The visualization commonly used to depict the bending of space by mass in two dimensions is a useful analogy, but it does not accurately represent the actual behavior of space in three dimensions. In reality, space itself is not thought to bend into a fourth spatial dimension.
According to our current understanding of gravity, described by Einstein's theory of general relativity, massive objects like stars and planets create what is often referred to as a "gravitational field" or "curvature of spacetime." This curvature affects the motion of objects in their vicinity, causing them to follow curved paths.
In three-dimensional space, the curvature of spacetime is not typically visualized as a bending into a higher-dimensional space. Instead, it is represented mathematically as a deformation of the fabric of spacetime itself. This deformation affects the geometry of spacetime, altering the paths that objects travel along.
To illustrate this concept, imagine a flat rubber sheet representing space without any masses. If you place a heavy object like a bowling ball on the sheet, it will create a depression or curvature in the sheet. Smaller objects, such as marbles, placed on the sheet will tend to move towards the bowling ball along curved paths. This bending of the sheet represents the curvature of spacetime caused by mass.
It's important to note that this visualization is just an analogy and doesn't capture the full complexity of general relativity. In reality, the behavior of spacetime is described using mathematical equations that involve curvature and tensors, allowing us to make precise predictions about the motion of objects in the presence of gravitational fields.