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Q & A: picturing curved space

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Most recent answer: 01/24/2013
Q:
in science videos it is often shown & told space is like a stretched rubber sheet & it is curved by the presence of matter . But it seems very confusing as space is 3 dimensional & therefore can't be ( i suppose) represented as a two dimensional sheet. then what is the exact concept?
- Shantonu Mukherjee (age 19)
Hooghly
A:
We can't explain the "exact concept" of general relativity here, but we can say something about those pictures.

We picture things in flat 3-D Euclidean space because our brains seem to be hardwired that way. That means that if we want to picture a curved space we need to picture something lower-dimensional, like those 2-D sheets. Let's look at what some of the intrinsic mathematical properties are of those curved sheets, because those are the properties that can apply in higher dimensions even if we can't picture the spaces.

In flat 2-D the circumference of a circle is 2π times the radius, r, where by circle we mean the set of points at distance r from the center. In a curved surface like a sphere the circumference is less than 2πr. If you make a saddle-shaped surface, the circumference wobbles and is larger than 2πr. 

What would the analogy be in 3-D? You could have a region where the area of a sphere was bigger than 4πr2 or less than 4πr2. In fact, for regions with mass in them the area is less than 4πr2.

These remarks should be taken with one precaution. In assigning "areas" and "radii" I've assumed a particular common-sense way of choosing coordinates. Really the space is a sort of 4-D space, including time, which is unlike the other dimensions. There are different ways of looking at a given physical situation that mix up effects in space and time. But I hope the simple spatial description I've given helps you get started.

Mike W.

(published on 01/24/2013)

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