important exact solutions include the Schwarzschild solution and the Friedman-Lemaître-Robertson–Walker solution. The EIH approximation plus other references (e.g. Geroch and Jang, 1975 - 'Motion of a body in general relativity', JMP, Vol. 16 Issue 1). The geodesic equations Once the EFE are solved to obtain a metric, it remains to determine the motion of inertial objects in the spacetime. In general relativity, it is assumed that inertial motion occurs along timelike and null geodesics of spacetime as parameterized by proper time. Geodesics are curves that parallel transport their own tangent vector ; i.e., . This condition, the geodesic equation, can be written in terms of a coordinate system with the tangent vector : where denotes the derivative by proper time, , with τ parametrising proper time along the curve and making manifest the presence of the Christoffel symbols. A principal feature of general relativity is to determine the paths of particles and radiation in gravitational fields. This is accomplished by solving the geodesic equations. The EFE relate the total matter (energy) distribution to the curvature of spacetime. Their nonlinearity leads to a problem in determining the precise motion of matter in the resultant spacetime. For example, in a system composed of one planet orbiting a star, the motion of the planet is determined by solving the field equations with the energy–momentum tensor the sum of that for the planet and the star. The gravitational field of the planet affects the total spacetime geometry and hence the motion of objects. It is therefore reasonable to suppose that the field equations can be used to derive the geodesic equations. When the energy–momentum tensor for a system is that of dust, it may be shown by using the local conservation law for the energy–momentum tensor that the geodesic equations are satisfied exactly. Lagrangian formulation The issue of deriving the equations of motion or the field equations in any physical theory is considered by many researchers to be appealing. A fairly universal way of performing these derivations is by using the techniques of variational calculus, the main objects used in this being Lagrangians. Many consider this approach to be an elegant way of constructing a theory, others as merely a formal way of expressing a theory (usually, the Lagrangian construction is performed after the theory has been developed). Mathematical techniques for analysing spacetimes Having outlined the basic mathematical structures used in formulating the theory,