De Sitter Cosmology

Abstract

The equation u ≡ t − z = constant in Minkowskian space-time with line element $$\displaystyle \begin{array}{@{}rcl@{}} ds^2=dt^2-dx^2-dy^2-dz^2=\eta _{ij}\,dx^i\,dx^j , \end{array} $$ is an example of a null hyperplane. It represents the history of a 2-plane, parallel to the x, y-plane in three dimensional Euclidean space, moving with the speed of light in the positive z-direction. Thus the family of null hyperplanes u = constant could be the histories of the wave fronts of plane electromagnetic waves travelling in the positive z-direction in Euclidean space. The vector field normal to the hyperplanes is ki = u,i and with ki = ηijkj this has the properties that kiki = 0 and ki,j = 0. Thus ki is a null vector field (and therefore tangent to u = constant) and covariantly constant (i.e. a constant vector field in the coordinates xi = (t, x, y, z) with i = 0, 1, 2, 3). To generalise this notion of null hyperplanes to space-times of non-zero constant curvature, the first obstacle one encounters is the non-existence in such a space-time of covariantly constant vector fields.