Why Can't There Be Gravity in 2 Spatial Dimensions?

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Gravity cannot exist in two spatial dimensions due to the constraints of Einstein's General Relativity (GR). In two or three dimensions, it is impossible for the Einstein field equations to be satisfied in a vacuum while allowing for a non-zero gravitational field. This is because a gravitational field requires space to be curved, which is not feasible in lower dimensions. In four dimensions, the complexity of the curvature tensor allows for curvature in empty space, enabling the existence of gravity.

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Why do they say there can there be no gravity in 2 spatial dimensions (and 1 temporal dimension)?

wouldn't the gravitational law be an inverse-law instead of an inverse-square law?
 
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lolgarithms said:
Why can there be no gravity in 2 spatial dimensions (and 1 temporal dimension)?

I don't think anyone clearly proved that there couldn't be (as far as I know). Not to mention, gravity is still relatively mysterious to us, so we don't know enough to assert whether there is or is not any gravity in 2 spatial dimensions.
 
According to GR there can not be a non zero gravitational field in empty space in two or three dimensions. For a gravitational field in a region it is necessary that space is curved. But in 2 or 3 dimensions it is not possible for the Einstein field equations to be satisfied in a vacuum and for space to be curved. In 4 dimensions there are a lot more independent components of the curvature tensor which means that space can be curved whilst still satisfying the field equations in empty space.
 

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