Lorentzian spaces

In ordinary Euclidean space distance is given by Sqrt[x^{2} + y^{2} + z^{2}]. In setting up relativity theory it is convenient (see page 1042) to define an analog of distance (so-called proper time) in 4D spacetime by Sqrt[c^{2} t^{2} - x^{2} - y^{2} - z^{2}]. And in terms of differential geometry such Minkowski space can be specified by the metric DiagonalMatrix[{+1, -1, -1, -1}] (now taking c = 1). To set up general relativity one then considers not Riemannian manifolds but instead Lorentzian ones in which the metric is not positive definite, but instead has the signature of Minkowski space.

In such Lorentzian spaces, however, there is no useful immediate analog of a sphere. For given any point, even the light cone that corresponds to points at zero spacetime distance from it has an infinite volume. But with an appropriate definition one can still set up cones that have finite volume. To do this in general one starts by picking a vector e in a timelike direction, then normalizes it to be a unit vector so that e . g . e -1. Then one defines a cone of height t whose apex is a given point to be those points whose displacement vector v satisfies 0 > e . g . v > -t (and 0 > v . g. v). And the volume of such a cone then turns out to be

s[d]t^{d + 1}(1 - t^{2}(d + 1)(d RicciScalar + 2(d + 1)(RicciTensor . e . e))/((d + 2)(d + 3)) + …)/(d + 1)