Curvature of space and spacetime

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Discussion Overview

The discussion revolves around the relationship between the curvature of space and spacetime, particularly in the context of general relativity (GR). Participants explore concepts such as the Ricci scalar, local versus global curvature, and the implications of massive bodies on curvature. The conversation includes theoretical considerations and interpretations of spacetime geometry.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions whether spacetime is flat on a global scale, suggesting that if it is, the Ricci scalar would be zero.
  • Another participant asserts that while a flat spacetime implies a zero Ricci scalar, the reverse is not necessarily true.
  • There is a discussion about the curvature of space alone at a global level, with some participants indicating that it is not necessarily flat.
  • Participants note that the concepts of "space" and "space curvature" lack unambiguous meaning in GR, as different observers may slice spacetime differently.
  • Local curvature due to massive bodies is acknowledged, with the understanding that curvature may also arise from other stress-energy-momentum sources.
  • Participants agree that the Ricci tensor and scalar apply to spacetime, but there is a question about analogous concepts for space alone.
  • One participant emphasizes that the choice of coordinates does not affect the physical reality of the curvature.
  • There is contention regarding the flatness of the universe, with one participant stating that the actual spacetime of the universe is not flat, while another suggests that spatial slices can be flat.
  • Discussions arise about the implications of a changing universe on the flatness of spacetime, with references to mass density and the Ricci tensor.
  • Clarifications are made regarding the distinction between non-static and stationary spacetimes.

Areas of Agreement / Disagreement

Participants express differing views on the global curvature of spacetime and space, with no consensus reached on whether the universe is flat or not. The discussion remains unresolved regarding the implications of curvature and the definitions of space in the context of GR.

Contextual Notes

Participants note that the interpretation of curvature can depend on how spacetime is divided into space and time, and that the definitions of curvature may vary based on observer perspectives.

qtm912
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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.
 
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qtm912 said:
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.

qtm912 said:
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".

qtm912 said:
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.
qtm912 said:
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.

qtm912 said:
Are any of the above related to the choice of coordinates?
No.
 
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qtm912 said:
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.

qtm912 said:
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.
 
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PeterDonis said:
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"?
 
A.T. said:
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)
 
A.T. said:
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.
 

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