The observations described in Chapter 17 show our universe to be approximately homogeneous and isotropic on spatial distance scales above several hundred megaparsecs. The simplest cosmological models enforce these symmetries exactly as a first approximation. For instance, the matter in galaxies and the radiation are approximated by smooth density distributions that are exactly uniform in space. Similarly, the geometry of spacetime incorporates the homogeneity and isotropy of space exactly. These simplifying assumptions define the Friedman–Robertson–Walker (FRW) family of cosmological models, which are the subject of this chapter.

Secure and predictable access to markets is essential for international trade. However, access of goods and services from other countries to the market of a WTO Member is frequently impeded or restricted. These restrictions can be in the form of either tariffs or non-tariff barriers to trade. The most common tariff barriers to trade are – at least for goods – customs duties.

Gravitational waves provide a window on the universe of astronomical phenomena that is different from any in the electromagnetic spectrum. Mass in many different varieties of motion is a source of propagating ripples in spacetime curvature. In order to interpret the observations of gravitational wave detectors on Earth and in space, it is necessary to solve the Einstein equation for the gravitational radiation produced by given sources. Predicting the gravitational radiation from strong-curvature, rapidly varying sources is a problem generally tractable only by numerical simulation of the fully nonlinear Einstein equation – a subject well beyond the scope of this book. However, some insight into the production of gravitational waves can be obtained from examining the more tractable problem of the small ripples in spacetime emitted by weak, nonrelativistic sources.

This chapter discusses the geometry of space and the notion of time assumed in Newtonian mechanics. This discussion will also serve to review aspects of mechanics and special relativity that will be important for later developments. Newtonian mechanics assumes a geometry for space and a particular idea for time. The laws of Newtonian mechanics take their standard and simplest forms in inertial frames. Using the laws of mechanics, an observer in an inertial frame can construct a clock that measures the time. Coordinate transformations can make the connection between different inertial frames. Newtonian mechanics assumes there is a single notion of time for all inertial observers. We explore Newtonian gravity and the Principle of Relativity: that identical experiments carried out in different inertial frames give identical results.

The relation between local spacetime curvature and matter energy density is given by the Einstein equation – it is the field equation of general relativity in the way that Maxwell’s equations are the field equations of electromagnetism. Maxwell’s equations relate the electromagnetic field to its sources – charges and currents. Einstein’s equation relates spacetime curvature to its source – the mass-energy of matter. This chapter gives a very brief introduction to the Einstein equation; we consider the equation in the absence of matter sources (the vacuum Einstein equation) and will include matter sources in Chapter 22. Even the vacuum Einstein equation has important implications. Just as the field of a static point charge and electromagnetic waves are solutions of the source-free Maxwell’s equations, the Schwarzschild geometry and gravitational waves are solutions of the vacuum Einstein equation.

At the time of writing, international trade is in deep crisis as a result of the COVID-19 pandemic and this calamity’s impact on the global economy. In September 2020, it was estimated that in 2020 international trade in goods would, in volume terms, be 9.2 per cent lower than in 2019, and this estimate is subject to much uncertainty since it depends on the unpredictable trajectory of the COVID-19 pandemic and government responses to it. It is important to note, however, that international trade, and the multilateral trading system that facilitates it, was in crisis before anyone had heard of the COVID-19 virus. Already, in 2019, global trade in goods declined and growth in trade in commercial services was paltry compared with preceding years.