The metric for a flat Minkowski spacetime in spherical coordinates is d s 2 = − − d t 2 + d r 2 + r 2 d Ω Ω 2 . Conformal compactification induces a transformation which preserves angles, but changes the local structure of the metric and adds the boundary of the manifold, thus making it compact. For a given metric g i j , a conformal compactification scales the entire metric by some conformal factor such that g i j ¯ ¯ = Ω Ω 2 g i j such that all of the points at infinity are scaled down to a finite value. Typically, the radial and time coordinates are transformed into null coordinates u = t + r and v = t − − r . These are then transformed as p = tan − − 1 ⁡ ⁡ u and q = tan − − 1 ⁡ ⁡ v in order to use the properties of the inverse tangent function to map infinity to a finite value. The typical time and space coordinates may be introduced as T = p + q and R = p − − q . After these coordinate transformations, a conformal factor is introduced, leading to a new unphysical metric for Minkowski space: