The Schwarzschild geometry that underlies much of the physics in previous chapters is exactly spherically symmetric. It is an excellent approximation to the geometry outside a nonrotating star, and is the exact geometry outside a nonrotating black hole. However, no body in nature is exactly nonrotating. The Sun, for example, is rotating at the equator with a period of approximately 27 days, and it is not exactly spherically symmetric, but is slightly squashed along the rotation axis (it is less than 1 part in 100,000 longer than a diameter along the rotation axis). The small value of that difference is why the Schwarzschild geometry is an excellent approximation to the curved spacetime geometry outside the Sun. The curved spacetimes produced by rotating bodies have a richer and more complex structure than the Schwarzschild geometry. This chapter explores one simple example of a gravitomagnetic effect – the dragging of inertial frames by a slowly rotating body.

We will trace out some parts of the path that led Einstein to a new theory of gravity that is, unlike Newtonian gravity, consistent with the principle of relativity. The result will be general relativity, a theory that is qualitatively different from Newtonian gravity. In general relativity, gravitational phenomena arise not from forces and fields, but from the curvature of four-dimensional spacetime. The starting point for these considerations is the equality of gravitational and inertial mass, one of the most accurately tested principles in all physics. This leads to Einstein’s equivalence principle, the idea that there is no experiment that can distinguish a uniform acceleration from a uniform gravitational field – the two are fully equivalent.

Which of the four-parameter family of Friedman–Robertson–Walker (FRW) cosmological models best fits our universe and why? This chapter addresses these two central questions for observation and theory in cosmology. Of the four parameters that define an FRW model, only two are determined by observations so far: the Hubble constant; and the ratio of energy density in radiation to the critical density. To determine the others, the spacetime geometry of the universe must be measured on large scales through a study of how matter moves through it. We describe two illustrative ways of doing that – one based on observations of distant supernovae, and the other on observations of the cosmic background radiation. Remarkably, the best cosmological parameter values are consistent with the universe being spatially flat – right on the borderline between positive and negative spatial curvature.

The prohibition of discrimination is a key concept in WTO law and is often the subject of trade disputes between WTO Members. This prohibition finds expression in twoobligations

Notwithstanding the recent surge in customs duties (in the context of trade wars) and the introduction of quotas (in response to the COVID-19 pandemic), customs duties and quotas have been decreasing in importance as barriers to trade in goods in recent decades. Instead, regulatory barriers to trade have gained steadily in significance. While technical regulations and standards, as well as sanitary and phytosanitary measures, are essential for the protection of, inter alia, public health, consumer safety, the environment, and public morals, they can pose significant obstacles to trade and may be (mis)used to protect domestic products against competition from imported products. The TBT Agreement and the SPS Agreement aim to prevent such misuse and to minimise the trade-restrictive impact of legitimate regulation. Also, inadequate protection of intellectual property (IP) rights negatively affects trade in goods and services incorporating IP. Therefore, the TRIPS Agreement aims to ensure that the relevant regulations of WTO Members provide a minimum level of effective protection of IP rights.

The laws of Newtonian mechanics have to be changed to be consistent with the principles of special relativity introduced in the previous chapter. This chapter describes special relativistic mechanics from a four-dimensional, spacetime point of view. Newtonian mechanics is an approximation to this mechanics of special relativity that is appropriate when motion is at speeds much less than the velocity of light in a particular inertial frame. We begin with the central idea of four-vectors, defined as a directed line segment in four-dimensional flat spacetime, and how to manipulate them. Special relativistic kinematics shows how four-vectors are used for describing the motion of a particle in spacetime terms. Concepts such as four-velocity and four-momentum are introduced. We will posit the principle of extremal proper time for a free particle in curved spacetime, and use it to derive the free particle equation of motion.