general relativity - Can gravity be interpreted as the acceleration of spacetime towards an object?

Greetings StackExchange!

We were having a conversation with a peer about stupid ways of interpreting theories. We would go to and fro with an interpretations, but we would always find a way to disprove them. There was one, however, that we couldn't refute:

Gravity on a plane in a bidimensional space can be interpreted as the acceleration of spacetime towards an object.

We tried to invalidate in saying that the inertial mass and gravitational mass would be different, thus violating GR, but after some thought this couldn't be. The gravitational mass would accelerate spacetime towards it. If that object we were to be pushed, the pulling of spacetime would make it difficult for it to move.

We know for a fact this can't be true, and we would greatly appreciate any refutation of this thought. I can clarify if needed, or do a sketch. I apologise in advance as I'm not a native speaker.

Answer

Spacetime is not a thing so it can't accelerate. Spacetime is a manifold equipped with a metric.

However, in order to describe events in spacetime we construct coordinate systems, and coordinate systems can be accelerating. For example the Gullstrand-Painlevé coordinates describe the geometry around a black hole and they accelerate towards the black hole. So if you are stationary in the GP coordinates you are accelerating towards the black hole, or conversely if you are stationary with respect to the black hole you are accelerating outwards in the GP coordinates.

An analogy is often made with an observer flowing in a river - if you are stationary wrt the water you are moving wrt the river banks as the river carries you along. For this reason the GP coordinates are often described as the river model.

But, but, but, it is essential to be clear what is going on here. In the river model we are using an accelerating coordinate system i.e. it's the coordinate system that is accelerating not spacetime. As I mentioned at the outset, spacetime is not a thing and to attempt to ascribe properties like acceleration to it is meaningless.