Space Curvature: Friedmann Models Explained

[SteveDC]

Currently reading Peter Coles, Cosmology a very short introduction. There is a bit I don't understand. In a section discussing Friedmann Models, and how going on the cosmological principle density of the universe is the same in every place, and therefore space must be warped in the same way at every point.

One of the ways of doing this is to have a flat universe and have the warped space caused by mass to be exactly counterbalanced by energy contained in the expansion of the Universe. Then it says, "even though space may be flat, space-time is still curved."

It's that last bit that I don't understand because I thought that space, and space time are both warped by mass and energy and so how can they be warped differently, and how is there even a distinction between the two?

 

[Orodruin]

It's that last bit that I don't understand because I thought that space, and space time are both warped by mass and energy and so how can they be warped differently, and how is there even a distinction between the two?

The Einstein equations describe the relation between energy-momentum and the Riemann curvature tensor in space-time. There is no demand that a sub-manifold of a manifold must have the same curvature as the manifold itself. For example, a sphere can be embedded in R^3 and has curvature although R^3 does not, nothing strange about this. The statement is that the spatial part of the FRW universe does not need to have a curvature, but the space-time, including the time coordinate and the evolution of the Universe, does.