A river model of space

Abstract

We expand on Hamilton and Lisle’s idea of a river model of black holes (Am. J. Phys. 76, 519 (2008)) by presenting a reformulation of the river description in terms of an orthonormal basis field attached to static observers and then apply it to the Schwarzschild-de Sitter, the Schwarzschild and the de Sitter spacetimes. The physical space is defined as a specified set of freely moving reference particles. Using a combination of orthonormal basis fields and the usual formalism in a coordinate basis we calculate the physical velocity field of these reference particles. In this way one obtains a vivid description of space in which space behaves like a river flowing radially toward the singularity in the Schwarzschild spacetime and radially toward infinity in the de Sitter spacetime. In the Schwarzschild-de Sitter spacetime there exists a three-dimensional timelike surface which divides the flow directions of the river of space into two distinct regions: the region outside this surface where space flows radially outwards toward infinity and the region inside this surface where space flows radially toward the central singularity. We also consider the effect of the river of space upon light rays and material particles in the Schwarzschild and the de Sitter spacetimes and show that the river model of space provides an intuitive explanation for the behavior of light and particles at and beyond the event horizons associated with these spacetimes. Finally, a local description of the kinematics of the river of space in terms of the expansion and shear of its velocity field is given. We find that the flow of the river of space in the Scwarzschild-de Sitter spacetime violates the geodesic focusing theorem outside the dividing three-surface.