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As I understand it, you can have time + flat space = curved spacetime.

  1. So, when one is trying to emphasise that there is a curvature to the space, is it more technically correct to say curved space in favour of curved spacetime.
  2. Does anyone do this?
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2 Answers 2

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Well, for starters, time + flat space, as you put it, isn't necessarily curved. The obvious example being Minkowski space, of course.

It can be curved though, depending on how you join the two to form a spacetime. For example, flat space can be joined with a time coordinate to form a hyperbolic space with metric $$ds^2 = \frac{-dt^2 + dx^2 + dy^2 +dz^2}{t^2},$$ such that for every $t$ fixed, we get a flat spacial slice, but the total space is most definitely curved.

I can't speak for the entire physics community out there, but I am under the impression that "curved spacetime" would be the commonly used expression.

Edit: Oh dear, I think I finally understood your question correctly. I suppose "curved space" would be an adequate description, though I can think of very few situation where you would want to explicitly emphasize the spacial curvature. Mind you that this is a very coordinate-dependent statement, to boot: if your curved spacetime is of the form "time + flat space" in one set of coordinates, in another set of coordinates, the spatial part may very well be curved, and the other way around.

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The only example I can think of is if you're studying the embeddings of curved spaces in flat higher-dimensional spacetimes. –  Jerry Schirmer Jan 17 '13 at 22:27
    
Definitely worth emphasizing: time + flat space = spacetime, which can very well be flat. Just because GR can handle curvature doesn't mean everything in GR is curved. –  Chris White Jan 18 '13 at 4:57

They're both technically correct, it depends on what you are talking about. It is possible to have the space be curved, the spacetime be curved, or to have neither be curved.

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