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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.

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  • $\begingroup$ You might think of space-time as flowing toward an object. As water going downhill, gathering acceleration as it approaches the object. $\endgroup$
    – Ernie
    May 14, 2015 at 1:58
  • $\begingroup$ It is wholly unclear what you mean by "Gravity on a plane in a bidimensional space can be interpreted as the acceleration of spacetime towards an object. " $\endgroup$
    – ACuriousMind
    May 14, 2015 at 12:29

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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.

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We were having a conversation with a peer about stupid ways of interpreting theories.

Once you understand a few things about gravity and relativity, you understand just how stupid it is.

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

Like John Rennie said, spacetime is not a thing. There is no spacetime around the Earth. Spacetime is a "mathematical space". It combines the spatial dimensions with the time dimension, and because the time dimension is folded into the mix, there's no motion through spacetime. You can draw worldlines in it to represent motion through space over time, and then you get into coordinate systems and geometry and curvilinear motion. But that motion is motion through space, not through spacetime, and it is not the motion of spacetime or space.

The gravitational mass would accelerate spacetime towards it.

Where are you getting this stuff from? A concentration of energy in the guise of a star "conditions" the surrounding space, altering its properties. As a result motion through this space is curved rather than linear. See Einstein talking about this here.

If that object we were to be pushed, the pulling of spacetime would make it difficult for it to move.

That's wrong too. See Einstein's E=mc² paper. The mass of a body is a measure of its energy-content. Not a measure of "the pulling of spacetime".

We know for a fact this can't be true, and we would greatly appreciate any refutation of this thought.

See what Einstein said: "According to this theory the metrical qualities of the continuum of space-time differ in the environment of different points of space-time, and are partly conditioned by the matter existing outside of the territory under consideration. This space-time variability of the reciprocal relations of the standards of space and time, or, perhaps, the recognition of the fact that "empty space" in its physical relation is neither homogeneous nor isotropic, compelling us to describe its state by ten functions (the gravitation potentials g$_{μν}$), has, I think, finally disposed of the view that space is physically empty". Also see Baez: "Similarly, in general relativity gravity is not really a `force', but just a manifestation of the curvature of spacetime. Note: not the curvature of space, but of spacetime. The distinction is crucial". In a gravitational field spacetime isn't moving, and nor is space. It isn't moving up, and it isn't falling down like some river or waterfall. We do not live in some Chicken-Little world where the sky is falling in. Instead, in a gravitational field, space is inhomogeneous. See this answer for more.

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