Can we see gravity as expanding spacetime near a massive object? Is it possible to see gravity as expanding spacetime near massive objects? I don't mean an accelerating frame through space, but a really expanding space, like the expansion of space that takes place on a cosmic scale. In that way, an object standing still with respect to space is dragged along by the expansion to the object. Like the stars in an expanding space fly apart not by moving but by the space that gets larger between them. 
 A: It is a fundamental principle of general relativity that all coordinate systems are equally valid, so when analysing some spacetime geometry we are free to choose any coordinate system we want. We generally choose the coordinates that make the maths simplest.
For an isotropic homogeneous universe there is a convenient choice of coordinates in which all the observers are stationary. These are the comoving coordinates. If we use these coordinates then we find that the distance between stationary observers increases with time, and it is in this sense that the universe is expanding.
Suppose we are interested instead in the spacetime geometry around a massive object. We could attempt a similar strategy. Suppose we consider observers that start at infinity and fall inwards towards the massive object, and we look for coordinates in wich these observers are stationary. However experience tells that the radial spacing of such observers increases as they fall inwards because the inner observer accelerates faster. We also find the tangential spacing decreases because the trajectories converge on the centre of the mass.
So this isn't like an expanding universe. What happens is that space gets squeezed instead. If we start with some spherical region of space then as that region falls inwards it is deformed into an ellipsoid but the volume doesn't change i.e. doesn't expand or contract.
However there is some utility in using coordinates like this as they are one way to remove the coordinate singularity at the event horizon. An example of this approach is the Gullstrand-Painlevé coordinate system. This is also known as the river model because in these coordinates spacetime flows inwards towards the mass.
