Abstract
Abstract Chapter 13 is the first of two chapters that look at how Einstein’s equations may be recast as a system of partial differential equations. In this chapter, the Cauchy initial value problem (also known as the Cauchy problem) is considered. This involves specifying as initial data the 3-metric and its rate of change on some initial hypersurface at some initial time and then showing that this allows one to determine the 4-metric at subsequent times. This shows that Einstein’s equations can be considered as a system of evolution equations for the geometry of space-time. The chapter also considers the special case of ‘harmonic coordinates’, which considerably simplifies the mathematics. The chapter also considers the so-called hole problem and the equivalence problem in general relativity and includes a discussion of the status of exact solutions.
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