some important mathematical techniques that are employed in investigating spacetimes will now be discussed. Frame fields A frame field is an orthonormal set of 4 vector fields (1 timelike, 3 spacelike) defined on a spacetime. Each frame field can be thought of as representing an observer in the spacetime moving along the integral curves of the timelike vector field. Every tensor quantity can be expressed in terms of a frame field, in particular, the metric tensor takes on a particularly convenient form. When allied with coframe fields, frame fields provide a powerful tool for analysing spacetimes and physically interpreting the mathematical results. Symmetry vector fields Some modern techniques in analysing spacetimes rely heavily on using spacetime symmetries, which are infinitesimally generated by vector fields (usually defined locally) on a spacetime that preserve some feature of the spacetime. The most common type of such symmetry vector fields include Killing vector fields (which preserve the metric structure) and their generalisations called generalised Killing vector fields. Symmetry vector fields find extensive application in the study of exact solutions in general relativity and the set of all such vector fields usually forms a finite-dimensional Lie algebra. The Cauchy problem The Cauchy problem (sometimes called the initial value problem) is the attempt at finding a solution to a differential equation given initial conditions. In the context of general relativity, it means the problem of finding solutions to Einstein's field equations - a system of hyperbolic partial differential equations - given some initial data on a hypersurface. Studying the Cauchy problem allows one to formulate the concept of causality in general relativity, as well as 'parametrising' solutions of the field equations. Ideally, one desires global solutions, but usually local solutions are the best that can be hoped for. Typically, solving this initial value problem requires selection of particular coordinate conditions. Spinor formalism Spinors find several important applications in relativity. Their use as a method of analysing spacetimes using tetrads, in particular, in the Newman–Penrose formalism is important. Another appealing feature of spinors in general relativity is the condensed way in which some tensor equations may be written using the spinor formalism. For example, in classifying the Weyl tensor, determining the various Petrov types becomes much easier when compared with the tensorial counterpart. Regge calculus Regge calculus is a formalism which chops up a Lorentzian manifold into discrete 'chunks' (four-dimensional simplicial blocks) and the block edge lengths