solution of the field equations that describes the universe is the metric tensor called the Friedmann–Lemaître–Robertson–Walker metric, where (r, θ, φ) correspond to a spherical coordinate system. This metric has only two undetermined parameters. An overall dimensionless length scale factor R describes the size scale of the universe as a function of time (an increase in R is the expansion of the universe), and a curvature index k describes the geometry. The index k is defined so that it can take only one of three values: 0, corresponding to flat Euclidean geometry; 1, corresponding to a space of positive curvature; or −1, corresponding to a space of positive or negative curvature. The value of R as a function of time t depends upon k and the cosmological constant Λ. The cosmological constant represents the energy density of the vacuum of space and could be related to dark energy. The equation describing how R varies with time is known as the Friedmann equation after its inventor, Alexander Friedmann. The solutions for R(t) depend on k and Λ, but some qualitative features of such solutions are general. First and most importantly, the length scale R of the universe can remain constant only if the universe is perfectly isotropic with positive curvature (k=1) and has one precise value of density everywhere, as first noted by Albert Einstein. However, this equilibrium is unstable: because the universe is inhomogeneous on smaller scales, R must change over time. When R changes, all the spatial distances in the universe change in tandem; there is an overall expansion or contraction of space itself. This accounts for the observation that galaxies appear to be flying apart; the space between them is stretching. The stretching of space also accounts for the apparent paradox that two galaxies can be 40 billion light-years apart, although they started from the same point 13.8 billion years ago and never moved faster than the speed of light. Second, all solutions suggest that there was a gravitational singularity in the past, when R went to zero and matter and energy were infinitely dense. It may seem that this conclusion is uncertain because it is based on the questionable assumptions of perfect homogeneity and isotropy (the cosmological principle) and that only the gravitational interaction is significant. However, the Penrose–Hawking singularity theorems show that a singularity should exist for very general conditions. Hence, according to Einstein's field equations,