# If the universe is very nearly flat how can space be curved?

As the WMAP and other satellite data have shown that universe is very nearly flat. But the theories like general theory of relativity assume curved space to describe gravity. So how can it be? Does it mean that space is curved around massive bodies and the whole universe is flat. (I know we do not know the shape of the universe for sure but current data show it to be nearly flat.)

• Nearly flat means nearly flat on a large scale. Stars, galaxies, etc. create small amounts of local curvature on top of that flatness. Like a sheet of paper with bumps on it. Jan 8 '17 at 15:23
• To extend the previous comment, cosmology takes place on insane scales. The black hole binary that generated the gravitational wave that was first detected by LIGO is extremely far away, even by astronomical standards, but it is extremely close by cosmological standards. When we say in cosmology that the universe is approximately flat means that when viewed from such a large scale, you can foliate spacetime into spacelike slices that are nearly flat. This is like, when you look at a table up-close, it has lots of ridges on it, but if you look at it from two meter away, it seems very smooth Jan 8 '17 at 15:38

As the WMAP and other satellite data have shown that universe is very nearly flat. But the theories like general theory of relativity assume curved space to describe gravity.

That is not correct. GR describes gravity as curved spaceTIME. The curvature of space itself is a completely different story!

Does it mean that space is curved around massive bodies and the whole universe is flat.

It means that the whole universe is measured to be practically flat, so you can go in any direction and will never end up where you started like you would on the 2d analogy, the surface of a 3 dimensional sphere.

Around massive bodies you have curved spaceTIME which manifests itself as acceleration (which is also a curve on the time:position-plot) while curved space alone without the time component is an analogue to a 2d surface of a 3d object.

• Good answer. It's important to make sure that the difference is noted and clear. Space is (almost or apprimately) flat, but spacetime is curved. Space is simply the space-like slices of the 4d spacetime, for a given comoving (I.e., cosmological) time. And it is fair that people can get confused, we don't spend too much time clarifying it when they are stated in popular cosmology descriptions. The writer doesn't know half the time. Jan 9 '17 at 6:57
• Its really an eye-opening answer to me. I have just started reading the details of GR (have just stepped a little bit further from the popular cosmology descriptions) so there are many confusions. Jan 9 '17 at 15:30

A flat space is just an example of a curved space, that is, flat space-times are contained in GR inasmuch this theory describes curved space-times. Assuming that the manifold is curved does not exclude the possibility of it being flat; the latter is just a particular case of the former.

Moreover, the universe is only spatially flat at cosmological scales; locally, the curvature can be arbitrarily large.

• But we still don't know for sure that space is actually flat, right? It can have any other shape, a doughnut or a sphere? Jan 8 '17 at 15:29
• @shivani in physics we can never know anything for sure. But current observations are compatible with a spatially flat universe. That's the best we can do: to put upper bounds on the curvature. Jan 8 '17 at 15:31
• @Accidental Fouirer Transform. Space are slices of a spatial hypersurface in spacetime. The fact that spacetime is curved does not mean that some parts of it are flat because that's a special case. Spacetime is curved, has curvature. Spatial slices for given comoving time are flat. Space would NOT be flat in the curved spacetime with a higher energy density, it would not happen as a special case of curved. Jan 9 '17 at 6:59

Occasioned by your comment in the previous answer: 'But we still don't know for sure that space is actually flat, right? It can have any other shape, a doughnut or a sphere?' , i would like to point out that the surface of a 'doughnut' or a Torus in geometry is indeed flat.

So there's a difference between the terms 'flat' and 'plane'. A 'flat' space can be 'curved'. Another example is the surface of a cylinder.

Finally about the possible shapes and curvature of our universe , i am giving you the terminology as stated in the wikipedia:

'The curvature of space is a mathematical description of whether or not the Pythagorean theorem is valid for spatial coordinates. There are three possible curvatures the universe can have:

Flat (A drawn triangle's angles add up to 180°)

Positively curved (A drawn triangle's angles add up to more than 180°)

Negatively curved (A drawn triangle's angles add up to less than 180°)'

• A torus can be flat OR have nonzero curvature. An example with nonzero curvature is the usual embedding in 3 dimensions (for a 2-dimensional torus). Jan 8 '17 at 21:53

Universe is a 3-dimentional.To us it's like infinity. According to researchers universe has been created some 13.4 billion years ago.So light now can reach us from 13.4 billion light years, but we don't know that it's the periphery of the universe.Its so vast that being flat is meaningless

• universe is 4 diamensional Jan 9 '17 at 15:14