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The FLRW metric starts with the assumption of homogeneity and isotropy of space.(Wikipedia)

FLRW metrics of the universe have no or only very weak curvature - It is curved space. In contrast, local curvature of metrics by gravitation is strongly perceivable. It is curved spacetime.

Is the spatial curvature of the universe of the same kind as local spacetime curvature by gravity which both just can be added one to the other, or are they belonging to two incompatible metrics which cannot be represented in the same coordinate system?

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  • $\begingroup$ Have you seen the most general perturbed FLRW metric? It includes local gravity sources and the curvature of the universe into one metric. As for whether they are the same type of curvature, the answer is assuredly no. The curvature from gravitating sources produces drastically different effects from the overall curvature of the universe $\endgroup$ – Jim Feb 6 '15 at 18:14
  • $\begingroup$ @Jim: Can we add them up? If they were not addable, they could not affect homogeneity and isotropy. $\endgroup$ – Moonraker Feb 6 '15 at 18:45
  • $\begingroup$ Can we add them up? In some ways we can, in others we can't. A perturbed metric has inhomogeneities and (depending on how it is perturbed) those can affect the isotropy as well. But the curvature of space does not, by itself, affect homogeneity and isotropy. $\endgroup$ – Jim Feb 6 '15 at 19:05
  • $\begingroup$ They are both an intrinsic curvature, but they don't carry the same meaning or effects. I suppose one could, in theory, add the two together, but since they are so different in scope and influence, to do so would be unnecessarily complicated. It is much easier and more practical to leave them as separate entities in the metric, one as a background part of space and the other as perturbative blobs inhabiting it. $\endgroup$ – Jim Feb 6 '15 at 19:26

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