General relativity (GR) is a theory that describes the behavior of gravity in terms of the curvature of spacetime. It uses a mathematical framework based on Riemannian geometry to represent the curvature of spacetime.
In GR, the curvature of spacetime is described by the metric tensor, which encodes the relationships between spacetime coordinates. Riemannian geometry provides a mathematical language to express these relationships. The metric tensor is a mathematical object that accounts for both the spatial and temporal aspects of spacetime.
When we talk about the "space density gradient" in the context of Riemannian coordinates, it refers to the curvature of spacetime itself. The curvature of spacetime is not limited to spatial dimensions alone; it affects the behavior of time as well. In GR, the curvature of spacetime is a unified concept that encompasses both space and time.
One of the consequences of this curvature is that it leads to the phenomenon of time dilation. Time dilation refers to the difference in the passage of time experienced by observers in different gravitational fields or under different accelerations. The curvature of spacetime affects the geometry of worldlines (the paths that objects follow in spacetime), leading to variations in the flow of time.
In GR, the equations are formulated using Riemannian geometry and tensor calculus, which provide a precise mathematical description of the curvature of spacetime. These mathematical tools allow us to calculate the effects of gravity, such as the bending of light, the motion of celestial bodies, and the phenomenon of time dilation.
While Riemannian coordinates describe the curvature of spacetime in terms of spatial gradients, it is important to note that the resulting predictions of GR encompass both spatial and temporal aspects. Time dilation is a consequence of the overall curvature of spacetime, not just the spatial gradients described by Riemannian coordinates.
So, although Riemannian coordinates emphasize the spatial curvature, the final results of GR are viewed in terms of time dilation because time is an integral part of the unified spacetime framework described by the theory.