Curvature of space and spacetime

[qtm912]

i am trying to understand the relationship between the two on a local and global scale and how these two concepts are related to the Ricci scalar.

Is it correct to say that as far as we know on a global scale, spacetime is flat so that the Ricci scalar is zero. If so, what can be said about the curvature of space alone at a global level, is it also flat?

Locally my understanding is that spacetime may be curved due to the presence of massive bodies. In this case is it true to say that both space as well as space time is curved near these bodies? I would assume than the nature of the curvature would depend on the form of the metric that applies locally.

Finally, my understanding is that the Ricci tensor and scalar is applicable to spacetime. (is this true?) If so is there an analagous concept in relation to space alone, if that means anything.

Are any of the above related to the choice of coordinates? I would have thought not as choice of coordinates should not change the physical reality.

 

[WannabeNewton]

Is it correct to say that as far as we know on a global scale, spacetime is flat so that the Ricci scalar is zero.

If space-time is flat then the Ricci scalar vanishes. The converse however doesn't hold.

If so, what can be said about the curvature of space alone at a global level, is it also flat?

No. Also keep in mind that the concepts of "space" and "space curvature" have no unambiguous meaning in GR: there are many different ways to slice up space-time into surfaces representing "space at a given instant of time" and these correspond usually to the different ways distinct families of observers perceive "space at a given instant of time".

Locally my understanding is that spacetime may be curved due to the presence of massive bodies. In this case is it true to say that both space as well as space time is curved near these bodies? I would assume than the nature of the curvature would depend on the form of the metric that applies locally.

In general yes. Keep in mind that space-time may be curved due to any stress-energy-momentum source, not just massive bodies.

Finally, my understanding is that the Ricci tensor and scalar is applicable to spacetime. (is this true?) If so is there an analagous concept in relation to space alone, if that means anything.

Yes to both questions. One just goes from the Lorentzian signature to the usual signature.

Are any of the above related to the choice of coordinates?

No.

 

[PeterDonis]

Is it correct to say that as far as we know on a global scale, spacetime is flat

If by "spacetime" you mean the actual spacetime of our universe, no, this is not correct. The spacetime of the universe is not flat. See below.

If so, what can be said about the curvature of space alone at a global level, is it also flat?

Our best current models indicate that the universe is spatially flat--more correctly, spacelike hypersurfaces in which the universe appears homogeneous and isotropic are flat. As WannabeNewton pointed out, whether "space" is flat depends on how you divide up spacetime into space and time.

The reason spacetime is not flat, even though we can cut flat spatial slices out of it, is that spacetime includes time, and the universe is changing in time: it is expanding.

 

[A.T.]

The reason spacetime is not flat, even though we can cut flat spatial slices out of it, is that spacetime includes time, and the universe is changing in time: it is expanding.

Changing over time is usally called "not static". But how does this imply "not flat"?

 

[Bill_K]

Changing over time is usally called "not static". But how does this imply "not flat"?

It's got a nonzero mass density, hence a nonzero Ricci tensor. However it is conformally flat. (Easy to see this just by symmetry)

 

[WannabeNewton]

Changing over time is usally called "not static". But how does this imply "not flat"?

Just a small clarification: a space-time can be non-static but still be time translation invariant as long as it is stationary.