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.

Einstein’s 1905 special theory of relativity requires a profound revision of the Newtonian ideas of space and time that were reviewed in the previous chapter. In special relativity, the Newtonian ideas of Euclidean space and a separate absolute time are subsumed into a single four-dimensional union of space and time, called spacetime. This chapter reviews the basic principles of special relativity, starting from the non-Euclidean geometry of its spacetime. Einstein’s 1905 successful modification of Newtonian mechanics, which we call special relativity, assumed that the velocity of light had the same value, c, in all inertial frames, which requires a reexamination, and ultimately the abandonment, of the Newtonian idea of absolute time. Instead, he found a new connection between inertial frames that is consistent with the same value of the velocity of light in all of them. The defining assumption of special relativity is a geometry for four-dimensional spacetime.