A river model of space

Abstract

Within the theory of general relativity gravitational phenomena are usually attributed to the curvature of four-dimensional spacetime. In this context we are often confronted with the question of how the concept of ordinary physical three-dimensional space fits into this picture. In this work we present a simple and intuitive model of space for both the Schwarzschild spacetime and the de Sitter spacetime in which 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. Thus we obtain 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. We also consider the effect of the river of space upon light rays and material particles 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.

On the Principle of Synchronization

“…It is interesting that the static de-Sitter coordinate frame represents the perspective of a non-co-moving observer, for whom there is a velocity of the "river of space" [19]. Yet, the metric looks entirely static.…”

A Solution of the Mitra Paradox

“…the 4-acceleration of a particle permanently at rest vanishes [9]. Hence the reference particles of the inertial 3-space are at rest at this surface.…”

“…This is the preferred 3-space of the relativistic universe models. In [9] it was shown that the 3-velocity of the inertial 3-space as given with respect to the orthonormal basis of an observer at rest in RF is…”

The influence of spacetime curvature on quantum emission in optical analogues to gravity

“…with the Newtonian escape velocity β = (2Gm/ζ) 1/2 , G the gravitational constant, m the mass of the black hole, ζ the spatial coordinate, t the proper time, and c the speed of light in vacuum [47][48][49]. In this so-called "River Model" of the black hole, the metric describes ordinary flat space (and curved time), with space itself flowing towards ζ = 0 (the spacetime singularity) at increasing velocity β [50,51]. This acceleration of the flow velocity of space towards ζ = 0 is the manifestation of the curvature of spacetime around the black hole: before the horizon, the flow velocity is subluminal, β = c at the horizon (when ζ = ζ Schw , the Schwarzschild radius) and the flow velocity of space is superluminal inside the horizon.…”