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What is the difference between the Space-time curvature and Space curvature?

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Good question! But, we may need to add another independent variable of acceleration/gravitational field. –  Alan Rominger Apr 19 '14 at 2:47
interesting ... May I ask ...? –  user44629 Apr 19 '14 at 2:48
So , the space is without scalar fields and the spacetime is with scalar fields ... is this right ? –  user44629 Apr 19 '14 at 2:50
there is no real distinction between space and time. It does not really make sense to ask what the difference is between space and spacetime. Space is simply the old, outdated way of referring to what we now call spacetime –  ACuriousJim Apr 19 '14 at 4:10
alternatively, one could say that spacetime is what physicists deal with; space is what astronauts and interior decorators deal with –  ACuriousJim Apr 19 '14 at 4:12

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Spacetime curvature is mathematically equivalent to the presence of so-called geodesical deviation of timelike geodesics. In other words there are freely falling bodies starting from points close to each other and with similar velocities which measure a nonvanishing relative acceleration. This is the most direct physical meaning of a nonvanishig Riemann curvature tensor in a spacetime.

Space curvature has a similar, but not identical, interpretation in the extended rest space of an observer (assuming that the metric induced on that space from the one in spacetime is stationary with respect to the notion of time adopted by the observer). There is a "relative acceleration" referred to the natural length parameter (instead of proper time) between geodesics. Here geodesics can be defined in terms of their variational definition, since the metric is positively defined. They are the shortest lines joining pairs of given points.

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"Space curvature" refers to the geometry of a spatial-slice of space-time, with a constant time coordinate (so the slice has no time dimension). "Space curvature" is what the common rubber-sheet-analogy refers to, mimicking Flamm's paraboloid, which represents the geometry of a spatial slice through the Schwarzschild metric:


"Space-time curvature" refers to the geometry of the 4D space-time, or a slice with a non-constant time coordinate (so the slice has a time dimension). "Space-time curvature" is rarely shown, but it looks something like this:


Here another comparison of the two types of slices of spaces-time. One with time, and one purely spatial:


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