# Gravity: Is there curved space besides curved spacetime?

Wikipedia:

Curved spaces play an essential role in General Relativity where gravity is often visualized as curved space.

Is the Wikipedia article "curved space" talking about curved space or about curved spacetime? As far as I know gravity is curving spacetime, not space.

I tried here a little proof of the possibility to reduce any curved space to flat space (for simplification I consider 2D space instead of 3D space, but it might also work for 3D space):

Lets start with an isolated group of mass objects. Gravitation acts all over the universe, but we consider only a zone on the border of which curvature is infinitesimally small because of the absence of mass near the border. We represent this zone by a sort of stamped plastic sheet (the blue sheet), and we place it upon the green flat sheet of paper. Supposing that there is no curvature with an angle of 90° or more, we can assign to each point of the green sheet one point on the blue sheet, so that we get flat coordinates.

Question: Should the title of the Wikipedia article be "Curved spacetime" instead of "Curved space"?

• The linked Wiki article does not seem to invoke time at all, so I don't see why it should be curved spacetime instead. – Kyle Kanos Feb 5 '15 at 20:52
• @Kyle Kanos: It seems to me that curved space does not make sense when talking about general relativity of gravity. - I do not want to change the Wiki article, but in a general way I would like to know if the concept of curved space does make any sense in the context of gravity. – Moonraker Feb 5 '15 at 21:00
• – Kyle Kanos Feb 5 '15 at 21:10
• Moonraker, one can have curved spatial hyperslices. For example, in the FLRW metric, the spatial geometry can be flat or curved (hyperbolic or elliptical): en.wikipedia.org/wiki/… – Alfred Centauri Feb 5 '15 at 21:39
• @AlfredCentauri: in GR, there isn't really such a thing as spatial geometry - a more appropriate way to think about it is as the matter distribution being layered – Christoph Feb 5 '15 at 22:10

Your argument is incorrect: The curvature of a spatial slice is coordinate-independent.

What is true is that in general relativity, there is a priori no preferred spatial slicing. For example, de Sitter spacetime (a universe dominated by cosmological constant) can be sliced into positively curved, negatively curved or flat spaces.

When we say that our universe appears to be spatially flat, we're talking about a particular slicing selected by the matter distribution, which we believe does come in layers of constant cosmological time.

• To expand on that first sentence, the curvature of any space or spacetime is defined in differential geometry in terms of a metric which gives you a coordinate-independent notion of "distance" along any arbitrary path in that space or spacetime--in curved spacetime this can be proper time or proper distance depending on whether the curve is space-like or time-like, but in a given 3D space-like slice of the 4D spacetime, the curvature is just defined by the proper distance along any space-like curve that lies in that slice. – Hypnosifl Feb 5 '15 at 22:36
• @Christoph: You are deriving space slices from spacetime. So what you are calling space is still sliced spacetime. But I am asking if under each of your (blue) slices there is underlying the (green) metric of 3D space of FLRW metrics of the universe which are flat (or very slightly curved). – Moonraker Feb 6 '15 at 17:58

You ask the question of whether the title for the Wikipedia article should be "Curved Spacetime" instead of "Curved Space".

The answer is a resounding no, leave it as is. The article itself covers strictly the mathematics of any curved space and is not specific to physics contexts. As is, the usage of "Space" does not mean purely spatial and not temporal dimensions as it does in physics. Furthermore, from the definition given, it is clear that even in a physics context, they are referring to principally spatial curvature in a metric.

It is curious that they wouldn't call this article "Intrinsic Curvature", however as I am not a mathematician, I won't comment on their choices.

But the fact is that there is no inclusion of time and that this is more a treatise of the mathematics, for which the label "Curved Space" is very much appropriate. So no, the title should not be changed.