates of stellar evolution, but we can isolate the essential role of gravitational physics and discuss the overall structure of these stars. To an excellent approximation, the properties of the matter relevant for the gross structure of neutron stars and white dwarfs can be summarized by an equation of state relating the pressure of an ideal matter fluid to its energy density.

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.

Black holes are the outcome of unhalted gravitational collapse. Gravitational collapse to a black hole occurs on a wide range of mass scales in the universe because gravity is an attractive and universal force. This chapter describes black holes of three different origins, with three different mass scales, how they have or could be identified, and sketches how they are at the heart of some of the most energetic phenomena in astrophysics. These are black holes in X-ray binaries, black holes in galaxy centers, and exploding primordial black holes. Black holes are not only interesting because they check general relativity, they also contribute to the explanation of frontier astrophysical phenomena.

Chapter 9’s analysis of the orbits of test particles and light rays in the Schwarzschild geometry identified four effects of general relativity that can be tested in the solar system: the gravitational redshift, the deflection of light by the Sun, the precession of the perihelion of a planetary orbit, and the time delay of light. This list does not exhaust the tests that can be carried out in the solar system, but describes some of the more important ones. Experiments that measure these effects confirm the predictions of general relativity in the solar system to a typical accuracy of a fraction of 1 percent. The discussion in this chapter is not a review of the experimental situation in general relativity either in the past or at the time of writing. Instead, it presents a discussion of representative experiments that are currently among the most accurate, but are not necessarily the most accurate.

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WTO law provides for detailed rules with respect to dumping and subsidisation – two specific trade practices commonly considered to be unfair. The following sections will briefly examine the WTO rules concerning these trade practices.