A river model of space

2021

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 sp...

American Journal of Physics, 2008

This paper presents an under-appreciated way to conceptualize stationary black holes, which we call the river model. The river model is mathematically sound, yet simple enough that the basic picture can be understood by non-experts. In the river model, space itself flows like a river through a flat background, while objects move through the river according to the rules of special relativity. In a spherical black hole, the river of space falls into the black hole at the Newtonian escape velocity, hitting the speed of light at the horizon. Inside the horizon, the river flows inward faster than light, carrying everything with it. We show that the river model works also for rotating (Kerr-Newman) black holes, though with a surprising twist. As in the spherical case, the river of space can be regarded as moving through a flat background. However, the river does not spiral inward, as one might have anticipated, but rather falls inward with no azimuthal swirl at all. Instead, the river has at each point not only a velocity but also a rotation, or twist. That is, the river has a Lorentz structure, characterized by six numbers (velocity and rotation), not just three (velocity). As an object moves through the river, it changes its velocity and rotation in response to tidal changes in the velocity and twist of the river along its path. An explicit expression is given for the river field, a six-component bivector field that encodes the velocity and twist of the river at each point, and that encapsulates all the properties of a stationary rotating black hole.

2003

We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a re-formulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in the orbit of Mercury is calculated on the alternative hypothesis that its worldline is an auto-parallel of such a connection. If scalar fields couple significantly to matter and spinless test particles move on such worldlines, current time keeping methods based on the conventional geodesic hypothesis may need refinement.

2001

We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a reformulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in the orbit of Mercury is calculated on the alternative hypothesis that its worldline is an auto-parallel of such a connection. If scalar fields couple significantly to matter and spinless test particles move on such worldlines, current time keeping methods based on the conventional geodesic hypothesis may need refinement.

The basic characteristics of space-time as a requisite dualistic condition of creation has been explicated. It is emphasized that space and time represent a set of conjugate phenomena that are symbiotic in nature in a sense that one cannot exist without the other. Furthermore, it is elucidated that in reality space represents congealed time that otherwise progresses in free flow. Moreover, space and time may be considered as being the inverse of one another in the sense that the propensity of congealed time (space) diminishes the free-flowing nature of time. It is further proposed that consistent with the measure of space at a given point in the Cosmos, time should be quantified in terms of a vectorial, and not a mere scalar quantity, as is erroneously and commonly perceived. In fact, both the measures of space and time at any point within the Cosmos represent free vectors, which are mutually orthogonal to one another at the given point in the Cosmos. Furthermore, as time defines the very essence of spinning about the SOURCE, the vector of time is rotational in nature. In fact, space-time may be depicted as a curvilinear (spherical) layer with its surface being reminiscent of its space aspect and its curvature epitomizing its time feature. It is, thus, clear and reaffirms the fact that at any point within the Cosmos, the free vector of space (position vector), being tangent to the surface, is perpendicular to the vector of time that is directed along the radius of the said spheroidal surface. Furthermore, it is clarified that similar to the vector of position, time is directional in nature that may be either positive or negative. It can proceed along the radius of the spherical space-time layer either towards or away from the SOURCE.